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"planck-treatise-on-thermodynamics-1903/eq-68575e9c54", "planck-treatise-on-thermodynamics-1903/eq-dc220dd2d2", "planck-treatise-on-thermodynamics-1903/eq-c6f498da01", "planck-treatise-on-thermodynamics-1903/eq-6c807eb9d2", "planck-treatise-on-thermodynamics-1903/eq-58ddde3223", "planck-treatise-on-thermodynamics-1903/eq-f391bcebd0", "planck-treatise-on-thermodynamics-1903/eq-c87f0b8dfd", "planck-treatise-on-thermodynamics-1903/eq-08b35d6a70", "planck-treatise-on-thermodynamics-1903/eq-89526f2f3b", "planck-treatise-on-thermodynamics-1903/eq-26b5709b36", "planck-treatise-on-thermodynamics-1903/eq-29babe90e0", "planck-treatise-on-thermodynamics-1903/eq-bab8e54140", "planck-treatise-on-thermodynamics-1903/eq-9d5d6f0cda", "planck-treatise-on-thermodynamics-1903/eq-4997d20cab", "planck-treatise-on-thermodynamics-1903/eq-58f8c0a837", "planck-treatise-on-thermodynamics-1903/eq-0902be4d32", "planck-treatise-on-thermodynamics-1903/eq-e828309c86", "planck-treatise-on-thermodynamics-1903/eq-a1879ab372", "planck-treatise-on-thermodynamics-1903/eq-075c9d643e", "planck-treatise-on-thermodynamics-1903/eq-aaafaa01bc", "planck-treatise-on-thermodynamics-1903/eq-e87e0b12cb", "planck-treatise-on-thermodynamics-1903/eq-d35bef4ef9", "planck-treatise-on-thermodynamics-1903/eq-17b4d2a2ad", "planck-treatise-on-thermodynamics-1903/eq-8024063bd3", "planck-treatise-on-thermodynamics-1903/eq-c2de726761", "planck-treatise-on-thermodynamics-1903/eq-69179b653b", "planck-treatise-on-thermodynamics-1903/eq-fc1d6acce4" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-introduction", "number": "Introduction", "title": "Introduction", "name": "Planck 1903, Introduction", "pages": [ "77", "86" ], "concepts": [ "concept/centre-of-gravity", "concept/conduction-of-heat", "concept/energetics", "concept/friction", "concept/heat", "concept/heat-reservoir", "concept/irreversible-process", "concept/isolated-system", "concept/mechanical-equilibrium", "concept/perfect-gas", "concept/reversible-process", "concept/temperature", "concept/transformability-of-heat-into-work", "experiment/joule-s-experiments", "law/conservation-of-energy", "law/first-law-of-thermodynamics", "law/second-law-of-thermodynamics", "quantity/internal-energy", "quantity/kinetic-energy", "quantity/mechanical-equivalent-of-heat", "quantity/potential-energy", "quantity/temperature" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-52549a3ef9", "planck-treatise-on-thermodynamics-1903/x-cdf2beea30", "planck-treatise-on-thermodynamics-1903/x-42a1fde2ca", "planck-treatise-on-thermodynamics-1903/x-a01a488b6c", "planck-treatise-on-thermodynamics-1903/x-72d86b068f", "planck-treatise-on-thermodynamics-1903/x-3de90059e7", "planck-treatise-on-thermodynamics-1903/x-81f8f59d12", "planck-treatise-on-thermodynamics-1903/x-c3b340d42b", "planck-treatise-on-thermodynamics-1903/x-da96c23daf" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-b561ee08c8" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-proof", "number": "Proof", "title": "Proof", "name": "Planck 1903, Proof", "pages": [ "86", "105" ], "concepts": [ "concept/adiabatic-process", "concept/clausius-equation", "concept/conduction-of-heat", "concept/cycle-of-operations", "concept/dissipation-of-energy", "concept/entropy-of-a-system", "concept/free-expansion-of-a-gas", "concept/friction", "concept/heat", "concept/heat-engine", "concept/heat-reservoir", "concept/irreversible-process", "concept/perfect-gas", "concept/perpetual-motion", "concept/quasi-static-process", "concept/reversible-process", "concept/temperature", "concept/work", "law/conservation-of-energy", "law/first-law-of-thermodynamics", "law/second-law-of-thermodynamics", "person/james-clerk-maxwell", "person/rudolf-clausius", "person/wilhelm-ostwald", "person/william-thomson", "quantity/entropy", "quantity/internal-energy" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-19161c0218", "planck-treatise-on-thermodynamics-1903/x-1f072e70a5", "planck-treatise-on-thermodynamics-1903/x-a93d34a2eb", "planck-treatise-on-thermodynamics-1903/x-0e0d97db1c", "planck-treatise-on-thermodynamics-1903/x-b8f37c7ff9", "planck-treatise-on-thermodynamics-1903/x-90e705d634", "planck-treatise-on-thermodynamics-1903/x-fc6f56c954", "planck-treatise-on-thermodynamics-1903/x-0e16568dd4", "planck-treatise-on-thermodynamics-1903/x-8a2fb8b5ad", "planck-treatise-on-thermodynamics-1903/x-52909a0434", "planck-treatise-on-thermodynamics-1903/x-7373396881", "planck-treatise-on-thermodynamics-1903/x-f0e3920eeb", "planck-treatise-on-thermodynamics-1903/x-1bd3cf2985", "planck-treatise-on-thermodynamics-1903/x-23128a9bcf", "planck-treatise-on-thermodynamics-1903/x-c3daaad29c", "planck-treatise-on-thermodynamics-1903/x-54c5d37511" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-dd8269b127", "planck-treatise-on-thermodynamics-1903/eq-baf5e6ad97", "planck-treatise-on-thermodynamics-1903/eq-ba24371b22", "planck-treatise-on-thermodynamics-1903/eq-bab4a8c510", "planck-treatise-on-thermodynamics-1903/eq-c1d65fe52e", "planck-treatise-on-thermodynamics-1903/eq-ad3d941ab7", "planck-treatise-on-thermodynamics-1903/eq-742e0cfea6", "planck-treatise-on-thermodynamics-1903/eq-d42a342646", "planck-treatise-on-thermodynamics-1903/eq-a36e82dd9d", "planck-treatise-on-thermodynamics-1903/eq-d2339480c5", "planck-treatise-on-thermodynamics-1903/eq-fd0de5c196", "planck-treatise-on-thermodynamics-1903/eq-15461fd21a", "planck-treatise-on-thermodynamics-1903/eq-2a94b0480b", "planck-treatise-on-thermodynamics-1903/eq-4a40bfb19e", "planck-treatise-on-thermodynamics-1903/eq-5bbc3f5dc7", "planck-treatise-on-thermodynamics-1903/eq-6fb45885ba", "planck-treatise-on-thermodynamics-1903/eq-863ead6ae6", "planck-treatise-on-thermodynamics-1903/eq-45f847b5bb", "planck-treatise-on-thermodynamics-1903/eq-63290a0a99", "planck-treatise-on-thermodynamics-1903/eq-eb734c0854", "planck-treatise-on-thermodynamics-1903/eq-5ac1e4955a", "planck-treatise-on-thermodynamics-1903/eq-3448721ed2", "planck-treatise-on-thermodynamics-1903/eq-856c171dcb", "planck-treatise-on-thermodynamics-1903/eq-c7b1f71a97", "planck-treatise-on-thermodynamics-1903/eq-71ad805282", "planck-treatise-on-thermodynamics-1903/eq-75d40f792e", "planck-treatise-on-thermodynamics-1903/eq-7d92ce9621", "planck-treatise-on-thermodynamics-1903/eq-08681809d4", "planck-treatise-on-thermodynamics-1903/eq-0d8f61ef4a", "planck-treatise-on-thermodynamics-1903/eq-a311cc7376" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "number": "General Deductions", "title": "General Deductions", "name": "Planck 1903, General Deductions", "pages": [ "105", "119" ], "concepts": [ "concept/adiabatic-process", "concept/carnot-cycle", "concept/chemical-affinity", "concept/chemical-reaction", "concept/conduction", "concept/cycle-of-operations", "concept/exact-differential", "concept/heat-reservoir", "concept/inertia-resistance", "concept/irreversible-process", "concept/isothermal-isopiestic-process", "concept/isothermal-process", "concept/maximum", "concept/maximum-work", "concept/perfect-gas", "concept/reversible-process", "concept/stability-of-equilibrium", "concept/thermodynamic-equilibrium", "concept/virtual-change", "concept/work", "law/first-law-of-thermodynamics", "law/second-law-of-thermodynamics", "method/carnot-cycle", "person/berthelot", "person/hermann-von-helmholtz", "person/pierre-duhem", "quantity/entropy", "quantity/free-energy", "quantity/heat-effect", "quantity/internal-energy", "quantity/latent-energy", "quantity/psi-function", "quantity/thermodynamic-potential", "theorem/berthelot-s-principle" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-bf35243829", "planck-treatise-on-thermodynamics-1903/x-ceefef0202", "planck-treatise-on-thermodynamics-1903/x-2548d13012", "planck-treatise-on-thermodynamics-1903/x-bf7b8ee747", "planck-treatise-on-thermodynamics-1903/x-514d3c7197", "planck-treatise-on-thermodynamics-1903/x-ebd4106d72", "planck-treatise-on-thermodynamics-1903/x-7a946b39ce", "planck-treatise-on-thermodynamics-1903/x-9dea7c9d12", "planck-treatise-on-thermodynamics-1903/x-715292d9b8", "planck-treatise-on-thermodynamics-1903/x-43698ee333", "planck-treatise-on-thermodynamics-1903/x-4d7ac252cd", "planck-treatise-on-thermodynamics-1903/x-d9cffa220f", "planck-treatise-on-thermodynamics-1903/x-4a3486fe52", "planck-treatise-on-thermodynamics-1903/x-0d69322d5d", "planck-treatise-on-thermodynamics-1903/x-2d70f48d9d", "planck-treatise-on-thermodynamics-1903/x-cb514a3342" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-3ae8f63545", "planck-treatise-on-thermodynamics-1903/eq-c8503d68e9", "planck-treatise-on-thermodynamics-1903/eq-7cc98df91a", "planck-treatise-on-thermodynamics-1903/eq-acabf38579", "planck-treatise-on-thermodynamics-1903/eq-f003fe1084", "planck-treatise-on-thermodynamics-1903/eq-fc0a546426", "planck-treatise-on-thermodynamics-1903/eq-2f7542b1c4", "planck-treatise-on-thermodynamics-1903/eq-c67635035f", "planck-treatise-on-thermodynamics-1903/eq-54ff2f2157", "planck-treatise-on-thermodynamics-1903/eq-f4cb016daf", "planck-treatise-on-thermodynamics-1903/eq-e30dfb0d5e", "planck-treatise-on-thermodynamics-1903/eq-3f572b1d00", "planck-treatise-on-thermodynamics-1903/eq-df10b04ea3", "planck-treatise-on-thermodynamics-1903/eq-03350b4e50", "planck-treatise-on-thermodynamics-1903/eq-1feca011d2", "planck-treatise-on-thermodynamics-1903/eq-af58ce4572", "planck-treatise-on-thermodynamics-1903/eq-c61a7f871f", "planck-treatise-on-thermodynamics-1903/eq-28f91b322e", "planck-treatise-on-thermodynamics-1903/eq-b29a20472d", "planck-treatise-on-thermodynamics-1903/eq-274e2abc97", "planck-treatise-on-thermodynamics-1903/eq-c403953b9f", "planck-treatise-on-thermodynamics-1903/eq-718f9195c2", "planck-treatise-on-thermodynamics-1903/eq-28059cbaac", "planck-treatise-on-thermodynamics-1903/eq-9419f48d5a", "planck-treatise-on-thermodynamics-1903/eq-6fbf277d92", "planck-treatise-on-thermodynamics-1903/eq-d5b1ed871a", "planck-treatise-on-thermodynamics-1903/eq-a4c9292168", "planck-treatise-on-thermodynamics-1903/eq-953e98471a", "planck-treatise-on-thermodynamics-1903/eq-eb36f42371", "planck-treatise-on-thermodynamics-1903/eq-e8422977e1", "planck-treatise-on-thermodynamics-1903/eq-e34ae472cf", "planck-treatise-on-thermodynamics-1903/eq-df6cf39e19", "planck-treatise-on-thermodynamics-1903/eq-705171867c", "planck-treatise-on-thermodynamics-1903/eq-2d45e60f0c", "planck-treatise-on-thermodynamics-1903/eq-d9e7a6e00e", "planck-treatise-on-thermodynamics-1903/eq-6edd246f27", "planck-treatise-on-thermodynamics-1903/eq-0a1d30c910", "planck-treatise-on-thermodynamics-1903/eq-63290a0a99", "planck-treatise-on-thermodynamics-1903/eq-10633e48da", "planck-treatise-on-thermodynamics-1903/eq-98fe49b50b", "planck-treatise-on-thermodynamics-1903/eq-35a5246563", "planck-treatise-on-thermodynamics-1903/eq-fb8fa599dd", "planck-treatise-on-thermodynamics-1903/eq-50057181f1", "planck-treatise-on-thermodynamics-1903/eq-a8b2b09b7c", "planck-treatise-on-thermodynamics-1903/eq-423a81a84c", "planck-treatise-on-thermodynamics-1903/eq-fe81ffe3a2", "planck-treatise-on-thermodynamics-1903/eq-30d368b06e", "planck-treatise-on-thermodynamics-1903/eq-d50bdf4ffa", "planck-treatise-on-thermodynamics-1903/eq-2f39329449" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "number": "Homogeneous Systems", "title": "Homogeneous Systems", "name": "Planck 1903, Homogeneous Systems", "pages": [ "119", "132" ], "concepts": [ "concept/characteristic-equation", "concept/constant-of-integration", "concept/difference-of-specific-heats", "concept/differential", "concept/homogeneous-system", "concept/mercury", "concept/partial-derivative", "concept/perfect-gas", "concept/pressure", "concept/temperature", "concept/work", "experiment/joule-s-experiments", "experiment/joule-thomson-experiment", "instrument/gas-thermometer", "law/first-law-of-thermodynamics", "law/gay-lussac-s-law", "law/second-law-of-thermodynamics", "person/james-prescott-joule", "person/william-thomson", "quantity/absolute-temperature", "quantity/coefficient-of-compressibility", "quantity/coefficient-of-expansion", "quantity/entropy", "quantity/internal-energy", "quantity/mass", "quantity/mechanical-equivalent-of-heat", "quantity/pressure", "quantity/ratio-of-specific-heats", "quantity/specific-energy", "quantity/specific-heat", "quantity/specific-volume", "unit/atmosphere", "unit/calorie" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-1cae37aaab", "planck-treatise-on-thermodynamics-1903/x-93cca2f397", "planck-treatise-on-thermodynamics-1903/x-5fcc9968f4", "planck-treatise-on-thermodynamics-1903/x-6232a89d50", "planck-treatise-on-thermodynamics-1903/x-0cdfc8bb69", "planck-treatise-on-thermodynamics-1903/x-4ee133e4be", "planck-treatise-on-thermodynamics-1903/x-6584d7ae2c", "planck-treatise-on-thermodynamics-1903/x-57908afb82", "planck-treatise-on-thermodynamics-1903/x-4247e40124", "planck-treatise-on-thermodynamics-1903/x-6f509e7d29", "planck-treatise-on-thermodynamics-1903/x-dbdcc94393", "planck-treatise-on-thermodynamics-1903/x-24ab25f3b2", "planck-treatise-on-thermodynamics-1903/x-0236c67f8a", "planck-treatise-on-thermodynamics-1903/x-1417ae5822", "planck-treatise-on-thermodynamics-1903/x-2acf688f4a" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-768f20f649", "planck-treatise-on-thermodynamics-1903/eq-1fa697f9f9", "planck-treatise-on-thermodynamics-1903/eq-5893c35fbe", "planck-treatise-on-thermodynamics-1903/eq-2b8108f944", "planck-treatise-on-thermodynamics-1903/eq-20db95c796", "planck-treatise-on-thermodynamics-1903/eq-f826ba780d", "planck-treatise-on-thermodynamics-1903/eq-1b5e9694e6", "planck-treatise-on-thermodynamics-1903/eq-9994c48efa", "planck-treatise-on-thermodynamics-1903/eq-9fc151304c", "planck-treatise-on-thermodynamics-1903/eq-943dcd1692", "planck-treatise-on-thermodynamics-1903/eq-caf2fa48bd", "planck-treatise-on-thermodynamics-1903/eq-a4afe4cffb", "planck-treatise-on-thermodynamics-1903/eq-304f06735f", "planck-treatise-on-thermodynamics-1903/eq-b16c6bcf63", "planck-treatise-on-thermodynamics-1903/eq-f6ca83f8c0", "planck-treatise-on-thermodynamics-1903/eq-be48d98703", "planck-treatise-on-thermodynamics-1903/eq-f4beb4ab67", "planck-treatise-on-thermodynamics-1903/eq-41b6ba2ef4", "planck-treatise-on-thermodynamics-1903/eq-0a9298afa0", "planck-treatise-on-thermodynamics-1903/eq-9f22c6d0dd", "planck-treatise-on-thermodynamics-1903/eq-930cf79e4f", "planck-treatise-on-thermodynamics-1903/eq-8e95b578e1", "planck-treatise-on-thermodynamics-1903/eq-bf2b77f35c", "planck-treatise-on-thermodynamics-1903/eq-a22e92b01f", "planck-treatise-on-thermodynamics-1903/eq-62d10c7412", "planck-treatise-on-thermodynamics-1903/eq-715f54862f", "planck-treatise-on-thermodynamics-1903/eq-e08af2a16f" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "number": "System in Different States of Aggregation", "title": "System in Different States of Aggregation", "name": "Planck 1903, System in Different States of Aggregation", "pages": [ "148", "168" ], "concepts": [ "concept/adiabatic-process", "concept/carnot-s-theory-of-heat", "concept/cartesian-coordinates", "concept/characteristic-equation", "concept/clausius-equation", "concept/corresponding-point", "concept/critical-point", "concept/developable-surface", "concept/discontinuous-function", "concept/energy", 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"person/mile-clapeyron", "person/rudolf-clausius", "person/thomas-andrews", "person/watt", "person/william-thomson", "quantity/absolute-temperature", "quantity/entropy", "quantity/free-energy", "quantity/latent-heat", "quantity/mass", "quantity/pressure", "quantity/specific-heat", "quantity/specific-heat-at-constant-volume", "quantity/specific-heat-of-saturated-vapour", "quantity/specific-volume", "quantity/volume", "theorem/clapeyron-equation", "theorem/taylor-s-theorem" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-b430526c8f", "planck-treatise-on-thermodynamics-1903/x-4bf41f0ef0", "planck-treatise-on-thermodynamics-1903/x-d6714cf392", "planck-treatise-on-thermodynamics-1903/x-e5e16f18ad", "planck-treatise-on-thermodynamics-1903/x-a6ad026515", "planck-treatise-on-thermodynamics-1903/x-52efad7777", "planck-treatise-on-thermodynamics-1903/x-4b6def4bd2", "planck-treatise-on-thermodynamics-1903/x-e05586e1cd", "planck-treatise-on-thermodynamics-1903/x-fa8b56442e", "planck-treatise-on-thermodynamics-1903/x-2cf607f4a1", "planck-treatise-on-thermodynamics-1903/x-5b615b0880", "planck-treatise-on-thermodynamics-1903/x-ab901604c4", "planck-treatise-on-thermodynamics-1903/x-93a41a8154", "planck-treatise-on-thermodynamics-1903/x-789c47ab04", "planck-treatise-on-thermodynamics-1903/x-18026c3026", "planck-treatise-on-thermodynamics-1903/x-6269520229", "planck-treatise-on-thermodynamics-1903/x-6c79695e6d", "planck-treatise-on-thermodynamics-1903/x-8dfb074ac8", "planck-treatise-on-thermodynamics-1903/x-16659e19ff", "planck-treatise-on-thermodynamics-1903/x-12be07168f", "planck-treatise-on-thermodynamics-1903/x-ebcf478661", "planck-treatise-on-thermodynamics-1903/x-ac42f74819", "planck-treatise-on-thermodynamics-1903/x-55f3643342", "planck-treatise-on-thermodynamics-1903/x-adb7b613ae", "planck-treatise-on-thermodynamics-1903/x-c024ed0e9c", "planck-treatise-on-thermodynamics-1903/x-f4015effd1", "planck-treatise-on-thermodynamics-1903/x-257e7d770f", "planck-treatise-on-thermodynamics-1903/x-bb39362914", "planck-treatise-on-thermodynamics-1903/x-8eea8e5b65", "planck-treatise-on-thermodynamics-1903/x-50dcfeef8c", "planck-treatise-on-thermodynamics-1903/x-cb51ca0bfe", "planck-treatise-on-thermodynamics-1903/x-20f2daf53d", "planck-treatise-on-thermodynamics-1903/x-2025b6be93", "planck-treatise-on-thermodynamics-1903/x-8382a91a0c", "planck-treatise-on-thermodynamics-1903/x-8e4dbfe46d", "planck-treatise-on-thermodynamics-1903/x-ccd09d75e9", "planck-treatise-on-thermodynamics-1903/x-caaea4eaef" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-cc1a2bbf03", "planck-treatise-on-thermodynamics-1903/eq-4c9492ab70", "planck-treatise-on-thermodynamics-1903/eq-f69d70f052", "planck-treatise-on-thermodynamics-1903/eq-862a1aa452", "planck-treatise-on-thermodynamics-1903/eq-16f627a1ce", "planck-treatise-on-thermodynamics-1903/eq-be4970fddc", "planck-treatise-on-thermodynamics-1903/eq-96bf2f9e25", "planck-treatise-on-thermodynamics-1903/eq-1b759eb264", "planck-treatise-on-thermodynamics-1903/eq-63c03e0514", "planck-treatise-on-thermodynamics-1903/eq-7e48c39cf6", "planck-treatise-on-thermodynamics-1903/eq-51eabeb049", "planck-treatise-on-thermodynamics-1903/eq-58bcda4bf2", "planck-treatise-on-thermodynamics-1903/eq-e80287e960", "planck-treatise-on-thermodynamics-1903/eq-1553950291", "planck-treatise-on-thermodynamics-1903/eq-05b493a4fc", "planck-treatise-on-thermodynamics-1903/eq-c0e9d20b7c", "planck-treatise-on-thermodynamics-1903/eq-e797c03eda", "planck-treatise-on-thermodynamics-1903/eq-7537f40422", "planck-treatise-on-thermodynamics-1903/eq-9d5123db3f", "planck-treatise-on-thermodynamics-1903/eq-410dc688b1", "planck-treatise-on-thermodynamics-1903/eq-7e7c68c575", "planck-treatise-on-thermodynamics-1903/eq-af02b771c3", "planck-treatise-on-thermodynamics-1903/eq-32276a2a3d", "planck-treatise-on-thermodynamics-1903/eq-37da518888", "planck-treatise-on-thermodynamics-1903/eq-207e4ca161", "planck-treatise-on-thermodynamics-1903/eq-9a264085fa", "planck-treatise-on-thermodynamics-1903/eq-38120d1693", "planck-treatise-on-thermodynamics-1903/eq-31c9571ea4", "planck-treatise-on-thermodynamics-1903/eq-cc53aa4623", "planck-treatise-on-thermodynamics-1903/eq-44bd0fd06a", "planck-treatise-on-thermodynamics-1903/eq-e82a962627", "planck-treatise-on-thermodynamics-1903/eq-f9d0f5989c", "planck-treatise-on-thermodynamics-1903/eq-f5f66c9c80", "planck-treatise-on-thermodynamics-1903/eq-d2969e0824", "planck-treatise-on-thermodynamics-1903/eq-cbf43437fe", "planck-treatise-on-thermodynamics-1903/eq-cc4f77f7f5", "planck-treatise-on-thermodynamics-1903/eq-8eab142a41", "planck-treatise-on-thermodynamics-1903/eq-020abfefa5", "planck-treatise-on-thermodynamics-1903/eq-bb15d0ce16", "planck-treatise-on-thermodynamics-1903/eq-e840de8f24", "planck-treatise-on-thermodynamics-1903/eq-eb59fef41f", "planck-treatise-on-thermodynamics-1903/eq-aa13c73477", "planck-treatise-on-thermodynamics-1903/eq-7216db55c2", "planck-treatise-on-thermodynamics-1903/eq-c35c28f7dc", "planck-treatise-on-thermodynamics-1903/eq-a853c4143d", "planck-treatise-on-thermodynamics-1903/eq-096537c457", "planck-treatise-on-thermodynamics-1903/eq-216ee73835", "planck-treatise-on-thermodynamics-1903/eq-d51c825f14", "planck-treatise-on-thermodynamics-1903/eq-e1c8b41c66", "planck-treatise-on-thermodynamics-1903/eq-3e68783718", "planck-treatise-on-thermodynamics-1903/eq-29a3789d2a", "planck-treatise-on-thermodynamics-1903/eq-3e97d4f3ce", "planck-treatise-on-thermodynamics-1903/eq-d912d81fbb", "planck-treatise-on-thermodynamics-1903/eq-bbbff32189", "planck-treatise-on-thermodynamics-1903/eq-42f682eb81", "planck-treatise-on-thermodynamics-1903/eq-a0564c7b66", "planck-treatise-on-thermodynamics-1903/eq-8aa7e87c09", "planck-treatise-on-thermodynamics-1903/eq-144dcc2370", "planck-treatise-on-thermodynamics-1903/eq-da7931c349", "planck-treatise-on-thermodynamics-1903/eq-33db367916", "planck-treatise-on-thermodynamics-1903/eq-c239bcbfbc", "planck-treatise-on-thermodynamics-1903/eq-9de0df905f", "planck-treatise-on-thermodynamics-1903/eq-46502a4bf4", "planck-treatise-on-thermodynamics-1903/eq-3ea07f64af", "planck-treatise-on-thermodynamics-1903/eq-903d77ccfb", "planck-treatise-on-thermodynamics-1903/eq-a670e7d455", "planck-treatise-on-thermodynamics-1903/eq-b88a6d8656", "planck-treatise-on-thermodynamics-1903/eq-0f0b2f3608", "planck-treatise-on-thermodynamics-1903/eq-1d2aa1d4ab", "planck-treatise-on-thermodynamics-1903/eq-21276ad018", "planck-treatise-on-thermodynamics-1903/eq-92ebe5a75d", "planck-treatise-on-thermodynamics-1903/eq-b731185761", "planck-treatise-on-thermodynamics-1903/eq-b4323c460a", "planck-treatise-on-thermodynamics-1903/eq-b600679148", "planck-treatise-on-thermodynamics-1903/eq-a80d3df803", "planck-treatise-on-thermodynamics-1903/eq-1e8aa2a618", "planck-treatise-on-thermodynamics-1903/eq-61f8624ddc", "planck-treatise-on-thermodynamics-1903/eq-5e15281954", "planck-treatise-on-thermodynamics-1903/eq-d5d509585f", "planck-treatise-on-thermodynamics-1903/eq-088141d2af" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "number": "System of any Number of Independent Constituents", "title": "System of any Number of Independent Constituents", "name": "Planck 1903, System of any Number of Independent Constituents", "pages": [ "173", "189" ], "concepts": [ "concept/boiling-point-elevation", "concept/chemical-element", "concept/concentration", "concept/condensed-system", "concept/dissociation", "concept/dissolved-substance", "concept/divariant-system", "concept/emulsion", "concept/external-conditions-of-equilibrium", "concept/freezing-point-depression", "concept/heat-of-removal", "concept/homogeneous-function", "concept/homogeneous-system", "concept/independent-constituent", "concept/internal-conditions-of-equilibrium", "concept/lowering-of-vapour-pressure", "concept/non-variant-system", "concept/perfect-gas", "concept/phase", "concept/quadruple-point", "concept/quintuple-point", "concept/semipermeable-membrane", "concept/solution", "concept/solvent", "concept/state-of-aggregation", "concept/thermodynamic-equilibrium", "concept/triple-point", "concept/univariant-system", "law/babo-s-law", "law/boyle-s-law", "law/first-law-of-thermodynamics", "law/w-llner-s-law", "person/bakhuis-roozeboom", "person/gibbs", "person/gustav-robert-kirchhoff", "person/leonhard-euler", "quantity/entropy", "quantity/heat-effect", "quantity/heat-of-dilution", "quantity/internal-energy", "quantity/latent-heat", "quantity/osmotic-pressure", "quantity/pressure", "quantity/psi-function", "quantity/solution-characteristic-function-phi", "quantity/specific-volume", "quantity/volume", "theorem/euler-s-theorem-for-homogeneous-functions", "theorem/kirchhoff-s-formula", "theorem/lowering-of-vapour-pressure", "theorem/phase-rule" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-53ec4848ce", "planck-treatise-on-thermodynamics-1903/x-65e22f9071", "planck-treatise-on-thermodynamics-1903/x-02d92ed8a6", "planck-treatise-on-thermodynamics-1903/x-9319911293", "planck-treatise-on-thermodynamics-1903/x-d139671e87", "planck-treatise-on-thermodynamics-1903/x-620df8855f", "planck-treatise-on-thermodynamics-1903/x-d38baf88b5", "planck-treatise-on-thermodynamics-1903/x-38f99da6b1", "planck-treatise-on-thermodynamics-1903/x-e58310ee9c", "planck-treatise-on-thermodynamics-1903/x-534eff6066", "planck-treatise-on-thermodynamics-1903/x-81f958bbb8", "planck-treatise-on-thermodynamics-1903/x-36c6eef31e", "planck-treatise-on-thermodynamics-1903/x-45f113a3bd", "planck-treatise-on-thermodynamics-1903/x-c93fa15efe", "planck-treatise-on-thermodynamics-1903/x-e18b91218a", "planck-treatise-on-thermodynamics-1903/x-540191e557", "planck-treatise-on-thermodynamics-1903/x-ad4aaa4fd3", "planck-treatise-on-thermodynamics-1903/x-4873cc04e9", "planck-treatise-on-thermodynamics-1903/x-726d0019ab", "planck-treatise-on-thermodynamics-1903/x-e1b331b927", "planck-treatise-on-thermodynamics-1903/x-8fabe345be" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-cd704b4aa5", "planck-treatise-on-thermodynamics-1903/eq-6dcded4acb", "planck-treatise-on-thermodynamics-1903/eq-c31d64130f", "planck-treatise-on-thermodynamics-1903/eq-b11f7960a4", "planck-treatise-on-thermodynamics-1903/eq-6922eedbc9", "planck-treatise-on-thermodynamics-1903/eq-a508b10cf9", "planck-treatise-on-thermodynamics-1903/eq-9d17b9e0d6", "planck-treatise-on-thermodynamics-1903/eq-70db3afef3", "planck-treatise-on-thermodynamics-1903/eq-80d55a3b17", "planck-treatise-on-thermodynamics-1903/eq-a419dde12a", "planck-treatise-on-thermodynamics-1903/eq-a73da99b73", "planck-treatise-on-thermodynamics-1903/eq-4598c83ef4", "planck-treatise-on-thermodynamics-1903/eq-d94e45e2ac", "planck-treatise-on-thermodynamics-1903/eq-c74fd04a22", "planck-treatise-on-thermodynamics-1903/eq-f64240e7c6", "planck-treatise-on-thermodynamics-1903/eq-aac5b85949", "planck-treatise-on-thermodynamics-1903/eq-2d6bda1396", "planck-treatise-on-thermodynamics-1903/eq-96aea18a0c", "planck-treatise-on-thermodynamics-1903/eq-e1820622d2", "planck-treatise-on-thermodynamics-1903/eq-803d5eaa28", "planck-treatise-on-thermodynamics-1903/eq-1fb2ee7621", "planck-treatise-on-thermodynamics-1903/eq-596f9caf74", "planck-treatise-on-thermodynamics-1903/eq-15b391c81d", "planck-treatise-on-thermodynamics-1903/eq-e80ab95979", "planck-treatise-on-thermodynamics-1903/eq-e4e744706e", "planck-treatise-on-thermodynamics-1903/eq-5482c8baa1", "planck-treatise-on-thermodynamics-1903/eq-f1e6a049fd", "planck-treatise-on-thermodynamics-1903/eq-89590a0259", "planck-treatise-on-thermodynamics-1903/eq-3d7f12c290", "planck-treatise-on-thermodynamics-1903/eq-607065685d", "planck-treatise-on-thermodynamics-1903/eq-9fa770a29d", "planck-treatise-on-thermodynamics-1903/eq-5f3e4dda36", "planck-treatise-on-thermodynamics-1903/eq-60973405e7", "planck-treatise-on-thermodynamics-1903/eq-7f591321af", "planck-treatise-on-thermodynamics-1903/eq-58c67cad38", "planck-treatise-on-thermodynamics-1903/eq-b4360c8e14", "planck-treatise-on-thermodynamics-1903/eq-4a988168aa", "planck-treatise-on-thermodynamics-1903/eq-56c9e829d2", "planck-treatise-on-thermodynamics-1903/eq-b5a63bc1dc", "planck-treatise-on-thermodynamics-1903/eq-dbf91c6b02", "planck-treatise-on-thermodynamics-1903/eq-5bb5d6a375", "planck-treatise-on-thermodynamics-1903/eq-7461956fd9", "planck-treatise-on-thermodynamics-1903/eq-c80a64fa97", "planck-treatise-on-thermodynamics-1903/eq-012b14d2bf", "planck-treatise-on-thermodynamics-1903/eq-9f2feaa4bb" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "number": "Gaseous System", "title": "Gaseous System", "name": "Planck 1903, Gaseous System", "pages": [ "207", "223" ], "concepts": [ "concept/concentration", "concept/constant-of-integration", "concept/diffusion", "concept/dissociation-of-hydriodic-acid", "concept/dissociation-of-iodine-vapour", "concept/gas-mixture", "concept/graded-dissociation", "concept/perfect-gas", "concept/reversible-process", "concept/semipermeable-membrane", "law/gibbs-s-proposition", "quantity/atomic-heat", "quantity/heat-effect", "quantity/partial-pressure", "theorem/condition-of-chemical-equilibrium-in-a-gas-mixture" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-50464fe9d5", "planck-treatise-on-thermodynamics-1903/x-1bccad4a68", "planck-treatise-on-thermodynamics-1903/x-0414ca9294", "planck-treatise-on-thermodynamics-1903/x-f700b7e73f", "planck-treatise-on-thermodynamics-1903/x-75852028b9", "planck-treatise-on-thermodynamics-1903/x-5c9d422e7b", "planck-treatise-on-thermodynamics-1903/x-12097863e9", "planck-treatise-on-thermodynamics-1903/x-930b63679d" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-e80ab95979", "planck-treatise-on-thermodynamics-1903/eq-b1eb743090", "planck-treatise-on-thermodynamics-1903/eq-ec089e71f7", "planck-treatise-on-thermodynamics-1903/eq-694651c1cd", "planck-treatise-on-thermodynamics-1903/eq-75d40f792e", "planck-treatise-on-thermodynamics-1903/eq-f5250d52e3", "planck-treatise-on-thermodynamics-1903/eq-6af7933b49", "planck-treatise-on-thermodynamics-1903/eq-9ec19c7703", "planck-treatise-on-thermodynamics-1903/eq-f9704b4c3c", "planck-treatise-on-thermodynamics-1903/eq-659a2ae639", "planck-treatise-on-thermodynamics-1903/eq-4b9c9201a5", "planck-treatise-on-thermodynamics-1903/eq-165068cebd", "planck-treatise-on-thermodynamics-1903/eq-42966d88eb", "planck-treatise-on-thermodynamics-1903/eq-838ae84f1d", "planck-treatise-on-thermodynamics-1903/eq-3d9f7e917f", "planck-treatise-on-thermodynamics-1903/eq-b87721437a", "planck-treatise-on-thermodynamics-1903/eq-60cb399748", "planck-treatise-on-thermodynamics-1903/eq-e4e744706e", "planck-treatise-on-thermodynamics-1903/eq-8c9fa502a4", "planck-treatise-on-thermodynamics-1903/eq-6ce771b120", "planck-treatise-on-thermodynamics-1903/eq-d434911f3d", "planck-treatise-on-thermodynamics-1903/eq-89dea9bc90", "planck-treatise-on-thermodynamics-1903/eq-cff68b7335", "planck-treatise-on-thermodynamics-1903/eq-7077c547d3", "planck-treatise-on-thermodynamics-1903/eq-e869c35663", "planck-treatise-on-thermodynamics-1903/eq-110d8ba3fd", "planck-treatise-on-thermodynamics-1903/eq-8d705882b5", "planck-treatise-on-thermodynamics-1903/eq-a9c58ed43f", "planck-treatise-on-thermodynamics-1903/eq-b10a5af58f", "planck-treatise-on-thermodynamics-1903/eq-5430b27408", "planck-treatise-on-thermodynamics-1903/eq-de0938ee8b", "planck-treatise-on-thermodynamics-1903/eq-0bfd1e9695", "planck-treatise-on-thermodynamics-1903/eq-7619cd1f67", "planck-treatise-on-thermodynamics-1903/eq-803c6efb62", "planck-treatise-on-thermodynamics-1903/eq-92428de629", "planck-treatise-on-thermodynamics-1903/eq-3b05497970", "planck-treatise-on-thermodynamics-1903/eq-2b7b2536cd", "planck-treatise-on-thermodynamics-1903/eq-fc8f50bb1a", "planck-treatise-on-thermodynamics-1903/eq-dfec285d1e", "planck-treatise-on-thermodynamics-1903/eq-0fb43ebc55", "planck-treatise-on-thermodynamics-1903/eq-90b3d8b451" ], "exercise_sets": [] }, { "id": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "number": "Dilute Solutions", "title": "Dilute Solutions", "name": "Planck 1903, Dilute Solutions", "pages": [ "240", "258" ], "concepts": [ "concept/boiling-point-elevation", "concept/characteristic-equation", "concept/chemical-reaction", "concept/common-ion", "concept/common-ion-effect", "concept/concentration", "concept/dilute-solution", "concept/dissociation", "concept/dissolved-substance", "concept/electrolyte", "concept/equilibrium-constant", "concept/exact-differential", "concept/freezing-point-depression", "concept/heat-of-solidification", "concept/homogeneous-function", "concept/independent-constituent", "concept/ion", "concept/isohydric-solution", "concept/isothermal-isopiestic-change", "concept/linear-function", "concept/molecule", "concept/perfect-gas", "concept/phase", "concept/phase-rule", "concept/semipermeable-membrane", "concept/solubility", "concept/solubility-product", "concept/solution", "concept/solvent", "concept/state-of-aggregation", "concept/thermodynamic-equilibrium", "law/distribution-law", "law/first-law-of-thermodynamics", "law/henry-s-law", "law/law-of-dissociation-of-an-electrolyte", "law/ostwald-s-law-of-dilution", "person/arrhenius", "person/berthelot", "person/gibbs", "person/j-thomsen", "person/jahn", "person/kohlrausch", "person/naccari", "person/nernst", "person/noyes", "person/pagliani", "person/raoult", "person/van-t-hoff", "quantity/conductivity", "quantity/entropy", "quantity/heat-effect", "quantity/internal-energy", "quantity/latent-heat", "quantity/osmotic-pressure", "quantity/partial-pressure", "quantity/psi-function", "quantity/solubility", "theorem/lowering-of-vapour-pressure", "theorem/phase-rule", "theorem/taylor-s-series", "unit/calorie" ], "excerpts": [ "planck-treatise-on-thermodynamics-1903/x-08e4f44d03", "planck-treatise-on-thermodynamics-1903/x-fd5296f13a", "planck-treatise-on-thermodynamics-1903/x-671bfa87ee", "planck-treatise-on-thermodynamics-1903/x-c6d01847e2", "planck-treatise-on-thermodynamics-1903/x-82ac73f2a2", "planck-treatise-on-thermodynamics-1903/x-a1c3bb8843", "planck-treatise-on-thermodynamics-1903/x-c51ff4a18e", "planck-treatise-on-thermodynamics-1903/x-68f69cf0c7", "planck-treatise-on-thermodynamics-1903/x-f72481da90", "planck-treatise-on-thermodynamics-1903/x-26d051c382", "planck-treatise-on-thermodynamics-1903/x-db640a6d11", "planck-treatise-on-thermodynamics-1903/x-49d6e4168e", "planck-treatise-on-thermodynamics-1903/x-92bcce3b9a", "planck-treatise-on-thermodynamics-1903/x-80526b2ea4", "planck-treatise-on-thermodynamics-1903/x-3bcdc5df6f", "planck-treatise-on-thermodynamics-1903/x-aa1bfbba58", "planck-treatise-on-thermodynamics-1903/x-b9e0190466", "planck-treatise-on-thermodynamics-1903/x-75048e68ce", "planck-treatise-on-thermodynamics-1903/x-89c5e481ad", "planck-treatise-on-thermodynamics-1903/x-5c3ee09825", "planck-treatise-on-thermodynamics-1903/x-a7e4a4faed", "planck-treatise-on-thermodynamics-1903/x-bc98e0aaf8", "planck-treatise-on-thermodynamics-1903/x-dd02ad3dbd", "planck-treatise-on-thermodynamics-1903/x-52c41cb80a" ], "equations": [ "planck-treatise-on-thermodynamics-1903/eq-88bce78dd2", "planck-treatise-on-thermodynamics-1903/eq-aa1712a81e", "planck-treatise-on-thermodynamics-1903/eq-704f940bd9", 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"planck-treatise-on-thermodynamics-1903/eq-b59bb3a5bd" ], "exercise_sets": [] } ], "excerpts": [ { "id": "planck-treatise-on-thermodynamics-1903/x-f0ed71f2a8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "2", "location": "Temperature", "latex": "Two bodies of equal temperature are, therefore, in thermal equilibrium, and \\textit{vice versâ}.", "markdown": "Two bodies of equal temperature are, therefore, in thermal equilibrium, and *vice versâ*.", "why": "It states in one line the link that lets a thermometer compare two bodies without bringing them into contact.", "use": [ "lesson" ], "concepts": [ "concept/temperature", "concept/thermal-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8f23357591", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "2", "location": "Temperature", "latex": "If a body,~$A$, be in thermal equilibrium with two other bodies, $B$~and~$C$, then $B$~and $C$ are in thermal equilibrium with one another.", "markdown": "If a body, $A$, be in thermal equilibrium with two other bodies, $B$ and $C$, then $B$ and $C$ are in thermal equilibrium with one another.", "why": "It gives the transitivity of thermal equilibrium that makes temperature a usable comparison, stated as a proposition a learner can test.", "use": [ "lesson" ], "concepts": [ "concept/temperature", "concept/thermal-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-bbf798c0fd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "3", "location": "Temperature", "latex": "The definition of temperature remains arbitrary in cases where the requirements of accuracy cannot be satisfied by the agreement between the readings of the different gas thermometers, for there is no sufficient reason for the preference of any one of these gases.", "markdown": "The definition of temperature remains arbitrary in cases where the requirements of accuracy cannot be satisfied by the agreement between the readings of the different gas thermometers, for there is no sufficient reason for the preference of any one of these gases.", "why": "It shows learners that the gas scale is a practical convention, not a final definition, which is a good check on overconfidence about what temperature 'is'.", "use": [ "lesson", "history" ], "concepts": [ "concept/temperature", "instrument/gas-thermometer" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-777a55ea24", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "7", "location": "Temperature", "latex": "\\emph{Coefficient of elasticity} is the ratio of an\ninfinitely small increase of pressure to the resulting contraction\nof unit volume of the substance.", "markdown": "*Coefficient of elasticity* is the ratio of an infinitely small increase of pressure to the resulting contraction of unit volume of the substance.", "why": "It gives a precise, self-contained definition of a compressibility-related coefficient that learners can apply to a real substance.", "use": [ "lesson" ], "concepts": [ "quantity/coefficient-of-elasticity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a62ec68d6b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "13", "location": "Temperature", "latex": "For lower pressures (\\ie\\ to the left of the minimum), the volume decreases at a more rapid rate, with increasing pressure, than in the case of perfect gases; for higher pressures (to the right of the minimum), at a slower rate.", "markdown": "For lower pressures (*i.e.* to the left of the minimum), the volume decreases at a more rapid rate, with increasing pressure, than in the case of perfect gases; for higher pressures (to the right of the minimum), at a slower rate.", "why": "It describes concretely how real gases depart from the perfect gas law, which helps learners read the isotherm diagrams.", "use": [ "lesson" ], "concepts": [ "concept/isotherm", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-d70c2f9ee5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "17", "location": "Temperature", "latex": "Above the critical temperature and critical pressure, condensation does not exist, as the diagram plainly shows.", "markdown": "Above the critical temperature and critical pressure, condensation does not exist, as the diagram plainly shows.", "why": "It states the key fact that explains why early attempts to liquefy hydrogen, oxygen and nitrogen failed.", "use": [ "lesson", "website" ], "concepts": [ "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-bb8953ede5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "18", "location": "Temperature", "latex": "The earlier fundamental distinction between liquids, vapours, and gases should therefore be dropped as no longer tenable.", "markdown": "The earlier fundamental distinction between liquids, vapours, and gases should therefore be dropped as no longer tenable.", "why": "It is a vivid turn in the argument that shows how a physical classification was overturned by new evidence, which suits a history reader.", "use": [ "history", "website" ], "concepts": [ "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-945f66236b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "20", "location": "Temperature", "latex": "Only for gases and vapours does Dalton's law hold, at least with great approximation, that the total pressure of a mixture is the sum of the partial pressures which each gas would exert if it alone filled the total volume at the given temperature.", "markdown": "Only for gases and vapours does Dalton’s law hold, at least with great approximation, that the total pressure of a mixture is the sum of the partial pressures which each gas would exert if it alone filled the total volume at the given temperature.", "why": "It gives the precise scope of Dalton's law, including its limit to gases, so learners do not apply it to liquids or solids.", "use": [ "lesson" ], "concepts": [ "law/dalton-s-law", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-b6c47d5855", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "1", "location": "Temperature", "latex": "This direct sensation, however, furnishes no quantitative scientific measure of a body's state with regard to heat; it yields only qualitative results, which vary according to external circumstances. For quantitative purposes we utilize the change of volume which takes place in all bodies when heated under constant pressure, for this admits of exact measurement.", "markdown": "This direct sensation, however, furnishes no quantitative scientific measure of a body’s state with regard to heat; it yields only qualitative results, which vary according to external circumstances. For quantitative purposes we utilize the change of volume which takes place in all bodies when heated under constant pressure, for this admits of exact measurement.", "why": "Shows why touch cannot serve as a measure and why a measurable change such as volume is needed.", "use": [ "lesson", "website" ], "concepts": [ "quantity/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0f68ff6468", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "2", "location": "Temperature", "latex": "From this follows the important proposition: \\emph{If a body,~$A$, be in thermal equilibrium with two other bodies, $B$~and~$C$, then $B$~and $C$ are in thermal equilibrium with one another.}", "markdown": "From this follows the important proposition: *If a body, $A$, be in thermal equilibrium with two other bodies, $B$ and $C$, then $B$ and $C$ are in thermal equilibrium with one another.*", "why": "States the transitivity of thermal equilibrium, the logical basis for using a thermometer at all.", "use": [ "lesson", "website" ], "concepts": [ "concept/thermal-equilibrium", "law/zeroth-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-fa99663a57", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "3", "location": "Temperature", "latex": "The definition of temperature is therefore somewhat arbitrary. This we may remedy to a certain extent by taking gases, in particular those hard to condense, such as hydrogen, oxygen, nitrogen, and carbon monoxide, as thermometric substances.", "markdown": "The definition of temperature is therefore somewhat arbitrary. This we may remedy to a certain extent by taking gases, in particular those hard to condense, such as hydrogen, oxygen, nitrogen, and carbon monoxide, as thermometric substances.", "why": "Shows that a temperature scale is a choice, and why gases are chosen to make it less arbitrary.", "use": [ "lesson" ], "concepts": [ "instrument/gas-thermometer", "quantity/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-5d210ebfc1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "4", "location": "Temperature", "latex": "The pressure of an atmosphere is the weight of a column of mercury at~$0°$~C., $76~\\Unit{cm.}$ high, and $1~\\Unit{sq.}\\ \\Unit{cm.}$ in cross-section, when placed in mean geographical latitude. This latter condition must be added, because the weight, \\ie\\ the force of the earth's attraction, varies with the locality.", "markdown": "The pressure of an atmosphere is the weight of a column of mercury at $0°$ C., $76~\\Unit{cm.}$ high, and $1~\\Unit{sq.}\\ \\Unit{cm.}$ in cross-section, when placed in mean geographical latitude. This latter condition must be added, because the weight, *i.e.* the force of the earth’s attraction, varies with the locality.", "why": "Defines the atmosphere precisely and explains why even a 'standard' unit needs a stated location.", "use": [ "lesson", "website" ], "concepts": [ "quantity/pressure", "unit/atmosphere" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-075c98ac33", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "16", "location": "Temperature", "latex": "Of these three values (indicated on the figure by $\\alpha$,~$\\beta$,~$\\gamma$, for instance) only the smallest~($\\alpha$) and the largest~($\\gamma$) represent practically realizable states, for at the middle point~($\\beta$) the pressure along the isotherm would increase with increasing volume, and the compressibility would accordingly be negative. Such a state has, therefore, only a theoretical signification.", "markdown": "Of these three values (indicated on the figure by $\\alpha$, $\\beta$, $\\gamma$, for instance) only the smallest ($\\alpha$) and the largest ($\\gamma$) represent practically realizable states, for at the middle point ($\\beta$) the pressure along the isotherm would increase with increasing volume, and the compressibility would accordingly be negative. Such a state has, therefore, only a theoretical signification.", "why": "Teaches how to read a cubic isotherm and why only some mathematical solutions correspond to real states.", "use": [ "lesson" ], "concepts": [ "concept/clausius-equation", "concept/isotherm", "quantity/coefficient-of-compressibility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-75704d8b86", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "18", "location": "Temperature", "latex": "Condensation nowhere occurs in this process, which leads, nevertheless, to a region of purely liquid states. The earlier fundamental distinction between liquids, vapours, and gases should therefore be dropped as no longer tenable.", "markdown": "Condensation nowhere occurs in this process, which leads, nevertheless, to a region of purely liquid states. The earlier fundamental distinction between liquids, vapours, and gases should therefore be dropped as no longer tenable.", "why": "A striking conclusion: gas and liquid are connected continuously by a path around the critical point.", "use": [ "lesson", "website" ], "concepts": [ "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e19e38468c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "20", "location": "Temperature", "latex": "Here the water vapour cannot be supposed to be subject to a pressure of $1~\\Unit{atm.}$, since at $0°$~C. no water vapour exists at this pressure. The only choice remaining is to assign to the air and water vapour a common volume (that of the mixture) and different pressures (partial pressures).", "markdown": "Here the water vapour cannot be supposed to be subject to a pressure of $1~\\Unit{atm.}$, since at $0°$ C. no water vapour exists at this pressure. The only choice remaining is to assign to the air and water vapour a common volume (that of the mixture) and different pressures (partial pressures).", "why": "Gives a concrete argument for why each gas in a mixture fills the whole volume at its own partial pressure.", "use": [ "lesson" ], "concepts": [ "concept/gas-mixture", "law/dalton-s-law", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-296dd10b51", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "23", "location": "Molecular Weight", "latex": "Such a weight is called an \\emph{equivalent weight}. It is arbitrarily fixed for one element---generally for hydrogen at $1~\\Unit{gr.}$---and then the equivalent weight of any other element (\\eg~oxygen) is that weight which will combine with $1~\\Unit{gr.}$ of hydrogen.", "markdown": "Such a weight is called an *equivalent weight*. It is arbitrarily fixed for one element---generally for hydrogen at $1~\\Unit{gr.}$---and then the equivalent weight of any other element (*e.g.* oxygen) is that weight which will combine with $1~\\Unit{gr.}$ of hydrogen.", "why": "It gives the learner the operational definition of equivalent weight by reference to hydrogen, which the rest of the chapter builds on.", "use": [ "lesson", "website" ], "concepts": [ "quantity/equivalent-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8b48c85f26", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "25", "location": "Molecular Weight", "latex": "Hence, \\emph{equal volumes of perfect gases at the same temperature and pressure contain an equal number of molecules} (Avogadro's law).", "markdown": "Hence, *equal volumes of perfect gases at the same temperature and pressure contain an equal number of molecules* (Avogadro’s law).", "why": "It states Avogadro's law in the form a student can apply directly to gas volumes and molecule counts.", "use": [ "lesson" ], "concepts": [ "law/avogadro-s-law", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-cde24d1332", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "26", "location": "Molecular Weight", "latex": "Hence half a molecule of hydrogen is called an atom of hydrogen,~\\ce{H}; similarly, half a molecule of oxygen an atom of oxygen,~\\ce{O}; and half a molecule of nitrogen an atom of nitrogen,~\\ce{N}.", "markdown": "Hence half a molecule of hydrogen is called an atom of hydrogen, H; similarly, half a molecule of oxygen an atom of oxygen, O; and half a molecule of nitrogen an atom of nitrogen, N.", "why": "It shows how the diatomic molecule and the atom are distinguished, a common source of confusion for beginners.", "use": [ "lesson" ], "concepts": [ "concept/atom", "concept/molecule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-dd853603ed", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "24", "location": "Molecular Weight", "latex": "In the definition of the molecular weight as a quite definite quantity depending only on the particular state of a substance, and independent of possible chemical reactions with other substances, lies one of the most important and most fruitful achievements of theoretical chemistry.", "markdown": "In the definition of the molecular weight as a quite definite quantity depending only on the particular state of a substance, and independent of possible chemical reactions with other substances, lies one of the most important and most fruitful achievements of theoretical chemistry.", "why": "It explains why molecular weight was a landmark idea and what makes it a property of the substance rather than of a reaction.", "use": [ "history", "website" ], "concepts": [ "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2f551712bf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "22", "location": "Molecular Weight", "latex": "In a word, we may, in a certain sense, say, that physical changes take place continuously, chemical ones, on the other hand, discontinuously. In consequence, the science of physics deals, primarily, with continuously varying numbers, the science of chemistry, on the contrary, with whole, or with simple rational numbers.", "markdown": "In a word, we may, in a certain sense, say, that physical changes take place continuously, chemical ones, on the other hand, discontinuously. In consequence, the science of physics deals, primarily, with continuously varying numbers, the science of chemistry, on the contrary, with whole, or with simple rational numbers.", "why": "Gives a memorable contrast between continuous physical change and whole-number chemical change, and shows how the book framed the two sciences in 1903.", "use": [ "lesson", "history", "website" ], "concepts": [ "concept/chemical-reaction", "concept/physical-change" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2e95a8871b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "23", "location": "Molecular Weight", "latex": "Thus $16$~parts by weight of oxygen combine with $28$~parts by weight of nitrogen to form nitrous oxide, or with $14$~parts to form \\index{Nitrogen!oxides}% \\index{Oxides of nitrogen}% nitric oxide, or with $9\\frac{1}{3}$~parts to form nitrous anhydride, or with $7$~parts to form nitrogen tetroxide, or with $5\\frac{3}{5}$~parts to form nitric anhydride.", "markdown": "Thus $16$ parts by weight of oxygen combine with $28$ parts by weight of nitrogen to form nitrous oxide, or with $14$ parts to form % % nitric oxide, or with $9\\frac{1}{3}$ parts to form nitrous anhydride, or with $7$ parts to form nitrogen tetroxide, or with $5\\frac{3}{5}$ parts to form nitric anhydride.", "why": "A concrete worked case showing why one element can have several equivalent weights, all in simple ratios.", "use": [ "lesson" ], "concepts": [ "law/law-of-multiple-proportions", "quantity/equivalent-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-b0fb634285", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "24", "location": "Molecular Weight", "latex": "The ambiguity is, however, removed by putting all these ratios $= 1$, \\ie\\ by establishing the condition that equal volumes of different gases shall contain an equal number of equivalents.", "markdown": "The ambiguity is, however, removed by putting all these ratios $= 1$, *i.e.* by establishing the condition that equal volumes of different gases shall contain an equal number of equivalents.", "why": "Shows how a definition is chosen to remove ambiguity, which is how molecular weight and Avogadro's law arise here.", "use": [ "lesson", "history" ], "concepts": [ "law/avogadro-s-law", "quantity/molecular-weight", "quantity/number-of-equivalents" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-16706281af", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "25", "location": "Molecular Weight", "latex": "Thus Avogadro's law enables us to give in quite definite numbers the molecular quantities of each constituent present in the molecule of any chemically homogeneous gas, provided we know its density and its chemical composition.", "markdown": "Thus Avogadro’s law enables us to give in quite definite numbers the molecular quantities of each constituent present in the molecule of any chemically homogeneous gas, provided we know its density and its chemical composition.", "why": "States what the law is good for: finding the composition of a molecule from density and analysis.", "use": [ "lesson", "website" ], "concepts": [ "law/avogadro-s-law", "quantity/density", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-942a3425f4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "which means that at a given temperature and pressure the volume of a quantity of gas depends only on the number of the molecules present, and not at all on the nature of the gas.", "markdown": "which means that at a given temperature and pressure the volume of a quantity of gas depends only on the number of the molecules present, and not at all on the nature of the gas.", "why": "Puts the meaning of V = (R theta/p) n into plain words.", "use": [ "lesson", "website" ], "concepts": [ "concept/perfect-gas", "quantity/absolute-gas-constant", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-f5107d0413", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "31", "location": "Molecular Weight", "latex": "In doubtful cases it is safest, in general, to leave this question open, and to admit both chemical and physical changes as causes for the deviations from the laws of perfect gases.", "markdown": "In doubtful cases it is safest, in general, to leave this question open, and to admit both chemical and physical changes as causes for the deviations from the laws of perfect gases.", "why": "Models scientific caution when the evidence cannot decide between explanations.", "use": [ "lesson", "history" ], "concepts": [ "concept/abnormal-vapour-densities", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-51261c2f1d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "31", "location": "Molecular Weight", "latex": "The molecular weight of sulphur vapour below $800°$, for instance, \\index{Sulphur}% is generally assumed to be $\\ce{S6} = 192$; but some assume a mixture of molecules $\\ce{S8} = 256$ and $\\ce{S2} = 64$, and others still different mixtures.", "markdown": "The molecular weight of sulphur vapour below $800°$, for instance, % is generally assumed to be $\\ce{S6} = 192$; but some assume a mixture of molecules $\\ce{S8} = 256$ and $\\ce{S2} = 64$, and others still different mixtures.", "why": "A historical example of real disagreement among chemists about molecular composition.", "use": [ "history", "website" ], "concepts": [ "concept/abnormal-vapour-densities", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8686f4354a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "32", "location": "Quantity of Heat", "latex": "It was, in general, customary to take as the unit of heat that quantity which must be added to $1~\\Unit{gr.}$ of water to raise its temperature from $0°$~C. to $1°$~C. (zero calorie).", "markdown": "It was, in general, customary to take as the unit of heat that quantity which must be added to $1~\\Unit{gr.}$ of water to raise its temperature from $0°$ C. to $1°$ C. (zero calorie).", "why": "It states the calorie as a clear, reproducible unit defined by a temperature rise in water.", "use": [ "lesson" ], "concepts": [ "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-802b149728", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "33", "location": "Quantity of Heat", "latex": "This, in general, varies with temperature, but very slowly for most substances. It is usually permissible to put the specific heat at a certain temperature equal to the mean specific heat of an adjoining interval of moderate size.", "markdown": "This, in general, varies with temperature, but very slowly for most substances. It is usually permissible to put the specific heat at a certain temperature equal to the mean specific heat of an adjoining interval of moderate size.", "why": "It explains why a specific heat can be treated as nearly constant over a small range, a useful habit when working problems.", "use": [ "lesson" ], "concepts": [ "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-5ba60dbc81", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "35", "location": "Quantity of Heat", "latex": "A gas at~$0°$ and atmospheric pressure can be brought to a state where its temperature is~$100°$ and its pressure $10$~atmospheres, either by heating to~$100°$ under constant pressure, and then compressing at constant temperature; or by compressing isothermally to $10$~atmospheres, and then heating isopiestically to~$100°$; or, finally, by compressing and heating simultaneously or alternately in a variety of ways.", "markdown": "A gas at $0°$ and atmospheric pressure can be brought to a state where its temperature is $100°$ and its pressure $10$ atmospheres, either by heating to $100°$ under constant pressure, and then compressing at constant temperature; or by compressing isothermally to $10$ atmospheres, and then heating isopiestically to $100°$; or, finally, by compressing and heating simultaneously or alternately in a variety of ways.", "why": "It is a concrete worked comparison of several paths between the same two states, showing why the heat absorbed is path-dependent.", "use": [ "lesson" ], "concepts": [ "concept/heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-fe8c8c8404", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "37", "location": "Quantity of Heat", "latex": "It is reckoned \\emph{positive} when heat is set free or developed, \\ie\\ given out by the body (exothermal processes); \\emph{negative}, when heat is absorbed, or rendered latent, \\ie\\ taken up by the body (endothermal processes).", "markdown": "It is reckoned *positive* when heat is set free or developed, *i.e.* given out by the body (exothermal processes); *negative*, when heat is absorbed, or rendered latent, *i.e.* taken up by the body (endothermal processes).", "why": "It fixes the sign convention for heat effects, which learners often mix up.", "use": [ "lesson" ], "concepts": [ "concept/endothermal-process", "concept/exothermal-process", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-7d907c885c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "37", "location": "Quantity of Heat", "latex": "Latent heat, as in the case of specific heat, is best referred, not to unit mass, but to molecular or atomic weight.", "markdown": "Latent heat, as in the case of specific heat, is best referred, not to unit mass, but to molecular or atomic weight.", "why": "It links latent heat to specific heat and points out that referring to molecular weight makes comparisons clearer.", "use": [ "lesson" ], "concepts": [ "quantity/latent-heat", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-26284431f6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "32", "location": "Quantity of Heat", "latex": "If we plunge a piece of iron and a piece of lead, both\nof equal weight and at the same temperature ($100°$~C.), into\ntwo precisely similar vessels containing equal quantities of\nwater at $0°$~C., we find that, after thermal equilibrium has\nbeen established in each case, the vessel containing the iron\nhas increased in temperature much more than that containing\nthe lead. Conversely, a quantity of water at~$100°$ is\ncooled to a much lower temperature by a piece of iron at~$0°$,\nthan by an equal weight of lead at the same temperature.", "markdown": "If we plunge a piece of iron and a piece of lead, both of equal weight and at the same temperature ($100°$ C.), into two precisely similar vessels containing equal quantities of water at $0°$ C., we find that, after thermal equilibrium has been established in each case, the vessel containing the iron has increased in temperature much more than that containing the lead. Conversely, a quantity of water at $100°$ is cooled to a much lower temperature by a piece of iron at $0°$, than by an equal weight of lead at the same temperature.", "why": "A concrete experiment that shows why temperature and quantity of heat are different things.", "use": [ "lesson", "website" ], "concepts": [ "concept/heat", "concept/temperature", "quantity/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-22cd08d8cc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "35", "location": "Quantity of Heat", "latex": "It would be absurd to define the heat contained\nin a body of given temperature, density, etc., as the\nnumber of calories absorbed by the body in its passage from\nsome normal state into its present state, for the quantity\nthus defined would assume different values according to the\nway in which the change was effected.", "markdown": "It would be absurd to define the heat contained in a body of given temperature, density, etc., as the number of calories absorbed by the body in its passage from some normal state into its present state, for the quantity thus defined would assume different values according to the way in which the change was effected.", "why": "Warns learners that heat is not a stored amount determined by a body's state, because it depends on the path.", "use": [ "lesson" ], "concepts": [ "concept/heat", "concept/heat-contained-in-a-body", "method/calorimetry" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a657c2f83d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "36", "location": "Quantity of Heat", "latex": "To explain the rise of temperature which\ntakes place notwithstanding, it was necessary to make the\nassumption that compression and friction so diminish the\nbody's heat capacity, that the same amount of heat now\nproduces a higher temperature, just as, for example, a\nmoist sponge appears more moist if compressed, although\nthe quantity of liquid in the sponge remains the same.", "markdown": "To explain the rise of temperature which takes place notwithstanding, it was necessary to make the assumption that compression and friction so diminish the body’s heat capacity, that the same amount of heat now produces a higher temperature, just as, for example, a moist sponge appears more moist if compressed, although the quantity of liquid in the sponge remains the same.", "why": "Gives the old theory's sponge analogy, a vivid historical picture of heat as a substance.", "use": [ "history", "website" ], "concepts": [ "concept/carnot-s-theory-of-heat", "concept/indestructibility-of-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6448dd8215", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "36", "location": "Quantity of Heat", "latex": "Each one of\nthese experimental results would by itself be sufficient to\ndisprove the hypothesis of the indestructibility of heat, and\nto overthrow the older theory.", "markdown": "Each one of these experimental results would by itself be sufficient to disprove the hypothesis of the indestructibility of heat, and to overthrow the older theory.", "why": "Shows how experimental evidence can overturn an accepted theory.", "use": [ "history", "lesson" ], "concepts": [ "concept/carnot-s-theory-of-heat", "concept/indestructibility-of-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2027d66c8c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "34", "location": "Quantity of Heat", "latex": "That the heat capacities of different substances\nshould be referred to unit mass is quite arbitrary. It arises\nfrom the fact that quantities of matter can be most easily\ncompared by weighing them.", "markdown": "That the heat capacities of different substances should be referred to unit mass is quite arbitrary. It arises from the fact that quantities of matter can be most easily compared by weighing them.", "why": "Tells the learner that the choice of unit mass is a convenience and not a law of nature.", "use": [ "lesson" ], "concepts": [ "quantity/atomic-heat", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ec3eee26b7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "34", "location": "Quantity of Heat", "latex": "It cannot be claimed that\nthis law is rigorously true, since the heat capacity depends\non the molecular constitution, as in the case of carbon, and\non the state of aggregation, as in the case of mercury, as\nwell as on the temperature.", "markdown": "It cannot be claimed that this law is rigorously true, since the heat capacity depends on the molecular constitution, as in the case of carbon, and on the state of aggregation, as in the case of mercury, as well as on the temperature.", "why": "Models honest scientific caution by stating why an empirical law only holds approximately.", "use": [ "lesson", "history" ], "concepts": [ "concept/state-of-aggregation", "law/dulong-and-petit-s-law", "quantity/atomic-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ec33a55743", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "37", "location": "Quantity of Heat", "latex": "At such temperatures\nthe heat absorbed no longer affects the entire body,\nbut only one of the parts into which it has split; and it no\nlonger serves to increase the temperature, but simply to\nalter the state of aggregation, \\ie\\ to melt, evaporate, or\nsublime.", "markdown": "At such temperatures the heat absorbed no longer affects the entire body, but only one of the parts into which it has split; and it no longer serves to increase the temperature, but simply to alter the state of aggregation, *i.e.* to melt, evaporate, or sublime.", "why": "Explains why heat can be added without a temperature rise, which is the idea behind latent heat.", "use": [ "lesson", "website" ], "concepts": [ "concept/singular-value", "concept/state-of-aggregation", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-52ea77912e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "38", "location": "General Exposition", "latex": "\\emph{it is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion}, \\ie\\ it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing.", "markdown": "*it is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion*, *i.e.* it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing.", "why": "Gives the learner the plain experimental fact from which the energy principle is built, stated without any appeal to the mechanical view of nature.", "use": [ "lesson", "history" ], "concepts": [ "concept/perpetual-motion", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-68c365f056", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "44", "location": "General Exposition", "latex": "The mechanical equivalent of the external effects is zero, or the external heat effect is equal in magnitude and opposite in sign to the external work.", "markdown": "The mechanical equivalent of the external effects is zero, or the external heat effect is equal in magnitude and opposite in sign to the external work.", "why": "States in one line why a complete cycle returns no net energy, the core of the impossibility of perpetual motion.", "use": [ "lesson" ], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/perpetual-motion", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6665179cf6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "39", "location": "General Exposition", "latex": "The energy of the system in a given state, referred to the arbitrarily selected normal state, is then equal to \\emph{the algebraic sum of the mechanical equivalents of all the effects produced outside the system when it passes in any way from the given to the normal state}.", "markdown": "The energy of the system in a given state, referred to the arbitrarily selected normal state, is then equal to *the algebraic sum of the mechanical equivalents of all the effects produced outside the system when it passes in any way from the given to the normal state*.", "why": "Gives the book's operational definition of energy as measured against a reference state.", "use": [ "lesson", "website" ], "concepts": [ "concept/energy", "concept/external-effect", "concept/normal-state" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c5846c329b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "41", "location": "General Exposition", "latex": "The only point of importance is that the state produced in the liquid by friction is identical with a state produced by the absorption of a definite number of calories.", "markdown": "The only point of importance is that the state produced in the liquid by friction is identical with a state produced by the absorption of a definite number of calories.", "why": "Shows that the experiment depends only on the resulting state, not on a theory of what heat is.", "use": [ "lesson" ], "concepts": [ "concept/heat", "experiment/joule-s-experiments", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-fabbf3eb9b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "41", "location": "General Exposition", "latex": "That all his experiments with different weights, different calorimetric substances, and different temperatures, led to the same value, goes to prove the correctness of the principle of the conservation of energy.", "markdown": "That all his experiments with different weights, different calorimetric substances, and different temperatures, led to the same value, goes to prove the correctness of the principle of the conservation of energy.", "why": "Shows how varied experiments that agree on one value serve as evidence for a general law.", "use": [ "lesson", "history" ], "concepts": [ "experiment/joule-s-experiments", "law/conservation-of-energy", "person/james-prescott-joule", "quantity/mechanical-equivalent-of-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-3ecebb60f2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "42", "location": "General Exposition", "latex": "The determination of the mechanical equivalent of heat enables us to express quantities of heat in ergs directly, instead of calories. The advantage of this is, that a quantity of heat is not only proportional to, but directly equal to its mechanical equivalent, whereby the mathematical expression for the energy is greatly simplified.", "markdown": "The determination of the mechanical equivalent of heat enables us to express quantities of heat in ergs directly, instead of calories. The advantage of this is, that a quantity of heat is not only proportional to, but directly equal to its mechanical equivalent, whereby the mathematical expression for the energy is greatly simplified.", "why": "Explains why heat is measured in work units from here on.", "use": [ "lesson" ], "concepts": [ "concept/heat", "quantity/mechanical-equivalent-of-heat", "unit/erg" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c035c86d4f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "44", "location": "General Exposition", "latex": "In the case of a gas which is being compressed by a weight sinking to a lower level, if the gas by itself be the system considered, the external effect on it is equal to the work done by the weight. The energy of the system accordingly increases. If, however, the weight and the earth be considered parts of the system, all external effects are eliminated, and the energy of this system remains constant.", "markdown": "In the case of a gas which is being compressed by a weight sinking to a lower level, if the gas by itself be the system considered, the external effect on it is equal to the work done by the weight. The energy of the system accordingly increases. If, however, the weight and the earth be considered parts of the system, all external effects are eliminated, and the energy of this system remains constant.", "why": "Shows that choosing the system's boundary decides what counts as an external effect.", "use": [ "lesson", "website" ], "concepts": [ "concept/external-effect", "concept/perfect-system", "concept/work", "law/conservation-of-energy", "quantity/potential-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6f7376fe8b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "40", "location": "General Exposition", "latex": "Special care must be taken, however, that the initial state of the system is the same each time, and that none of the external effects is overlooked or taken into account more than once.", "markdown": "Special care must be taken, however, that the initial state of the system is the same each time, and that none of the external effects is overlooked or taken into account more than once.", "why": "Warns of the two common errors in checking the energy principle by experiment.", "use": [ "lesson" ], "concepts": [ "concept/energy", "concept/external-effect", "law/conservation-of-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-fb08ba28de", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "38", "location": "General Exposition", "latex": "It will be different, however, in the case of the \\emph{second law} of thermodynamics, the proof of which, at the present stage of the development of our subject, cannot be too carefully presented. The general validity of this law is still contested from time to time, and its significance variously interpreted, even by the adherents of the principle.", "markdown": "It will be different, however, in the case of the *second law* of thermodynamics, the proof of which, at the present stage of the development of our subject, cannot be too carefully presented. The general validity of this law is still contested from time to time, and its significance variously interpreted, even by the adherents of the principle.", "why": "A historical remark showing that the second law was still disputed in 1903, unlike the first.", "use": [ "history", "website" ], "concepts": [ "law/first-law-of-thermodynamics", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0f43f92ed5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "38", "location": "General Exposition", "latex": "Or we may, as is done in this work, leave open the question concerning the possibility of reducing all natural processes to those of motion, and start from the fact which has been tested by centuries of human experience, and repeatedly verified, viz.\\ that \\emph{it is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion}, \\ie\\ it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing.", "markdown": "Or we may, as is done in this work, leave open the question concerning the possibility of reducing all natural processes to those of motion, and start from the fact which has been tested by centuries of human experience, and repeatedly verified, viz. that *it is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion*, *i.e.* it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing.", "why": "Shows the book's starting point: a plain experimental fact, not a theory of nature, is the foundation of the energy principle.", "use": [ "lesson", "website" ], "concepts": [ "concept/perpetual-motion", "law/conservation-of-energy", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-f13b977df3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "48", "location": "Applications to Homogeneous Systems", "latex": "The results of the experiment show that when the flow has become steady there is, for air, a very small change of temperature, and, for hydrogen, a still smaller, hardly appreciable change.", "markdown": "The results of the experiment show that when the flow has become steady there is, for air, a very small change of temperature, and, for hydrogen, a still smaller, hardly appreciable change.", "why": "It shows how a careful measurement narrows a conclusion, which makes a good historical account of the porous plug experiment.", "use": [ "history", "website" ], "concepts": [ "concept/perfect-gas", "experiment/porous-plug-experiment", "person/william-thomson" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a22359d3fb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "49", "location": "Applications to Homogeneous Systems", "latex": "Strictly speaking, this expression is vague, since a process presupposes changes, and, therefore, disturbances of equilibrium.", "markdown": "Strictly speaking, this expression is vague, since a process presupposes changes, and, therefore, disturbances of equilibrium.", "why": "It warns the learner that an idealised slow process is an approximation, and explains why the limit is taken.", "use": [ "lesson" ], "concepts": [ "concept/quasi-static-process" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-1a8e89ac78", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "50", "location": "Applications to Homogeneous Systems", "latex": "Wherever external pressure enters---as, for instance, in the calculation of the work of compression---a very small error will then be committed, if the pressure of the gas be substituted for the external pressure.", "markdown": "Wherever external pressure enters---as, for instance, in the calculation of the work of compression---a very small error will then be committed, if the pressure of the gas be substituted for the external pressure.", "why": "It explains why the gas's own pressure can stand in for the external pressure in work calculations, which is a common point of confusion.", "use": [ "lesson" ], "concepts": [ "concept/pressure", "concept/quasi-static-process", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8e460c2cc5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "53", "location": "Applications to Homogeneous Systems", "latex": "Only for infinitesimal changes, \\ie\\ when $1$~and $2$ are infinitely near one another and $\\alpha$~shrinks to a curve element, is $W$~determined by the initial and final points of the curve alone.", "markdown": "Only for infinitesimal changes, *i.e.* when $1$ and $2$ are infinitely near one another and $\\alpha$ shrinks to a curve element, is $W$ determined by the initial and final points of the curve alone.", "why": "It tells the learner exactly when work becomes a function of state alone, which prepares the idea of an exact differential.", "use": [ "lesson" ], "concepts": [ "concept/exact-differential", "concept/infinitesimal", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-664fa2f591", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "65", "location": "Applications to Homogeneous Systems", "latex": "Carnot's cycle, performed with a perfect gas, thus affords a means of drawing heat from a body and of gaining work in its stead, without introducing any changes in nature except the transference of a certain quantity of heat from a body of higher temperature to one of lower temperature.", "markdown": "Carnot’s cycle, performed with a perfect gas, thus affords a means of drawing heat from a body and of gaining work in its stead, without introducing any changes in nature except the transference of a certain quantity of heat from a body of higher temperature to one of lower temperature.", "why": "It sums up the Carnot cycle's effect in plain words, giving the learner the physical meaning before the algebra.", "use": [ "lesson", "website" ], "concepts": [ "concept/carnot-cycle", "concept/heat", "concept/heat-reservoir", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-39dd5ad811", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "46", "location": "Applications to Homogeneous Systems", "latex": "The term \\emph{homogeneous} is used here in the sense of \\emph{physically\nhomogeneous}, and is applied to any system which appears\nof completely uniform structure throughout.", "markdown": "The term *homogeneous* is used here in the sense of *physically homogeneous*, and is applied to any system which appears of completely uniform structure throughout.", "why": "Fixes what 'homogeneous' means in thermodynamics, which differs from the chemical sense.", "use": [ "lesson", "website" ], "concepts": [ "concept/homogeneous-system" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e0d8002041", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "47", "location": "Applications to Homogeneous Systems", "latex": "He put the two\ncommunicating vessels, one filled with air at high pressure,\nthe other exhausted, into a common water-bath at the\nsame temperature, and found that, after the air had expanded\nand equilibrium had been established, the change\nof temperature of the water-bath was inappreciable.", "markdown": "He put the two communicating vessels, one filled with air at high pressure, the other exhausted, into a common water-bath at the same temperature, and found that, after the air had expanded and equilibrium had been established, the change of temperature of the water-bath was inappreciable.", "why": "A clear, concrete description of Joule's free-expansion experiment that a learner can picture.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/perfect-gas", "experiment/joule-s-experiments" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e2b31ca6d8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "48", "location": "Applications to Homogeneous Systems", "latex": "In other\nwords, \\emph{the internal energy of a perfect gas depends only on the\ntemperature, and not on the volume}.", "markdown": "In other words, *the internal energy of a perfect gas depends only on the temperature, and not on the volume*.", "why": "States the key result about perfect gases that the later derivations rely on.", "use": [ "lesson", "website" ], "concepts": [ "concept/perfect-gas", "quantity/internal-energy", "theorem/joule-s-law" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0dadecce8b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "50", "location": "Applications to Homogeneous Systems", "latex": "Thus, a gas may be compressed very slowly to\nany fraction of its original volume, by making the external\npressure, at each moment, just a trifle greater than the\ninternal pressure of the gas.", "markdown": "Thus, a gas may be compressed very slowly to any fraction of its original volume, by making the external pressure, at each moment, just a trifle greater than the internal pressure of the gas.", "why": "Gives an intuitive picture of an infinitely slow process, the basis of reversibility.", "use": [ "lesson" ], "concepts": [ "concept/quasi-static-process", "concept/reversible-process" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-19e84b6e56", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "53", "location": "Applications to Homogeneous Systems", "latex": "In fact, it assumes an entirely different value along\na different curve,~$\\beta$, joining $1$~and~$2$. Therefore $p\\, dV$~is\nnot a perfect differential.", "markdown": "In fact, it assumes an entirely different value along a different curve, $\\beta$, joining $1$ and $2$. Therefore $p\\, dV$ is not a perfect differential.", "why": "Shows why work depends on the path between two states and not just on the end points.", "use": [ "lesson" ], "concepts": [ "concept/exact-differential", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ffcdcd56e9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "54", "location": "Applications to Homogeneous Systems", "latex": "In isothermal\nchanges $C$~is evidently $= \\pm\\infty$, because $d\\theta = 0$, and the\nheat added or withdrawn is a finite quantity. In adiabatic\nchanges $C = 0$, for here the temperature may change in\nany way, while no heat is added or withdrawn.", "markdown": "In isothermal changes $C$ is evidently $= \\pm\\infty$, because $d\\theta = 0$, and the heat added or withdrawn is a finite quantity. In adiabatic changes $C = 0$, for here the temperature may change in any way, while no heat is added or withdrawn.", "why": "Explains why heat capacity has no single value at a given state, so heat is unlike work.", "use": [ "lesson" ], "concepts": [ "concept/adiabatic-process", "concept/heat", "concept/isothermal-process" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-693de0de1c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "55", "location": "Applications to Homogeneous Systems", "latex": "It is usual to follow the example of Clausius, and denote this quantity\n by~$dQ$, to indicate that it is infinitely small. This notation, however, has\n frequently given rise to misunderstanding, for $dQ$~has been repeatedly\n regarded as the differential of a known finite quantity~$Q$.", "markdown": "It is usual to follow the example of Clausius, and denote this quantity by $dQ$, to indicate that it is infinitely small. This notation, however, has frequently given rise to misunderstanding, for $dQ$ has been repeatedly regarded as the differential of a known finite quantity $Q$.", "why": "Warns against reading dQ as the differential of a state quantity, a common mistake.", "use": [ "lesson", "history" ], "concepts": [ "concept/exact-differential", "concept/heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-24e1b2c8b0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "66", "location": "Applications to Homogeneous Systems", "latex": "By reversing Carnot's cycle, we have,\nthen, a means of transferring heat from a colder to a hotter\nbody without introducing any other changes in nature than\nthe transformation of a certain amount of mechanical work\ninto heat.", "markdown": "By reversing Carnot’s cycle, we have, then, a means of transferring heat from a colder to a hotter body without introducing any other changes in nature than the transformation of a certain amount of mechanical work into heat.", "why": "Shows what running the Carnot cycle backwards achieves, a preview of the second law.", "use": [ "lesson", "website" ], "concepts": [ "concept/reversible-process", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a3a269a53e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "69", "location": "Applications to Non-Homogeneous Systems", "latex": "This means that the internal energy of lead and sulphur, when separate, is $18,400$ calories greater than that of their combination at the same temperature.", "markdown": "This means that the internal energy of lead and sulphur, when separate, is $18,400$ calories greater than that of their combination at the same temperature.", "why": "It shows a learner how to read a thermochemical equation as an energy statement, with the sign of Q fixed by the convention.", "use": [ "lesson" ], "concepts": [ "quantity/heat-effect", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8955f873bc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "71", "location": "Applications to Non-Homogeneous Systems", "latex": "It, therefore, depends on the initial and final states only, and not on the intermediate steps of the process.", "markdown": "It, therefore, depends on the initial and final states only, and not on the intermediate steps of the process.", "why": "It states plainly why a heat effect can be found by any convenient path between the same two states.", "use": [ "lesson" ], "concepts": [ "quantity/heat-effect", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-524fc04014", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "69", "location": "Applications to Non-Homogeneous Systems", "latex": "The pressure has, however, very little influence on the internal energy; in fact, none at all in the case of perfect gases [equation~\\Eq{(35)}].", "markdown": "The pressure has, however, very little influence on the internal energy; in fact, none at all in the case of perfect gases [equation % [eqn:(35)](35)%].", "why": "It tells the learner when pressure can be ignored in an energy calculation and gives the ideal-gas exception.", "use": [ "lesson" ], "concepts": [ "concept/perfect-gas", "quantity/internal-energy", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-cc77ecc37f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "68", "location": "Applications to Non-Homogeneous Systems", "latex": "In our equations we shall therefore use~$Q$ (the heat absorbed) with the negative sign, in processes with positive heat effect (\\eg\\ combustion); with the positive sign, in those with negative heat effect (\\eg\\ evaporation, fusion, dissociation).", "markdown": "In our equations we shall therefore use $Q$ (the heat absorbed) with the negative sign, in processes with positive heat effect (*e.g.* combustion); with the positive sign, in those with negative heat effect (*e.g.* evaporation, fusion, dissociation).", "why": "It sets out the sign convention for Q that a learner must keep fixed to avoid errors in thermochemical signs.", "use": [ "lesson", "website" ], "concepts": [ "concept/dissociation", "concept/exothermal-process", "concept/heat", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-20f37cee70", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "67", "location": "Applications to Non-Homogeneous Systems", "latex": "For this reason we classify substances according to their physical and not according to their chemical homogeneity.", "markdown": "For this reason we classify substances according to their physical and not according to their chemical homogeneity.", "why": "Explains why the chapter's key distinction rests on what can actually be observed.", "use": [ "lesson", "website" ], "concepts": [ "concept/non-homogeneous-system" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4c6b96e5ba", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "68", "location": "Applications to Non-Homogeneous Systems", "latex": "Furthermore, most chemical processes are accompanied by a rise in temperature, or, if the initial temperature be re-established, by an external yield of heat (exothermal processes).", "markdown": "Furthermore, most chemical processes are accompanied by a rise in temperature, or, if the initial temperature be re-established, by an external yield of heat (exothermal processes).", "why": "Introduces exothermal processes and why heat given out counts as a positive heat effect.", "use": [ "lesson" ], "concepts": [ "concept/exothermal-process", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c4131ec1d2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "68", "location": "Applications to Non-Homogeneous Systems", "latex": "He denoted by the formulæ for the atomic or molecular weight of the substances enclosed in brackets, the internal energy of a corresponding weight referred to an arbitrary zero of energy.", "markdown": "He denoted by the formulæ for the atomic or molecular weight of the substances enclosed in brackets, the internal energy of a corresponding weight referred to an arbitrary zero of energy.", "why": "Describes Thomsen's historic notation, which the rest of the chapter relies on.", "use": [ "lesson", "history" ], "concepts": [ "concept/thermochemical-notation", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-01219dea82", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "69", "location": "Applications to Non-Homogeneous Systems", "latex": "These symbols may be treated like algebraic quantities, whereby considerations, which would otherwise present considerable complications, may be materially shortened.", "markdown": "These symbols may be treated like algebraic quantities, whereby considerations, which would otherwise present considerable complications, may be materially shortened.", "why": "Shows learners how chemical symbols can be manipulated like algebra to save effort.", "use": [ "lesson", "website" ], "concepts": [ "concept/thermochemical-notation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-f42e7b5c9c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "71", "location": "Applications to Non-Homogeneous Systems", "latex": "But since its change of energy $U_{2} - U_{1}$ depends on the initial and final states only, a greater amount of work done against the external forces necessitates a smaller heat effect for the process, and \\textit{vice versâ}.", "markdown": "But since its change of energy $U_{2} - U_{1}$ depends on the initial and final states only, a greater amount of work done against the external forces necessitates a smaller heat effect for the process, and *vice versâ*.", "why": "Explains why the external conditions must be known to find a heat effect.", "use": [ "lesson" ], "concepts": [ "quantity/heat-effect", "quantity/internal-energy", "quantity/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-733c94fae5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "72", "location": "Applications to Non-Homogeneous Systems", "latex": "The heat effect, however, is not equal to the difference of the internal energies~$U$, but to the difference of the values of the quantity $(U + p_{0} V)$ at the beginning and end of the process.", "markdown": "The heat effect, however, is not equal to the difference of the internal energies $U$, but to the difference of the values of the quantity $(U + p_{0} V)$ at the beginning and end of the process.", "why": "Warns learners that at constant pressure the heat effect tracks U + p0 V, not U alone.", "use": [ "lesson" ], "concepts": [ "quantity/enthalpy", "quantity/heat-effect", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-658bddb1e3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "73", "location": "Applications to Non-Homogeneous Systems", "latex": "Frequently, of two ways of transition, one is better adapted for calorimetric measurements than the other. Thus, the heat effect of the decomposition of hydrogen peroxide into water and oxygen cannot readily be measured directly.", "markdown": "Frequently, of two ways of transition, one is better adapted for calorimetric measurements than the other. Thus, the heat effect of the decomposition of hydrogen peroxide into water and oxygen cannot readily be measured directly.", "why": "Motivates finding a heat effect indirectly by choosing a measurable route.", "use": [ "lesson" ], "concepts": [ "method/calorimetry", "method/indirect-determination-of-heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-7e142d1546", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "76", "location": "Applications to Non-Homogeneous Systems", "latex": "The difference between these values is~$-7.71$, and, therefore, the heat of combustion of a gram molecule of hydrogen decreases with rising temperature by $7.7~\\Unit{cal.}$ per degree Centigrade.", "markdown": "The difference between these values is $-7.71$, and, therefore, the heat of combustion of a gram molecule of hydrogen decreases with rising temperature by $7.7~\\Unit{cal.}$ per degree Centigrade.", "why": "A worked conclusion showing how heat capacities give the temperature dependence of a heat effect.", "use": [ "lesson" ], "concepts": [ "concept/temperature", "quantity/heat-effect", "quantity/specific-heat", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-52549a3ef9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "80", "location": "Introduction", "latex": "If a perfect gas be allowed to expand, doing external work, and be prevented from cooling by connecting it with a heat-reservoir of higher temperature, the temperature of the gas, and at the same time its internal energy, remains unchanged, and it may be said that the amount of heat given out by the reservoir is completely changed into work without an exchange of energy taking place anywhere.", "markdown": "If a perfect gas be allowed to expand, doing external work, and be prevented from cooling by connecting it with a heat-reservoir of higher temperature, the temperature of the gas, and at the same time its internal energy, remains unchanged, and it may be said that the amount of heat given out by the reservoir is completely changed into work without an exchange of energy taking place anywhere.", "why": "This apparent counterexample to the usual heat-into-work statement is the hook that makes the second law's subtlety concrete.", "use": [ "lesson" ], "concepts": [ "concept/heat-reservoir", "concept/perfect-gas", "concept/transformability-of-heat-into-work", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-cdf2beea30", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "81", "location": "Introduction", "latex": "Such a proposition cannot be proved \\textit{a~priori}, neither does it amount to a definition, but it contains a definite assertion, to be stated precisely in each case, which may be verified by actual experiment.", "markdown": "Such a proposition cannot be proved *a priori*, neither does it amount to a definition, but it contains a definite assertion, to be stated precisely in each case, which may be verified by actual experiment.", "why": "It warns learners that a physical law is an experimental claim, not a definition or an a priori truth.", "use": [ "lesson", "website" ], "concepts": [ "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-42a1fde2ca", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "77", "location": "Introduction", "latex": "Not every change which is consistent with the principle of the conservation of energy satisfies also the additional conditions which the second law imposes upon the processes, which actually take place in nature. In other words, the principle of the conservation of energy does not suffice for a unique determination of natural processes.", "markdown": "Not every change which is consistent with the principle of the conservation of energy satisfies also the additional conditions which the second law imposes upon the processes, which actually take place in nature. In other words, the principle of the conservation of energy does not suffice for a unique determination of natural processes.", "why": "It states plainly what the second law adds beyond energy conservation, which is the direction of natural processes.", "use": [ "lesson", "website" ], "concepts": [ "law/conservation-of-energy", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a01a488b6c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "77", "location": "Introduction", "latex": "If, for instance, an exchange of heat by conduction takes place between two bodies of different temperature, the first law, or the principle of the conservation of energy, merely demands that the quantity of heat given out by the one body shall be equal to that taken up by the other. Whether the flow of heat, however, takes place from the colder to the hotter body, or \\textit{vice versâ}, cannot be answered by the energy principle alone.", "markdown": "If, for instance, an exchange of heat by conduction takes place between two bodies of different temperature, the first law, or the principle of the conservation of energy, merely demands that the quantity of heat given out by the one body shall be equal to that taken up by the other. Whether the flow of heat, however, takes place from the colder to the hotter body, or *vice versâ*, cannot be answered by the energy principle alone.", "why": "A concrete example showing that energy balance alone cannot tell which way heat flows.", "use": [ "lesson", "website" ], "concepts": [ "concept/conduction-of-heat", "concept/temperature", "law/conservation-of-energy", "law/first-law-of-thermodynamics", "law/second-law-of-thermodynamics", "quantity/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-72d86b068f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "78", "location": "Introduction", "latex": "If a heavy liquid be initially at rest at different levels in two communicating tubes, then motion will set in, so as to equalize the levels, for the centre of gravity of the system is thereby lowered, and the potential energy diminished. Equilibrium exists when the centre of gravity is at its lowest, and therefore the potential energy at a minimum, \\ie\\ when the liquid stands at the same level in both tubes.", "markdown": "If a heavy liquid be initially at rest at different levels in two communicating tubes, then motion will set in, so as to equalize the levels, for the centre of gravity of the system is thereby lowered, and the potential energy diminished. Equilibrium exists when the centre of gravity is at its lowest, and therefore the potential energy at a minimum, *i.e.* when the liquid stands at the same level in both tubes.", "why": "A vivid mechanical picture of a direction of change set by an energy minimum, which contrasts with thermal phenomena.", "use": [ "lesson", "website" ], "concepts": [ "concept/centre-of-gravity", "concept/mechanical-equilibrium", "quantity/potential-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-3de90059e7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "81", "location": "Introduction", "latex": "There is but one way of clearly showing the significance of the second law, and that is to base it on facts by formulating propositions which may be proved or disproved by experiment.", "markdown": "There is but one way of clearly showing the significance of the second law, and that is to base it on facts by formulating propositions which may be proved or disproved by experiment.", "why": "It shows the second law as a claim about the world that experiment could test, not a definition.", "use": [ "lesson", "website" ], "concepts": [ "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-81f8f59d12", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "82", "location": "Introduction", "latex": "A process which can in no way be completely reversed is termed \\emph{irreversible}, all other processes \\emph{reversible}. That a process may be irreversible, it is not sufficient that it cannot be directly reversed.", "markdown": "A process which can in no way be completely reversed is termed *irreversible*, all other processes *reversible*. That a process may be irreversible, it is not sufficient that it cannot be directly reversed.", "why": "It gives the definition of irreversibility and warns that failing to reverse a process directly is not enough to call it irreversible.", "use": [ "lesson" ], "concepts": [ "concept/irreversible-process", "concept/reversible-process" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c3b340d42b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "84", "location": "Introduction", "latex": "Consequently, either all or none of these processes are irreversible. There is no third possibility. If those processes are not irreversible, the entire edifice of the second law will crumble.", "markdown": "Consequently, either all or none of these processes are irreversible. There is no third possibility. If those processes are not irreversible, the entire edifice of the second law will crumble.", "why": "It shows how much the second law rests on one fact, and in a memorable way.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/irreversible-process", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-da96c23daf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "85", "location": "Introduction", "latex": "Since there exists in nature no process entirely free from friction or heat-conduction, all processes which actually take place in nature, if the second law be correct, are in reality irreversible; reversible processes form only an ideal limiting case.", "markdown": "Since there exists in nature no process entirely free from friction or heat-conduction, all processes which actually take place in nature, if the second law be correct, are in reality irreversible; reversible processes form only an ideal limiting case.", "why": "It makes clear that reversible processes are an idealization and real processes are irreversible.", "use": [ "lesson" ], "concepts": [ "concept/conduction-of-heat", "concept/friction", "concept/irreversible-process", "concept/reversible-process" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-19161c0218", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "86", "location": "Proof", "latex": "It would, it is true, not be equivalent to perpetual motion, for it does not produce work from nothing, but from the heat, which it draws from the reservoir. It would not, therefore, like perpetual motion, contradict the principle of energy, but would, nevertheless, possess for man the essential advantage of perpetual motion, the supply of work without cost; for the inexhaustible supply of heat in the earth, in the atmosphere, and in the sea, would, like the oxygen of the atmosphere, be at everybody's immediate disposal.", "markdown": "It would, it is true, not be equivalent to perpetual motion, for it does not produce work from nothing, but from the heat, which it draws from the reservoir. It would not, therefore, like perpetual motion, contradict the principle of energy, but would, nevertheless, possess for man the essential advantage of perpetual motion, the supply of work without cost; for the inexhaustible supply of heat in the earth, in the atmosphere, and in the sea, would, like the oxygen of the atmosphere, be at everybody’s immediate disposal.", "why": "It shows how a machine can obey energy conservation and still be impossible, which separates the two kinds of perpetual motion.", "use": [ "lesson", "website" ], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/perpetual-motion", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-1f072e70a5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "87", "location": "Proof", "latex": "For supposing it were not so, \\ie\\ supposing a method could be found by which a process involving generation of heat by friction could be completely reversed, this very method would produce what is identically perpetual motion of the second kind: viz.\\ a change which consists of nothing but the production of work, and the absorption of an equivalent amount of heat.", "markdown": "For supposing it were not so, *i.e.* supposing a method could be found by which a process involving generation of heat by friction could be completely reversed, this very method would produce what is identically perpetual motion of the second kind: viz. a change which consists of nothing but the production of work, and the absorption of an equivalent amount of heat.", "why": "It is a short model proof by contradiction, showing how irreversibility follows from the founding proposition.", "use": [ "lesson" ], "concepts": [ "concept/friction", "concept/irreversible-process", "concept/perpetual-motion", "concept/reversible-process", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a93d34a2eb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "86", "location": "Proof", "latex": "As soon as a phenomenon is found to contradict any legitimate conclusions from the second law, this contradiction must arise from an inaccuracy in our first assumption, and the phenomenon could be used for the construction of the above-described engine.", "markdown": "As soon as a phenomenon is found to contradict any legitimate conclusions from the second law, this contradiction must arise from an inaccuracy in our first assumption, and the phenomenon could be used for the construction of the above-described engine.", "why": "It explains the logic of testing the second law: any counterexample would reveal an error in the starting assumption.", "use": [ "lesson" ], "concepts": [ "concept/heat-engine", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0e0d97db1c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "89", "location": "Proof", "latex": "It increases or decreases according as heat is absorbed or evolved.", "markdown": "It increases or decreases according as heat is absorbed or evolved.", "why": "It gives the sign rule for entropy change on heating, a short statement a learner can check against experience.", "use": [ "lesson" ], "concepts": [ "concept/heat", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-b8f37c7ff9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "87", "location": "Proof", "latex": "Not a single really rational proof of the second law has thus far been advanced which does not require this fundamental principle, however numerous the attempts in this direction may have been in recent times, nor do I believe that such an attempt will ever meet with success.", "markdown": "Not a single really rational proof of the second law has thus far been advanced which does not require this fundamental principle, however numerous the attempts in this direction may have been in recent times, nor do I believe that such an attempt will ever meet with success.", "why": "It shows the author's view of the logical status of the second law in his own period, which makes a good historical remark.", "use": [ "history" ], "concepts": [ "concept/perpetual-motion", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-90e705d634", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "86", "location": "Proof", "latex": "\\emph{It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.}", "markdown": "*It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.*", "why": "It gives the single experimental postulate from which the whole second law is deduced, stated as a clear, testable claim a learner can argue about.", "use": [ "lesson" ], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/heat-engine", "concept/heat-reservoir", "concept/perpetual-motion", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-fc6f56c954", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "86", "location": "Proof", "latex": "Such an engine could be used simultaneously as a motor and a refrigerator without any waste of energy or material, and would in any case be the most profitable engine ever made. It would, it is true, not be equivalent to perpetual motion, for it does not produce work from nothing, but from the heat, which it draws from the reservoir. It would not, therefore, like perpetual motion, contradict the principle of energy, but would, nevertheless, possess for man the essential advantage of perpetual motion, the supply of work without cost; for the inexhaustible supply of heat in the earth, in the atmosphere, and in the sea, would, like the oxygen of the atmosphere, be at everybody's immediate disposal.", "markdown": "Such an engine could be used simultaneously as a motor and a refrigerator without any waste of energy or material, and would in any case be the most profitable engine ever made. It would, it is true, not be equivalent to perpetual motion, for it does not produce work from nothing, but from the heat, which it draws from the reservoir. It would not, therefore, like perpetual motion, contradict the principle of energy, but would, nevertheless, possess for man the essential advantage of perpetual motion, the supply of work without cost; for the inexhaustible supply of heat in the earth, in the atmosphere, and in the sea, would, like the oxygen of the atmosphere, be at everybody’s immediate disposal.", "why": "It shows why the forbidden engine would be so valuable, and why it differs from perpetual motion of the first kind, which helps a learner keep the two laws apart.", "use": [ "website", "lesson" ], "concepts": [ "concept/heat", "concept/heat-engine", "concept/heat-reservoir", "concept/perpetual-motion", "concept/work", "law/conservation-of-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0e16568dd4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "87", "location": "Proof", "latex": "From the impossibility of perpetual motion of the second kind, it follows, in the first place, that the generation of heat by friction is \\emph{irreversible} (\\cf\\ def.~\\SecRef{112}).", "markdown": "From the impossibility of perpetual motion of the second kind, it follows, in the first place, that the generation of heat by friction is *irreversible* (*cf.* def. 112).", "why": "It gives a familiar everyday case, friction, to show what an irreversible process is.", "use": [ "lesson" ], "concepts": [ "concept/friction", "concept/perpetual-motion", "concept/reversible-process" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8a2fb8b5ad", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "89", "location": "Proof", "latex": "The entropy of the gas, therefore, remains constant during the described adiabatic change of state.", "markdown": "The entropy of the gas, therefore, remains constant during the described adiabatic change of state.", "why": "It states plainly that entropy stays fixed in a reversible adiabatic change, the anchor for everything that follows.", "use": [ "lesson" ], "concepts": [ "concept/adiabatic-process", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-52909a0434", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "97", "location": "Proof", "latex": "The entropy of a body in a given state, like the internal energy, is completely determined up to an additive constant, whose value depends on the zero state.", "markdown": "The entropy of a body in a given state, like the internal energy, is completely determined up to an additive constant, whose value depends on the zero state.", "why": "It explains that entropy, like energy, is fixed only up to a constant chosen by a zero state, which removes a common source of confusion.", "use": [ "lesson", "history" ], "concepts": [ "quantity/entropy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-7373396881", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "89", "location": "Proof", "latex": "It should, however, be emphasized that equation~\\Eq{(53)} is by no means generally true. It holds only in the particular case where the external work performed by the gas is expressed by~$p\\, dV$.", "markdown": "It should, however, be emphasized that equation % [eqn:(53)](53)% is by no means generally true. It holds only in the particular case where the external work performed by the gas is expressed by $p\\, dV$.", "why": "It warns the learner that Q = theta dPhi is not a general law of heat but holds only under a stated condition.", "use": [ "lesson" ], "concepts": [ "concept/heat", "concept/work", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-f0e3920eeb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "86", "location": "Proof", "latex": "We, therefore, put forward the following proposition as being given directly by experience: \\emph{It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.} Such an engine could be used simultaneously as a motor and a refrigerator without any waste of energy or material, and would in any case be the most profitable engine ever made.", "markdown": "We, therefore, put forward the following proposition as being given directly by experience: *It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.* Such an engine could be used simultaneously as a motor and a refrigerator without any waste of energy or material, and would in any case be the most profitable engine ever made.", "why": "It states the single experience-based proposition on which the whole proof rests, and says why such an engine would be so valuable.", "use": [ "lesson", "website" ], "concepts": [ "concept/perpetual-motion", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-1bd3cf2985", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "100", "location": "Proof", "latex": "\\emph{Every physical or chemical process in nature takes place in such a way as to increase the sum of the entropies of all the bodies taking any part in the process. In the limit, \\ie\\ for reversible processes, the sum of the entropies remains unchanged.} This is the most general statement of the second law of Thermodynamics.", "markdown": "*Every physical or chemical process in nature takes place in such a way as to increase the sum of the entropies of all the bodies taking any part in the process. In the limit, *i.e.* for reversible processes, the sum of the entropies remains unchanged.* This is the most general statement of the second law of Thermodynamics.", "why": "It gives the second law in its widest form, with the reversible case as the limit.", "use": [ "lesson", "website" ], "concepts": [ "concept/reversible-process", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-23128a9bcf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "100", "location": "Proof", "latex": "It should be emphasized, however, that the form here given is the only one of unrestricted applicability to any finite process, and that no other universal measure of the irreversibility of processes exists than the amount of the increase of the entropy to which they lead.", "markdown": "It should be emphasized, however, that the form here given is the only one of unrestricted applicability to any finite process, and that no other universal measure of the irreversibility of processes exists than the amount of the increase of the entropy to which they lead.", "why": "It tells the learner why entropy, rather than lost work or dissipated energy, is the general measure of irreversibility.", "use": [ "lesson" ], "concepts": [ "concept/dissipation-of-energy", "concept/irreversible-process", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c3daaad29c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "103", "location": "Proof", "latex": "It would be absurd to assume that the validity of the second law depends in any way on the skill of the physicist or chemist in observing or experimenting. The gist of the second law has nothing to do with experiment; the law asserts briefly that \\emph{there exists in nature a quantity which changes always in the same sense in all natural processes}.", "markdown": "It would be absurd to assume that the validity of the second law depends in any way on the skill of the physicist or chemist in observing or experimenting. The gist of the second law has nothing to do with experiment; the law asserts briefly that *there exists in nature a quantity which changes always in the same sense in all natural processes*.", "why": "It gives a plain, memorable statement of what kind of law the second law is, independent of the observer.", "use": [ "lesson", "website" ], "concepts": [ "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-54c5d37511", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "100", "location": "Proof", "latex": "As the impossibility of perpetual motion of the first kind leads to the first law of Thermodynamics, or the principle of the conservation of energy; so the impossibility of perpetual motion of the second kind has led to the second law, properly designated as the \\emph{principle of the increase of the entropy}.", "markdown": "As the impossibility of perpetual motion of the first kind leads to the first law of Thermodynamics, or the principle of the conservation of energy; so the impossibility of perpetual motion of the second kind has led to the second law, properly designated as the *principle of the increase of the entropy*.", "why": "It sets the first and second laws side by side, each resting on an impossibility.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/perpetual-motion", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-bf35243829", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "105", "location": "General Deductions", "latex": "\\First{Our} first application of the principle of the entropy which was expressed in its most general form in the preceding chapter, will be to Carnot's cycle, described in detail for perfect gases in~\\SecRef{90}.", "markdown": "Our first application of the principle of the entropy which was expressed in its most general form in the preceding chapter, will be to Carnot’s cycle, described in detail for perfect gases in 90.", "why": "It states the chapter's aim: the entropy principle is first applied to Carnot's cycle, now for any substance.", "use": [ "lesson" ], "concepts": [ "concept/carnot-cycle", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ceefef0202", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "106", "location": "General Deductions", "latex": "Observe, however, that the expressions~\\Eq{(64)} for the change of the entropy of the reservoirs are still correct, provided we assume that any changes of volume of the substances used as reservoirs are reversible.", "markdown": "Observe, however, that the expressions % [eqn:(64)](64)% for the change of the entropy of the reservoirs are still correct, provided we assume that any changes of volume of the substances used as reservoirs are reversible.", "why": "It warns that the reservoir entropy formulas remain valid for irreversible cycles only under a stated assumption about the reservoirs.", "use": [ "lesson" ], "concepts": [ "concept/heat-reservoir", "concept/reversible-process", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2548d13012", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "107", "location": "General Deductions", "latex": "In this case the cyclic process results in the transference of heat~($Q_{2}$) from the reservoir of temperature~$\\theta_{2}$ to that of temperature~$\\theta_{1}$, and the inequality means that this flow of heat is always directed from the hotter to the colder reservoir.", "markdown": "In this case the cyclic process results in the transference of heat ($Q_{2}$) from the reservoir of temperature $\\theta_{2}$ to that of temperature $\\theta_{1}$, and the inequality means that this flow of heat is always directed from the hotter to the colder reservoir.", "why": "It gives the learner the second-law meaning of the inequality: with no work done, heat flows only from hot to cold.", "use": [ "lesson", "website" ], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/heat-reservoir", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-bf7b8ee747", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "107", "location": "General Deductions", "latex": "This means that the amount of work,~$W'$, to be gained by means of a cyclic process from the transference of the heat,~$Q_{1}'$, from a hotter to a colder reservoir, is always smaller for an irreversible process than for a reversible one. Consequently the equation~\\Eq{(66)} represents the maximum amount of work to be gained from any cyclic process between heat-reservoirs at the temperatures $\\theta_{2}$ and~$\\theta_{1}$.", "markdown": "This means that the amount of work, $W'$, to be gained by means of a cyclic process from the transference of the heat, $Q_{1}'$, from a hotter to a colder reservoir, is always smaller for an irreversible process than for a reversible one. Consequently the equation % [eqn:(66)](66)% represents the maximum amount of work to be gained from any cyclic process between heat-reservoirs at the temperatures $\\theta_{2}$ and $\\theta_{1}$.", "why": "It states plainly that reversibility sets the ceiling on the work any engine can get from a given heat flow.", "use": [ "lesson", "website" ], "concepts": [ "concept/carnot-cycle", "concept/cycle-of-operations", "concept/heat", "concept/irreversible-process", "concept/maximum-work", "concept/reversible-process", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-514d3c7197", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "105", "location": "General Deductions", "latex": "This time, the system operated upon may be of any character whatsoever, and chemical reactions, too, may take place, provided they are reversible.", "markdown": "This time, the system operated upon may be of any character whatsoever, and chemical reactions, too, may take place, provided they are reversible.", "why": "It tells the learner that the entropy argument applies to any substance, including reacting ones, as long as the steps are reversible.", "use": [ "lesson", "website" ], "concepts": [ "concept/chemical-reaction", "concept/reversible-process", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ebd4106d72", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "113", "location": "General Deductions", "latex": "This leads to the proposition that chemical reactions, in which there is no external work, take place in such a manner as to give the greatest heat effects (Berthelot's principle).", "markdown": "This leads to the proposition that chemical reactions, in which there is no external work, take place in such a manner as to give the greatest heat effects (Berthelot’s principle).", "why": "It gives a memorable rule of thumb for chemical reactions and shows when it fails, which helps the learner judge it.", "use": [ "lesson", "history" ], "concepts": [ "concept/chemical-reaction", "quantity/heat-effect", "theorem/berthelot-s-principle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-7a946b39ce", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "117", "location": "General Deductions", "latex": "Among all the states of the system which can proceed from one another by adiabatic processes, the state of equilibrium is distinguished by a maximum of the entropy.", "markdown": "Among all the states of the system which can proceed from one another by adiabatic processes, the state of equilibrium is distinguished by a maximum of the entropy.", "why": "It turns the entropy principle into a clear test for equilibrium, which is the key idea of the chapter's later sections.", "use": [ "lesson" ], "concepts": [ "concept/adiabatic-process", "concept/maximum", "concept/stability-of-equilibrium", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-9dea7c9d12", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "It cannot in general be integrated, since the left-hand side is not, in general, a perfect differential.", "markdown": "It cannot in general be integrated, since the left-hand side is not, in general, a perfect differential.", "why": "It warns the learner that the second law alone gives no finite-change result unless the external conditions are known.", "use": [ "lesson" ], "concepts": [ "concept/exact-differential", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-715292d9b8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "has been called by H.~v.~Helmholtz the \\emph{free energy} (freie Energie) of the system.", "markdown": "has been called by H. v. Helmholtz the *free energy* (freie Energie) of the system.", "why": "It gives the historical origin of the term free energy, so a reader can connect the name to its source and meaning.", "use": [ "history" ], "concepts": [ "person/hermann-von-helmholtz", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-43698ee333", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "The second law, then, does not lead to a general statement with regard to finite changes of a system taken by itself unless something be known of the external conditions to which it is subject.", "markdown": "The second law, then, does not lead to a general statement with regard to finite changes of a system taken by itself unless something be known of the external conditions to which it is subject.", "why": "It warns the learner that the second law alone does not settle finite changes; the external conditions must be specified.", "use": [ "lesson" ], "concepts": [ "concept/exact-differential", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4d7ac252cd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "For finite reversible isothermal changes the total work done on the system is equal to the increase of~$F$; or, the entire work performed by the system is equal to the decrease of~$F$, and, therefore, depends only on the initial and final states of the system. Where $F_{1} = F_{2}$, as in cyclic processes, the external work is zero.", "markdown": "For finite reversible isothermal changes the total work done on the system is equal to the increase of $F$; or, the entire work performed by the system is equal to the decrease of $F$, and, therefore, depends only on the initial and final states of the system. Where $F_{1} = F_{2}$, as in cyclic processes, the external work is zero.", "why": "It shows why free energy matters: in reversible isothermal changes the work depends only on the end states.", "use": [ "lesson", "website" ], "concepts": [ "concept/isothermal-process", "concept/reversible-process", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-d9cffa220f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "111", "location": "General Deductions", "latex": "Hence, any reversible transformation of the system from one state to another yields the maximum amount of work that can be gained by any isothermal process between those two states.", "markdown": "Hence, any reversible transformation of the system from one state to another yields the maximum amount of work that can be gained by any isothermal process between those two states.", "why": "It gives the key result linking reversibility to maximum obtainable work.", "use": [ "lesson", "website" ], "concepts": [ "concept/isothermal-process", "concept/maximum-work", "concept/reversible-process", "concept/work", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4a3486fe52", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "112", "location": "General Deductions", "latex": "Such a process, as here described, is composed only of states of equilibrium. Hence it is reversible, and the external work thereby gained represents at the same time the decrease of the free energy, $F_{2} - F_{1}$, which takes place on directly mixing the solution and the water.", "markdown": "Such a process, as here described, is composed only of states of equilibrium. Hence it is reversible, and the external work thereby gained represents at the same time the decrease of the free energy, $F_{2} - F_{1}$, which takes place on directly mixing the solution and the water.", "why": "It ends a worked thought-experiment (diluting a salt solution) that shows how a decrease of free energy can be measured as work.", "use": [ "lesson" ], "concepts": [ "concept/chemical-affinity", "concept/reversible-process", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0d69322d5d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "113", "location": "General Deductions", "latex": "Dividing this quantity by the number of oxidized molecules of hydrogen, we obtain a measure of the force with which a molecule of hydrogen tends to become oxidized. This definition of chemical force, however, has only a meaning in so far as it is connected with that work.", "markdown": "Dividing this quantity by the number of oxidized molecules of hydrogen, we obtain a measure of the force with which a molecule of hydrogen tends to become oxidized. This definition of chemical force, however, has only a meaning in so far as it is connected with that work.", "why": "It shows how an abstract thing, chemical force, is given an operational meaning through work.", "use": [ "lesson", "history" ], "concepts": [ "concept/chemical-affinity", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2d70f48d9d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "116", "location": "General Deductions", "latex": "States of equilibrium of this description are always unstable. Often a very small disturbance, not comparable in size with the quantities within the system, suffices to produce the change, which under these conditions often occurs with great violence. We have examples of this in overcooled liquids, supersaturated vapour, supersaturated solutions, explosive substances, etc.", "markdown": "States of equilibrium of this description are always unstable. Often a very small disturbance, not comparable in size with the quantities within the system, suffices to produce the change, which under these conditions often occurs with great violence. We have examples of this in overcooled liquids, supersaturated vapour, supersaturated solutions, explosive substances, etc.", "why": "It gives vivid everyday examples of unstable equilibrium held in place by passive resistance.", "use": [ "lesson", "website" ], "concepts": [ "concept/inertia-resistance", "concept/stability-of-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-cb514a3342", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "117", "location": "General Deductions", "latex": "\\ie\\ among the states which can proceed from one another by isothermal processes, without the performance of external work, the state of most stable equilibrium is distinguished by an absolute minimum of the free energy.", "markdown": "*i.e.* among the states which can proceed from one another by isothermal processes, without the performance of external work, the state of most stable equilibrium is distinguished by an absolute minimum of the free energy.", "why": "It states the equilibrium criterion chemists use: minimum free energy at constant temperature without external work.", "use": [ "lesson", "website" ], "concepts": [ "concept/stability-of-equilibrium", "concept/thermodynamic-equilibrium", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-1cae37aaab", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "119", "location": "Homogeneous Systems", "latex": "For the present, besides~$M$, let $\\theta$~and $v$ be the independent\nvariables.", "markdown": "For the present, besides $M$, let $\\theta$ and $v$ be the independent variables.", "why": "It tells the learner which variables fix the state of a homogeneous system before any equation is written.", "use": [ "lesson" ], "concepts": [ "concept/homogeneous-system", "concept/temperature", "concept/variable", "quantity/mass", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-93cca2f397", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "119", "location": "Homogeneous Systems", "latex": "Then the pressure~$p$, the specific energy $u = \\dfrac{U}{M}$,\nand the specific entropy $\\phi = \\dfrac{\\Phi}{M}$ are functions of $\\theta$~and~$v$,", "markdown": "Then the pressure $p$, the specific energy $u = \\dfrac{U}{M}$, and the specific entropy $\\phi = \\dfrac{\\Phi}{M}$ are functions of $\\theta$ and $v$,", "why": "It shows that pressure, specific energy and specific entropy all depend on the same two independent variables.", "use": [ "lesson" ], "concepts": [ "concept/function", "concept/pressure", "quantity/entropy", "quantity/specific-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-5fcc9968f4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "119", "location": "Homogeneous Systems", "latex": "Therefore, since $d\\theta$~and $dv$ are independent of each other,", "markdown": "Therefore, since $d\\theta$ and $dv$ are independent of each other,", "why": "It gives the step that lets a learner equate the coefficients of dθ and dv separately.", "use": [ "lesson" ], "concepts": [ "concept/differential", "concept/partial-derivative" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6232a89d50", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "These two equations lead to an experimental test of the\nsecond law;", "markdown": "These two equations lead to an experimental test of the second law;", "why": "It states that the two entropy equations can be checked by experiment, which connects the mathematics to the second law.", "use": [ "website", "history" ], "concepts": [ "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0cdfc8bb69", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "124", "location": "Homogeneous Systems", "latex": "This expression vanishes in the case of perfect gases, since then the temperature remains constant.", "markdown": "This expression vanishes in the case of perfect gases, since then the temperature remains constant.", "why": "It gives the simple physical reason why a perfect gas shows no temperature change in the throttling experiment.", "use": [ "lesson" ], "concepts": [ "concept/perfect-gas", "experiment/joule-s-experiments" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4ee133e4be", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "128", "location": "Homogeneous Systems", "latex": "The numerator of this expression may be found directly from the characteristic equation of the substance. The denominator, however, depends on the amount of heat which the substance absorbs during isothermal reversible expansion.", "markdown": "The numerator of this expression may be found directly from the characteristic equation of the substance. The denominator, however, depends on the amount of heat which the substance absorbs during isothermal reversible expansion.", "why": "It separates a formula into a part read from an equation of state and a part that needs a heat measurement, which teaches where each piece of data comes from.", "use": [ "lesson" ], "concepts": [ "concept/characteristic-equation", "concept/heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6584d7ae2c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "122", "location": "Homogeneous Systems", "latex": "For gases, $\\gamma$~is large; and, in fact, the fewer the number of atoms in a molecule of the gas, the larger does it become.", "markdown": "For gases, $\\gamma$ is large; and, in fact, the fewer the number of atoms in a molecule of the gas, the larger does it become.", "why": "It links the size of the ratio of specific heats to molecular structure, giving the learner a physical picture to remember.", "use": [ "lesson", "website" ], "concepts": [ "quantity/ratio-of-specific-heats" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-57908afb82", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "As $\\left(\\dfrac{\\dd p}{\\dd v}\\right)_{\\theta}$ is necessarily negative, $c_{p}$~is always greater than~$c_{v}$, except in the limiting case, when the coefficient of expansion is $= 0$, as in the case of water at $4°$~C.\\Chg{,}{}; then $c_{p} - c_{v} = 0$.", "markdown": "As $\\left(\\dfrac{\\dd p}{\\dd v}\\right)_{\\theta}$ is necessarily negative, $c_{p}$ is always greater than $c_{v}$, except in the limiting case, when the coefficient of expansion is $= 0$, as in the case of water at $4°$ C.,; then $c_{p} - c_{v} = 0$.", "why": "Shows learners that c_p exceeds c_v always, and names the one exception, water at 4°C.", "use": [ "lesson", "website" ], "concepts": [ "concept/difference-of-specific-heats", "quantity/coefficient-of-expansion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4247e40124", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "122", "location": "Homogeneous Systems", "latex": "It follows that, for solids and liquids, the difference $c_{p} - c_{v}$ depends rather on the relation between the energy and the volume than on the external work of expansion.", "markdown": "It follows that, for solids and liquids, the difference $c_{p} - c_{v}$ depends rather on the relation between the energy and the volume than on the external work of expansion.", "why": "Contrasts what controls c_p - c_v in solids and liquids with what controls it in perfect gases.", "use": [ "lesson" ], "concepts": [ "concept/difference-of-specific-heats", "concept/work", "quantity/internal-energy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6f509e7d29", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "122", "location": "Homogeneous Systems", "latex": "This means that in solids and liquids the energy depends far more on the temperature than on the volume. For gases, $\\gamma$~is large; and, in fact, the fewer the number of atoms in a molecule of the gas, the larger does it become.", "markdown": "This means that in solids and liquids the energy depends far more on the temperature than on the volume. For gases, $\\gamma$ is large; and, in fact, the fewer the number of atoms in a molecule of the gas, the larger does it become.", "why": "Links the ratio of specific heats to the structure of the substance, from solids up to monatomic gases.", "use": [ "lesson", "website" ], "concepts": [ "quantity/internal-energy", "quantity/ratio-of-specific-heats" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-dbdcc94393", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "123", "location": "Homogeneous Systems", "latex": "This equation contains only quantities that can be directly measured, and establishes a relation between the rate of change of the coefficient of thermal expansion of the substance with temperature (\\ie\\ the deviation from Gay-Lussac's law), and the rate of change of the specific heat with pressure.", "markdown": "This equation contains only quantities that can be directly measured, and establishes a relation between the rate of change of the coefficient of thermal expansion of the substance with temperature (*i.e.* the deviation from Gay-Lussac’s law), and the rate of change of the specific heat with pressure.", "why": "Shows how the second law yields a relation that can be checked by experiment.", "use": [ "lesson" ], "concepts": [ "law/gay-lussac-s-law", "law/second-law-of-thermodynamics", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-24ab25f3b2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "125", "location": "Homogeneous Systems", "latex": "If, under constant pressure, $v$~were proportional to~$\\theta$, as in Gay-Lussac's law, then, by equation~\\Eq{(86)}, $\\Delta \\theta = 0$, as is really the case for perfect gases.", "markdown": "If, under constant pressure, $v$ were proportional to $\\theta$, as in Gay-Lussac’s law, then, by equation % [eqn:(86)](86)%, $\\Delta \\theta = 0$, as is really the case for perfect gases.", "why": "Explains why a perfect gas shows no temperature change in the Joule-Thomson experiment.", "use": [ "lesson" ], "concepts": [ "concept/perfect-gas", "experiment/joule-thomson-experiment", "law/gay-lussac-s-law" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0236c67f8a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "127", "location": "Homogeneous Systems", "latex": "In \\SecRef{4} we defined temperature by means of the gas thermometer, but had to confine that definition to the cases in which the readings of the different gas thermometers (hydrogen, air, etc.)\\ agree as nearly as the desired accuracy of the result requires.", "markdown": "In 4 we defined temperature by means of the gas thermometer, but had to confine that definition to the cases in which the readings of the different gas thermometers (hydrogen, air, etc.) agree as nearly as the desired accuracy of the result requires.", "why": "Explains why the gas-thermometer definition of temperature is limited and why an absolute scale is needed.", "use": [ "lesson", "history" ], "concepts": [ "instrument/gas-thermometer", "quantity/absolute-temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-1417ae5822", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "126", "location": "Homogeneous Systems", "latex": "Equations \\Eq{(88)}~and \\Eq{(89)}, like Thomson and Joule's formula, are valid only within certain limits. It is, however, of theoretical interest to see how the different relations necessarily follow from one another.", "markdown": "Equations % [eqn:(88)](88)% and % [eqn:(89)](89)%, like Thomson and Joule’s formula, are valid only within certain limits. It is, however, of theoretical interest to see how the different relations necessarily follow from one another.", "why": "Models honest scientific caution: the derived formulas are approximate, yet worth following for how they connect.", "use": [ "lesson", "history" ], "concepts": [ "concept/characteristic-equation", "experiment/joule-thomson-experiment" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2acf688f4a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "131", "location": "Homogeneous Systems", "latex": "As soon as accurate measurement of even a single substance has determined $\\theta$~as a function of~$t$, the question regarding the value of the absolute temperature may be considered as solved for all cases.", "markdown": "As soon as accurate measurement of even a single substance has determined $\\theta$ as a function of $t$, the question regarding the value of the absolute temperature may be considered as solved for all cases.", "why": "States the payoff of the absolute-temperature method: one careful substance fixes the scale for all.", "use": [ "lesson", "website" ], "concepts": [ "quantity/absolute-temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-b430526c8f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "153", "location": "System in Different States of Aggregation", "latex": "In the critical state the compressibility is infinite; so are also the thermal coefficient of expansion and the specific heat at constant pressure; the heat of vaporization is zero.", "markdown": "In the critical state the compressibility is infinite; so are also the thermal coefficient of expansion and the specific heat at constant pressure; the heat of vaporization is zero.", "why": "It gives a concrete picture of what happens to the measurable quantities as a substance reaches its critical state.", "use": [ "lesson" ], "concepts": [ "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4bf41f0ef0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "155", "location": "System in Different States of Aggregation", "latex": "these curves will meet in one point, the \\emph{fundamental point}, also called the \\emph{triple point}.", "markdown": "these curves will meet in one point, the *fundamental point*, also called the *triple point*.", "why": "It states plainly the geometric meaning of the triple point, where three pressure curves meet.", "use": [ "lesson", "website" ], "concepts": [ "concept/triple-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-d6714cf392", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "157", "location": "System in Different States of Aggregation", "latex": "The existence of a sharp bend in the curve, however, can only be inferred from theory.", "markdown": "The existence of a sharp bend in the curve, however, can only be inferred from theory.", "why": "It shows learners the difference between what a theory predicts and what has been measured.", "use": [ "lesson" ], "concepts": [ "concept/discontinuous-function", "concept/triple-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e5e16f18ad", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "150", "location": "System in Different States of Aggregation", "latex": "No off-hand statement can be made with regard to the value of~$h_{1}$; even its sign must in the mean time remain uncertain.", "markdown": "No off-hand statement can be made with regard to the value of $h_{1}$; even its sign must in the mean time remain uncertain.", "why": "It shows that a physical quantity's sign can depend on a competition between two effects, so a learner should not guess.", "use": [ "lesson" ], "concepts": [ "quantity/specific-heat-of-saturated-vapour" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a6ad026515", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "158", "location": "System in Different States of Aggregation", "latex": "A geometrical representation may facilitate a general survey of the problem.", "markdown": "A geometrical representation may facilitate a general survey of the problem.", "why": "It introduces the (v, u) diagram as a method for seeing all equilibrium states at once.", "use": [ "lesson" ], "concepts": [ "concept/stability-of-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-52efad7777", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "151", "location": "System in Different States of Aggregation", "latex": "Watt assumed this to be the case for steam.", "markdown": "Watt assumed this to be the case for steam.", "why": "It records an early engineering assumption about steam, showing how the adiabatic case was once used as a working model.", "use": [ "history" ], "concepts": [ "concept/adiabatic-process", "person/watt" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4b6def4bd2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "171", "location": "System in Different States of Aggregation", "latex": "The total mass~$M$, the volume~$V$, and the energy~$U$ of a system being given, its corresponding state of stable equilibrium is determined by the position of the point $v = \\dfrac{V}{M}$, $u = \\dfrac{U}{M}$, in the plane of \\Fig{4}.", "markdown": "The total mass $M$, the volume $V$, and the energy $U$ of a system being given, its corresponding state of stable equilibrium is determined by the position of the point $v = \\dfrac{V}{M}$, $u = \\dfrac{U}{M}$, in the plane of [fig:4]Fig. 4.", "why": "It states in one sentence how a learner moves from the given mass, volume and energy to the equilibrium state, which is the central idea of the section.", "use": [ "lesson", "website" ], "concepts": [ "concept/energy", "concept/stability-of-equilibrium", "quantity/mass", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e05586e1cd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "172", "location": "System in Different States of Aggregation", "latex": "In this way we may find that ice cannot exist in stable equilibrium at a higher temperature than the fundamental temperature ($0.0074°$~C.), no matter how the pressure may be reduced. Liquid water, on the other hand, may, under suitable pressure, be brought to any temperature without freezing or evaporating.", "markdown": "In this way we may find that ice cannot exist in stable equilibrium at a higher temperature than the fundamental temperature ($0.0074°$ C.), no matter how the pressure may be reduced. Liquid water, on the other hand, may, under suitable pressure, be brought to any temperature without freezing or evaporating.", "why": "Turns the abstract diagram into concrete statements about ice and liquid water.", "use": [ "lesson", "website" ], "concepts": [ "concept/stability-of-equilibrium", "concept/state-of-aggregation", "concept/temperature", "concept/triple-point", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-fa8b56442e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "172", "location": "System in Different States of Aggregation", "latex": "A question which may also be answered directly is the following. Through what stages will a body pass if subjected to a series of definite external changes?", "markdown": "A question which may also be answered directly is the following. Through what stages will a body pass if subjected to a series of definite external changes?", "why": "It poses the practical question a student can ask of the diagram: what happens to a body as it is heated or cooled.", "use": [ "lesson" ], "concepts": [ "concept/stability-of-equilibrium", "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2cf607f4a1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "171", "location": "System in Different States of Aggregation", "latex": "It will be seen that their ratio is that of the three triangles, which the point~$(v, u)$ makes with the three sides of the fundamental triangle.", "markdown": "It will be seen that their ratio is that of the three triangles, which the point $(v, u)$ makes with the three sides of the fundamental triangle.", "why": "It tells the learner how to read the proportions of solid, liquid and gas off the geometry of the diagram.", "use": [ "lesson" ], "concepts": [ "concept/fundamental-triangle", "concept/ratio", "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-5b615b0880", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "170", "location": "System in Different States of Aggregation", "latex": "This quantity is essentially positive, since $M_{12}$,~$M_{21}$, as well as~$c_{v}$, are always positive, and $\\dfrac{\\dd p}{\\dd v}$~always negative for states of equilibrium.", "markdown": "This quantity is essentially positive, since $M_{12}$, $M_{21}$, as well as $c_{v}$, are always positive, and $\\dfrac{\\dd p}{\\dd v}$ always negative for states of equilibrium.", "why": "It shows the sign argument that makes the stability proof work, and which physical facts it relies on.", "use": [ "lesson" ], "concepts": [ "concept/stability-of-equilibrium", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ab901604c4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "171", "location": "System in Different States of Aggregation", "latex": "The conditions of stable equilibrium of any substance can thus be found, provided its fundamental triangle, its vaporization, fusion, and sublimation curves have been drawn once for all.", "markdown": "The conditions of stable equilibrium of any substance can thus be found, provided its fundamental triangle, its vaporization, fusion, and sublimation curves have been drawn once for all.", "why": "States the practical payoff of the diagram: one drawing answers every equilibrium question for a substance.", "use": [ "lesson", "website" ], "concepts": [ "concept/fundamental-triangle", "concept/fusion-curve", "concept/stability-of-equilibrium", "concept/sublimation-curve", "concept/vaporization-curve" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-93a41a8154", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "133", "location": "System in Different States of Aggregation", "latex": "The entropy may in general, however, as we shall see, assume several relative maxima, under the given external conditions. Each maximum, which is not the absolute one, will correspond to a more or less unstable equilibrium.", "markdown": "The entropy may in general, however, as we shall see, assume several relative maxima, under the given external conditions. Each maximum, which is not the absolute one, will correspond to a more or less unstable equilibrium.", "why": "It explains in plain terms why a system can sit in a state that is in equilibrium yet not the most stable one.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/stability-of-equilibrium", "concept/thermodynamic-equilibrium", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-789c47ab04", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "133", "location": "System in Different States of Aggregation", "latex": "The system in a state of this kind (\\eg\\ as supersaturated vapour) may occasionally, upon appropriate, very slight disturbances, undergo a finite change, and pass into another state of equilibrium, which necessarily corresponds to a greater value of the entropy.", "markdown": "The system in a state of this kind (*e.g.* as supersaturated vapour) may occasionally, upon appropriate, very slight disturbances, undergo a finite change, and pass into another state of equilibrium, which necessarily corresponds to a greater value of the entropy.", "why": "It gives a vivid picture of metastable equilibrium: a tiny disturbance can push the system into a new state with higher entropy.", "use": [ "lesson" ], "concepts": [ "concept/stability-of-equilibrium", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-18026c3026", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "146", "location": "System in Different States of Aggregation", "latex": "This was first verified by the measurements of W.~Thomson (Lord Kelvin).", "markdown": "This was first verified by the measurements of W. Thomson (Lord Kelvin).", "why": "It credits the first experimental confirmation of the pressure dependence of the melting point of ice.", "use": [ "history" ], "concepts": [ "concept/melting-point", "person/william-thomson" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6269520229", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "137", "location": "System in Different States of Aggregation", "latex": "Experience immediately shows, however, that in any state of equilibrium $\\dfrac{\\dd p}{\\dd v}$~is negative, since the pressure, whether positive or negative, and the volume always change in opposite directions.", "markdown": "Experience immediately shows, however, that in any state of equilibrium $\\dfrac{\\dd p}{\\dd v}$ is negative, since the pressure, whether positive or negative, and the volume always change in opposite directions.", "why": "It gives the physical reason a stable equilibrium needs pressure to fall as volume rises, a common point of confusion.", "use": [ "lesson" ], "concepts": [ "concept/characteristic-equation", "concept/stability-of-equilibrium", "concept/thermodynamic-equilibrium", "quantity/pressure", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-6c79695e6d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "136", "location": "System in Different States of Aggregation", "latex": "The equations~\\Eq{(99)} might therefore be called the system's \\emph{internal} or \\emph{intrinsic} conditions of equilibrium", "markdown": "The equations % [eqn:(99)](99)% might therefore be called the system’s *internal* or *intrinsic* conditions of equilibrium", "why": "It separates the conditions set by the substance itself from those set by the given mass, volume and energy.", "use": [ "lesson" ], "concepts": [ "concept/external-conditions-of-equilibrium", "concept/internal-conditions-of-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8dfb074ac8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "133", "location": "System in Different States of Aggregation", "latex": "Each maximum, which is not the absolute one, will correspond to a more or less unstable equilibrium. The system in a state of this kind (\\eg\\ as supersaturated vapour) may occasionally, upon appropriate, very slight disturbances, undergo a finite change, and pass into another state of equilibrium, which necessarily corresponds to a greater value of the entropy.", "markdown": "Each maximum, which is not the absolute one, will correspond to a more or less unstable equilibrium. The system in a state of this kind (*e.g.* as supersaturated vapour) may occasionally, upon appropriate, very slight disturbances, undergo a finite change, and pass into another state of equilibrium, which necessarily corresponds to a greater value of the entropy.", "why": "Explains why a supersaturated vapour can linger and then suddenly change: it sits at a relative, not absolute, entropy maximum.", "use": [ "lesson", "website" ], "concepts": [ "concept/stability-of-equilibrium", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-16659e19ff", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "132", "location": "System in Different States of Aggregation", "latex": "It is still very uncertain whether the molecules of liquid water are the same as those of ice.", "markdown": "It is still very uncertain whether the molecules of liquid water are the same as those of ice.", "why": "Shows that thermodynamic results here do not depend on settling what the molecules are.", "use": [ "history", "website" ], "concepts": [ "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-12be07168f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "135", "location": "System in Different States of Aggregation", "latex": "These six equations represent necessary properties of any state, which corresponds to a maximum value of the entropy, \\ie\\ of any state of equilibrium. As the first four refer to equality of temperature and pressure, the main interest centres in the last two, which contain the thermodynamical theory of fusion, evaporation, and sublimation.", "markdown": "These six equations represent necessary properties of any state, which corresponds to a maximum value of the entropy, *i.e.* of any state of equilibrium. As the first four refer to equality of temperature and pressure, the main interest centres in the last two, which contain the thermodynamical theory of fusion, evaporation, and sublimation.", "why": "Tells a learner which equalities are obvious and where the new physics about phase change lies.", "use": [ "lesson" ], "concepts": [ "concept/state-of-aggregation", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ebcf478661", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "140", "location": "System in Different States of Aggregation", "latex": "We learn, therefore, from equation~\\Eq{(102)} that in every isotherm the pressure, under which two states of aggregation of the substance may be kept in lasting contact, is represented by the ordinate of the straight line parallel to the axis of abscissæ, which intercepts equal areas on both sides of the isotherm.", "markdown": "We learn, therefore, from equation % [eqn:(102)](102)% that in every isotherm the pressure, under which two states of aggregation of the substance may be kept in lasting contact, is represented by the ordinate of the straight line parallel to the axis of abscissæ, which intercepts equal areas on both sides of the isotherm.", "why": "Gives a geometric picture for the pressure at which liquid and vapour coexist.", "use": [ "lesson", "website" ], "concepts": [ "concept/characteristic-equation", "concept/isotherm", "concept/saturated-vapour", "concept/state-of-aggregation", "method/equal-area-construction" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ac42f74819", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "142", "location": "System in Different States of Aggregation", "latex": "It is the heat which must be added to unit mass of the liquid, in order to completely change it to vapour under the constant pressure of its saturated vapour.", "markdown": "It is the heat which must be added to unit mass of the liquid, in order to completely change it to vapour under the constant pressure of its saturated vapour.", "why": "A plain definition of the heat of vaporization.", "use": [ "lesson" ], "concepts": [ "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-55f3643342", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "143", "location": "System in Different States of Aggregation", "latex": "By direct observation Regnault found the heat of vaporization of water at $100°$~C. to be~$536$.", "markdown": "By direct observation Regnault found the heat of vaporization of water at $100°$ C. to be $536$.", "why": "Gives the measured value to compare with the 535 calories computed from the Clapeyron equation, a check of theory against experiment.", "use": [ "lesson", "history" ], "concepts": [ "person/henri-victor-regnault", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-adb7b613ae", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "144", "location": "System in Different States of Aggregation", "latex": "This shows that the external work forms only a small part of the value of the latent heat of vaporization.", "markdown": "This shows that the external work forms only a small part of the value of the latent heat of vaporization.", "why": "States the physical meaning of the 0.075 ratio: most of the heat of vaporization goes into energy, not work.", "use": [ "lesson" ], "concepts": [ "concept/work", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c024ed0e9c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "146", "location": "System in Different States of Aggregation", "latex": "The melting pressure, therefore, just as the pressure of evaporation, depends on the temperature only. Conversely, a change of pressure produces a change in the melting point:", "markdown": "The melting pressure, therefore, just as the pressure of evaporation, depends on the temperature only. Conversely, a change of pressure produces a change in the melting point:", "why": "Links the vapour-pressure curve to the melting curve before the ice example.", "use": [ "lesson" ], "concepts": [ "concept/melting-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-f4015effd1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "171", "location": "System in Different States of Aggregation", "latex": "If this point lie within one of the regions $(1)$,~$(2)$, or~$(3)$, the system behaves as a homogeneous gas, liquid, or solid. If it lie within $(12)$, $(23)$, or~$(31)$, the system splits into two different states of aggregation, indicated by the numbers used in the notation of the region.", "markdown": "If this point lie within one of the regions $(1)$, $(2)$, or $(3)$, the system behaves as a homogeneous gas, liquid, or solid. If it lie within $(12)$, $(23)$, or $(31)$, the system splits into two different states of aggregation, indicated by the numbers used in the notation of the region.", "why": "Shows a learner how the position of one point on the diagram tells whether a substance is one phase or two.", "use": [ "lesson", "website" ], "concepts": [ "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-257e7d770f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "171", "location": "System in Different States of Aggregation", "latex": "The masses of these three portions may then be determined by the equations~\\Eq{(121a)}. It will be seen that their ratio is that of the three triangles, which the point~$(v, u)$ makes with the three sides of the fundamental triangle.", "markdown": "The masses of these three portions may then be determined by the equations % [eqn:(121a)](121a)%. It will be seen that their ratio is that of the three triangles, which the point $(v, u)$ makes with the three sides of the fundamental triangle.", "why": "Gives a geometric picture of how much solid, liquid and gas coexist at the triple point.", "use": [ "lesson" ], "concepts": [ "concept/fundamental-triangle", "concept/triple-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-bb39362914", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "172", "location": "System in Different States of Aggregation", "latex": "For instance, the behaviour of a body of mass~$M$, when cooled or heated at constant volume~$V$, may be known by observing the line $v = \\dfrac{V}{M}$ parallel to the axis of ordinates. The regions which this line traverses show the states through which the body passes, \\eg\\ whether the substance melts during the process, or whether it sublimes, etc.", "markdown": "For instance, the behaviour of a body of mass $M$, when cooled or heated at constant volume $V$, may be known by observing the line $v = \\dfrac{V}{M}$ parallel to the axis of ordinates. The regions which this line traverses show the states through which the body passes, *e.g.* whether the substance melts during the process, or whether it sublimes, etc.", "why": "Shows how to read a heating or cooling process off the diagram as a line crossing regions.", "use": [ "lesson" ], "concepts": [ "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8eea8e5b65", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "170", "location": "System in Different States of Aggregation", "latex": "It follows that the plane area~$\\phi''$ rises everywhere above the surface~$\\phi'$, and that $\\phi'' - \\phi'$ is never negative. This proves that the third solution within its region of validity (the fundamental triangle of the substance) represents stable equilibrium.", "markdown": "It follows that the plane area $\\phi''$ rises everywhere above the surface $\\phi'$, and that $\\phi'' - \\phi'$ is never negative. This proves that the third solution within its region of validity (the fundamental triangle of the substance) represents stable equilibrium.", "why": "Gives the conclusion of the stability proof in the book's own words.", "use": [ "lesson", "history" ], "concepts": [ "concept/stability-of-equilibrium", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-50dcfeef8c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "148", "location": "System in Different States of Aggregation", "latex": "By direct measurement, Regnault found the mean specific heat of steam under atmospheric pressure for temperatures somewhat higher than $100°$~C. to be~$0.48$.", "markdown": "By direct measurement, Regnault found the mean specific heat of steam under atmospheric pressure for temperatures somewhat higher than $100°$ C. to be $0.48$.", "why": "Shows a computed value (0.47) being checked against an independent direct measurement, so the thermodynamic relation is tested against experiment.", "use": [ "lesson", "history" ], "concepts": [ "person/henri-victor-regnault", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-cb51ca0bfe", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "150", "location": "System in Different States of Aggregation", "latex": "For, if during a rise of temperature of~$1°$ the vapour is to remain just saturated, it must evidently be compressed while being heated, since the specific volume of the saturated vapour decreases as the temperature rises. This compression, however, generates heat, and the question is, whether the latter is so considerable that it must be in part withdrawn by conduction, so as not to superheat the vapour.", "markdown": "For, if during a rise of temperature of $1°$ the vapour is to remain just saturated, it must evidently be compressed while being heated, since the specific volume of the saturated vapour decreases as the temperature rises. This compression, however, generates heat, and the question is, whether the latter is so considerable that it must be in part withdrawn by conduction, so as not to superheat the vapour.", "why": "Gives a physical argument for why the sign of the specific heat of saturated vapour is not obvious.", "use": [ "lesson", "website" ], "concepts": [ "concept/saturated-vapour", "quantity/specific-heat-of-saturated-vapour" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-20f2daf53d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "152", "location": "System in Different States of Aggregation", "latex": "Water vapour at $100°$~C. represents the first of the cases described above, \\ie\\ saturated water vapour at~$100°$ is superheated by adiabatic compression. Conversely, saturated water vapour at~$100°$ becomes supersaturated by adiabatic expansion.", "markdown": "Water vapour at $100°$ C. represents the first of the cases described above, *i.e.* saturated water vapour at $100°$ is superheated by adiabatic compression. Conversely, saturated water vapour at $100°$ becomes supersaturated by adiabatic expansion.", "why": "States the surprising result that steam behaves opposite to Watt's assumption, with h_1 = -1.12.", "use": [ "lesson", "website" ], "concepts": [ "concept/adiabatic-process", "quantity/specific-heat-of-saturated-vapour" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-2025b6be93", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "152", "location": "System in Different States of Aggregation", "latex": "Then the two states which are in contact with one another are identical. Such a value of~$\\theta$ is called a \\emph{critical temperature} of the substance.", "markdown": "Then the two states which are in contact with one another are identical. Such a value of $\\theta$ is called a *critical temperature* of the substance.", "why": "A short, clear definition of the critical temperature as the point where two coexisting states become identical.", "use": [ "lesson", "website" ], "concepts": [ "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8382a91a0c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "154", "location": "System in Different States of Aggregation", "latex": "According to equations~\\Eq{(120)}, the fundamental temperature is characterized by the condition that at it the pressure of the saturated vapour is equal to the pressure of fusion. It necessarily follows, by addition of the last two equations, that this pressure is also equal to the pressure of sublimation.", "markdown": "According to equations % [eqn:(120)](120)%, the fundamental temperature is characterized by the condition that at it the pressure of the saturated vapour is equal to the pressure of fusion. It necessarily follows, by addition of the last two equations, that this pressure is also equal to the pressure of sublimation.", "why": "Explains what makes the triple point special: all three transition pressures coincide.", "use": [ "lesson" ], "concepts": [ "concept/triple-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8e4dbfe46d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "154", "location": "System in Different States of Aggregation", "latex": "Let us determine, \\eg, the fundamental state of water. $0°$~C. is not its fundamental temperature, for at $0°$~C. the maximum vapour pressure of water is $4.62~\\Unit{mm.}$, but the melting pressure of ice is $760~\\Unit{mm}$.", "markdown": "Let us determine, *e.g.*, the fundamental state of water. $0°$ C. is not its fundamental temperature, for at $0°$ C. the maximum vapour pressure of water is $4.62~\\Unit{mm.}$, but the melting pressure of ice is $760~\\Unit{mm}$.", "why": "Shows the common misconception that water's triple point is at 0°C and begins the calculation that puts it near 0.0074°C.", "use": [ "lesson", "website" ], "concepts": [ "concept/triple-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ccd09d75e9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "157", "location": "System in Different States of Aggregation", "latex": "\\ie\\ at $1°$~C. the maximum vapour pressure of ice is $0.045~\\Unit{mm.}$ less than that of water. This has been verified by experiment. The existence of a sharp bend in the curve, however, can only be inferred from theory.", "markdown": "*i.e.* at $1°$ C. the maximum vapour pressure of ice is $0.045~\\Unit{mm.}$ less than that of water. This has been verified by experiment. The existence of a sharp bend in the curve, however, can only be inferred from theory.", "why": "Separates what experiment confirmed from what only theory predicts; note the book's own text reads '1°' where the preceding line uses -1° below the triple point.", "use": [ "lesson" ], "concepts": [ "concept/sublimation-curve", "concept/triple-point", "concept/vaporization-curve" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-caaea4eaef", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "162", "location": "System in Different States of Aggregation", "latex": "If, as a rough approximation, we assume this same ratio to hold for much lower temperatures, the latent heat of fusion would be zero at about $-120°$~C., and this would be the critical point of the fusion curve. The pressure here would be about $17,000$~atmospheres, and water and ice would become identical. We might imagine this to be the result of a considerable increase in the viscosity of water and in the plasticity of ice, as they both approach this state.", "markdown": "If, as a rough approximation, we assume this same ratio to hold for much lower temperatures, the latent heat of fusion would be zero at about $-120°$ C., and this would be the critical point of the fusion curve. The pressure here would be about $17,000$ atmospheres, and water and ice would become identical. We might imagine this to be the result of a considerable increase in the viscosity of water and in the plasticity of ice, as they both approach this state.", "why": "A candid, speculative extrapolation that shows how the critical-point idea might apply even to melting ice.", "use": [ "website", "history" ], "concepts": [ "concept/critical-point", "concept/fusion-curve", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-53ec4848ce", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "174", "location": "System of any Number of Independent Constituents", "latex": "An aqueous solution of sulphuric acid forms a system of three chemical elements, \\ce{S},~\\ce{H}, and~\\ce{O}, but contains only two independent constituents, for, in each phase (\\eg\\ liquid, vapour, solid) the mass of~\\ce{O} depends on that of \\ce{S}~and~\\ce{H}, while the masses of \\ce{S}~and~\\ce{H} are not in each phase interdependent.", "markdown": "An aqueous solution of sulphuric acid forms a system of three chemical elements, S, H, and O, but contains only two independent constituents, for, in each phase (*e.g.* liquid, vapour, solid) the mass of O depends on that of S and H, while the masses of S and H are not in each phase interdependent.", "why": "A worked case that separates the number of elements from the number of independent constituents, which is the step learners most often confuse.", "use": [ "lesson" ], "concepts": [ "concept/chemical-element", "concept/independent-constituent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-65e22f9071", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "180", "location": "System of any Number of Independent Constituents", "latex": "The composition of all the phases is then completely determined by a single variable, \\eg\\ the temperature or the pressure. This case is generally called \\emph{perfect heterogeneous} equilibrium.", "markdown": "The composition of all the phases is then completely determined by a single variable, *e.g.* the temperature or the pressure. This case is generally called *perfect heterogeneous* equilibrium.", "why": "It names the univariant case and tells the learner that one variable fixes everything, which makes the phase rule concrete.", "use": [ "lesson" ], "concepts": [ "concept/univariant-system", "theorem/phase-rule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-02d92ed8a6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "174", "location": "System of any Number of Independent Constituents", "latex": "The question as to the number of the independent constituents has nothing at all to do with the chemical constitution of the substances in the different phases, in particular, with the number of different kinds of molecules.", "markdown": "The question as to the number of the independent constituents has nothing at all to do with the chemical constitution of the substances in the different phases, in particular, with the number of different kinds of molecules.", "why": "It warns against a common mistake: counting molecular species instead of independent constituents.", "use": [ "lesson" ], "concepts": [ "concept/independent-constituent", "concept/molecule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-9319911293", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "174", "location": "System of any Number of Independent Constituents", "latex": "Thus, a quantity of water in any number of states forms but one independent constituent, however many associations and dissociations of \\ce{H2O}~molecules may occur (it may be a mixture of hydrogen and oxygen or ions), for the mass of the oxygen in each phase is completely determined by that of the hydrogen, and \\textit{vice versâ}. Should, however, an excess of oxygen or hydrogen be present in the vapour, we have then two independent constituents.", "markdown": "Thus, a quantity of water in any number of states forms but one independent constituent, however many associations and dissociations of H2O molecules may occur (it may be a mixture of hydrogen and oxygen or ions), for the mass of the oxygen in each phase is completely determined by that of the hydrogen, and *vice versâ*. Should, however, an excess of oxygen or hydrogen be present in the vapour, we have then two independent constituents.", "why": "A worked example showing that the count depends on whether the masses are interlinked, not on the chemistry.", "use": [ "lesson", "website" ], "concepts": [ "concept/dissociation", "concept/independent-constituent", "concept/molecule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-d139671e87", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "179", "location": "System of any Number of Independent Constituents", "latex": "The number of the phases, therefore, cannot exceed the number of the independent constituents by more than two; or, a system of $\\alpha$~independent constituents will contain at most $(\\alpha + 2)$ phases.", "markdown": "The number of the phases, therefore, cannot exceed the number of the independent constituents by more than two; or, a system of $\\alpha$ independent constituents will contain at most $(\\alpha + 2)$ phases.", "why": "States the phase rule plainly as a limit on how many phases can coexist.", "use": [ "lesson", "website" ], "concepts": [ "concept/independent-constituent", "concept/phase", "theorem/phase-rule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-620df8855f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "187", "location": "System of any Number of Independent Constituents", "latex": "This means that the heat effect in a variation that leaves the composition of all phases unchanged, divided by the change of volume of the system and by the absolute temperature, gives the rate of change of the equilibrium pressure with the temperature. Where application of heat increases the volume, as in the case of evaporation, the equilibrium pressure increases with temperature; in the opposite case, as in the melting of ice, it decreases with increase of temperature.", "markdown": "This means that the heat effect in a variation that leaves the composition of all phases unchanged, divided by the change of volume of the system and by the absolute temperature, gives the rate of change of the equilibrium pressure with the temperature. Where application of heat increases the volume, as in the case of evaporation, the equilibrium pressure increases with temperature; in the opposite case, as in the melting of ice, it decreases with increase of temperature.", "why": "Puts the meaning of the pressure-temperature relation into words, with evaporation and melting ice as contrasting examples.", "use": [ "lesson", "website" ], "concepts": [ "concept/thermodynamic-equilibrium", "concept/univariant-system", "quantity/heat-effect", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-d38baf88b5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "190", "location": "System of any Number of Independent Constituents", "latex": "If, finally, we dissolve salt sufficient for saturation in the newly formed unit of water, at constant temperature~$\\theta$ and constant pressure~$p$, the sum of the heat and work is simply the heat of solution", "markdown": "If, finally, we dissolve salt sufficient for saturation in the newly formed unit of water, at constant temperature $\\theta$ and constant pressure $p$, the sum of the heat and work is simply the heat of solution", "why": "It states in one sentence how the heat of solution is defined as the total of heat and work in the final step of the cycle.", "use": [ "lesson" ], "concepts": [ "concept/solution", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-38f99da6b1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "199", "location": "System of any Number of Independent Constituents", "latex": "This means that the relative decrease of the vapour pressure is proportional to the concentration of the solution (Wüllner's law).", "markdown": "This means that the relative decrease of the vapour pressure is proportional to the concentration of the solution (Wüllner’s law).", "why": "It states the lowering of vapour pressure in a form a student can test against measurements.", "use": [ "lesson" ], "concepts": [ "concept/concentration", "concept/lowering-of-vapour-pressure", "law/w-llner-s-law" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e58310ee9c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "203", "location": "System of any Number of Independent Constituents", "latex": "The error committed in putting the rate of diffusion of a salt through such a membrane equal to zero, falls below all measurable limits.", "markdown": "The error committed in putting the rate of diffusion of a salt through such a membrane equal to zero, falls below all measurable limits.", "why": "It shows the reader why an idealised semipermeable membrane is still a sound approximation, a point that is easy to misjudge.", "use": [ "lesson" ], "concepts": [ "concept/semipermeable-membrane" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-534eff6066", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "199", "location": "System of any Number of Independent Constituents", "latex": "This proposition furnishes a means of distinguishing between a solution and an emulsion. In an emulsion the number of particles suspended in the solution has no influence on the vapour pressure.", "markdown": "This proposition furnishes a means of distinguishing between a solution and an emulsion. In an emulsion the number of particles suspended in the solution has no influence on the vapour pressure.", "why": "It gives a concrete experimental test that separates a true solution from an emulsion.", "use": [ "lesson", "history" ], "concepts": [ "concept/lowering-of-vapour-pressure", "concept/solution" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-81f958bbb8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "198", "location": "System of any Number of Independent Constituents", "latex": "Since $\\Delta$~is \\emph{small} for small values of~$c$ (dilute solutions, \\SecRef{97}), then, according to~\\Eq{(178)}, the ratio of the vapour pressure of a dilute solution of fixed concentration to the vapour pressure of the pure solvent is practically independent of the temperature (Babo's law).", "markdown": "Since $\\Delta$ is *small* for small values of $c$ (dilute solutions, 97), then, according to % [eqn:(178)](178)%, the ratio of the vapour pressure of a dilute solution of fixed concentration to the vapour pressure of the pure solvent is practically independent of the temperature (Babo’s law).", "why": "It links Babo's law to the heat of dilution, so the learner sees why the law holds for dilute solutions.", "use": [ "history", "lesson" ], "concepts": [ "law/babo-s-law", "quantity/heat-of-dilution", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-36c6eef31e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "189", "location": "System of any Number of Independent Constituents", "latex": "This process may be accomplished directly, or in two steps, viz.\\ by condensing unit mass of water vapour into pure water, and then dissolving the salt in the water.", "markdown": "This process may be accomplished directly, or in two steps, viz. by condensing unit mass of water vapour into pure water, and then dissolving the salt in the water.", "why": "It shows the key trick behind Kirchhoff's formula: reach the same end state by two routes and equate the energy accounts.", "use": [ "lesson" ], "concepts": [ "law/first-law-of-thermodynamics", "theorem/kirchhoff-s-formula" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-45f113a3bd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "199", "location": "System of any Number of Independent Constituents", "latex": "Since $\\varphi$~is always positive (\\SecRef{217}), the vapour pressure must decrease with increasing concentration. This proposition furnishes a means of distinguishing between a solution and an emulsion. In an emulsion the number of particles suspended in the solution has no influence on the vapour pressure.", "markdown": "Since $\\varphi$ is always positive (217), the vapour pressure must decrease with increasing concentration. This proposition furnishes a means of distinguishing between a solution and an emulsion. In an emulsion the number of particles suspended in the solution has no influence on the vapour pressure.", "why": "It turns an abstract sign result into a practical test separating true solutions from suspensions.", "use": [ "lesson", "website" ], "concepts": [ "concept/emulsion", "theorem/lowering-of-vapour-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c93fa15efe", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "202", "location": "System of any Number of Independent Constituents", "latex": "Our last equations, therefore, connect in a perfectly general way the laws regarding the lowering of the vapour pressure, the elevation of the boiling temperature, the depression of the freezing point, and the change of the saturation point. Only one of these phenomena need be experimentally investigated in order to calculate~$\\varphi$, and by means of the value thus determined the others may be deduced for the same solution.", "markdown": "Our last equations, therefore, connect in a perfectly general way the laws regarding the lowering of the vapour pressure, the elevation of the boiling temperature, the depression of the freezing point, and the change of the saturation point. Only one of these phenomena need be experimentally investigated in order to calculate $\\varphi$, and by means of the value thus determined the others may be deduced for the same solution.", "why": "It shows the unifying idea of the chapter: one measured quantity predicts all the dilute-solution effects.", "use": [ "lesson", "website" ], "concepts": [ "concept/boiling-point-elevation", "concept/freezing-point-depression", "quantity/solution-characteristic-function-phi", "theorem/lowering-of-vapour-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e18b91218a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "203", "location": "System of any Number of Independent Constituents", "latex": "It is true that for no solution can perfectly \\emph{semipermeable} membranes of this character be manufactured. In fact, the further development of this theory (\\SecRef{259}) will exclude them as a matter of principle, for in every case the dissolved substance will also diffuse through the membrane, though possibly at an extremely slow rate.", "markdown": "It is true that for no solution can perfectly *semipermeable* membranes of this character be manufactured. In fact, the further development of this theory (259) will exclude them as a matter of principle, for in every case the dissolved substance will also diffuse through the membrane, though possibly at an extremely slow rate.", "why": "It candidly admits that an idealisation used in the argument cannot be realised exactly.", "use": [ "lesson", "history" ], "concepts": [ "concept/semipermeable-membrane" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-540191e557", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "205", "location": "System of any Number of Independent Constituents", "latex": "Since $\\varphi$~is positive, the osmotic pressure increases with increasing concentration, and also, since $p' - p''$ vanishes when $c = 0$, the osmotic pressure is necessarily positive.", "markdown": "Since $\\varphi$ is positive, the osmotic pressure increases with increasing concentration, and also, since $p' - p''$ vanishes when $c = 0$, the osmotic pressure is necessarily positive.", "why": "It shows how a sign argument gives two qualitative facts about osmotic pressure at once.", "use": [ "lesson" ], "concepts": [ "quantity/osmotic-pressure", "quantity/solution-characteristic-function-phi" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-ad4aaa4fd3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "205", "location": "System of any Number of Independent Constituents", "latex": "A better insight into the nature of these quantities is gained by extending to the liquid state the idea of the molecule, hitherto applied only to gases.", "markdown": "A better insight into the nature of these quantities is gained by extending to the liquid state the idea of the molecule, hitherto applied only to gases.", "why": "It marks a historical turning point where the theory is about to give φ a molecular meaning for liquids.", "use": [ "history" ], "concepts": [ "quantity/solution-characteristic-function-phi" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-4873cc04e9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "173", "location": "System of any Number of Independent Constituents", "latex": "The number of phases as well as the states of aggregation is quite arbitrary, although we at once recognize the fact that a system in equilibrium may consist of any number of solid and liquid phases, but only one single \\emph{gaseous} phase, for two different gases in contact are never in equilibrium with one another.", "markdown": "The number of phases as well as the states of aggregation is quite arbitrary, although we at once recognize the fact that a system in equilibrium may consist of any number of solid and liquid phases, but only one single *gaseous* phase, for two different gases in contact are never in equilibrium with one another.", "why": "Tells the learner what freedom a system has in its phases and why gases are the exception.", "use": [ "lesson", "website" ], "concepts": [ "concept/phase", "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-726d0019ab", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "173", "location": "System of any Number of Independent Constituents", "latex": "We define the number of independent constituents as follows. First find the number of elements contained in the system, and from these discard, as dependent constituents, all those whose quantity is determined in each phase by the remaining ones. The number of the remaining elements will be the number of independent constituents of the system.", "markdown": "We define the number of independent constituents as follows. First find the number of elements contained in the system, and from these discard, as dependent constituents, all those whose quantity is determined in each phase by the remaining ones. The number of the remaining elements will be the number of independent constituents of the system.", "why": "Gives a clear step-by-step recipe for counting independent constituents.", "use": [ "lesson" ], "concepts": [ "concept/independent-constituent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-e1b331b927", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "177", "location": "System of any Number of Independent Constituents", "latex": "There are for each independent constituent $(\\beta - 1)$ equations, which must be satisfied, and therefore for all the $\\alpha$~independent constituents $\\alpha (\\beta - 1)$ conditions.", "markdown": "There are for each independent constituent $(\\beta - 1)$ equations, which must be satisfied, and therefore for all the $\\alpha$ independent constituents $\\alpha (\\beta - 1)$ conditions.", "why": "Shows the equation count that, with the variable count, leads to the phase rule.", "use": [ "lesson" ], "concepts": [ "concept/internal-conditions-of-equilibrium", "theorem/phase-rule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-8fabe345be", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "180", "location": "System of any Number of Independent Constituents", "latex": "For water it was shown in~\\SecRef{187}, that at the triple point the temperature is $0.0074°$~C., and the pressure $4.62~\\Unit{mm.}$ of mercury.", "markdown": "For water it was shown in 187, that at the triple point the temperature is $0.0074°$ C., and the pressure $4.62~\\Unit{mm.}$ of mercury.", "why": "Gives a concrete measured triple point for the phase rule's simplest non-variant case.", "use": [ "lesson", "website" ], "concepts": [ "concept/non-variant-system", "concept/triple-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-50464fe9d5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "212", "location": "Gaseous System", "latex": "\\emph{The entropy of a mixture of gases is the sum of the entropies which the individual gases would have, if each at the same temperature occupied a volume equal to the total volume of the mixture.} This proposition was first established by Gibbs.", "markdown": "*The entropy of a mixture of gases is the sum of the entropies which the individual gases would have, if each at the same temperature occupied a volume equal to the total volume of the mixture.* This proposition was first established by Gibbs.", "why": "States the central result of the chapter and credits its source.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/gas-mixture", "law/gibbs-s-proposition" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-1bccad4a68", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "210", "location": "Gaseous System", "latex": "Experience shows that a gas on both sides of a membrane permeable to it is in equilibrium when its partial pressures (\\SecRef{18}) are the same on both sides, quite independent of the other gases present.", "markdown": "Experience shows that a gas on both sides of a membrane permeable to it is in equilibrium when its partial pressures (18) are the same on both sides, quite independent of the other gases present.", "why": "Gives the equilibrium rule for a gas across a membrane, which the later separation argument depends on.", "use": [ "lesson" ], "concepts": [ "concept/semipermeable-membrane", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-0414ca9294", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "210", "location": "Gaseous System", "latex": "Platinum foil at a white heat is permeable to hydrogen, but impermeable to air. If a vessel having a platinum wall be filled with pure hydrogen, and hermetically sealed, and the platinum be then heated, the hydrogen must completely diffuse out against atmospheric pressure. As the air cannot enter, the vessel must finally become completely exhausted.", "markdown": "Platinum foil at a white heat is permeable to hydrogen, but impermeable to air. If a vessel having a platinum wall be filled with pure hydrogen, and hermetically sealed, and the platinum be then heated, the hydrogen must completely diffuse out against atmospheric pressure. As the air cannot enter, the vessel must finally become completely exhausted.", "why": "A concrete experiment showing the membrane rule at work.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/diffusion", "concept/semipermeable-membrane" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-f700b7e73f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "214", "location": "Gaseous System", "latex": "It also appears that the increase of the entropy depends solely on the number of the molecules $n_{1}$,~$n_{2}$, and not on the nature---\\eg\\ the molecular weight, of the diffusing gases. The increase of the entropy does not depend on whether the gases are chemically alike or not.", "markdown": "It also appears that the increase of the entropy depends solely on the number of the molecules $n_{1}$, $n_{2}$, and not on the nature---*e.g.* the molecular weight, of the diffusing gases. The increase of the entropy does not depend on whether the gases are chemically alike or not.", "why": "Shows that the entropy gain from mixing depends only on how many molecules there are, not on what they are.", "use": [ "lesson" ], "concepts": [ "concept/diffusion", "concept/reversible-process" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-75852028b9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "214", "location": "Gaseous System", "latex": "It follows that the chemical difference of two gases, or, in general, of two substances, cannot be represented by a continuous variable; but that here we can speak only of a discontinuous relation, either of equality or inequality. This fact involves a fundamental distinction between chemical and physical properties, since the latter may always be regarded as continuous.", "markdown": "It follows that the chemical difference of two gases, or, in general, of two substances, cannot be represented by a continuous variable; but that here we can speak only of a discontinuous relation, either of equality or inequality. This fact involves a fundamental distinction between chemical and physical properties, since the latter may always be regarded as continuous.", "why": "A reflective remark on how chemical difference differs in kind from physical properties.", "use": [ "website", "history" ], "concepts": [ "concept/diffusion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-5c9d422e7b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "217", "location": "Gaseous System", "latex": "Where the volume remains unchanged, as, \\eg, in the dissociation of hydriodic acid, considered below, the equilibrium is independent of the pressure.", "markdown": "Where the volume remains unchanged, as, *e.g.*, in the dissociation of hydriodic acid, considered below, the equilibrium is independent of the pressure.", "why": "Tells learners when pressure matters to a gas equilibrium and when it does not.", "use": [ "lesson" ], "concepts": [ "concept/dissociation-of-hydriodic-acid", "theorem/condition-of-chemical-equilibrium-in-a-gas-mixture" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-12097863e9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "218", "location": "Gaseous System", "latex": "The term containing~$b$ refers to the heat spent in the increase of the internal energy; the term containing~$\\theta$ to that spent in external work.", "markdown": "The term containing $b$ refers to the heat spent in the increase of the internal energy; the term containing $\\theta$ to that spent in external work.", "why": "Splits the heat of a reaction into its internal-energy and external-work parts.", "use": [ "lesson" ], "concepts": [ "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-930b63679d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "222", "location": "Gaseous System", "latex": "There is always present a finite, though perhaps a very small number of all possible kinds of molecules. Thus, in water vapour at any temperature at least a trace of oxygen and hydrogen must be present (see also \\SecRef{259}).", "markdown": "There is always present a finite, though perhaps a very small number of all possible kinds of molecules. Thus, in water vapour at any temperature at least a trace of oxygen and hydrogen must be present (see also 259).", "why": "A surprising consequence of the equilibrium equation: dissociation is never complete and never entirely absent.", "use": [ "lesson", "website" ], "concepts": [ "concept/dissociation", "theorem/condition-of-chemical-equilibrium-in-a-gas-mixture" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-08e4f44d03", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "242", "location": "Dilute Solutions", "latex": "\\emph{the concentration of the dissolved gas is proportional to the pressure of the free gas on the solution} (Henry's law).", "markdown": "*the concentration of the dissolved gas is proportional to the pressure of the free gas on the solution* (Henry’s law).", "why": "States Henry's law as proportionality of dissolved concentration to gas pressure, and flags that the chapter's temperature equation (225) carries an erratum (θ2 corrected to θ²) that the later temperature-dependence line uses.", "use": [ "lesson" ], "concepts": [ "concept/concentration", "law/henry-s-law", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-fd5296f13a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "243", "location": "Dilute Solutions", "latex": "Van't Hoff was the first to calculate~$L$ by means of this equation from the solubility of succinic acid at $0°$~C.", "markdown": "Van’t Hoff was the first to calculate $L$ by means of this equation from the solubility of succinic acid at $0°$ C.", "why": "Shows the method of obtaining a heat effect from temperature variation of solubility; flag that the next line prints the factor as 0.4494 where the stated 6600 cal requires 0.04494, a print error.", "use": [ "lesson", "history" ], "concepts": [ "person/van-t-hoff", "quantity/heat-effect", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-671bfa87ee", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "259", "location": "Dilute Solutions", "latex": "Conversely, the dissociation of the sodium acetate increases on the addition of water, but the concentration of the free ions decreases, because they are distributed over a larger quantity of water.", "markdown": "Conversely, the dissociation of the sodium acetate increases on the addition of water, but the concentration of the free ions decreases, because they are distributed over a larger quantity of water.", "why": "It explains the opposite effects of dilution on dissociation and on ion concentration, the point learners most often confuse.", "use": [ "lesson" ], "concepts": [ "concept/concentration", "concept/dissociation", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c6d01847e2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "261", "location": "Dilute Solutions", "latex": "Since the addition of silver nitrate increases the number of the \\ce{Ag+}-ions, it diminishes the number of the \\ce{BrO3-}-ions, and thereby the solubility of the bromate, which is evidently measured by the sum $c_{1} + c_{4}$.", "markdown": "Since the addition of silver nitrate increases the number of the Ag+-ions, it diminishes the number of the BrO3--ions, and thereby the solubility of the bromate, which is evidently measured by the sum $c_{1} + c_{4}$.", "why": "It shows how adding a common ion lowers solubility, a result a learner can predict before doing any algebra.", "use": [ "lesson", "website" ], "concepts": [ "concept/ion", "concept/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-82ac73f2a2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "259", "location": "Dilute Solutions", "latex": "In order to distribute the\nsolvent so that the concentration of the common ion\n\\ce{CH3- . COO} may be the same in both solutions, some water\nmust be withdrawn from the less dissociated electrolyte\n(acetic acid), and added to the more strongly dissociated\n(\\ce{Na}-acetate).", "markdown": "In order to distribute the solvent so that the concentration of the common ion CH3- . COO may be the same in both solutions, some water must be withdrawn from the less dissociated electrolyte (acetic acid), and added to the more strongly dissociated (Na-acetate).", "why": "Shows how to reason about which way water must be moved to make two solutions isohydric.", "use": [ "lesson" ], "concepts": [ "concept/common-ion", "concept/dissociation", "concept/electrolyte", "concept/isohydric-solution", "concept/solvent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a1c3bb8843", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "260", "location": "Dilute Solutions", "latex": "The concentration\nof the \\ce{Ag+}-ions is inversely proportional to the\nconcentration of the \\ce{BrO3-}-ions. Since the addition of silver\nnitrate increases the number of the \\ce{Ag+}-ions, it diminishes\nthe number of the \\ce{BrO3-}-ions, and thereby the solubility of\nthe bromate, which is evidently measured by the sum $c_{1} + c_{4}$.", "markdown": "The concentration of the Ag+-ions is inversely proportional to the concentration of the BrO3--ions. Since the addition of silver nitrate increases the number of the Ag+-ions, it diminishes the number of the BrO3--ions, and thereby the solubility of the bromate, which is evidently measured by the sum $c_{1} + c_{4}$.", "why": "Explains why adding a salt with a common ion lowers solubility.", "use": [ "lesson", "website" ], "concepts": [ "concept/common-ion", "concept/equilibrium-constant", "concept/ion", "concept/solubility-product", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-c51ff4a18e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "258", "location": "Dilute Solutions", "latex": "or, \\emph{the two solutions are isohydric if the concentration of\nthe common ion \\emph{\\ce{CH3- COO}} is the same in both}.", "markdown": "or, *the two solutions are isohydric if the concentration of the common ion *CH3- COO* is the same in both*.", "why": "States in one sentence the practical test for isohydric solutions.", "use": [ "lesson", "website" ], "concepts": [ "concept/common-ion", "concept/isohydric-solution" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-68f69cf0c7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "259", "location": "Dilute Solutions", "latex": "It\nfollows, then, that when two equally diluted solutions of\nbinary electrolytes are mixed, the dissociation of the more\nweakly dissociated recedes, while that of the more strongly\ndissociated increases still further.", "markdown": "It follows, then, that when two equally diluted solutions of binary electrolytes are mixed, the dissociation of the more weakly dissociated recedes, while that of the more strongly dissociated increases still further.", "why": "Gives the general result of mixing two electrolytes with a common ion.", "use": [ "lesson", "website" ], "concepts": [ "concept/dissociation", "concept/electrolyte" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-f72481da90", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "261", "location": "Dilute Solutions", "latex": "It assigns to\neach kind of molecule in the two phases a constant ratio of\ndistribution, which is independent of the presence of other\ndissolved molecules.", "markdown": "It assigns to each kind of molecule in the two phases a constant ratio of distribution, which is independent of the presence of other dissolved molecules.", "why": "Gives the plain content of Nernst's distribution law.", "use": [ "lesson" ], "concepts": [ "law/distribution-law" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-26d051c382", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "262", "location": "Dilute Solutions", "latex": "When chemical interchanges between the different substances\nin solution are possible, as, \\eg, in a solution of dissociating\nsalts and acids with common ions, the term \\emph{degree\nof dissociation} has no meaning, for the ions may be combined\narbitrarily into dissociated molecules.", "markdown": "When chemical interchanges between the different substances in solution are possible, as, *e.g.*, in a solution of dissociating salts and acids with common ions, the term *degree of dissociation* has no meaning, for the ions may be combined arbitrarily into dissociated molecules.", "why": "Warns learners that a familiar quantity stops being well defined when ions are shared.", "use": [ "lesson" ], "concepts": [ "concept/common-ion", "concept/dissociation", "concept/ion", "concept/molecule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-db640a6d11", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "223", "location": "Dilute Solutions", "latex": "On account of incomplete experimental data, however, the calculation of~$\\Psi$ can be performed, besides for a gaseous phase, only for a \\emph{dilute solution}, \\ie\\ for a phase in which one kind of molecule far outnumbers all the others in the phase.", "markdown": "On account of incomplete experimental data, however, the calculation of $\\Psi$ can be performed, besides for a gaseous phase, only for a *dilute solution*, *i.e.* for a phase in which one kind of molecule far outnumbers all the others in the phase.", "why": "Gives a clear working definition of a dilute solution and explains why the theory is restricted to it.", "use": [ "lesson", "website" ], "concepts": [ "concept/dilute-solution", "quantity/psi-function" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-49d6e4168e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "224", "location": "Dilute Solutions", "latex": "Physically speaking, this means that the properties of a dilute solution, besides depending on the interactions between the molecules of the solvent, necessarily depend only on the interactions between the molecules of the solvent and the molecules of the dissolved substances, but not on the interactions of the dissolved substances among themselves, for these are small quantities of a higher order.", "markdown": "Physically speaking, this means that the properties of a dilute solution, besides depending on the interactions between the molecules of the solvent, necessarily depend only on the interactions between the molecules of the solvent and the molecules of the dissolved substances, but not on the interactions of the dissolved substances among themselves, for these are small quantities of a higher order.", "why": "Turns the mathematical linearity result into a physical picture of which molecular interactions matter.", "use": [ "lesson" ], "concepts": [ "concept/dilute-solution", "concept/dissolved-substance", "concept/linear-function", "concept/solvent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-92bcce3b9a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "226", "location": "Dilute Solutions", "latex": "We may therefore enunciate the following proposition: \\emph{Further dilution of a dilute solution, if no chemical changes accompany the process, produces neither an appreciable change of volume nor an appreciable heat effect}; or, in other words, \\emph{any change of volume or any heat effect produced by further dilution of a dilute solution is due to chemical transformations among the molecules of the dissolved substances}.", "markdown": "We may therefore enunciate the following proposition: *Further dilution of a dilute solution, if no chemical changes accompany the process, produces neither an appreciable change of volume nor an appreciable heat effect*; or, in other words, *any change of volume or any heat effect produced by further dilution of a dilute solution is due to chemical transformations among the molecules of the dissolved substances*.", "why": "A testable prediction of the theory, stated twice in complementary forms.", "use": [ "lesson", "website" ], "concepts": [ "concept/chemical-reaction", "concept/dilute-solution", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-80526b2ea4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "232", "location": "Dilute Solutions", "latex": "The influence of the temperature on~$K$, and therewith on the condition of equilibrium towards a certain chemical reaction, is controlled by the heat effect of that reaction, and the influence of the pressure is controlled by the corresponding change of volume of the system.", "markdown": "The influence of the temperature on $K$, and therewith on the condition of equilibrium towards a certain chemical reaction, is controlled by the heat effect of that reaction, and the influence of the pressure is controlled by the corresponding change of volume of the system.", "why": "States in words what the equations for the temperature and pressure dependence of K mean.", "use": [ "lesson" ], "concepts": [ "concept/chemical-reaction", "concept/equilibrium-constant", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-3bcdc5df6f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "233", "location": "Dilute Solutions", "latex": "It also follows from it that absolutely semipermeable membranes are non-existent, for the substance of any membrane would, in time, become saturated with the molecules of all the various kinds of substances in contact with one side of it, and thus give up each kind of substance to the other side.", "markdown": "It also follows from it that absolutely semipermeable membranes are non-existent, for the substance of any membrane would, in time, become saturated with the molecules of all the various kinds of substances in contact with one side of it, and thus give up each kind of substance to the other side.", "why": "Shows a surprising consequence of the equilibrium theory that challenges an idealization.", "use": [ "lesson", "website" ], "concepts": [ "concept/semipermeable-membrane" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-aa1bfbba58", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "233", "location": "Dilute Solutions", "latex": "We must not neglect any kind of molecule until we have ascertained by a particular experiment that its quantity is inappreciable.", "markdown": "We must not neglect any kind of molecule until we have ascertained by a particular experiment that its quantity is inappreciable.", "why": "A methodological warning against dropping species from a calculation on assumption.", "use": [ "lesson" ], "concepts": [ "concept/chemical-reaction", "concept/dissociation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-b9e0190466", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "234", "location": "Dilute Solutions", "latex": "This number represents the ratio of the number of dissociated molecules to the total number of molecules.", "markdown": "This number represents the ratio of the number of dissociated molecules to the total number of molecules.", "why": "Clarifies what the degree of dissociation of water, 14.3 × 10^-10 at 18 °C, measures.", "use": [ "lesson" ], "concepts": [ "concept/dissociation", "quantity/conductivity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-75048e68ce", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "242", "location": "Dilute Solutions", "latex": "\\ie\\ \\emph{the concentration of the dissolved gas is proportional to the\npressure of the free gas on the solution} (Henry's law).", "markdown": "*i.e.* *the concentration of the dissolved gas is proportional to the pressure of the free gas on the solution* (Henry’s law).", "why": "States Henry's law plainly as a result derived from the equilibrium condition, not just asserted.", "use": [ "lesson", "website" ], "concepts": [ "law/henry-s-law", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-89c5e481ad", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "254", "location": "Dilute Solutions", "latex": "\\emph{For every kind of molecule, which possesses the saute molecular\nweight in both phases, there is a constant ratio of distribution,\nwhich is independent of the presence of other molecules}\n(Nernst's law of distribution).", "markdown": "*For every kind of molecule, which possesses the saute molecular weight in both phases, there is a constant ratio of distribution, which is independent of the presence of other molecules* (Nernst’s law of distribution).", "why": "Gives the distribution law in one sentence (the source prints 'saute' where 'same' is meant, kept verbatim).", "use": [ "lesson", "website" ], "concepts": [ "law/distribution-law", "person/nernst" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-5c3ee09825", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "250", "location": "Dilute Solutions", "latex": "It may be well therefore to emphasize\nthis fact, that nothing concerning the molecular weight of\nthe solvent can be inferred from the relative lowering of the\nvapour pressure, any more than from its boiling point, freezing\npoint, or osmotic pressure.", "markdown": "It may be well therefore to emphasize this fact, that nothing concerning the molecular weight of the solvent can be inferred from the relative lowering of the vapour pressure, any more than from its boiling point, freezing point, or osmotic pressure.", "why": "Warns learners what these measurements cannot reveal: they count dissolved molecules only.", "use": [ "lesson" ], "concepts": [ "concept/boiling-point-elevation", "concept/freezing-point-depression", "quantity/osmotic-pressure", "theorem/lowering-of-vapour-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-a7e4a4faed", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "251", "location": "Dilute Solutions", "latex": "Should the number calculated from such a measurement\ndisagree with the number calculated from the percentage\ncomposition of the solution on the assumption of normal\nmolecules, some chemical change of the dissolved molecules\nmust have taken place by dissociation, association, hydrolysis,\nor the like.", "markdown": "Should the number calculated from such a measurement disagree with the number calculated from the percentage composition of the solution on the assumption of normal molecules, some chemical change of the dissolved molecules must have taken place by dissociation, association, hydrolysis, or the like.", "why": "Shows how a mismatch between measured and expected molecule counts reveals chemical change.", "use": [ "lesson", "website" ], "concepts": [ "concept/dilute-solution", "concept/dissociation", "concept/molecule", "concept/solution" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-bc98e0aaf8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "252", "location": "Dilute Solutions", "latex": "Conversely, a disagreement\nbetween the depression of the freezing point as calculated\nfrom the conductivity, and as observed, is not in itself an\nobjection to the theory, but rather to the assumptions made\nin the calculation concerning the kinds of molecules present.", "markdown": "Conversely, a disagreement between the depression of the freezing point as calculated from the conductivity, and as observed, is not in itself an objection to the theory, but rather to the assumptions made in the calculation concerning the kinds of molecules present.", "why": "Teaches that a failed prediction may point to a faulty assumption rather than a faulty theory.", "use": [ "lesson" ], "concepts": [ "concept/electrolyte", "concept/freezing-point-depression", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-dd02ad3dbd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "252", "location": "Dilute Solutions", "latex": "Raoult was the first to establish rigorously by experiment\nthe relation between the depression of the freezing\npoint and the number of the molecules of the dissolved\nsubstance; and van't Hoff gave a thermodynamical explanation\nand generalization of it by means of his theory of\nosmotic pressure. Application to electrolytes was rendered\npossible by Arrhenius' theory of electrolytic dissociation.", "markdown": "Raoult was the first to establish rigorously by experiment the relation between the depression of the freezing point and the number of the molecules of the dissolved substance; and van’t Hoff gave a thermodynamical explanation and generalization of it by means of his theory of osmotic pressure. Application to electrolytes was rendered possible by Arrhenius’ theory of electrolytic dissociation.", "why": "A short history of who contributed what to the theory of dilute solutions.", "use": [ "history", "website" ], "concepts": [ "concept/freezing-point-depression", "person/arrhenius", "person/raoult", "person/van-t-hoff", "quantity/osmotic-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/x-52c41cb80a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "242", "location": "Dilute Solutions", "latex": "Thomsen found the heat effect of the absorption of one gram\nmolecule of carbon dioxide to be $5880~\\Unit{cal}$.", "markdown": "Thomsen found the heat effect of the absorption of one gram molecule of carbon dioxide to be $5880~\\Unit{cal}$.", "why": "Gives a measured value to set beside the 4700 cal. computed from solubility, an honest case of two methods disagreeing.", "use": [ "lesson" ], "concepts": [ "law/henry-s-law", "person/j-thomsen", "quantity/heat-effect" ] } ], "equations": [ { "id": "planck-treatise-on-thermodynamics-1903/eq-9690d8638e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "5", "location": "Temperature", "latex": "\\frac{V}{M} = v", "name": "specific volume", "statement": "The specific volume v is the volume V divided by the mass M of the substance.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "M", "meaning": "mass" }, { "unit": null, "symbol": "v", "meaning": "specific volume (volume of unit mass)" } ], "sympy": "Eq(v, V/M)", "physics": true, "states": [ "quantity/specific-volume" ], "concepts": [ "quantity/mass", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-53ff5ff8ae", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "5", "location": "Temperature", "latex": "p = f(v, t)", "name": "characteristic equation", "statement": "Every substance has a characteristic relation giving its pressure as a function of specific volume and temperature.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree Centigrade", "symbol": "t", "meaning": "temperature in degrees Centigrade" } ], "sympy": "Eq(p, f(v, t))", "physics": true, "states": [ "concept/characteristic-equation" ], "concepts": [ "concept/temperature", "quantity/pressure", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-eddb0eef33", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "5", "location": "Temperature", "latex": "pv = T", "name": "Boyle's law", "statement": "At constant temperature the product of pressure and specific volume of a perfect gas is constant; T depends only on the temperature for a given gas.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "T", "meaning": "temperature function of the gas (depends only on temperature for a given gas)" } ], "sympy": "Eq(p*v, T)", "physics": true, "states": [ "law/boyle-s-law" ], "concepts": [ "concept/isothermal-process", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c9fe1f44ed", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "5", "location": "Temperature", "latex": "t = (v - v_{0})P", "name": null, "statement": "At constant pressure the temperature is proportional to the difference between the present specific volume and the specific volume at 0 degrees C.", "kind": "formula", "symbols": [ { "unit": "degree Centigrade", "symbol": "t", "meaning": "temperature in degrees Centigrade" }, { "unit": null, "symbol": "v", "meaning": "present specific volume" }, { "unit": null, "symbol": "v_{0}", "meaning": "specific volume at 0 degrees C" }, { "unit": null, "symbol": "P", "meaning": "function depending only on the pressure p" } ], "sympy": "Eq(t, (v - v0)*P)", "physics": true, "states": [], "concepts": [ "concept/isobaric-change", "concept/temperature", "quantity/pressure", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f166490237", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "5", "location": "Temperature", "latex": "pv_{0} = T_{0}", "name": null, "statement": "The product of pressure and the specific volume at 0 degrees C equals the value of the temperature function T at t = 0 degrees C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v_{0}", "meaning": "specific volume at 0 degrees C" }, { "unit": null, "symbol": "T_{0}", "meaning": "value of the function T when t = 0 degrees C" } ], "sympy": "Eq(p*v0, T0)", "physics": true, "states": [], "concepts": [ "concept/temperature", "law/boyle-s-law", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-ac192091cd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "6", "location": "Temperature", "latex": "v - v_{0} = \\alpha v_{0}", "name": "Gay-Lussac's law", "statement": "Heating a permanent gas from 0 to 1 degree C expands it by the same fraction alpha of its volume at 0 degrees C.", "kind": "law", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume at t = 1 degree C" }, { "unit": null, "symbol": "v_{0}", "meaning": "specific volume at 0 degrees C" }, { "unit": null, "symbol": "\\alpha", "meaning": "fractional expansion per degree C (about 1/273)" } ], "sympy": "Eq(v - v0, alpha*v0)", "physics": true, "states": [ "law/gay-lussac-s-law" ], "concepts": [ "concept/temperature", "quantity/coefficient-of-expansion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bde3d0cdce", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "6", "location": "Temperature", "latex": "1 = \\alpha v_{0} P\\Add{.}", "name": null, "statement": "Substituting t = 1 into the temperature relation gives a condition on the constant P, the coefficient alpha and the specific volume at 0 degrees C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "fractional expansion per degree C (about 1/273)" }, { "unit": null, "symbol": "v_{0}", "meaning": "specific volume at 0 degrees C" }, { "unit": null, "symbol": "P", "meaning": "function depending only on the pressure p" } ], "sympy": "Eq(1, alpha*v0*P)", "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/coefficient-of-expansion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b3510bc128", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "6", "location": "Temperature", "latex": "T = T_{0} (1 + \\alpha t)", "name": null, "statement": "The temperature function of a perfect gas is a linear function of the temperature t in degrees C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "T", "meaning": "temperature function of the gas" }, { "unit": null, "symbol": "T_{0}", "meaning": "value of T at t = 0 degrees C" }, { "unit": null, "symbol": "\\alpha", "meaning": "fractional expansion per degree C (about 1/273)" }, { "unit": "degree Centigrade", "symbol": "t", "meaning": "temperature in degrees Centigrade" } ], "sympy": "Eq(T, T0*(1 + alpha*t))", "physics": true, "states": [], "concepts": [ "concept/linear-function", "concept/perfect-gas", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d341be6921", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "6", "location": "Temperature", "latex": "p = \\frac{T_{0}}{v} (1 + \\alpha t)", "name": null, "statement": "The characteristic equation of a perfect gas at constant volume-pressure relation in degrees C, with pressure as a linear function of t at fixed specific volume.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "T_{0}", "meaning": "value of T at t = 0 degrees C" }, { "unit": null, "symbol": "\\alpha", "meaning": "fractional expansion per degree C (about 1/273)" }, { "unit": "degree Centigrade", "symbol": "t", "meaning": "temperature in degrees Centigrade" } ], "sympy": "Eq(p, T0*(1 + alpha*t)/v)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/perfect-gas", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2d719e3138", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "6", "location": "Temperature", "latex": "t + \\dfrac{1}{\\alpha} = \\theta", "name": "absolute temperature", "statement": "The absolute temperature theta is the Centigrade temperature shifted by 1/alpha degrees, so that the zero of theta lies about 273 degrees below the melting point of ice.", "kind": "definition", "symbols": [ { "unit": "degree Centigrade", "symbol": "t", "meaning": "temperature in degrees Centigrade" }, { "unit": null, "symbol": "\\alpha", "meaning": "fractional expansion per degree C (about 1/273)" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" } ], "sympy": "Eq(t + 1/alpha, theta)", "physics": true, "states": [ "quantity/absolute-temperature" ], "concepts": [ "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-66a0b8bd64", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "6", "location": "Temperature", "latex": "\\alpha T_{0} = C", "name": null, "statement": "The constant C is defined as alpha times T_0, the characteristic constant for the perfect gas under consideration.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "fractional expansion per degree C (about 1/273)" }, { "unit": null, "symbol": "T_{0}", "meaning": "value of T at t = 0 degrees C" }, { "unit": null, "symbol": "C", "meaning": "characteristic constant of the perfect gas" } ], "sympy": "Eq(alpha*T0, C)", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-dc3e77bfb0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "6", "location": "Temperature", "latex": "p = \\frac{C}{v} \\theta = \\frac{CM}{V} \\theta\\Add{.}", "name": "characteristic equation of a perfect gas", "statement": "In absolute temperature the characteristic equation of a perfect gas is pressure equal to C times theta over specific volume, or C M theta over V.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "C", "meaning": "characteristic constant of the perfect gas" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "M", "meaning": "mass" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": "Eq(p, C*theta/v)", "physics": true, "states": [ "law/characteristic-equation-of-a-perfect-gas" ], "concepts": [ "concept/characteristic-equation", "concept/perfect-gas", "quantity/absolute-temperature", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0225f4d7d6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "7", "location": "Temperature", "latex": "\\frac{1}{273} = \\alpha", "name": "coefficient of expansion", "statement": "The coefficient of expansion of a perfect gas equals 1/273, the fraction of its volume at 0 degrees C gained per degree.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "coefficient of expansion (fractional expansion per degree C)" } ], "sympy": "Eq(1/273, alpha)", "physics": true, "states": [ "quantity/coefficient-of-expansion" ], "concepts": [ "concept/perfect-gas", "quantity/pressure-coefficient" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1aebc0e362", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "8", "location": "Temperature", "latex": "dV = \\frac{CM\\theta}{p^{2}}\\, dp = \\frac{V}{p}\\, dp", "name": null, "statement": "At constant temperature an infinitely small increase of pressure dp contracts the volume of a perfect gas by dV, which equals V dp over p.", "kind": "result", "symbols": [ { "unit": null, "symbol": "dV", "meaning": "infinitely small change of volume (contraction)" }, { "unit": null, "symbol": "dp", "meaning": "infinitely small increase of pressure" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "C", "meaning": "characteristic constant of the perfect gas" }, { "unit": null, "symbol": "M", "meaning": "mass" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/infinitesimal", "concept/isothermal-process", "quantity/coefficient-of-elasticity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5f34473412", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "8", "location": "Temperature", "latex": "-\\frac{dV}{V} = \\frac{dp}{p}", "name": null, "statement": "The fractional contraction of unit volume of a perfect gas equals the fractional increase of pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "dV", "meaning": "infinitely small change of volume" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "dp", "meaning": "infinitely small increase of pressure" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/infinitesimal", "concept/isothermal-process", "quantity/coefficient-of-compressibility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-25549c8500", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "8", "location": "Temperature", "latex": "\\frac{\\;\\;dp\\;\\;}{\\dfrac{dp}{p}} = p", "name": "coefficient of elasticity of a perfect gas", "statement": "The coefficient of elasticity of a perfect gas equals its pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "dp", "meaning": "infinitely small increase of pressure" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [ "theorem/coefficient-of-elasticity-of-a-perfect-gas" ], "concepts": [ "concept/perfect-gas", "quantity/coefficient-of-elasticity", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1d0ee8e408", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "8", "location": "Temperature", "latex": "dp = \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} d\\theta + \\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} dv", "name": null, "statement": "The total differential of the characteristic equation: a change of pressure is the sum of its changes with temperature at constant volume and with volume at constant temperature.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinitesimal", "method/differentiation", "quantity/pressure-coefficient" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-506a29afad", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "8", "location": "Temperature", "latex": "\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} = -\\frac{\\left(\\dfrac{\\dd p}{\\dd \\theta}\\right)_{v}}{\\left(\\dfrac{\\dd p}{\\dd v}\\right)_{\\theta}}\\Add{.}", "name": null, "statement": "The change of volume with temperature at constant pressure equals minus the ratio of the pressure coefficient to the derivative of pressure with volume.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "method/differentiation", "quantity/coefficient-of-compressibility", "quantity/coefficient-of-expansion", "quantity/pressure-coefficient" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7c3f6b8493", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "9", "location": "Temperature", "latex": "\\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} = -\\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} · \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}", "name": null, "statement": "The pressure coefficient equals minus the product of the compressibility-type derivative and the expansion-type derivative, so the three coefficients are linked.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "quantity/coefficient-of-compressibility", "quantity/coefficient-of-expansion", "quantity/pressure-coefficient" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b8684b27cc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "9", "location": "Temperature", "latex": "\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} · \\frac{1}{v_{0}} = 0.00018", "name": null, "statement": "For mercury at 0 degrees C and atmospheric pressure the coefficient of expansion is 0.00018.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume of mercury" }, { "unit": null, "symbol": "v_{0}", "meaning": "specific volume at 0 degrees C" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/mercury", "quantity/coefficient-of-expansion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9fea9ca451", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "9", "location": "Temperature", "latex": "-\\left(\\frac{\\dd v}{\\dd p}\\right)_{\\theta} · \\frac{1}{v_{0}} = 0.000003", "name": null, "statement": "For mercury at 0 degrees C and atmospheric pressure the coefficient of compressibility, in atmospheres, is 0.000003.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume of mercury" }, { "unit": null, "symbol": "v_{0}", "meaning": "specific volume at 0 degrees C" }, { "unit": "atmosphere", "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/mercury", "quantity/coefficient-of-compressibility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1a780895d3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "10", "location": "Temperature", "latex": "p = \\frac{C_{1}M_{1} \\theta}{V_{1}}", "name": null, "statement": "Before diffusion, the first gas alone in volume V_1 at absolute temperature theta has pressure C_1 M_1 theta over V_1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure of the gas before diffusion" }, { "unit": null, "symbol": "C_{1}", "meaning": "characteristic constant of gas 1" }, { "unit": null, "symbol": "M_{1}", "meaning": "mass of gas 1" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "V_{1}", "meaning": "volume of gas 1 before diffusion" } ], "sympy": "Eq(p, C1*M1*theta/V1)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/gas-mixture", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7d83b5837e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "10", "location": "Temperature", "latex": "V = V_{1} + V_{2} + \\dots", "name": null, "statement": "The total volume of a gas mixture before diffusion is the sum of the volumes of its constituents and remains constant during diffusion.", "kind": "law", "symbols": [ { "unit": null, "symbol": "V", "meaning": "total volume of the mixture" }, { "unit": null, "symbol": "V_{1}", "meaning": "volume of gas 1" }, { "unit": null, "symbol": "V_{2}", "meaning": "volume of gas 2" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/diffusion", "concept/gas-mixture", "concept/sum", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b59dacb538", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "10", "location": "Temperature", "latex": "p_{1} = \\frac{C_{1}M_{1} \\theta}{V} = \\frac{V_{1}}{V} p", "name": null, "statement": "After diffusion each gas fills the total volume, so its partial pressure p_1 is the fraction V_1/V of the total pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p_{1}", "meaning": "partial pressure of gas 1" }, { "unit": null, "symbol": "p", "meaning": "total pressure" }, { "unit": null, "symbol": "V_{1}", "meaning": "volume of gas 1 before diffusion" }, { "unit": null, "symbol": "V", "meaning": "total volume" }, { "unit": null, "symbol": "C_{1}", "meaning": "characteristic constant of gas 1" }, { "unit": null, "symbol": "M_{1}", "meaning": "mass of gas 1" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" } ], "sympy": "Eq(p1, V1*p/V)", "physics": true, "states": [], "concepts": [ "concept/diffusion", "concept/gas-mixture", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e2a61a8415", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "10", "location": "Temperature", "latex": "p_{1} + p_{2} + \\dots = \\frac{V_{1} + V_{2} + \\dots}{V} p = p\\Add{.}", "name": "Dalton's law", "statement": "In a homogeneous mixture of gases the total pressure equals the sum of the partial pressures of the gases.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p_{1}", "meaning": "partial pressure of gas 1" }, { "unit": null, "symbol": "p_{2}", "meaning": "partial pressure of gas 2" }, { "unit": null, "symbol": "p", "meaning": "total pressure of the mixture" }, { "unit": null, "symbol": "V_{1}", "meaning": "volume of gas 1 before diffusion" }, { "unit": null, "symbol": "V", "meaning": "total volume" } ], "sympy": null, "physics": true, "states": [ "law/dalton-s-law" ], "concepts": [ "concept/gas-mixture", "concept/sum", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9595a4c42d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "11", "location": "Temperature", "latex": "p_{1} : p_{2} : \\dots = V_{1} : V_{2} : \\dots = C_{1}M_{1} : C_{2}M_{2} : \\dots\\Add{,}", "name": null, "statement": "The partial pressures of the gases are proportional to their volumes before diffusion and to C_i M_i.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p_{1}", "meaning": "partial pressure of gas 1" }, { "unit": null, "symbol": "V_{1}", "meaning": "volume of gas 1 before diffusion" }, { "unit": null, "symbol": "C_{1}", "meaning": "characteristic constant of gas 1" }, { "unit": null, "symbol": "M_{1}", "meaning": "mass of gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/common-ratio", "concept/gas-mixture", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-302e25586a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "11", "location": "Temperature", "latex": "p = (C_{1}M_{1} + C_{2}M_{2} + \\dots) \\frac{\\theta}{V}", "name": "characteristic equation of a gas mixture", "statement": "The characteristic equation of a mixture of perfect gases has the form of that of a single perfect gas with summed constants.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "total pressure of the mixture" }, { "unit": null, "symbol": "C_{1}", "meaning": "characteristic constant of gas 1" }, { "unit": null, "symbol": "M_{1}", "meaning": "mass of gas 1" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "V", "meaning": "total volume" } ], "sympy": null, "physics": true, "states": [ "theorem/characteristic-equation-of-a-gas-mixture" ], "concepts": [ "concept/characteristic-equation", "concept/gas-mixture", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-10b71dd3eb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "11", "location": "Temperature", "latex": "C = \\frac{C_{1}M_{1} + C_{2}M_{2} + \\dots}{M_{1} + M_{2} + \\dots}", "name": "characteristic constant", "statement": "The characteristic constant of a gas mixture is the mass-weighted average of the constants of its constituents.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "C", "meaning": "characteristic constant of the mixture" }, { "unit": null, "symbol": "C_{1}", "meaning": "characteristic constant of gas 1" }, { "unit": null, "symbol": "M_{1}", "meaning": "mass of gas 1" } ], "sympy": null, "physics": true, "states": [ "theorem/characteristic-constant" ], "concepts": [ "concept/arithmetical-mean", "concept/gas-mixture" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-24a672e192", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "11", "location": "Temperature", "latex": "0.0014291 : 0.0012571 : 0.0012930 = \\frac{1}{C_{1}} : \\frac{1}{C_{2}} : \\frac{1}{C_{3}}", "name": null, "statement": "The ratio of the densities of oxygen, atmospheric nitrogen and air equals the ratio of the reciprocals of their characteristic constants.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C_{1}", "meaning": "characteristic constant of oxygen" }, { "unit": null, "symbol": "C_{2}", "meaning": "characteristic constant of atmospheric nitrogen" }, { "unit": null, "symbol": "C_{3}", "meaning": "characteristic constant of air" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/common-ratio", "quantity/density" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4f736f3404", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "12", "location": "Temperature", "latex": "C = \\frac{C_{1}M_{1} + C_{2}M_{2}}{M_{1} + M_{2}}", "name": "characteristic constant", "statement": "The characteristic constant of a two-gas mixture is the mass-weighted average of the constants of the two gases.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "C", "meaning": "characteristic constant of the mixture" }, { "unit": null, "symbol": "C_{1}", "meaning": "characteristic constant of oxygen" }, { "unit": null, "symbol": "C_{2}", "meaning": "characteristic constant of atmospheric nitrogen" }, { "unit": null, "symbol": "M_{1}", "meaning": "mass of oxygen" }, { "unit": null, "symbol": "M_{2}", "meaning": "mass of atmospheric nitrogen" } ], "sympy": "Eq(C, (C1*M1 + C2*M2)/(M1 + M2))", "physics": true, "states": [ "theorem/characteristic-constant" ], "concepts": [ "concept/arithmetical-mean", "concept/gas-mixture" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-defb0bc16e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "12", "location": "Temperature", "latex": "M_{1} : M_{2} = 0.2998", "name": null, "statement": "The mass ratio of oxygen to atmospheric nitrogen in air is 0.2998, i.e. 23.1 per cent oxygen by weight.", "kind": "result", "symbols": [ { "unit": null, "symbol": "M_{1}", "meaning": "mass of oxygen" }, { "unit": null, "symbol": "M_{2}", "meaning": "mass of atmospheric nitrogen" } ], "sympy": "Eq(M1/M2, 0.2998)", "physics": true, "states": [], "concepts": [ "concept/air", "concept/common-ratio", "concept/composition-of-gas-mixture" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c76c2da803", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "12", "location": "Temperature", "latex": "C_{1}M_{1} : C_{2}M_{2} = p_{1} : p_{2} = V_{1} : V_{2} = 0.2637", "name": null, "statement": "The volume (and partial pressure) ratio of oxygen to nitrogen in air is 0.2637, i.e. 20.9 per cent oxygen by volume.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p_{1}", "meaning": "partial pressure of oxygen" }, { "unit": null, "symbol": "p_{2}", "meaning": "partial pressure of atmospheric nitrogen" }, { "unit": null, "symbol": "V_{1}", "meaning": "partial volume of oxygen" }, { "unit": null, "symbol": "V_{2}", "meaning": "partial volume of atmospheric nitrogen" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/common-ratio", "concept/composition-of-gas-mixture", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-58dd8f0b82", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "13", "location": "Temperature", "latex": "pv = \\const", "name": "Boyle's law", "statement": "At constant temperature the product of pressure and volume of a perfect gas is constant, so its isotherms are equilateral hyperbolae.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(p*v, C)", "physics": true, "states": [ "law/boyle-s-law" ], "concepts": [ "concept/constant", "concept/isothermal-curve", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b1f744f3ab", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "13", "location": "Temperature", "latex": "p = \\frac{R\\theta}{v - b} - \\frac{a}{v^{2}}", "name": "van der Waals' equation", "statement": "An approximate characteristic equation for gases and liquids, which reduces to that of a perfect gas for large specific volumes.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "R", "meaning": "constant depending on the nature of the substance" }, { "unit": null, "symbol": "a", "meaning": "constant depending on the nature of the substance" }, { "unit": null, "symbol": "b", "meaning": "constant depending on the nature of the substance" } ], "sympy": "Eq(p, R*theta/(v - b) - a/v**2)", "physics": true, "states": [ "concept/van-der-waals-equation" ], "concepts": [ "concept/approximation", "concept/characteristic-equation", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-df53eb51b4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "14", "location": "Temperature", "latex": "p = \\frac{R\\theta}{v - a} - \\frac{c}{\\theta(v + b)^{2}}", "name": "Clausius' equation", "statement": "Clausius' characteristic equation, an improvement on van der Waals' equation with an extra constant c, approaches the perfect-gas form for large specific volumes.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "R", "meaning": "constant depending on the nature of the substance" }, { "unit": null, "symbol": "a", "meaning": "constant depending on the nature of the substance" }, { "unit": null, "symbol": "b", "meaning": "constant depending on the nature of the substance" }, { "unit": null, "symbol": "c", "meaning": "additional constant depending on the nature of the substance" } ], "sympy": "Eq(p, R*theta/(v - a) - c/(theta*(v + b)**2))", "physics": true, "states": [ "concept/clausius-equation" ], "concepts": [ "concept/approximation", "concept/characteristic-equation", "concept/perfect-gas" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4c2e4e323f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "18", "location": "Temperature", "latex": "\\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} = 0", "name": null, "statement": "At the critical point the tangent to the isotherm is parallel to the axis of abscissae, so the slope of pressure with volume vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/critical-point", "concept/isotherm", "concept/slope-of-a-curve", "concept/tangent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a7054d6a70", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "18", "location": "Temperature", "latex": "\\left(\\frac{\\dd^{2} p}{\\dd v^{2}}\\right)_{\\theta} = 0", "name": null, "statement": "At the critical point the isotherm has a point of inflection, so the second derivative of pressure with volume vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "\\theta", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/critical-point", "concept/isotherm", "method/differentiation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e83cbe875a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "19", "location": "Temperature", "latex": "\\theta^{2} = \\frac{8c}{27(a + b)R}", "name": null, "statement": "From Clausius' equation the critical absolute temperature squared equals 8c over 27(a+b)R.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "\\theta", "meaning": "critical absolute temperature" }, { "unit": null, "symbol": "c", "meaning": "additional constant in Clausius' equation" }, { "unit": null, "symbol": "a", "meaning": "constant in Clausius' equation" }, { "unit": null, "symbol": "b", "meaning": "constant in Clausius' equation" }, { "unit": null, "symbol": "R", "meaning": "gas constant in Clausius' equation" } ], "sympy": "Eq(theta**2, 8*c/(27*(a + b)*R))", "physics": true, "states": [], "concepts": [ "concept/clausius-equation", "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9bfc6ab936", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "19", "location": "Temperature", "latex": "p^{2} = \\frac{cR}{216(a + b)^{3}}", "name": null, "statement": "From Clausius' equation the square of the critical pressure equals cR over 216(a+b)^3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "critical pressure" }, { "unit": null, "symbol": "c", "meaning": "additional constant in Clausius' equation" }, { "unit": null, "symbol": "R", "meaning": "gas constant in Clausius' equation" }, { "unit": null, "symbol": "a", "meaning": "constant in Clausius' equation" }, { "unit": null, "symbol": "b", "meaning": "constant in Clausius' equation" } ], "sympy": "Eq(p**2, c*R/(216*(a + b)**3))", "physics": true, "states": [], "concepts": [ "concept/clausius-equation", "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f4e79629cc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-temperature", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "19", "location": "Temperature", "latex": "v = 3a + 2b", "name": null, "statement": "From Clausius' equation the critical specific volume equals 3a + 2b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "critical specific volume" }, { "unit": null, "symbol": "a", "meaning": "constant in Clausius' equation" }, { "unit": null, "symbol": "b", "meaning": "constant in Clausius' equation" } ], "sympy": "Eq(v, 3*a + 2*b)", "physics": true, "states": [], "concepts": [ "concept/clausius-equation", "concept/critical-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-304dbefb1c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "26", "location": "Molecular Weight", "latex": "p = \\frac{C_{0} \\theta}{v_{0}}", "name": null, "statement": "At a given temperature and pressure, hydrogen's pressure equals its constant C_0 times the temperature divided by its specific volume v_0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "C_0", "meaning": "characteristic constant of hydrogen (the constant C for hydrogen)" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "v_0", "meaning": "specific volume of hydrogen (volume per unit mass)" } ], "sympy": "Eq(p, C0*theta/v0)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/perfect-gas", "concept/temperature", "law/characteristic-equation-of-a-perfect-gas", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-00e2a9c407", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "26", "location": "Molecular Weight", "latex": "p = \\frac{C\\theta}{v}", "name": null, "statement": "For any other gas at the same temperature and pressure, the pressure equals its constant C times the temperature divided by its specific volume v.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "C", "meaning": "characteristic constant of the gas" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume of the gas (volume per unit mass)" } ], "sympy": "Eq(p, C*theta/v)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/perfect-gas", "concept/temperature", "law/characteristic-equation-of-a-perfect-gas", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cd1f212d42", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "26", "location": "Molecular Weight", "latex": "C = \\frac{m_{0}C_{0}}{m}", "name": null, "statement": "The characteristic constant C of a gas is inversely proportional to its molecular weight m, scaled from the hydrogen constant C_0 and hydrogen molecular weight m_0 (equation 13).", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "characteristic constant of the gas" }, { "unit": null, "symbol": "m_0", "meaning": "molecular weight of hydrogen" }, { "unit": null, "symbol": "C_0", "meaning": "characteristic constant of hydrogen" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the gas" } ], "sympy": "Eq(C, m0*C0/m)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/common-ratio", "concept/proportion", "law/characteristic-equation-of-a-perfect-gas", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9293f7771e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "C = \\frac{m_{0}C_{0}}{m} = \\frac{m_{0}}{m} · \\frac{pv_{0}}{\\theta} = \\frac{2 · 1013650}{m · 273 · 0.00008988} = \\frac{82600000}{m}", "name": null, "statement": "Substituting the measured density of hydrogen at 0 °C and atmospheric pressure (with m_0 = 2) gives the characteristic constant C as approximately 82600000 divided by m.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "characteristic constant of the gas" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the gas" }, { "unit": null, "symbol": "m_0", "meaning": "molecular weight of hydrogen (equal to 2)" }, { "unit": null, "symbol": "C_0", "meaning": "characteristic constant of hydrogen" }, { "unit": null, "symbol": "p", "meaning": "pressure (atmospheric, at 0 °C)" }, { "unit": null, "symbol": "v_0", "meaning": "specific volume of hydrogen at 0 °C and atmospheric pressure" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature (273)" } ], "sympy": "Eq(C, 82600000/m)", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/characteristic-equation", "quantity/absolute-gas-constant", "quantity/density", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b75812a8d3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "82600000 = R", "name": "absolute gas constant", "statement": "The constant R is defined as the number 82600000 for brevity, and it is independent of the nature of the individual gas.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "absolute gas constant" } ], "sympy": "Eq(R, 82600000)", "physics": true, "states": [ "quantity/absolute-gas-constant" ], "concepts": [ "concept/constant" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bab4a8c510", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "p = \\frac{R}{m} · \\frac{\\theta}{v}", "name": null, "statement": "The characteristic equation of a chemically homogeneous perfect gas of molecular weight m: pressure equals the absolute gas constant R divided by m, times temperature divided by specific volume (equation 14).", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "R", "meaning": "absolute gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the gas" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume of the gas (volume per unit mass)" } ], "sympy": "Eq(p, R*theta/(m*v))", "physics": true, "states": [ "law/characteristic-equation-of-a-perfect-gas" ], "concepts": [ "concept/characteristic-equation", "concept/perfect-gas", "concept/pressure", "concept/temperature", "law/characteristic-equation-of-a-perfect-gas", "quantity/absolute-gas-constant", "quantity/molecular-weight", "quantity/pressure", "quantity/specific-volume", "quantity/volume" ], "pages": [ "27", "57", "88" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "planck-treatise-on-thermodynamics-1903/ch-proof" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c4eb3de3f8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "m = \\frac{R}{C}", "name": null, "statement": "The molecular weight of a gas can be deduced from its characteristic equation as the absolute gas constant divided by the characteristic constant C (equation 15).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "m", "meaning": "molecular weight of the gas" }, { "unit": null, "symbol": "R", "meaning": "absolute gas constant" }, { "unit": null, "symbol": "C", "meaning": "characteristic constant of the gas" } ], "sympy": "Eq(m, R/C)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "quantity/absolute-gas-constant", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-32276a2a3d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "v = \\dfrac{V}{M}", "name": null, "statement": "The specific volume v is the total volume V divided by the mass M of the gas.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume (volume per unit mass)" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" } ], "sympy": "Eq(v, V/M)", "physics": true, "states": [], "concepts": [ "quantity/density", "quantity/entropy", "quantity/mass", "quantity/specific-volume", "quantity/volume" ], "pages": [ "27", "157", "171" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8afd7f3492", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "V = \\frac{R\\theta}{p} · \\frac{M}{m}", "name": null, "statement": "The volume of a gas equals R times temperature over pressure, times the ratio of its mass to its molecular weight.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "R", "meaning": "absolute gas constant" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the gas" } ], "sympy": "Eq(V, R*theta/p*M/m)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "law/characteristic-equation-of-a-perfect-gas", "quantity/absolute-gas-constant", "quantity/molecular-weight", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6443a8be92", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "\\dfrac{M}{m} = n", "name": null, "statement": "The number of molecules n in a quantity of gas is defined as its mass M divided by its molecular weight m.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of molecules in the quantity of gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the gas" } ], "sympy": "Eq(n, M/m)", "physics": false, "states": [], "concepts": [ "quantity/mass", "quantity/molecular-weight", "quantity/number-of-equivalents", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-90a28a87ce", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "27", "location": "Molecular Weight", "latex": "V = \\frac{R\\theta}{p} · n", "name": null, "statement": "At given temperature and pressure, the volume of a quantity of gas depends only on the number of molecules n present, not on the nature of the gas.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "R", "meaning": "absolute gas constant" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "n", "meaning": "number of molecules" } ], "sympy": "Eq(V, R*theta*n/p)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "law/avogadro-s-law", "quantity/absolute-gas-constant", "quantity/number-of-molecules", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-513dae5b74", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "28", "location": "Molecular Weight", "latex": "p_{1} : p_{2} : \\dots = C_{1}M_{1} : C_{2}M_{2}", "name": null, "statement": "In a mixture, the ratio of the partial pressures of the constituent gases is the ratio of their characteristic constants times their masses (taken from equation 9).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p_1, p_2", "meaning": "partial pressures of the constituent gases" }, { "unit": null, "symbol": "C_1, C_2", "meaning": "characteristic constants of the constituent gases" }, { "unit": null, "symbol": "M_1, M_2", "meaning": "masses of the constituent gases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/common-ratio", "concept/gas-mixture", "concept/proportion", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-089de8f735", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "28", "location": "Molecular Weight", "latex": "p_{1} : p_{2} : \\dots = \\frac{M_{1}}{m_{1}} : \\frac{M_{2}}{m_{2}} : \\dots = n_{1} : n_{2} : \\dots", "name": null, "statement": "The ratio of the partial pressures of the gases in a mixture equals the ratio of the numbers of molecules of each gas present.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p_1, p_2", "meaning": "partial pressures of the constituent gases" }, { "unit": null, "symbol": "M_1, M_2", "meaning": "masses of the constituent gases" }, { "unit": null, "symbol": "m_1, m_2", "meaning": "molecular weights of the constituent gases" }, { "unit": null, "symbol": "n_1, n_2", "meaning": "numbers of molecules of the constituent gases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/common-ratio", "concept/gas-mixture", "concept/proportion", "law/avogadro-s-law", "quantity/number-of-molecules", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-42878e3e41", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "28", "location": "Molecular Weight", "latex": "\\frac{M_{1} + M_{2} + \\dots}{m} = \\frac{M_{1}}{m_{1}} + \\frac{M_{2}}{m_{2}} + \\dots", "name": null, "statement": "The apparent molecular weight m of a mixture is defined so that the total mass divided by it equals the sum of the masses of each constituent divided by its molecular weight.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "m", "meaning": "apparent molecular weight of the mixture" }, { "unit": null, "symbol": "M_1, M_2", "meaning": "masses of the constituent gases" }, { "unit": null, "symbol": "m_1, m_2", "meaning": "molecular weights of the constituent gases" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/gas-mixture", "quantity/apparent-molecular-weight", "quantity/molecular-weight", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-219b4db4b3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "28", "location": "Molecular Weight", "latex": "m = \\frac{M_{1} + M_{2} + \\dots}{\\dfrac{M_{1}}{m_{1}} + \\dfrac{M_{2}}{m_{2}} + \\dots}", "name": null, "statement": "The apparent molecular weight of a mixture is the total mass divided by the total number of molecules, which is the sum of each constituent's mass over its molecular weight.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "m", "meaning": "apparent molecular weight of the mixture" }, { "unit": null, "symbol": "M_1, M_2", "meaning": "masses of the constituent gases" }, { "unit": null, "symbol": "m_1, m_2", "meaning": "molecular weights of the constituent gases" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/gas-mixture", "quantity/apparent-molecular-weight", "quantity/molecular-weight", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bc49f96784", "chapter": "planck-treatise-on-thermodynamics-1903/ch-molecular-weight", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "30", "location": "Molecular Weight", "latex": "\\ce{C5H11Br} = \\ce{C5H10 + HBr}", "name": null, "statement": "The dissociation of amylene hydrobromide into amylene and hydrogen bromide, which doubles the number of molecules when the reaction is complete.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "C5H11Br", "meaning": "amylene hydrobromide (book's erratum-corrected name for the substance)" }, { "unit": null, "symbol": "C5H10", "meaning": "amylene" }, { "unit": null, "symbol": "HBr", "meaning": "hydrogen bromide" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/abnormal-vapour-densities", "concept/chemical-reaction", "concept/dissociation", "concept/molecule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c6ff0b5f02", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "33", "location": "Quantity of Heat", "latex": "\\frac{Q}{\\Delta\\theta} = c_{m}.", "name": null, "statement": "The mean specific heat (mean heat capacity) of 1 gram of a substance is the heat it receives divided by the corresponding increase of temperature between the initial and final temperatures of the process.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "quantity of heat each gram of the substance receives" }, { "unit": null, "symbol": "\\Delta\\theta", "meaning": "corresponding increase of temperature" }, { "unit": null, "symbol": "c_{m}", "meaning": "mean specific heat (mean heat capacity) of 1 gram of the substance between the initial and final temperatures" } ], "sympy": "Eq(Q/Delta_theta, c_m)", "physics": true, "states": [], "concepts": [ "concept/common-ratio", "concept/heat", "concept/temperature", "quantity/quantity", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-66bf3ac1cb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "33", "location": "Quantity of Heat", "latex": "\\frac{Q}{d\\theta} = c.", "name": null, "statement": "The specific heat of a substance at temperature theta is the quantity of heat received divided by the infinitely small increase of temperature.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "quantity of heat each gram of the substance receives" }, { "unit": null, "symbol": "d\\theta", "meaning": "infinitely small increase of temperature" }, { "unit": null, "symbol": "c", "meaning": "specific heat of the substance at temperature theta" } ], "sympy": "Eq(Q/d_theta, c)", "physics": true, "states": [], "concepts": [ "concept/common-ratio", "concept/heat", "concept/infinitesimal", "concept/temperature", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-acad5749c4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "43", "location": "General Exposition", "latex": "U_{2} - U_{1} = Q + W\\Add{.}", "name": null, "statement": "The increase of the energy of a system undergoing a change equals the mechanical equivalent of the heat absorbed plus the work expended on it.", "kind": "law", "symbols": [ { "unit": null, "symbol": "U_{2}", "meaning": "energy of the system in state 2" }, { "unit": null, "symbol": "U_{1}", "meaning": "energy of the system in state 1" }, { "unit": "erg", "symbol": "Q", "meaning": "mechanical equivalent of the heat absorbed by the system" }, { "unit": null, "symbol": "W", "meaning": "amount of work expended on the system (positive when the change takes place in the direction of the external forces)" } ], "sympy": "Eq(U_2 - U_1, Q + W)", "physics": true, "states": [], "concepts": [ "concept/change-of-energy", "concept/heat", "concept/work", "law/conservation-of-energy", "law/first-law-of-thermodynamics", "quantity/internal-energy", "quantity/mechanical-equivalent-of-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9588d311c6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "44", "location": "General Exposition", "latex": "U_{2} = U_{1}", "name": null, "statement": "For a cycle of operations, where the final state is the initial state, the energy of the system is the same at the end as at the start.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U_{2}", "meaning": "energy of the system in state 2 (here identical with state 1)" }, { "unit": null, "symbol": "U_{1}", "meaning": "energy of the system in state 1" } ], "sympy": "Eq(U_2, U_1)", "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/work", "law/conservation-of-energy", "law/first-law-of-thermodynamics", "quantity/internal-energy" ], "pages": [ "44", "47" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-ed3cf157b4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "44", "location": "General Exposition", "latex": "Q + W = 0\\Add{.}", "name": null, "statement": "For a cycle of operations the mechanical equivalent of the external heat effect equals in magnitude and opposite in sign the external work, so their sum is zero; this shows perpetual motion is impracticable.", "kind": "result", "symbols": [ { "unit": "erg", "symbol": "Q", "meaning": "mechanical equivalent of the heat absorbed by the system" }, { "unit": null, "symbol": "W", "meaning": "amount of work expended on the system" } ], "sympy": "Eq(Q + W, 0)", "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/perpetual-motion", "concept/work", "law/conservation-of-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b9dc081efd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "44", "location": "General Exposition", "latex": "U = \\const", "name": null, "statement": "If no external effects are produced by a change of state, the energy of the system remains constant.", "kind": "law", "symbols": [ { "unit": null, "symbol": "U", "meaning": "energy of the system" }, { "unit": null, "symbol": "C", "meaning": "a constant value of the energy" } ], "sympy": "Eq(U, C)", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/external-effect", "concept/perfect-system", "law/conservation-of-energy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-015a3f5d2c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "43", "location": "General Exposition", "latex": "U_{2} - U_{1} = U_{2}", "name": null, "statement": "When state 1 is taken as the normal state, the energy of state 2 referred to it equals the difference U_2 - U_1, since U_1 is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U_{2}", "meaning": "energy of the system in state 2" }, { "unit": null, "symbol": "U_{1}", "meaning": "energy of the system in state 1, taken as the normal state (zero energy)" } ], "sympy": "Eq(U_2 - U_1, U_2)", "physics": true, "states": [], "concepts": [ "concept/normal-state", "concept/normal-state-zero-of-energy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-833599be42", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-exposition", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "43", "location": "General Exposition", "latex": "U_{1} = 0", "name": null, "statement": "Taking state 1 as the normal state, its energy is zero, because no energy is needed to bring the system from state 1 to the normal state.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "U_{1}", "meaning": "energy of the system in state 1, taken as the normal state" } ], "sympy": "Eq(U_1, 0)", "physics": true, "states": [], "concepts": [ "concept/normal-state", "concept/zero-displacement", "concept/zero-energy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-02bd55e693", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "54", "location": "Applications to Homogeneous Systems", "latex": "Q = U_{2} - U_{1} + \\int_{1}^{2} p\\, dV", "name": null, "statement": "The heat absorbed along a reversible path equals the change of internal energy plus the integral of pressure over volume, so Q depends on the path.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": "Eq(Q, U2 - U1 + Integral(p, (V, V1, V2)))", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/integral", "concept/reversible-process", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5a433ac025", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "46", "location": "Applications to Homogeneous Systems", "latex": "U_{2} - U_{1} = Q + W", "name": "first law of thermodynamics", "statement": "The change in internal energy equals the heat added plus the work done, as stated for the homogeneous system (equation 17 applied).", "kind": "law", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "Q", "meaning": "heat absorbed" }, { "unit": null, "symbol": "W", "meaning": "external work" } ], "sympy": "Eq(U2 - U1, Q + W)", "physics": true, "states": [ "law/first-law-of-thermodynamics" ], "concepts": [ "concept/heat", "concept/work", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9754c8c091", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "47", "location": "Applications to Homogeneous Systems", "latex": "\\theta_{2} = \\theta_{1}", "name": null, "statement": "Joule's experiment found that for perfect gases the temperature is unchanged by free expansion.", "kind": "result", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": "Eq(theta2, theta1)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "experiment/joule-s-experiments" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-eb89a07b3e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "49", "location": "Applications to Homogeneous Systems", "latex": "\\left(\\frac{\\dd U}{\\dd V}\\right)_{\\theta} = 0\\Add{.}", "name": null, "statement": "For a perfect gas the internal energy does not change with volume at constant temperature, so it depends only on temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": "Eq(Derivative(U, V), 0)", "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/perfect-gas", "concept/temperature", "quantity/internal-energy", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2fff2c26d4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "49", "location": "Applications to Homogeneous Systems", "latex": "Q = 0", "name": null, "statement": "No heat passes through the non-conducting porous plug tube, so the heat term vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed" } ], "sympy": "Eq(Q, 0)", "physics": true, "states": [], "concepts": [ "concept/heat", "experiment/porous-plug-experiment" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5bc099843f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "49", "location": "Applications to Homogeneous Systems", "latex": "W = p_{1}V_{1} - p_{2}V_{2},", "name": null, "statement": "The external work done on a nearly perfect gas pushed through the porous plug is the difference of the pressure-volume products at the two ends.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work" }, { "unit": null, "symbol": "p_1", "meaning": "pressure at high-pressure side" }, { "unit": null, "symbol": "V_1", "meaning": "volume at high-pressure side" }, { "unit": null, "symbol": "p_2", "meaning": "pressure at low-pressure side" }, { "unit": null, "symbol": "V_2", "meaning": "volume at low-pressure side" } ], "sympy": "Eq(W, p1*V1 - p2*V2)", "physics": true, "states": [], "concepts": [ "concept/work", "experiment/porous-plug-experiment", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-88e72090c5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "52", "location": "Applications to Homogeneous Systems", "latex": "W = -\\int_{1}^{2} p\\, dV", "name": null, "statement": "The external work of a reversible process equals minus the integral of pressure over volume along the path from state 1 to state 2.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": "Eq(W, -Integral(p, (V, V1, V2)))", "physics": true, "states": [], "concepts": [ "concept/integral", "concept/reversible-process", "concept/work", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c7db2fdfe4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "54", "location": "Applications to Homogeneous Systems", "latex": "Q = -W", "name": null, "statement": "For a complete cycle returning to the initial state, the heat absorbed equals minus the external work.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed" }, { "unit": null, "symbol": "W", "meaning": "external work" } ], "sympy": "Eq(Q, -W)", "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/work", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1b72d07003", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "54", "location": "Applications to Homogeneous Systems", "latex": "W = -\\int_{1}^{1} p\\, dV", "name": null, "statement": "The external work over a complete cycle is minus the integral of pressure around the closed curve, which equals the area enclosed.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/integral", "concept/work", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d2339480c5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "55", "location": "Applications to Homogeneous Systems", "latex": "Q = dU + p\\, dV", "name": null, "statement": "For an infinitesimal reversible change, the heat absorbed equals the increment of internal energy plus pressure times the increment of volume.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": "Eq(Q, dU + p*dV)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/heat", "concept/pressure", "concept/work", "law/first-law-of-thermodynamics", "quantity/internal-energy", "quantity/pressure", "quantity/volume" ], "pages": [ "55", "89" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "planck-treatise-on-thermodynamics-1903/ch-proof" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8a9cfc9dca", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "51", "location": "Applications to Homogeneous Systems", "latex": "U_{2} - U_{1} = W + Q", "name": null, "statement": "Along a reversible curve from state 1 to state 2, the energy increase equals the work expended plus the heat absorbed.", "kind": "law", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "W", "meaning": "mechanical work expended on the substance" }, { "unit": null, "symbol": "Q", "meaning": "total heat absorbed" } ], "sympy": "Eq(U2 - U1, W + Q)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/work", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-baf5e6ad97", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "55", "location": "Applications to Homogeneous Systems", "latex": "q = du + p\\, dv", "name": null, "statement": "Per unit mass, the heat absorbed equals the increment of specific internal energy plus pressure times the increment of specific volume.", "kind": "law", "symbols": [ { "unit": null, "symbol": "q", "meaning": "heat absorbed per unit mass" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": "Eq(q, du + p*dv)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/heat", "concept/pressure", "concept/work", "law/first-law-of-thermodynamics", "quantity/entropy", "quantity/internal-energy", "quantity/pressure", "quantity/specific-heat", "quantity/specific-volume", "quantity/volume" ], "pages": [ "55", "88" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "planck-treatise-on-thermodynamics-1903/ch-proof" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f2f1cfc6b3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "55", "location": "Applications to Homogeneous Systems", "latex": "c = \\frac{q}{d\\theta} = \\frac{du}{d\\theta} + p\\, \\frac{dv}{d\\theta}", "name": null, "statement": "The specific heat for any heating process equals heat per unit mass divided by the temperature increment, expressed through internal energy and volume changes.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c", "meaning": "specific heat for the given heating process" }, { "unit": null, "symbol": "q", "meaning": "heat absorbed per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/heat", "concept/temperature", "quantity/internal-energy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-15e23bfe52", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "56", "location": "Applications to Homogeneous Systems", "latex": "c_{v} = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v}\\Add{,}", "name": null, "statement": "The specific heat at constant volume is the rate of change of specific internal energy with temperature at fixed volume.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/temperature", "quantity/internal-energy", "quantity/specific-heat", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f22de43403", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "56", "location": "Applications to Homogeneous Systems", "latex": "c_{v} = \\left(\\frac{\\dd u}{\\dd p}\\right)_{v} \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v}\\Add{.}", "name": null, "statement": "The specific heat at constant volume can be written as the product of internal energy change with pressure and pressure change with temperature, both at constant volume.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/temperature", "quantity/internal-energy", "quantity/pressure", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c8504d2ebe", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "56", "location": "Applications to Homogeneous Systems", "latex": "c_{p} = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{p} + p\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}\\Add{,}", "name": null, "statement": "The specific heat at constant pressure equals the internal energy rate plus pressure times the volume rate, both at constant pressure.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/internal-energy", "quantity/pressure", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2681fe33b6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "56", "location": "Applications to Homogeneous Systems", "latex": "c_{p} = \\left[\\left(\\frac{\\dd u}{\\dd v}\\right)_{p} + p\\right]\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}\\Add{.}", "name": null, "statement": "The specific heat at constant pressure equals the bracket of internal energy change with volume plus pressure, times the volume change with temperature at constant pressure.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/internal-energy", "quantity/pressure", "quantity/specific-heat", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5f422f0ea1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "56", "location": "Applications to Homogeneous Systems", "latex": "c_{p} = c_{v} + \\left[\\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} + p\\right]\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}", "name": null, "statement": "The difference between the specific heats at constant pressure and constant volume is given by the internal energy dependence on volume plus pressure, times the expansion coefficient at constant pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/internal-energy", "quantity/pressure", "quantity/specific-heat", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d3d3dd2e52", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "57", "location": "Applications to Homogeneous Systems", "latex": "(c_{p} - c_{v})\\, \\frac{\\dd^{2} \\theta}{\\dd p\\, \\dd v} + \\frac{\\dd c_{p}}{\\dd p} · \\frac{\\dd \\theta}{\\dd v} - \\frac{\\dd c_{v}}{\\dd v} · \\frac{\\dd \\theta}{\\dd p} = 1\\Add{.}", "name": null, "statement": "A relation among measurable quantities for any homogeneous substance, which tests the first law of thermodynamics by experiment.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": "Eq((cp - cv)*Derivative(theta, p, v) + Derivative(cp, p)*Derivative(theta, v) - Derivative(cv, v)*Derivative(theta, p), 1)", "physics": true, "states": [], "concepts": [ "concept/homogeneous-system", "concept/temperature", "law/first-law-of-thermodynamics", "quantity/pressure", "quantity/specific-heat", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5ac7f3372a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "57", "location": "Applications to Homogeneous Systems", "latex": "\\theta = \\frac{m}{R}\\, pv", "name": null, "statement": "Rearranging the characteristic equation, temperature equals molecular weight over the gas constant times pressure times volume.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": "Eq(theta, m/R*p*v)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/perfect-gas", "concept/temperature", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bc59b0790b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "57", "location": "Applications to Homogeneous Systems", "latex": "c_{p} - c_{v} + p\\, \\frac{\\dd c_{p}}{\\dd p} - v\\, \\frac{\\dd c_{v}}{\\dd v} = \\frac{R}{m}", "name": null, "statement": "For a perfect gas, the first-law test relation reduces to this condition involving the two specific heats and the gas constant per molecular weight.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" } ], "sympy": "Eq(cp - cv + p*Derivative(cp, p) - v*Derivative(cv, v), R/m)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "law/first-law-of-thermodynamics", "quantity/molecular-weight", "quantity/pressure", "quantity/specific-heat", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0169318754", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "57", "location": "Applications to Homogeneous Systems", "latex": "\\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} = 0", "name": null, "statement": "For a perfect gas the internal energy per unit mass is independent of volume at constant temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": "Eq(Derivative(u, v), 0)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "quantity/internal-energy", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-127731aa97", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "58", "location": "Applications to Homogeneous Systems", "latex": "du = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v} d\\theta + \\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} dv", "name": null, "statement": "The general differential of specific internal energy as a function of temperature and volume.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/differential", "concept/temperature", "quantity/internal-energy", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2525e532da", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "58", "location": "Applications to Homogeneous Systems", "latex": "du = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v} d\\theta", "name": null, "statement": "For a perfect gas the increment of specific internal energy depends on temperature alone.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/differential", "concept/perfect-gas", "concept/temperature", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-676a63c785", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "58", "location": "Applications to Homogeneous Systems", "latex": "du = c_{v} · d\\theta", "name": null, "statement": "For a perfect gas the increment of specific internal energy equals the specific heat at constant volume times the temperature increment.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": "Eq(du, cv*dtheta)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/perfect-gas", "concept/temperature", "quantity/internal-energy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bef2f09b64", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "58", "location": "Applications to Homogeneous Systems", "latex": "c_{p} = c_{v} + p \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}", "name": null, "statement": "For a perfect gas the difference of specific heats equals pressure times the volume change with temperature at constant pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "quantity/pressure", "quantity/specific-heat", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-545f113302", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "58", "location": "Applications to Homogeneous Systems", "latex": "c_{p} = c_{v} + \\frac{R}{m}", "name": null, "statement": "For a perfect gas the specific heat at constant pressure exceeds that at constant volume by the gas constant divided by molecular weight.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" } ], "sympy": "Eq(cp, cv + R/m)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "quantity/absolute-gas-constant", "quantity/molecular-weight", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f0138a7b70", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "58", "location": "Applications to Homogeneous Systems", "latex": "mc_{p} - mc_{v} = R", "name": "Mayer relation", "statement": "The molecular heats at constant pressure and at constant volume differ by the gas constant, independently of the nature of the gas.", "kind": "law", "symbols": [ { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": "Eq(m*cp - m*cv, R)", "physics": true, "states": [ "law/mayer-relation" ], "concepts": [ "concept/perfect-gas", "quantity/absolute-gas-constant", "quantity/atomic-heat", "quantity/molecular-weight", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-117de13736", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "59", "location": "Applications to Homogeneous Systems", "latex": "mc_{p} - mc_{v} = \\frac{R}{J} = \\frac{826 · 10^{5}}{419 · 10^{5}} = 1.971\\Add{.}", "name": null, "statement": "With molecular heats in calories, the difference of molecular heats equals the gas constant divided by Joule's equivalent, about 1.971.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": "erg per calorie", "symbol": "J", "meaning": "mechanical equivalent of heat (Joule's equivalent)" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "experiment/joule-s-experiments", "quantity/absolute-gas-constant", "quantity/atomic-heat", "quantity/mechanical-equivalent-of-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-009c704d4e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "59", "location": "Applications to Homogeneous Systems", "latex": "u = c_{v} \\theta + \\const", "name": null, "statement": "Over a range where specific heat is constant, specific internal energy is proportional to temperature plus a constant set by the zero of energy.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": "Eq(u, cv*theta + C)", "physics": true, "states": [], "concepts": [ "concept/constant-of-integration", "concept/temperature", "quantity/internal-energy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d41722a44f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "59", "location": "Applications to Homogeneous Systems", "latex": "0 = du + p\\, dv", "name": "adiabatic condition", "statement": "In an adiabatic process no heat is absorbed, so the increment of internal energy plus pressure times volume increment is zero.", "kind": "law", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": "Eq(0, du + p*dv)", "physics": true, "states": [ "law/adiabatic-condition" ], "concepts": [ "concept/adiabatic-process", "concept/heat", "quantity/internal-energy", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-30ae3b28f6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "60", "location": "Applications to Homogeneous Systems", "latex": "0 = c_{v}\\, d\\theta + \\frac{R}{m} · \\frac{\\theta}{v}\\, dv", "name": null, "statement": "For a perfect gas undergoing an adiabatic change, the specific heat times the temperature increment plus the gas-law term times the volume increment vanishes.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/perfect-gas", "concept/temperature", "quantity/specific-heat", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-36a05f1f5c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "60", "location": "Applications to Homogeneous Systems", "latex": "\\log \\theta + (\\gamma - 1) \\log v = \\const", "name": null, "statement": "During an adiabatic change of a perfect gas, the log of temperature plus (gamma minus one) times the log of volume is constant.", "kind": "law", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats c_p/c_v" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/constant-of-integration", "concept/logarithm", "concept/temperature", "quantity/ratio-of-specific-heats", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2bf7733693", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "60", "location": "Applications to Homogeneous Systems", "latex": "-\\gamma \\log \\theta + (\\gamma - 1) \\log p = \\const", "name": null, "statement": "During adiabatic compression of a perfect gas the temperature rises, as expressed in terms of temperature and pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/logarithm", "concept/temperature", "quantity/pressure", "quantity/ratio-of-specific-heats" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-776e64e106", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "60", "location": "Applications to Homogeneous Systems", "latex": "\\log p + \\gamma \\log v = \\const", "name": null, "statement": "For an adiabatic change of a perfect gas, log pressure plus gamma times log volume is constant.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/logarithm", "quantity/pressure", "quantity/ratio-of-specific-heats", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a90d199fed", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "60", "location": "Applications to Homogeneous Systems", "latex": "pv^{\\gamma} = \\const", "name": "adiabatic law", "statement": "For a perfect gas in an adiabatic change, pressure times volume raised to gamma is constant, so adiabatic curves are steeper than isotherms.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats c_p/c_v" } ], "sympy": "Eq(p*v**gamma, C)", "physics": true, "states": [ "law/adiabatic-law" ], "concepts": [ "concept/adiabatic-process", "concept/constant-of-integration", "law/boyle-s-law", "quantity/pressure", "quantity/ratio-of-specific-heats", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-46f3e29ccf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "61", "location": "Applications to Homogeneous Systems", "latex": "\\frac{p}{\\rho^{\\gamma}} = \\const", "name": null, "statement": "In the adiabatic compressions of sound waves in a perfect gas, pressure over density raised to gamma is constant.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "ρ", "meaning": "density, equal to 1/v" }, { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats" } ], "sympy": "Eq(p/rho**gamma, C)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "quantity/density", "quantity/pressure", "quantity/ratio-of-specific-heats", "quantity/velocity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e964dbc77d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "61", "location": "Applications to Homogeneous Systems", "latex": "\\frac{dp}{d\\rho} = \\frac{\\gamma p}{\\rho} = \\gamma pv", "name": null, "statement": "Differentiating the adiabatic relation gives the rate of change of pressure with density as gamma times pressure over density, equal to gamma times pressure times volume.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "ρ", "meaning": "density" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass" }, { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats" } ], "sympy": "Eq(dp/drho, gamma*p/rho)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/coefficient", "quantity/density", "quantity/pressure", "quantity/ratio-of-specific-heats" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d1a107c4a1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "61", "location": "Applications to Homogeneous Systems", "latex": "\\frac{dp}{d\\rho} = \\gamma \\frac{R}{m} \\theta", "name": null, "statement": "For a perfect gas the rate of change of pressure with density equals gamma times the gas constant over molecular weight times temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "ρ", "meaning": "density" }, { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "θ", "meaning": "temperature" } ], "sympy": "Eq(dp/drho, gamma*R/m*theta)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "quantity/density", "quantity/molecular-weight", "quantity/pressure", "quantity/ratio-of-specific-heats" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6f9c23c2cc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "61", "location": "Applications to Homogeneous Systems", "latex": "\\gamma = \\frac{m}{R\\theta} · \\frac{dp}{d\\rho}", "name": null, "statement": "The ratio of specific heats can be determined from the velocity of sound, using molecular weight, gas constant, temperature and the pressure-density derivative.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "θ", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "ρ", "meaning": "density" } ], "sympy": "Eq(gamma, m/(R*theta)*dp/drho)", "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/density", "quantity/molecular-weight", "quantity/pressure", "quantity/ratio-of-specific-heats", "quantity/velocity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e47eff52a0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "61", "location": "Applications to Homogeneous Systems", "latex": "\\sqrt{\\dfrac{dp}{d\\rho}}", "name": null, "statement": "The velocity of sound in a fluid equals the square root of the rate of change of pressure with density.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "ρ", "meaning": "density" } ], "sympy": "Eq(c, sqrt(dp/drho))", "physics": true, "states": [], "concepts": [ "concept/physical-analogy", "quantity/density", "quantity/pressure", "quantity/velocity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c3be53c89a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "61", "location": "Applications to Homogeneous Systems", "latex": "\\gamma = \\frac{28.8}{826 · 10^{5}} · \\frac{33280^{2}}{273} = 1.41", "name": null, "statement": "From the measured velocity of sound in air at 0°, the ratio of specific heats for air comes out at 1.41, agreeing with the calorimetric value.", "kind": "result", "symbols": [ { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats of air" }, { "unit": null, "symbol": "R", "meaning": "gas constant (826 × 10^5)" }, { "unit": "kelvin", "symbol": "θ", "meaning": "temperature (273 at 0°)" } ], "sympy": "Eq(gamma, 28.8/(826*10**5)*33280**2/273)", "physics": true, "states": [], "concepts": [ "concept/air", "concept/temperature", "person/robert-meyer", "quantity/ratio-of-specific-heats", "quantity/velocity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3bb3ecfc81", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "64", "location": "Applications to Homogeneous Systems", "latex": "W = -\\frac{R}{m} \\left(\\theta_{2} \\log \\frac{v_{2}'}{v_{2}} + \\theta_{1} \\log \\frac{v_{1}'}{v_{1}}\\right)", "name": null, "statement": "For the Carnot cycle with a perfect gas, the external work is the sum of the isothermal contributions at the two reservoir temperatures.", "kind": "result", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work over the cycle" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "θ_1", "meaning": "lower reservoir temperature" }, { "unit": null, "symbol": "θ_2", "meaning": "upper reservoir temperature" }, { "unit": null, "symbol": "v", "meaning": "volume per unit mass at the cycle corners" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/isothermal-process", "concept/perfect-gas", "concept/temperature", "concept/work", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b82c946383", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "64", "location": "Applications to Homogeneous Systems", "latex": "\\frac{v_{2}'}{v_{2}} = \\frac{v_{1}'}{v_{1}}", "name": null, "statement": "The two adiabatic branches of the Carnot cycle give equal volume ratios for the isothermal expansions.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v_2", "meaning": "volume at start of the upper isothermal expansion" }, { "unit": null, "symbol": "v_2'", "meaning": "volume at end of the upper isothermal expansion" }, { "unit": null, "symbol": "v_1", "meaning": "volume at start of the lower isothermal compression" }, { "unit": null, "symbol": "v_1'", "meaning": "volume at end of the lower adiabatic expansion" } ], "sympy": "Eq(v2p/v2, v1p/v1)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/common-ratio", "concept/isothermal-process", "method/carnot-cycle", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-832337b05e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "64", "location": "Applications to Homogeneous Systems", "latex": "W = -\\frac{R}{m} (\\theta_{2} - \\theta_{1}) \\log \\frac{v_{1}'}{v_{1}}", "name": null, "statement": "The external work of the Carnot cycle with a perfect gas equals minus the gas constant term times the temperature difference times the log of the volume ratio.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work over the cycle" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "θ_2", "meaning": "upper reservoir temperature" }, { "unit": null, "symbol": "θ_1", "meaning": "lower reservoir temperature" }, { "unit": null, "symbol": "v_1", "meaning": "volume at start of the lower isothermal compression" }, { "unit": null, "symbol": "v_1'", "meaning": "volume at end of the lower adiabatic expansion" } ], "sympy": "Eq(W, -R/m*(theta2 - theta1)*log(v1p/v1))", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "concept/work", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8d8d8bffe2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "64", "location": "Applications to Homogeneous Systems", "latex": "Q = Q_{1} + Q_{2}= -W", "name": null, "statement": "Over the Carnot cycle the net heat absorbed equals minus the external work, so the net heat is positive when the cycle gains work.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "total heat absorbed over the cycle" }, { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed from the lower reservoir (negative)" }, { "unit": null, "symbol": "Q_2", "meaning": "heat absorbed from the upper reservoir" }, { "unit": null, "symbol": "W", "meaning": "external work over the cycle" } ], "sympy": "Eq(Q1 + Q2, -W)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/work", "law/first-law-of-thermodynamics", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-883aae3b4c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "64", "location": "Applications to Homogeneous Systems", "latex": "Q = Q_{1} + Q_{2} = \\frac{R}{m} (\\theta_{2} - \\theta_{1}) \\log \\frac{v_{1}'}{v_{1}}", "name": null, "statement": "The net heat taken in during the Carnot cycle of a perfect gas equals gas constant over molecular weight times the temperature difference times the log volume ratio.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "total heat absorbed over the cycle" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "θ_2", "meaning": "upper reservoir temperature" }, { "unit": null, "symbol": "θ_1", "meaning": "lower reservoir temperature" } ], "sympy": "Eq(Q1 + Q2, R/m*(theta2 - theta1)*log(v1p/v1))", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/isothermal-process", "concept/perfect-gas", "concept/temperature", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-eef029c36a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "65", "location": "Applications to Homogeneous Systems", "latex": "Q_{2} = \\frac{R}{m} \\theta_{2} \\log \\frac{v_{2}'}{v_{2}} = \\frac{R}{m} \\theta_{2} \\log \\frac{v_{1}'}{v_{1}}", "name": null, "statement": "The heat absorbed in the isothermal expansion at the upper reservoir equals the gas-law work of that expansion.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q_2", "meaning": "heat absorbed at the upper reservoir temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "θ_2", "meaning": "upper reservoir temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/isothermal-process", "concept/perfect-gas", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-511cc78739", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "65", "location": "Applications to Homogeneous Systems", "latex": "Q_{1} = \\frac{R}{m} \\theta_{1} \\log \\frac{v_{1}}{v_{1}'} = -\\frac{R}{m} \\theta_{1} \\log \\frac{v_{1}'}{v_{1}}", "name": null, "statement": "The heat absorbed in the isothermal compression at the lower reservoir is negative and equals the gas-law work of that compression.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed at the lower reservoir temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "θ_1", "meaning": "lower reservoir temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/isothermal-process", "concept/perfect-gas", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fc7b0cf825", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "65", "location": "Applications to Homogeneous Systems", "latex": "Q_{1} : Q_{2}: W = (-\\theta_{1}): \\theta_{2} : (\\theta_{1} - \\theta_{2})", "name": null, "statement": "For a reversible Carnot cycle of a perfect gas, the heats and work are in the ratio of the reservoir temperatures.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed at the lower reservoir" }, { "unit": null, "symbol": "Q_2", "meaning": "heat absorbed at the upper reservoir" }, { "unit": null, "symbol": "W", "meaning": "external work over the cycle" }, { "unit": null, "symbol": "θ_1", "meaning": "lower reservoir temperature" }, { "unit": null, "symbol": "θ_2", "meaning": "upper reservoir temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/common-ratio", "concept/heat", "concept/heat-reservoir", "concept/temperature", "concept/work", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-378b1bef73", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "62", "location": "Applications to Homogeneous Systems", "latex": "Q + W = 0", "name": null, "statement": "Over a complete cycle the sum of heat absorbed and work done on the system is zero.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed over the cycle" }, { "unit": null, "symbol": "W", "meaning": "work done on the system over the cycle" } ], "sympy": "Eq(Q + W, 0)", "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/work", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a0752067de", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "62", "location": "Applications to Homogeneous Systems", "latex": "Q = Q_{1} + Q_{2}", "name": null, "statement": "The total heat absorbed over the Carnot cycle is the sum of the heats taken from the two reservoirs, with Q_1 negative.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "total heat absorbed over the cycle" }, { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed at the lower reservoir (negative)" }, { "unit": null, "symbol": "Q_2", "meaning": "heat absorbed at the upper reservoir" } ], "sympy": "Eq(Q, Q1 + Q2)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cf373073f6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "68", "location": "Applications to Non-Homogeneous Systems", "latex": "Q + W = U_{2} - U_{1}", "name": "first law of thermodynamics", "statement": "The external effects (heat Q and external work W) together equal the change of internal energy from state 1 to state 2.", "kind": "law", "symbols": [ { "unit": "calorie", "symbol": "Q", "meaning": "heat absorbed by the system (taken with negative sign in processes with positive heat effect)" }, { "unit": null, "symbol": "W", "meaning": "external work" }, { "unit": null, "symbol": "U_{1}", "meaning": "internal energy of the system in the initial state" }, { "unit": null, "symbol": "U_{2}", "meaning": "internal energy of the system in the final state" } ], "sympy": "Eq(Q + W, U_2 - U_1)", "physics": true, "states": [ "law/first-law-of-thermodynamics" ], "concepts": [ "concept/change-of-energy", "concept/external-effect", "concept/heat", "concept/thermochemistry", "concept/work", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3392e5ed4a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "69", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{[Pb] + [S] - [PbS]} = 18,400~\\Unit{cal.}", "name": null, "statement": "Forming one molecule of lead sulphide from separate lead and sulphur atoms at the same temperature releases 18,400 calories, so the internal energy of the separate atoms exceeds that of the compound by this amount.", "kind": "result", "symbols": [ { "unit": null, "symbol": "[Pb]", "meaning": "internal energy of an atom of lead (referred to an arbitrary zero of energy)" }, { "unit": null, "symbol": "[S]", "meaning": "internal energy of an atom of sulphur" }, { "unit": null, "symbol": "[PbS]", "meaning": "internal energy of a molecule of lead sulphide" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/atom", "concept/chemical-element", "concept/chemical-reaction", "concept/molecule", "concept/thermochemistry", "quantity/heat-effect", "quantity/internal-energy", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-04e665298e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "69", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{[PbS]} = -18,400~\\Unit{cal.}", "name": null, "statement": "With the uncombined elements taken as the zero of energy, the energy of a molecule of lead sulphide is -18,400 calories.", "kind": "result", "symbols": [ { "unit": "calorie", "symbol": "[PbS]", "meaning": "internal energy of a molecule of lead sulphide, zero of energy at uncombined elements" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/molecule", "concept/normal-state", "quantity/internal-energy", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-483ffde56b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "69", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(H2O) - [H2O]} = 80 × 18 = 1440~\\Unit{cal.}", "name": null, "statement": "The fusion of ice at 0 degrees C absorbs 1440 calories per gram molecule of water (80 calories per gram times 18 grams).", "kind": "result", "symbols": [ { "unit": null, "symbol": "(H2O)", "meaning": "internal energy of a molecule of liquid water" }, { "unit": null, "symbol": "[H2O]", "meaning": "internal energy of a molecule of ice" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/molecule", "concept/state-of-aggregation", "concept/temperature", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6c66a6cd07", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "70", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(H2SO4) + 5(H2O) - (H2SO4 . 5H2O)} = 13,100~\\Unit{cal.}", "name": null, "statement": "Dissolving one molecule of sulphuric acid in five molecules of water gives out 13,100 calories of heat.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(H2SO4)", "meaning": "internal energy of a molecule of liquid sulphuric acid" }, { "unit": null, "symbol": "(H2O)", "meaning": "internal energy of a molecule of liquid water" }, { "unit": null, "symbol": "(H2SO4 . 5H2O)", "meaning": "internal energy of a solution of one molecule of sulphuric acid in five molecules of water" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/solution", "concept/solvent", "quantity/heat-effect", "quantity/internal-energy", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-60eec4b213", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "70", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(H2SO4) + 10(H2O) - (H2SO4 . 10H2O)} = 15,100~\\Unit{cal.}", "name": null, "statement": "Dissolving one molecule of sulphuric acid in ten molecules of water gives out 15,100 calories of heat.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(H2SO4 . 10H2O)", "meaning": "internal energy of a solution of one molecule of sulphuric acid in ten molecules of water" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/solution", "concept/solvent", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-196e28ac0a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "70", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(H2SO4 . 5H2O) + 5(H2O) - (H2SO4 . 10H2O)} = 2000~\\Unit{cal.}", "name": null, "statement": "Diluting the five-molecule solution of sulphuric acid by five more molecules of water gives out 2000 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(H2SO4 . 10H2O)", "meaning": "internal energy of a solution of one molecule of sulphuric acid in ten molecules of water" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/solution", "quantity/heat-effect", "quantity/heat-of-dilution", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-68575e9c54", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "70", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(H2SO4) + ($\\aq$) - (H2SO4 $\\aq$)} = 17,900~\\Unit{cal.}", "name": null, "statement": "The heat effect of infinite dilution of one molecule of sulphuric acid is 17,900 calories, with the solvent taken as any amount sufficient for an infinitely dilute solution.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(H2SO4 $\\aq$)", "meaning": "internal energy of one molecule of sulphuric acid in infinitely dilute aqueous solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dissociation", "concept/infinity", "concept/solution", "concept/solvent", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-dc220dd2d2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "71", "location": "Applications to Non-Homogeneous Systems", "latex": "U_{2} - U_{1} = Q", "name": null, "statement": "When volume changes are negligible (solids and liquids only), the heat effect alone is the change of energy, depending only on initial and final states.", "kind": "law", "symbols": [ { "unit": "calorie", "symbol": "Q", "meaning": "heat effect of the process" }, { "unit": null, "symbol": "U_{1}", "meaning": "internal energy in the initial state" }, { "unit": null, "symbol": "U_{2}", "meaning": "internal energy in the final state" } ], "sympy": "Eq(U_2 - U_1, Q)", "physics": true, "states": [], "concepts": [ "concept/change-of-energy", "concept/solution", "concept/work", "quantity/heat-effect", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c6f498da01", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "71", "location": "Applications to Non-Homogeneous Systems", "latex": "W = -\\int_{1}^{2} p_{0}\\, dV = p_{0} (V_{1} - V_{2})", "name": null, "statement": "At constant pressure the external work equals the pressure times the decrease of volume.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work" }, { "unit": null, "symbol": "p_{0}", "meaning": "constant external (atmospheric) pressure" }, { "unit": null, "symbol": "V_{1}", "meaning": "volume in the initial state" }, { "unit": null, "symbol": "V_{2}", "meaning": "volume in the final state" } ], "sympy": "Eq(W, p_0*(V_1 - V_2))", "physics": true, "states": [], "concepts": [ "concept/external-conditions-of-equilibrium", "concept/work", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6c807eb9d2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "71", "location": "Applications to Non-Homogeneous Systems", "latex": "U_{2} - U_{1} = Q + p_{0} (V_{1} - V_{2})", "name": null, "statement": "At constant pressure the change of internal energy equals the heat effect plus the pressure times the decrease of volume.", "kind": "law", "symbols": [ { "unit": "calorie", "symbol": "Q", "meaning": "heat effect of the process" }, { "unit": null, "symbol": "p_{0}", "meaning": "constant pressure" }, { "unit": null, "symbol": "V_{1}", "meaning": "volume in the initial state" }, { "unit": null, "symbol": "V_{2}", "meaning": "volume in the final state" } ], "sympy": "Eq(U_2 - U_1, Q + p_0*(V_1 - V_2))", "physics": true, "states": [], "concepts": [ "concept/change-of-energy", "law/first-law-of-thermodynamics", "quantity/heat-effect", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-58ddde3223", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "72", "location": "Applications to Non-Homogeneous Systems", "latex": "V_{1} - V_{2} = R \\frac{\\theta}{p_{0}} (n_{1} - n_{2})", "name": null, "statement": "The decrease of gaseous volume in a reaction is proportional to the change in the number of gas molecules, at fixed temperature and pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "R", "meaning": "gas constant (per molecule-based gas law as used in the book)" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature (degrees, as used in the book)" }, { "unit": null, "symbol": "p_{0}", "meaning": "constant pressure" }, { "unit": null, "symbol": "n_{1}", "meaning": "number of gas molecules present before the reaction" }, { "unit": null, "symbol": "n_{2}", "meaning": "number of gas molecules present after the reaction" } ], "sympy": "Eq(V_1 - V_2, R*theta/p_0*(n_1 - n_2))", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "quantity/number-of-molecules", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f391bcebd0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "72", "location": "Applications to Non-Homogeneous Systems", "latex": "\\frac{W}{J} = \\frac{p_{0} (V_{1} - V_{2})}{J}", "name": null, "statement": "The heat equivalent of the external work at constant pressure is the external work divided by the mechanical equivalent of heat.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work" }, { "unit": null, "symbol": "J", "meaning": "mechanical equivalent of heat" }, { "unit": null, "symbol": "p_{0}", "meaning": "constant pressure" } ], "sympy": "Eq(W/J, p_0*(V_1 - V_2)/J)", "physics": true, "states": [], "concepts": [ "concept/heat-equivalent", "concept/work", "quantity/pressure", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c87f0b8dfd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "72", "location": "Applications to Non-Homogeneous Systems", "latex": "\\frac{R}{J} \\theta (n_{1} - n_{2})\n = 1.97 \\theta (n_{1} - n_{2})~\\Unit{cal.}", "name": null, "statement": "The heat equivalent of the external work at constant pressure equals 1.97 calories per degree times the temperature times the change in number of gas molecules.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "J", "meaning": "mechanical equivalent of heat" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "n_{1}", "meaning": "number of gas molecules before the reaction" }, { "unit": null, "symbol": "n_{2}", "meaning": "number of gas molecules after the reaction" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/work", "quantity/heat-effect", "quantity/number-of-molecules", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-08b35d6a70", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "72", "location": "Applications to Non-Homogeneous Systems", "latex": "-Q = U_{1} - U_{2} + 1.97 \\theta (n_{1} - n_{2})~\\Unit{cal.}", "name": null, "statement": "The heat effect of a process at constant pressure equals the decrease of internal energy plus the heat equivalent of the external work.", "kind": "formula", "symbols": [ { "unit": "calorie", "symbol": "Q", "meaning": "heat absorbed by the system (negative for positive heat effect)" }, { "unit": null, "symbol": "U_{1}", "meaning": "internal energy before the process" }, { "unit": null, "symbol": "U_{2}", "meaning": "internal energy after the process" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "n_{1}", "meaning": "number of gas molecules before the reaction" }, { "unit": null, "symbol": "n_{2}", "meaning": "number of gas molecules after the reaction" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/change-of-energy", "concept/exothermal-process", "concept/perfect-gas", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-89526f2f3b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "72", "location": "Applications to Non-Homogeneous Systems", "latex": "-Q = \\ce{\\{H2\\} + $\\tfrac{1}{2}$ \\{O2\\}} - \\ce{(H2O)} + 860~\\Unit{cal.}", "name": null, "statement": "The heat of combustion of one gram molecule of hydrogen with half a gram molecule of oxygen at 18 degrees C and constant pressure is 860 calories more than the decrease of internal energy.", "kind": "result", "symbols": [ { "unit": null, "symbol": "{H2}", "meaning": "internal energy of a gram molecule of gaseous hydrogen" }, { "unit": null, "symbol": "{O2}", "meaning": "internal energy of gaseous oxygen" }, { "unit": null, "symbol": "(H2O)", "meaning": "internal energy of liquid water" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/molecule", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-26b5709b36", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "72", "location": "Applications to Non-Homogeneous Systems", "latex": "(U + p_{0} V)_{2} - (U + p_{0} V)_{1} = Q", "name": "heat function at constant pressure", "statement": "At constant pressure the heat effect equals the difference of the heat function U + p0 V between the final and initial states.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p_{0}", "meaning": "constant pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": "calorie", "symbol": "Q", "meaning": "heat effect of the process" } ], "sympy": "Eq(U_2 + p_0*V_2 - (U_1 + p_0*V_1), Q)", "physics": true, "states": [ "quantity/enthalpy" ], "concepts": [ "quantity/heat-effect", "quantity/internal-energy", "quantity/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-29babe90e0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "76", "location": "Applications to Non-Homogeneous Systems", "latex": "(U_{2} + p_{0} V_{2})_{\\theta} - (U_{1} + p_{0} V_{1})_{\\theta} = Q_{\\theta}", "name": null, "statement": "At temperature theta, the heat effect at constant pressure equals the difference of the heat function between final and initial states.", "kind": "law", "symbols": [ { "unit": "calorie", "symbol": "Q_{\\theta}", "meaning": "heat effect of the process at temperature theta" }, { "unit": "degree Centigrade", "symbol": "theta", "meaning": "temperature of the process" }, { "unit": null, "symbol": "p_{0}", "meaning": "constant pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/enthalpy", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bab8e54140", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "73", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(NaHCO3 $\\aq$) + (NaHO $\\aq$) - (Na2CO3 $\\aq$)} = 9200~\\Unit{cal.}", "name": null, "statement": "The heat of neutralization of a solution of sodium bicarbonate with caustic soda is 9200 calories (Thomsen).", "kind": "result", "symbols": [ { "unit": null, "symbol": "(NaHCO3 $\\aq$)", "meaning": "internal energy of sodium bicarbonate in aqueous solution" }, { "unit": null, "symbol": "(Na2CO3 $\\aq$)", "meaning": "internal energy of sodium carbonate in aqueous solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/solution", "person/j-thomsen", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9d5d6f0cda", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "73", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(CO2 $\\aq$) + 2(NaHO $\\aq$) - (Na2CO3 $\\aq$)} = 20,200~\\Unit{cal.}", "name": null, "statement": "The heat of neutralization of carbon dioxide by caustic soda to sodium carbonate is 20,200 calories (Thomsen).", "kind": "result", "symbols": [ { "unit": null, "symbol": "(CO2 $\\aq$)", "meaning": "internal energy of carbon dioxide in aqueous solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/solution", "person/j-thomsen", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4997d20cab", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "73", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(CO2 $\\aq$) + (NaHO $\\aq$) - (NaHCO3 $\\aq$)} = 11,000~\\Unit{cal.}", "name": null, "statement": "The direct combination of carbon dioxide and caustic soda to sodium bicarbonate releases 11,000 calories, obtained by subtracting the two neutralization results.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(NaHCO3 $\\aq$)", "meaning": "internal energy of sodium bicarbonate in aqueous solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/solution", "method/elimination", "person/berthelot", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-58f8c0a837", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "74", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(SnCl2 . 2HCl $\\aq$) + (H2O2 $\\aq$) - (SnCl4 $\\aq$)} = 88,800~\\Unit{cal.}", "name": null, "statement": "Oxidizing stannous chloride in hydrochloric acid by hydrogen peroxide in solution releases 88,800 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(H2O2 $\\aq$)", "meaning": "hydrogen peroxide in aqueous solution" }, { "unit": null, "symbol": "(SnCl4 $\\aq$)", "meaning": "stannic chloride in aqueous solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/solution", "person/j-thomsen", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0902be4d32", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "74", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(SnCl2 . 2HCl $\\aq$) + $\\tfrac{1}{2}$ \\{O2\\} - (SnCl4 $\\aq$)} = 65,700~\\Unit{cal.}", "name": null, "statement": "Oxidizing the same stannous chloride solution by oxygen gas releases 65,700 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "{O2}", "meaning": "oxygen gas" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/solution", "person/j-thomsen", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e828309c86", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "74", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{(H2O2 $\\aq$) - $\\tfrac{1}{2}$ \\{O2\\} - ($\\aq$)} = 23,100~\\Unit{cal.}", "name": null, "statement": "The decomposition of dissolved hydrogen peroxide into oxygen and water releases 23,100 calories, found by subtracting the two oxidation results.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(H2O2 $\\aq$)", "meaning": "hydrogen peroxide in aqueous solution" }, { "unit": null, "symbol": "{O2}", "meaning": "oxygen gas" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/dissociation", "concept/solution", "method/elimination", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a1879ab372", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "74", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{[C] + \\{O2\\} - \\{CO2\\}} = 97,000~\\Unit{cal.}", "name": null, "statement": "The complete combustion of solid carbon to carbon dioxide releases 97,000 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "[C]", "meaning": "internal energy of an atom of solid carbon" }, { "unit": null, "symbol": "{CO2}", "meaning": "internal energy of gaseous carbon dioxide" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-075c9d643e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "74", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{\\{CO\\} + $\\tfrac{1}{2}$ \\{O2\\} - \\{CO2\\}} = 68,000~\\Unit{cal.}", "name": null, "statement": "The combustion of carbon monoxide to carbon dioxide releases 68,000 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "{CO}", "meaning": "internal energy of gaseous carbon monoxide" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-aaafaa01bc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "74", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{[C] + $\\tfrac{1}{2}$ \\{O2\\} - \\{CO\\}} = 29,000~\\Unit{cal.}", "name": null, "statement": "The heat of formation of carbon monoxide from solid carbon and oxygen is 29,000 calories, found by subtraction.", "kind": "result", "symbols": [ { "unit": null, "symbol": "[C]", "meaning": "internal energy of an atom of solid carbon" }, { "unit": null, "symbol": "{CO}", "meaning": "internal energy of gaseous carbon monoxide" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "method/elimination", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e87e0b12cb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "75", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{[S] + \\{O2\\} - \\{SO2\\}} = 71,100~\\Unit{cal.}", "name": null, "statement": "The combustion of solid sulphur to sulphur dioxide gas releases 71,100 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "[S]", "meaning": "internal energy of an atom of solid sulphur" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d35bef4ef9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "75", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{\\{CS2\\} + 3\\{O2\\} - \\{CO2\\} - 2\\{SO2\\}} = 265,100~\\Unit{cal.}", "name": null, "statement": "The combustion of carbon bisulphide vapour to carbon dioxide and sulphur dioxide releases 265,100 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "{CS2}", "meaning": "internal energy of carbon bisulphide vapour" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-17b4d2a2ad", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "75", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{\\{CS2\\} - (CS2)} = 6400~\\Unit{cal.}", "name": null, "statement": "The condensation of carbon bisulphide vapour to liquid releases 6400 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(CS2)", "meaning": "internal energy of liquid carbon bisulphide" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8024063bd3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "75", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{[C] + 2[S] - (CS2)} = - 19,500~\\Unit{cal.}", "name": null, "statement": "The heat of formation of liquid carbon bisulphide from solid carbon and solid sulphur is -19,500 calories, hence negative, obtained by elimination.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(CS2)", "meaning": "internal energy of liquid carbon bisulphide" }, { "unit": null, "symbol": "[S]", "meaning": "internal energy of an atom of solid sulphur" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/negative-electrification", "method/elimination", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c2de726761", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "75", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{\\{CH4\\} + 2\\{O2\\} - \\{CO2\\} - 2(H2O)} &= 211,900~\\Unit{cal.}", "name": null, "statement": "The complete combustion of methane to carbon dioxide and liquid water releases 211,900 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "{CH4}", "meaning": "internal energy of gaseous methane" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/molecule", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-69179b653b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "75", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{\\{H2\\} + $\\tfrac{1}{2}$ \\{O2\\} - (H2O)} &= \\Z68,400~\\Unit{cal.}", "name": null, "statement": "The formation of liquid water from hydrogen and oxygen gas releases 68,400 calories.", "kind": "result", "symbols": [ { "unit": null, "symbol": "{H2}", "meaning": "internal energy of gaseous hydrogen" }, { "unit": null, "symbol": "(H2O)", "meaning": "internal energy of liquid water" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fc1d6acce4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-applications-to-non-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "75", "location": "Applications to Non-Homogeneous Systems", "latex": "\\ce{[C] + 2\\{H2\\} - \\{CH4\\}} = 21,900~\\Unit{cal.}", "name": null, "statement": "The heat of formation of methane from solid carbon and hydrogen gas is 21,900 calories, obtained by elimination.", "kind": "result", "symbols": [ { "unit": null, "symbol": "[C]", "meaning": "internal energy of an atom of solid carbon" }, { "unit": null, "symbol": "{CH4}", "meaning": "internal energy of gaseous methane" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "method/elimination", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b561ee08c8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-introduction", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "78", "location": "Introduction", "latex": "\\ce{\\{H2\\} + $\\tfrac{1}{2}$ \\{O2\\} - (H2O)} = 68,400~\\Unit{cal.}", "name": null, "statement": "Forming water from hydrogen and oxygen at constant pressure gives off 68,400 cal of heat, and decomposing water absorbs the same amount, so the reaction's heat effect is fixed at that value (the book's equation (50)).", "kind": "result", "symbols": [ { "unit": null, "symbol": "H2", "meaning": "hydrogen (molecule)" }, { "unit": null, "symbol": "O2", "meaning": "oxygen (molecule)" }, { "unit": null, "symbol": "H2O", "meaning": "water" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "law/conservation-of-energy", "law/first-law-of-thermodynamics", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-dd8269b127", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "95", "location": "Proof", "latex": "-\\frac{Q}{\\theta}", "name": null, "statement": "The entropy change of a heat-reservoir during an infinitely small time element equals minus the heat given to the substance divided by the reservoir temperature.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat given to the substance by the reservoir in an infinitely small element of time" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature of the reservoir at that moment" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-ba24371b22", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "88", "location": "Proof", "latex": "du = c_{v}\\, d\\theta", "name": null, "statement": "For a perfect gas the change of internal energy per unit mass is the specific heat at constant volume times the change of temperature.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "c_{v}", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(du, c_v*dtheta)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c1d65fe52e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "88", "location": "Proof", "latex": "q = c_{v}\\, d\\theta + \\frac{R}{m} · \\frac{\\theta}{v}\\, dv", "name": null, "statement": "For a perfect gas the heat received per unit mass is the sum of the internal-energy change and the work term, written in temperature and volume.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "q", "meaning": "heat received per unit mass" }, { "unit": null, "symbol": "c_{v}", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "mass of the gas" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/perfect-gas", "concept/temperature", "law/first-law-of-thermodynamics", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-ad3d941ab7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "88", "location": "Proof", "latex": "\\phi = c_{v} \\log \\theta + \\frac{R}{m} \\log v + \\const", "name": null, "statement": "The entropy of unit mass of a perfect gas is defined, up to an additive constant, by logarithms of temperature and specific volume.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "entropy of unit mass of the gas" }, { "unit": null, "symbol": "c_{v}", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "mass of the gas" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "\\const", "meaning": "additive constant fixed by the choice of zero state" } ], "sympy": "Eq(phi, c_v*log(theta) + R/m*log(v) + const)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/constant", "concept/logarithm", "concept/perfect-gas", "concept/temperature", "quantity/entropy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-742e0cfea6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "89", "location": "Proof", "latex": "\\Phi = M\\phi = M \\left(c_{v} \\log \\theta + \\frac{R}{m} \\log v + \\const\\right)", "name": null, "statement": "The entropy of a mass M of a perfect gas is M times the entropy of unit mass.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of mass M of the gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "\\phi", "meaning": "entropy of unit mass" }, { "unit": null, "symbol": "c_{v}", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(Phi, M*(c_v*log(theta) + R/m*log(v) + const))", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/logarithm", "concept/perfect-gas", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d42a342646", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "89", "location": "Proof", "latex": "d\\Phi = M \\left(c_{v}\\, \\frac{d\\theta}{\\theta} + \\frac{R}{m}\\, \\frac{dv}{v}\\right) = \\frac{M · q}{\\theta} = \\frac{Q}{\\theta}", "name": null, "statement": "On application of heat to a perfect gas the change of entropy equals the absorbed heat divided by the temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "c_{v}", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "mass of the gas" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "q", "meaning": "heat received per unit mass" }, { "unit": null, "symbol": "Q", "meaning": "absorbed heat" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/perfect-gas", "concept/reversible-process", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a36e82dd9d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "89", "location": "Proof", "latex": "d\\Phi = M \\left(c_{v}\\, \\frac{d\\theta}{\\theta} + \\frac{R}{m}\\, \\frac{dv}{v}\\right) = \\frac{dU + p\\, dV}{\\theta}", "name": null, "statement": "The change of entropy of a perfect gas equals the sum of internal-energy change and pressure-volume work, divided by temperature, for any process in which temperature and volume change.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fd0de5c196", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "89", "location": "Proof", "latex": "Q + W = dU", "name": null, "statement": "The heat absorbed plus the work done on the substance equals the change of its internal energy (first law).", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed" }, { "unit": null, "symbol": "W", "meaning": "work done on the substance" }, { "unit": null, "symbol": "U", "meaning": "internal energy" } ], "sympy": "Eq(Q + W, dU)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/work", "law/conservation-of-energy", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-15461fd21a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "90", "location": "Proof", "latex": "\\Phi_{1} + \\Phi_{2} = \\const", "name": null, "statement": "In a reversible process of two gases exchanging heat, the sum of their entropies remains constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi_{1}", "meaning": "entropy of the first gas" }, { "unit": null, "symbol": "\\Phi_{2}", "meaning": "entropy of the second gas" }, { "unit": null, "symbol": "\\const", "meaning": "constant" } ], "sympy": "Eq(Phi_1 + Phi_2, C)", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/heat", "concept/perfect-gas", "concept/reversible-process", "concept/sum", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2a94b0480b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "91", "location": "Proof", "latex": "\\Phi_{1} + \\Phi_{2} = \\Phi_{1}' + \\Phi_{2}'\\Add{.}", "name": null, "statement": "The two-gas system has the same total entropy in its initial and final states.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\Phi_{1}", "meaning": "entropy of the first gas in the initial state" }, { "unit": null, "symbol": "\\Phi_{2}", "meaning": "entropy of the second gas in the initial state" }, { "unit": null, "symbol": "\\Phi_{1}'", "meaning": "entropy of the first gas in the final state" }, { "unit": null, "symbol": "\\Phi_{2}'", "meaning": "entropy of the second gas in the final state" } ], "sympy": "Eq(Phi_1 + Phi_2, Phi_1p + Phi_2p)", "physics": true, "states": [], "concepts": [ "concept/equality", "concept/reversible-process", "concept/sum", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4a40bfb19e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "91", "location": "Proof", "latex": "\\Phi_{1} + \\Phi_{2} + \\dots + \\Phi_{n} = \\Phi_{1}' + \\Phi_{2}' + \\dots + \\Phi_{n}'", "name": null, "statement": "A system of n gases has the same total entropy in two states, which is the condition for reversible transformation between them.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "\\Phi_{i}", "meaning": "entropy of the i-th gas in the initial state" }, { "unit": null, "symbol": "\\Phi_{i}'", "meaning": "entropy of the i-th gas in the final state" }, { "unit": null, "symbol": "n", "meaning": "number of gases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/entropy-of-a-system", "concept/equality", "concept/reversible-process", "concept/sum", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5bbc3f5dc7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "93", "location": "Proof", "latex": "\\Phi_{1}' + \\Phi_{2}' + \\dots + \\Phi_{n}' < \\Phi_{1} + \\Phi_{2} + \\dots + \\Phi_{n}", "name": null, "statement": "The supposed final total entropy of the gas system is smaller than its initial total entropy, which leads to a contradiction.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "\\Phi_{i}", "meaning": "entropy of the i-th gas in the initial state" }, { "unit": null, "symbol": "\\Phi_{i}'", "meaning": "entropy of the i-th gas in the final state" }, { "unit": null, "symbol": "n", "meaning": "number of gases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/reversible-process", "concept/sum", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6fb45885ba", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "93", "location": "Proof", "latex": "(\\Phi_{1}' + \\Phi_{2}' + \\dots + \\Phi_{n}') - \\Phi_{1} - \\Phi_{2} - \\dots - \\Phi_{n-1}\\Add{.}", "name": null, "statement": "The entropy of the n-th gas is the total final entropy minus the entropies of the first n-1 gases.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi_{i}", "meaning": "entropy of the i-th gas in the initial state" }, { "unit": null, "symbol": "\\Phi_{i}'", "meaning": "entropy of the i-th gas in the final state" }, { "unit": null, "symbol": "n", "meaning": "number of gases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/difference", "concept/sum", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-863ead6ae6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "96", "location": "Proof", "latex": "-\\tsum \\frac{Q}{\\theta} \\geq 0", "name": null, "statement": "The total entropy change of all heat-reservoirs cannot be negative, since no change remains in other bodies.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat given to the substance by a reservoir" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature of the reservoir" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat-reservoir", "concept/inequality", "concept/sum", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-45f847b5bb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "96", "location": "Proof", "latex": "\\tsum \\frac{Q}{\\theta} \\leq 0", "name": null, "statement": "The sum of heat absorbed divided by reservoir temperatures over a cycle is not positive; this is the form in which Clausius first stated the second law.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat given to the substance by a reservoir" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature of the reservoir" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/clausius-equation", "concept/cycle-of-operations", "concept/heat-reservoir", "concept/sum", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-63290a0a99", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "96", "location": "Proof", "latex": "W = -p\\, dV", "name": null, "statement": "When the external pressure equals the pressure of the substance, the work done on it during compression is minus p dV.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "W", "meaning": "work done on the body" }, { "unit": null, "symbol": "p", "meaning": "pressure of the substance" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/differential", "concept/pressure", "concept/reversible-process", "concept/work", "quantity/volume" ], "pages": [ "96", "114" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-proof", "planck-treatise-on-thermodynamics-1903/ch-general-deductions" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-eb734c0854", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "96", "location": "Proof", "latex": "\\tsum \\frac{Q}{\\theta} = 0", "name": null, "statement": "For a cyclic process in which each heat-reservoir is at the temperature of the substance, the sum of Q over theta vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat given to the substance" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/equality", "concept/heat", "concept/reversible-process", "concept/sum" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5ac1e4955a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "96", "location": "Proof", "latex": "\\tsum \\frac{dU + p\\, dV}{\\theta} = 0", "name": null, "statement": "Over a reversible cyclic process of a homogeneous body, the summation of (dU + p dV)/theta vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/pressure", "concept/reversible-process", "concept/sum", "concept/temperature", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3448721ed2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "97", "location": "Proof", "latex": "\\int_{1}^{2} \\frac{dU + p\\, dV}{\\theta}", "name": null, "statement": "The integral of (dU + p dV)/theta from state 1 to state 2 depends only on the two states, not on the path.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/definite-integral", "concept/integral", "concept/thermodynamic-equilibrium", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-856c171dcb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "97", "location": "Proof", "latex": "\\int_{1\\; (\\alpha)}^{2} \\frac{dU + p\\, dV}{\\theta} + \\int_{2\\; (\\beta)}^{1} \\frac{dU + p\\, dV}{\\theta} = 0", "name": null, "statement": "Over the complete cycle formed by path alpha from 1 to 2 and path beta back to 1, the integral vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "\\alpha", "meaning": "first path from state 1 to state 2" }, { "unit": null, "symbol": "\\beta", "meaning": "second path from state 2 back to state 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/definite-integral", "concept/equality", "concept/integral", "concept/sum" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c7b1f71a97", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "97", "location": "Proof", "latex": "\\int_{1\\; (\\alpha)}^{2} \\frac{dU + p\\, dV}{\\theta} = \\int_{1\\; (\\beta)}^{2} \\frac{dU + p\\, dV}{\\theta}", "name": null, "statement": "The integral from state 1 to state 2 is the same along any two reversible paths between those states.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "\\alpha", "meaning": "first path" }, { "unit": null, "symbol": "\\beta", "meaning": "second path" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/definite-integral", "concept/equality", "concept/reversible-process", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-71ad805282", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "98", "location": "Proof", "latex": "\\Phi = \\int \\frac{dU + p\\, dV}{\\theta}", "name": null, "statement": "The entropy of a body in a state is the integral of (dU + p dV)/theta from the zero state, defined up to an additive constant.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the body" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/integral", "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-75d40f792e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "98", "location": "Proof", "latex": "d\\Phi = \\frac{dU + p\\, dV}{\\theta}", "name": null, "statement": "The differential of entropy equals (dU + p dV) divided by temperature, for any body.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the body" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/differential", "concept/energy", "concept/exact-differential", "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/volume" ], "pages": [ "98", "208", "226" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-proof", "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7d92ce9621", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "98", "location": "Proof", "latex": "d\\phi = \\frac{du + p\\, dv}{\\theta}\\Add{.}", "name": null, "statement": "The differential of entropy per unit mass equals (du + p dv) divided by temperature.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "entropy of unit mass" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/differential", "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-08681809d4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "99", "location": "Proof", "latex": "d\\Phi = \\frac{Q}{\\theta}\\Add{.}", "name": null, "statement": "For a reversible change of volume, the entropy change of a body equals the absorbed heat divided by temperature.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the body" }, { "unit": null, "symbol": "Q", "meaning": "absorbed heat" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(dPhi, Q/theta)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/reversible-process", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0d8f61ef4a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "98", "location": "Proof", "latex": "U = Mu", "name": null, "statement": "The energy of a body equals its mass times its energy per unit mass.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy of the body" }, { "unit": null, "symbol": "M", "meaning": "mass of the body" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" } ], "sympy": "Eq(U, M*u)", "physics": true, "states": [], "concepts": [ "quantity/internal-energy", "quantity/mass" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a311cc7376", "chapter": "planck-treatise-on-thermodynamics-1903/ch-proof", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "98", "location": "Proof", "latex": "V = Mv", "name": null, "statement": "The volume of a body equals its mass times its specific volume.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the body" }, { "unit": null, "symbol": "M", "meaning": "mass of the body" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(V, M*v)", "physics": true, "states": [], "concepts": [ "quantity/mass", "quantity/specific-volume", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3ae8f63545", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "106", "location": "General Deductions", "latex": "\\frac{Q_{1}}{\\theta_{1}} + \\frac{Q_{2}}{\\theta_{2}} < 0", "name": null, "statement": "For a cycle with reservoir entropy changes allowed to be reversible in volume, the sum of Q/theta over the reservoirs is negative for an irreversible cycle.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed from the colder reservoir" }, { "unit": null, "symbol": "Q_2", "meaning": "heat given out by the hotter reservoir" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": "Lt(Q1/theta1 + Q2/theta2, 0)", "physics": true, "states": [], "concepts": [ "concept/heat-reservoir", "concept/reversible-process", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c8503d68e9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "105", "location": "General Deductions", "latex": "Q_{2} = W' + Q_{1}'", "name": null, "statement": "In a cyclic process the heat given out by the hotter reservoir equals the work done by the system plus the heat received by the colder reservoir.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q_2", "meaning": "heat given out by the hotter reservoir" }, { "unit": null, "symbol": "W'", "meaning": "work done by the system" }, { "unit": null, "symbol": "Q_1'", "meaning": "heat received by the colder reservoir" } ], "sympy": "Eq(Q2, Wp + Q1p)", "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/heat-reservoir", "concept/work", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7cc98df91a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "105", "location": "General Deductions", "latex": "Q_{1} + Q_{2} + W = 0", "name": null, "statement": "Over a complete cycle the heat exchanged with the two reservoirs and the work done on the system sum to zero (energy equation).", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed from the colder reservoir" }, { "unit": null, "symbol": "Q_2", "meaning": "heat given out by the hotter reservoir" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Eq(Q1 + Q2 + W, 0)", "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/heat", "concept/work", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-acabf38579", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "105", "location": "General Deductions", "latex": "\\frac{Q_{1}}{\\theta_{1}} + \\frac{Q_{2}}{\\theta_{2}} = 0", "name": null, "statement": "For a reversible cycle the total entropy of the two heat reservoirs is unchanged.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed from the colder reservoir" }, { "unit": null, "symbol": "Q_2", "meaning": "heat given out by the hotter reservoir" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": "Eq(Q1/theta1 + Q2/theta2, 0)", "physics": true, "states": [], "concepts": [ "concept/cycle-of-operations", "concept/heat-reservoir", "concept/reversible-process", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f003fe1084", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "105", "location": "General Deductions", "latex": "Q_{1} : Q_{2} : W = (-\\theta_{1}) : \\theta_{2} : (\\theta_{1} - \\theta_{2})", "name": null, "statement": "For a reversible Carnot cycle the heats and work are in the ratio of the temperatures given by the second law, for any working substance.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed from the colder reservoir" }, { "unit": null, "symbol": "Q_2", "meaning": "heat given out by the hotter reservoir" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/reversible-process", "concept/temperature", "concept/work", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fc0a546426", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "106", "location": "General Deductions", "latex": "Q_{1}' = \\frac{\\theta_{1}}{\\theta_{2} - \\theta_{1}} W'", "name": null, "statement": "The heat that must pass from the hotter to the colder reservoir in a reversible Carnot cycle to gain a given work W' is fixed by the two temperatures.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Q_1'", "meaning": "heat passing from the hotter to the colder reservoir" }, { "unit": null, "symbol": "W'", "meaning": "mechanical work gained" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": "Eq(Q1p, theta1/(theta2 - theta1)*Wp)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/temperature", "concept/work", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2f7542b1c4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "106", "location": "General Deductions", "latex": "W' = \\frac{\\theta_{2} - \\theta_{1}}{\\theta_{1}} Q_{1}'", "name": null, "statement": "The work obtainable from heat transferred between two reservoirs by a reversible cycle is the maximum possible for any cyclic process between them.", "kind": "result", "symbols": [ { "unit": null, "symbol": "W'", "meaning": "mechanical work gained" }, { "unit": null, "symbol": "Q_1'", "meaning": "heat passing from the hotter to the colder reservoir" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": "Eq(Wp, (theta2 - theta1)/theta1*Q1p)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/maximum", "concept/temperature", "concept/work", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c67635035f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "106", "location": "General Deductions", "latex": "- \\frac{Q_{1}}{\\theta_{1}} - \\frac{Q_{2}}{\\theta_{2}} > 0", "name": null, "statement": "For an irreversible cycle the sum of the entropy changes of the reservoirs is positive.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q_1", "meaning": "heat absorbed from the colder reservoir" }, { "unit": null, "symbol": "Q_2", "meaning": "heat given out by the hotter reservoir" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": "Gt(-Q1/theta1 - Q2/theta2, 0)", "physics": true, "states": [], "concepts": [ "concept/heat-reservoir", "concept/reversible-process", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-54ff2f2157", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "106", "location": "General Deductions", "latex": "W' < \\frac{\\theta_{2} - \\theta_{1}}{\\theta_{1}} Q_{1}'", "name": null, "statement": "The work gained by an irreversible cycle is smaller than the reversible maximum for the same heat transfer.", "kind": "result", "symbols": [ { "unit": null, "symbol": "W'", "meaning": "mechanical work gained" }, { "unit": null, "symbol": "Q_1'", "meaning": "heat passing from the hotter to the colder reservoir" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": "Lt(Wp, (theta2 - theta1)/theta1*Q1p)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/maximum", "concept/reversible-process", "concept/work", "method/carnot-cycle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f4cb016daf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "107", "location": "General Deductions", "latex": "Q_{2} \\left(\\frac{1}{\\theta_{2}} - \\frac{1}{\\theta_{1}}\\right) < 0", "name": null, "statement": "In a cycle with no work, heat can only flow from the hotter to the colder reservoir.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q_2", "meaning": "heat given out by the hotter reservoir" }, { "unit": null, "symbol": "\\theta_1", "meaning": "temperature of the colder reservoir" }, { "unit": null, "symbol": "\\theta_2", "meaning": "temperature of the hotter reservoir" } ], "sympy": "Lt(Q2*(1/theta2 - 1/theta1), 0)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/inequality", "concept/temperature", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e30dfb0d5e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "107", "location": "General Deductions", "latex": "W + Q = 0", "name": null, "statement": "For a process using one reservoir of constant temperature, the work done on the system and the heat absorbed by it cancel.", "kind": "law", "symbols": [ { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "Q", "meaning": "heat absorbed by the system from the reservoir" } ], "sympy": "Eq(W + Q, 0)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/work", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3f572b1d00", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "108", "location": "General Deductions", "latex": "-\\frac{Q}{\\theta} \\geq 0", "name": null, "statement": "The entropy change of the constant-temperature reservoir is non-negative.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed by the system from the reservoir" }, { "unit": null, "symbol": "\\theta", "meaning": "constant temperature of the reservoir" } ], "sympy": "Ge(-Q/theta, 0)", "physics": true, "states": [], "concepts": [ "concept/heat-reservoir", "concept/temperature", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-df10b04ea3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "108", "location": "General Deductions", "latex": "Q \\leq 0", "name": null, "statement": "Heat is added to the reservoir, not taken from it, in any such process.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed by the system from the reservoir" } ], "sympy": "Le(Q, 0)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/heat-reservoir", "concept/inequality" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-03350b4e50", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "108", "location": "General Deductions", "latex": "W \\geq 0", "name": null, "statement": "Work must be expended on the system in any cycle using a single reservoir; in the reversible limit it vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Ge(W, 0)", "physics": true, "states": [], "concepts": [ "concept/heat-reservoir", "concept/inequality", "concept/reversible-process", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1feca011d2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "108", "location": "General Deductions", "latex": "dU = Q + W", "name": null, "statement": "For any infinitesimal change the increase of internal energy equals the heat absorbed plus the work done on the system.", "kind": "law", "symbols": [ { "unit": null, "symbol": "U", "meaning": "total internal energy of the system" }, { "unit": null, "symbol": "Q", "meaning": "heat absorbed by the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Eq(dU, Q + W)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/heat", "concept/work", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-af58ce4572", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "108", "location": "General Deductions", "latex": "d\\Phi + d\\Phi_{0} \\geq 0", "name": null, "statement": "The total entropy change of the system and its surroundings is non-negative for any natural process.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "\\Phi_0", "meaning": "entropy of the surrounding medium" } ], "sympy": "Ge(dPhi + dPhi0, 0)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/inequality", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c61a7f871f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "d\\Phi_{0} = -\\frac{Q}{\\theta}", "name": null, "statement": "The entropy change of the surrounding medium equals minus the heat absorbed by the system divided by the temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi_0", "meaning": "entropy of the surrounding medium" }, { "unit": null, "symbol": "Q", "meaning": "heat absorbed by the system" }, { "unit": null, "symbol": "\\theta", "meaning": "common temperature" } ], "sympy": "Eq(dPhi0, -Q/theta)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/heat", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-28f91b322e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "d\\Phi_{0} = -\\frac{dU - W}{\\theta}", "name": null, "statement": "Substituting the first law, the entropy change of the surroundings is minus the energy change less work, over the temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi_0", "meaning": "entropy of the surrounding medium" }, { "unit": null, "symbol": "U", "meaning": "total internal energy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "\\theta", "meaning": "common temperature" } ], "sympy": "Eq(dPhi0, -(dU - W)/theta)", "physics": true, "states": [], "concepts": [ "concept/work", "law/first-law-of-thermodynamics", "quantity/entropy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b29a20472d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "d\\Phi - \\frac{dU - W}{\\theta} \\geq 0", "name": null, "statement": "The criterion for any natural change at fixed temperature combining both laws.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "U", "meaning": "total internal energy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "\\theta", "meaning": "common temperature" } ], "sympy": "Ge(dPhi - (dU - W)/theta, 0)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/inequality", "law/first-law-of-thermodynamics", "law/second-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-274e2abc97", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "dU - \\theta\\, d\\Phi \\leq W", "name": null, "statement": "The fundamental relation (70) between energy, entropy, temperature and work for any change at a common temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "U", "meaning": "total internal energy of the system" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "\\theta", "meaning": "common temperature" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Le(dU - theta*dPhi, W)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/temperature", "concept/work", "law/second-law-of-thermodynamics", "quantity/entropy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c403953b9f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "dU = W", "name": null, "statement": "In an adiabatic process with no heat exchange the energy change equals the work done on the system.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "total internal energy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Eq(dU, W)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/work", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-718f9195c2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "109", "location": "General Deductions", "latex": "d\\Phi \\geq 0", "name": null, "statement": "In an adiabatic process the entropy of the system increases or remains constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" } ], "sympy": "Ge(dPhi, 0)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/inequality", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-28059cbaac", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "d(U - \\theta\\Phi) \\leq W", "name": null, "statement": "In an isothermal process the increment of U minus theta times the entropy is at most the work done on the system.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "total internal energy of the system" }, { "unit": null, "symbol": "\\theta", "meaning": "constant temperature" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Le(dU_minus_thetaPhi, W)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/isothermal-process", "quantity/entropy", "quantity/free-energy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9419f48d5a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "U - \\theta\\Phi = F", "name": "free energy", "statement": "Defines the free energy F as internal energy minus temperature times entropy.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the system" }, { "unit": null, "symbol": "U", "meaning": "total internal energy of the system" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" } ], "sympy": "Eq(F, U - theta*Phi)", "physics": true, "states": [ "quantity/free-energy" ], "concepts": [ "concept/isothermal-process", "concept/temperature", "quantity/entropy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6fbf277d92", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "dF = W", "name": null, "statement": "For a reversible isothermal change the increment of free energy equals the work done on the system.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Eq(dF, W)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/isothermal-process", "concept/reversible-process", "concept/work", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d5b1ed871a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "F_{2} - F_{1} = \\tsum W", "name": null, "statement": "For finite reversible isothermal changes the free energy change equals the total work done on the system.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F_1", "meaning": "free energy in the initial state" }, { "unit": null, "symbol": "F_2", "meaning": "free energy in the final state" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/isothermal-process", "concept/reversible-process", "concept/sum", "concept/work", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a4c9292168", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "F_{2} - F_{1} < \\tsum W", "name": null, "statement": "For irreversible isothermal changes the free energy increases by less than the work done on the system.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F_1", "meaning": "free energy in the initial state" }, { "unit": null, "symbol": "F_2", "meaning": "free energy in the final state" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/isothermal-process", "concept/reversible-process", "concept/work", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-953e98471a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "dF < W", "name": null, "statement": "In an irreversible isothermal process the free energy increment is less than the work done on the system.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Lt(dF, W)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/isothermal-process", "concept/reversible-process", "concept/work", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-eb36f42371", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "110", "location": "General Deductions", "latex": "U - F = \\theta\\Phi", "name": "latent energy", "statement": "Defines the latent energy as the difference of total energy and free energy, equal to temperature times entropy.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "U", "meaning": "total energy of the system" }, { "unit": null, "symbol": "F", "meaning": "free energy of the system" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" } ], "sympy": "Eq(U - F, theta*Phi)", "physics": true, "states": [ "quantity/latent-energy" ], "concepts": [ "concept/temperature", "quantity/entropy", "quantity/free-energy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e8422977e1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "111", "location": "General Deductions", "latex": "\\tsum W = 0", "name": null, "statement": "When the work during an isothermal process vanishes, the sum of work is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/isothermal-process", "concept/sum", "concept/work" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e34ae472cf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "111", "location": "General Deductions", "latex": "F_{2} - F_{1} < 0", "name": null, "statement": "With no external work in an isothermal process, the free energy decreases.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F_1", "meaning": "free energy in the initial state" }, { "unit": null, "symbol": "F_2", "meaning": "free energy in the final state" } ], "sympy": "Lt(F2 - F1, 0)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/isothermal-process", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-df6cf39e19", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "113", "location": "General Deductions", "latex": "dF = dU - \\theta\\, d\\Phi - \\Phi\\, d\\theta", "name": null, "statement": "The differential of the free energy in terms of the differentials of energy, entropy and temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the system" }, { "unit": null, "symbol": "U", "meaning": "total internal energy" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(dF, dU - theta*dPhi - Phi*dtheta)", "physics": true, "states": [], "concepts": [ "concept/differential", "concept/temperature", "quantity/entropy", "quantity/free-energy", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-705171867c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "113", "location": "General Deductions", "latex": "dF \\leq W - \\Phi\\, d\\theta", "name": null, "statement": "For any physical or chemical process the free energy change is bounded by the work less entropy times temperature change.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Le(dF, W - Phi*dtheta)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/temperature", "concept/work", "quantity/entropy", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2d45e60f0c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "113", "location": "General Deductions", "latex": "U = Mu = M(c_{v} \\theta + \\const)", "name": null, "statement": "The internal energy of a perfect gas is linear in temperature, up to an arbitrary constant.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy of the gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "\\const", "meaning": "arbitrary constant" } ], "sympy": "Eq(U, M*(c_v*theta + C))", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/perfect-gas", "concept/temperature", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d9e7a6e00e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "114", "location": "General Deductions", "latex": "\\Phi = M\\phi = M(c_{v} \\log \\theta + \\frac{R}{m} \\log v + \\const)", "name": null, "statement": "The entropy of a perfect gas as a function of temperature and specific volume, up to an arbitrary constant.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "\\phi", "meaning": "entropy per unit mass" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "\\const", "meaning": "arbitrary constant" } ], "sympy": "Eq(Phi, M*(c_v*log(theta) + R/m*log(v) + C))", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/logarithm", "concept/perfect-gas", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6edd246f27", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "114", "location": "General Deductions", "latex": "F = M\\{c_{v} \\theta (\\const - \\log \\theta) - \\frac{R\\theta}{m} \\log v + \\const\\}", "name": null, "statement": "The free energy of a perfect gas, which contains an arbitrary linear function of temperature.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "\\const", "meaning": "arbitrary constant" } ], "sympy": "Eq(F, M*(c_v*theta*(C - log(theta)) - R*theta/m*log(v) + C))", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/logarithm", "concept/perfect-gas", "concept/temperature", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0a1d30c910", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "114", "location": "General Deductions", "latex": "dF = -\\frac{M\\theta R}{m} · \\frac{dv}{v} = -p\\, dV \\leq W", "name": null, "statement": "For isothermal changes of a perfect gas the free energy change equals minus p dV and is bounded by the work done on the gas.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the gas" }, { "unit": null, "symbol": "M", "meaning": "mass of the gas" }, { "unit": null, "symbol": "\\theta", "meaning": "constant temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "W", "meaning": "work done on the gas" } ], "sympy": "Eq(dF, -M*theta*R/m*dv/v)", "physics": true, "states": [], "concepts": [ "concept/isothermal-process", "concept/perfect-gas", "concept/pressure", "concept/work", "quantity/free-energy", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-10633e48da", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "114", "location": "General Deductions", "latex": "d\\left(\\Phi - \\frac{U + pV}{\\theta}\\right) \\geq 0", "name": null, "statement": "At constant temperature and pressure the quantity Phi minus (U+pV)/theta cannot decrease in a natural change.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": "pressure unit (not stated)", "symbol": "p", "meaning": "external pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/differential", "concept/inequality", "concept/isothermal-isopiestic-process", "concept/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-98fe49b50b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "114", "location": "General Deductions", "latex": "\\Phi - \\frac{U + pV}{\\theta} = \\Psi", "name": null, "statement": "Defines the function Psi as entropy minus (internal energy plus pV) over temperature.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\Psi", "meaning": "Psi function" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "p", "meaning": "external pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(Psi, Phi - (U + p*V)/theta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/psi-function", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-35a5246563", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "117", "location": "General Deductions", "latex": "\\delta\\left(\\Phi - \\frac{U}{\\theta}\\right) + \\frac{W}{\\theta} = 0", "name": null, "statement": "The equilibrium condition at constant temperature, in virtual variations.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta", "meaning": "virtual infinitely small change" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/mechanical-equilibrium", "concept/temperature", "concept/thermodynamic-equilibrium", "concept/variation-of-sign" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fb8fa599dd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "117", "location": "General Deductions", "latex": "- \\delta F = -W", "name": null, "statement": "At constant temperature the virtual change of free energy equals the virtual work done on the system.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the system" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Eq(-deltaF, -W)", "physics": true, "states": [], "concepts": [ "concept/isothermal-process", "concept/variation-of-sign", "concept/work", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-50057181f1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "117", "location": "General Deductions", "latex": "\\delta F = 0", "name": null, "statement": "Equilibrium at constant temperature with no external work: the free energy is at a minimum.", "kind": "law", "symbols": [ { "unit": null, "symbol": "F", "meaning": "free energy of the system" } ], "sympy": "Eq(deltaF, 0)", "physics": true, "states": [], "concepts": [ "concept/isothermal-process", "concept/minimum", "concept/thermodynamic-equilibrium", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a8b2b09b7c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "118", "location": "General Deductions", "latex": "W = -p\\, \\delta V", "name": null, "statement": "Virtual external work at constant pressure equals minus p times the virtual volume change.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "p", "meaning": "constant external pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": "Eq(W, -p*deltaV)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/variation-of-sign", "concept/work", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-423a81a84c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "118", "location": "General Deductions", "latex": "\\delta\\Psi = 0", "name": null, "statement": "At constant temperature and pressure, equilibrium is an absolute maximum of the Psi function.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\Psi", "meaning": "Psi function" } ], "sympy": "Eq(deltaPsi, 0)", "physics": true, "states": [], "concepts": [ "concept/maximum", "concept/thermodynamic-equilibrium", "concept/variation-of-sign", "quantity/psi-function" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fe81ffe3a2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "115", "location": "General Deductions", "latex": "\\delta\\Phi - \\frac{\\delta U - W}{\\theta} \\leq 0", "name": null, "statement": "The general condition that equilibrium is maintained when only virtual changes satisfying it are permitted.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\delta", "meaning": "virtual infinitely small change" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Le(deltaPhi - (deltaU - W)/theta, 0)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/stability-of-equilibrium", "concept/thermodynamic-equilibrium", "concept/variation-of-sign" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-30d368b06e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "116", "location": "General Deductions", "latex": "\\delta\\Phi - \\frac{\\delta U - W}{\\theta} = 0", "name": null, "statement": "The condition for equilibrium (76): for every permitted virtual change the variation vanishes.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\delta", "meaning": "virtual infinitely small change" }, { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(deltaPhi - (deltaU - W)/theta, 0)", "physics": true, "states": [], "concepts": [ "concept/thermodynamic-equilibrium", "concept/variation-of-sign", "law/second-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d50bdf4ffa", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "117", "location": "General Deductions", "latex": "\\delta U = W", "name": null, "statement": "In the first case (no heat exchange) the virtual energy change equals the virtual work.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy" }, { "unit": null, "symbol": "W", "meaning": "work done on the system" } ], "sympy": "Eq(deltaU, W)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/variation-of-sign", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2f39329449", "chapter": "planck-treatise-on-thermodynamics-1903/ch-general-deductions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "117", "location": "General Deductions", "latex": "\\delta \\Phi = 0", "name": null, "statement": "Adiabatic equilibrium: the entropy is at a maximum among states reachable by adiabatic processes.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "entropy of the system" } ], "sympy": "Eq(deltaPhi, 0)", "physics": true, "states": [], "concepts": [ "concept/adiabatic-process", "concept/maximum", "concept/thermodynamic-equilibrium", "concept/variation-of-sign", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-768f20f649", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "119", "location": "Homogeneous Systems", "latex": "d\\phi = \\frac{du + p\\, dv}{\\theta} = \\frac{1}{\\theta} \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v} d\\theta + \\frac{\\left(\\dfrac{\\dd u}{\\dd v}\\right)_{\\theta} + p}{\\theta}\\, dv", "name": null, "statement": "The definition of specific entropy gives its differential as the heat term (du + p dv) divided by the absolute temperature, expanded in the independent variables θ and v.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "specific entropy (Φ/M)" }, { "unit": null, "symbol": "u", "meaning": "specific energy (U/M)" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume (V/M)" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/pressure", "law/second-law-of-thermodynamics", "quantity/absolute-temperature", "quantity/entropy", "quantity/specific-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1fa697f9f9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} = \\theta \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} - p", "name": null, "statement": "The rate of change of specific energy with volume at constant temperature equals θ times the rate of change of pressure with temperature at constant volume, minus the pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "specific energy" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/energy", "concept/pressure", "law/second-law-of-thermodynamics", "quantity/absolute-temperature", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5893c35fbe", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd \\phi}{\\dd \\theta}\\right)_{v} = \\frac{c_{v}}{\\theta}", "name": null, "statement": "The temperature rate of change of specific entropy at constant volume equals the specific heat at constant volume divided by absolute temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "specific entropy" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "quantity/absolute-temperature", "quantity/entropy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2b8108f944", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd \\phi}{\\dd v}\\right)_{\\theta} = \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v}", "name": null, "statement": "The volume rate of change of specific entropy at constant temperature equals the temperature rate of change of pressure at constant volume.", "kind": "result", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "specific entropy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/pressure", "quantity/absolute-temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-20db95c796", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "c_{p} - c_{v} = \\theta \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} · \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}", "name": null, "statement": "The difference of the specific heats at constant pressure and constant volume equals θ times the product of two rates of change of pressure and volume with temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/difference-of-specific-heats", "concept/pressure", "law/first-law-of-thermodynamics", "law/second-law-of-thermodynamics", "quantity/absolute-temperature", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f826ba780d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "c_{p} - c_{v} = -\\theta \\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} · \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}^{2}", "name": null, "statement": "The difference of the specific heats is written, using the relation between the derivatives of pressure and volume, as -θ times the volume-pressure rate of change times the square of the thermal expansion rate.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/difference-of-specific-heats", "concept/pressure", "quantity/absolute-temperature", "quantity/coefficient-of-expansion", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1b5e9694e6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "123", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd c_{p}}{\\dd p}\\right)_{\\theta} = -\\theta \\left(\\frac{\\dd^{2} v}{\\dd \\theta^{2}}\\right)_{p}", "name": null, "statement": "The rate of change of specific heat at constant pressure with pressure equals minus θ times the second temperature derivative of specific volume at constant pressure; it relates measurable quantities.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/pressure", "quantity/absolute-temperature", "quantity/coefficient-of-expansion", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9994c48efa", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "121", "location": "Homogeneous Systems", "latex": "c_{p} - c_{v} = \\left\\{\\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} + p\\right\\} \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}", "name": null, "statement": "The difference of specific heats, from the first law, equals the sum of the internal-energy term and the external-work term, multiplied by the thermal expansion rate.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "u", "meaning": "specific energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/difference-of-specific-heats", "concept/work", "law/first-law-of-thermodynamics", "quantity/internal-energy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9fc151304c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "121", "location": "Homogeneous Systems", "latex": "\\frac{\\theta}{p} · \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} - 1", "name": null, "statement": "The ratio of the internal-energy term to the external-work term, written in terms of the pressure-temperature rate of change, is θ/p times that rate minus one.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/work", "quantity/absolute-temperature", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-943dcd1692", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "122", "location": "Homogeneous Systems", "latex": "\\dfrac{c_{p}}{c_{v}} = \\gamma", "name": null, "statement": "The ratio of the specific heat at constant pressure to that at constant volume is denoted γ.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "γ", "meaning": "ratio of specific heats" } ], "sympy": "Eq(c_p/c_v, gamma)", "physics": true, "states": [], "concepts": [ "quantity/ratio-of-specific-heats", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-caf2fa48bd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "123", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd u}{\\dd p}\\right)_{\\theta} = -\\theta \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} - p\\left(\\frac{\\dd v}{\\dd p}\\right)_{\\theta}", "name": null, "statement": "The rate of change of specific energy with pressure at constant temperature equals minus θ times the thermal expansion rate minus p times the compressibility rate.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "specific energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/pressure", "quantity/absolute-temperature", "quantity/coefficient-of-compressibility", "quantity/coefficient-of-expansion", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a4afe4cffb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "123", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd \\phi}{\\dd p}\\right)_{\\theta} = -\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}", "name": null, "statement": "The pressure rate of change of specific entropy at constant temperature equals minus the thermal expansion rate at constant pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "specific entropy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/pressure", "quantity/absolute-temperature", "quantity/coefficient-of-expansion", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-304f06735f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "120", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} = -\\frac{1014000}{0.00000295 · v}", "name": null, "statement": "As used for mercury at 0° C, the pressure-volume rate of change at constant temperature is given numerically in atmospheres, with 0.00000295 the compressibility coefficient in atmospheres and 1014000 the absolute pressure of one atmosphere.", "kind": "formula", "symbols": [ { "unit": "atmosphere", "symbol": "p", "meaning": "pressure (in atmospheres for the numerator and compressibility)" }, { "unit": null, "symbol": "v", "meaning": "specific volume of mercury, here 1/13.6 (volume of 1 gram)" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/mercury", "concept/pressure", "quantity/coefficient-of-compressibility", "quantity/specific-heat", "unit/atmosphere" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b16c6bcf63", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "125", "location": "Homogeneous Systems", "latex": "\\Delta \\theta = \\frac{\\theta \\left(\\dfrac{\\dd v}{\\dd \\theta}\\right)_{p} - v}{c_{p}}\\, \\Delta p", "name": null, "statement": "For a small pressure drop in a throttling expansion, the temperature change equals the change in pressure times θ times the thermal expansion rate minus v, divided by c_p.", "kind": "law", "symbols": [ { "unit": "degree", "symbol": "Δθ", "meaning": "change of absolute temperature in the throttled gas" }, { "unit": "atmosphere", "symbol": "Δp", "meaning": "change of pressure (negative in the experiment)" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/pressure", "experiment/joule-s-experiments", "law/second-law-of-thermodynamics", "quantity/absolute-temperature", "quantity/coefficient-of-expansion", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f6ca83f8c0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "125", "location": "Homogeneous Systems", "latex": "\\Delta \\theta = \\frac{\\alpha}{\\theta^{2}}\\, \\Delta p", "name": null, "statement": "Thomson and Joule's empirical formula: the temperature change in the throttling experiment equals a constant α over θ squared times the pressure change.", "kind": "law", "symbols": [ { "unit": "degree", "symbol": "Δθ", "meaning": "change of absolute temperature" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": "atmosphere", "symbol": "Δp", "meaning": "change of pressure" }, { "unit": null, "symbol": "α", "meaning": "constant in the Thomson and Joule formula (for air 0.276 × 273²)" } ], "sympy": "Eq(Delta_theta, alpha/theta**2*Delta_p)", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/pressure", "experiment/joule-s-experiments", "quantity/absolute-temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-be48d98703", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "125", "location": "Homogeneous Systems", "latex": "\\theta \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} - v = c_{p} \\frac{\\alpha}{\\theta^{2}}", "name": null, "statement": "Combining the throttling relation with the Thomson and Joule formula gives θ times the thermal expansion rate minus v equal to c_p times α over θ squared.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": null, "symbol": "α", "meaning": "constant of the Thomson and Joule formula" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/approximation", "experiment/joule-s-experiments", "quantity/absolute-temperature", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f4beb4ab67", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "126", "location": "Homogeneous Systems", "latex": "c_{p} = \\frac{c_{p}^{(0)}}{\\left(1 - \\dfrac{3\\alpha p}{\\theta^{3}}\\right)^{\\efrac{2}{3}}}", "name": null, "statement": "The specific heat at constant pressure, deduced from the Thomson and Joule formula with the ideal-state limit, equals its zero-pressure value divided by a power of (1 - 3αp/θ³).", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": "calorie", "symbol": "c_p^(0)", "meaning": "specific heat at constant pressure at zero pressure (for air 0.238 calorie)" }, { "unit": null, "symbol": "α", "meaning": "constant of the Thomson and Joule formula" }, { "unit": "atmosphere", "symbol": "p", "meaning": "pressure" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": "Eq(c_p, c_p0/(1 - 3*alpha*p/theta**3)**(2/3))", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/pressure", "experiment/joule-s-experiments", "quantity/absolute-temperature", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-41b6ba2ef4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "126", "location": "Homogeneous Systems", "latex": "c_{p} = \\theta^{2} · f(\\theta^{3} - 3\\alpha p)", "name": null, "statement": "The general solution of the differential equation for c_p is θ² times an arbitrary function of θ³ - 3αp.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_p", "meaning": "specific heat at constant pressure" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "α", "meaning": "constant of the Thomson and Joule formula" }, { "unit": "atmosphere", "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "f", "meaning": "arbitrary function of its argument" } ], "sympy": "Eq(c_p, theta**2*f(theta**3 - 3*alpha*p))", "physics": false, "states": [], "concepts": [ "concept/differential-equation", "concept/function", "experiment/joule-s-experiments", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0a9298afa0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "126", "location": "Homogeneous Systems", "latex": "v = \\frac{c_{p}^{(0)} \\theta}{3p} \\left(\\sqrt[3]{1 - \\frac{3\\alpha p}{\\theta^{3}}} + \\beta\\right)", "name": null, "statement": "The characteristic equation of the gas deduced from the Thomson and Joule experiments gives specific volume in terms of θ and p, with β a constant of integration fixed from the density at 0° C and atmospheric pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "calorie", "symbol": "c_p^(0)", "meaning": "specific heat at constant pressure at zero pressure" }, { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": "atmosphere", "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "α", "meaning": "constant of the Thomson and Joule formula" }, { "unit": null, "symbol": "β", "meaning": "constant of integration" } ], "sympy": "Eq(v, c_p0*theta/(3*p)*((1 - 3*alpha*p/theta**3)**(1/3) + beta))", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/constant-of-integration", "concept/pressure", "experiment/joule-s-experiments", "quantity/absolute-temperature", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9f22c6d0dd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "124", "location": "Homogeneous Systems", "latex": "p_{1}v_{1} - p_{2}v_{2} = W", "name": null, "statement": "The external work per unit mass done in the throttled flow equals the difference of the pressure-volume products before and after the throttle.", "kind": "formula", "symbols": [ { "unit": "atmosphere", "symbol": "p_1", "meaning": "pressure before the throttle" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume before the throttle" }, { "unit": "atmosphere", "symbol": "p_2", "meaning": "pressure after the throttle" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume after the throttle" }, { "unit": null, "symbol": "W", "meaning": "external work per unit mass" } ], "sympy": "Eq(p1*v1 - p2*v2, W)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/work", "experiment/joule-s-experiments", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-930cf79e4f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "124", "location": "Homogeneous Systems", "latex": "W = -\\Delta(pv)", "name": null, "statement": "For small changes, the external work per unit mass in the throttled flow equals minus the change of the product pv.", "kind": "result", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work per unit mass" }, { "unit": null, "symbol": "Δ(pv)", "meaning": "change of the product of pressure and specific volume" } ], "sympy": "Eq(W, -Delta_pv)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/work", "experiment/joule-s-experiments", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8e95b578e1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "124", "location": "Homogeneous Systems", "latex": "\\Delta u = W + Q = -\\Delta(pv)", "name": null, "statement": "By the first law, with no heat added (Q = 0), the change of specific energy equals the work term, which is minus the change of pv.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Δu", "meaning": "change of specific energy" }, { "unit": null, "symbol": "W", "meaning": "external work per unit mass" }, { "unit": null, "symbol": "Q", "meaning": "heat added per unit mass (zero in the experiment)" }, { "unit": null, "symbol": "Δ(pv)", "meaning": "change of the product of pressure and specific volume" } ], "sympy": "Eq(Delta_u, -Delta_pv)", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/work", "experiment/joule-s-experiments", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bf2b77f35c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "128", "location": "Homogeneous Systems", "latex": "\\left(\\frac{q}{dv}\\right)_{t} = \\left(\\frac{\\dd u}{dv}\\right)_{t} + p", "name": null, "statement": "By the first law, the ratio of heat absorbed in isothermal reversible expansion to the change of volume equals the internal-energy rate of change plus the pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "q", "meaning": "heat absorbed during isothermal reversible expansion" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "t", "meaning": "temperature by an arbitrary thermometer" }, { "unit": null, "symbol": "u", "meaning": "specific energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/heat", "concept/isothermal-curve", "concept/pressure", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a22e92b01f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "129", "location": "Homogeneous Systems", "latex": "\\theta = \\frac{100 e^{J}}{e^{J_{1}} - 1}", "name": null, "statement": "The absolute temperature is found from J and J₁, which are integrals fixed by the readings of an arbitrary thermometer between the freezing and boiling points of water.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "J", "meaning": "integral over the arbitrary thermometer from the freezing point to t" }, { "unit": null, "symbol": "J_1", "meaning": "integral over the arbitrary thermometer from the freezing point to the boiling point" } ], "sympy": "Eq(theta, 100*exp(J)/(exp(J_1) - 1))", "physics": true, "states": [], "concepts": [ "concept/exponent", "concept/integral", "concept/limits-of-integration", "concept/temperature", "quantity/absolute-temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-62d10c7412", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "129", "location": "Homogeneous Systems", "latex": "\\alpha = \\frac{1}{\\theta_{0}} = \\frac{e^{J_{1}} - 1}{100}", "name": null, "statement": "The coefficient of thermal expansion of a perfect gas, independent of any gas thermometer, equals the reciprocal of the absolute freezing-point temperature, expressed through J₁.", "kind": "result", "symbols": [ { "unit": null, "symbol": "α", "meaning": "coefficient of expansion of a perfect gas" }, { "unit": "degree", "symbol": "θ_0", "meaning": "absolute temperature of water freezing under atmospheric pressure (273)" }, { "unit": null, "symbol": "J_1", "meaning": "integral from freezing to boiling point" } ], "sympy": "Eq(alpha, (exp(J_1) - 1)/100)", "physics": true, "states": [], "concepts": [ "concept/integral", "concept/perfect-gas", "quantity/absolute-temperature", "quantity/coefficient-of-expansion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-715f54862f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "130", "location": "Homogeneous Systems", "latex": "\\theta = t + \\frac{1}{\\alpha'}", "name": null, "statement": "For a perfect gas used as the arbitrary thermometer, the absolute temperature equals the thermometer reading t plus the reciprocal of the expansion coefficient α'.", "kind": "result", "symbols": [ { "unit": "degree", "symbol": "θ", "meaning": "absolute temperature" }, { "unit": "degree", "symbol": "t", "meaning": "temperature reading of the gas thermometer (from the melting point of ice)" }, { "unit": null, "symbol": "α'", "meaning": "coefficient of expansion referred to temperature t" } ], "sympy": "Eq(theta, t + 1/alpha_p)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "instrument/gas-thermometer", "quantity/absolute-temperature", "quantity/coefficient-of-expansion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e08af2a16f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "130", "location": "Homogeneous Systems", "latex": "\\left(\\frac{\\dd v}{\\dd t}\\right)_{p} = \\alpha' v_{0}", "name": null, "statement": "At constant pressure, the rate of change of specific volume with the thermometer reading t equals the expansion coefficient α' times v₀, the specific volume at the melting point of ice.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": "degree", "symbol": "t", "meaning": "temperature reading of the thermometer" }, { "unit": null, "symbol": "α'", "meaning": "coefficient of expansion referred to temperature t" }, { "unit": null, "symbol": "v_0", "meaning": "specific volume at the melting point of ice under atmospheric pressure" }, { "unit": "atmosphere", "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/pressure", "instrument/gas-thermometer", "quantity/coefficient-of-expansion", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cc1a2bbf03", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "149", "location": "System in Different States of Aggregation", "latex": "v_{1} = \\frac{R\\theta}{mp_{1}}", "name": null, "statement": "For the vapour, if the perfect-gas characteristic equation is applied, its specific volume is R times the temperature divided by m times its pressure.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "v_1", "meaning": "specific volume of the vapour (state 1)" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "m", "meaning": "mass" }, { "unit": null, "symbol": "p_1", "meaning": "pressure of the vapour" } ], "sympy": "Eq(v_1, R*theta/(m*p_1))", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/perfect-gas", "concept/pressure", "concept/saturated-vapour", "concept/temperature", "law/characteristic-equation-of-a-perfect-gas", "quantity/absolute-gas-constant", "quantity/molecular-weight", "quantity/specific-volume" ], "pages": [ "149", "144" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4c9492ab70", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "149", "location": "System in Different States of Aggregation", "latex": "\\left(\\frac{\\dd v_{1}}{\\dd \\theta}\\right)_{p} = \\frac{R}{mp_{1}}", "name": null, "statement": "Under the perfect-gas approximation, the rate of change of the vapour's specific volume with temperature at constant pressure is R divided by m times its pressure.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "v_1", "meaning": "specific volume of the vapour (state 1)" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "mass" }, { "unit": null, "symbol": "p_1", "meaning": "pressure of the vapour" } ], "sympy": "Eq(dv1_dtheta_p, R/(m*p_1))", "physics": true, "states": [], "concepts": [ "concept/partial-derivative", "concept/perfect-gas", "concept/pressure", "concept/temperature", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f69d70f052", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "149", "location": "System in Different States of Aggregation", "latex": "(c_{p})_{1} - (c_{p})_{2} = \\frac{dL}{d\\theta}", "name": null, "statement": "With the perfect-gas approximation the difference between the specific heats at constant pressure of vapour and liquid equals the rate of change of the latent heat with temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(c_p)_1", "meaning": "specific heat at constant pressure of the vapour" }, { "unit": null, "symbol": "(c_p)_2", "meaning": "specific heat at constant pressure of the liquid" }, { "unit": null, "symbol": "L", "meaning": "latent heat of vaporization" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(cp1 - cp2, dL_dtheta)", "physics": true, "states": [], "concepts": [ "concept/saturated-vapour", "concept/temperature", "quantity/latent-heat", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-862a1aa452", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "149", "location": "System in Different States of Aggregation", "latex": "\\frac{dL}{d\\theta} = (c_{p})_{1} - (c_{p})_{2} + \\frac{L}{\\theta} - \\frac{L}{v_{1} - v_{2}} \\left[\\left(\\frac{\\dd v_{1}}{\\dd \\theta}\\right)_{p} - \\left(\\frac{\\dd v_{2}}{\\dd \\theta}\\right)_{p}\\right]", "name": null, "statement": "The rate of change of the latent heat with temperature is expressed through the specific heats, the latent heat, the temperature, and the specific volumes and their thermal expansion rates at constant pressure.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "L", "meaning": "latent heat of the change of state" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature of the change" }, { "unit": null, "symbol": "(c_p)_1", "meaning": "specific heat at constant pressure of state 1" }, { "unit": null, "symbol": "(c_p)_2", "meaning": "specific heat at constant pressure of state 2" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of state 1" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of state 2" } ], "sympy": "Eq(dL_dtheta, cp1 - cp2 + L/theta - L/(v1 - v2)*(dv1_dtheta_p - dv2_dtheta_p))", "physics": true, "states": [], "concepts": [ "concept/partial-derivative", "concept/temperature", "quantity/latent-heat", "quantity/specific-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-16f627a1ce", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "150", "location": "System in Different States of Aggregation", "latex": "c = \\frac{du}{d\\theta} + p\\, \\frac{dv}{d\\theta}", "name": null, "statement": "The specific heat under any heating condition equals the rate of change of internal energy with temperature plus the pressure times the rate of change of specific volume (first law applied to heating).", "kind": "law", "symbols": [ { "unit": null, "symbol": "c", "meaning": "specific heat for the heating process considered" }, { "unit": null, "symbol": "u", "meaning": "specific internal energy" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(c, du_dtheta + p*dv_dtheta)", "physics": true, "states": [], "concepts": [ "concept/derivative", "concept/pressure", "law/first-law-of-thermodynamics", "quantity/internal-energy", "quantity/specific-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-be4970fddc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "150", "location": "System in Different States of Aggregation", "latex": "h_{1} = \\frac{du_{1}}{d\\theta} + p_{1}\\, \\frac{dv_{1}}{d\\theta}", "name": null, "statement": "The specific heat of the saturated vapour, defined for the process that keeps the vapour saturated, is the internal-energy rate plus pressure times volume rate for the vapour.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "h_1", "meaning": "specific heat of the saturated vapour (Clausius's 'saturated vapour' heat)" }, { "unit": null, "symbol": "u_1", "meaning": "specific internal energy of the vapour" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of the vapour" }, { "unit": null, "symbol": "p_1", "meaning": "pressure of the saturated vapour" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(h1, du1_dtheta + p1*dv1_dtheta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/saturated-vapour", "quantity/internal-energy", "quantity/specific-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-96bf2f9e25", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "151", "location": "System in Different States of Aggregation", "latex": "h_{2} = \\frac{du_{2}}{d\\theta} + p_{2}\\, \\frac{dv_{2}}{d\\theta}", "name": null, "statement": "The specific heat of the liquid kept under the pressure of its saturated vapour is the internal-energy rate plus pressure times volume rate for the liquid.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "h_2", "meaning": "specific heat of the liquid kept in saturation" }, { "unit": null, "symbol": "u_2", "meaning": "specific internal energy of the liquid" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of the liquid" }, { "unit": null, "symbol": "p_2", "meaning": "pressure on the liquid (saturated vapour pressure)" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(h2, du2_dtheta + p2*dv2_dtheta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/saturated-vapour", "quantity/internal-energy", "quantity/specific-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1b759eb264", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "151", "location": "System in Different States of Aggregation", "latex": "h_{2} = (c_{p})_{2}", "name": null, "statement": "Because external pressure has no appreciable effect on a liquid unless it is many atmospheres, the liquid's saturation specific heat practically equals its specific heat at constant pressure.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "h_2", "meaning": "specific heat of the liquid kept in saturation" }, { "unit": null, "symbol": "(c_p)_2", "meaning": "specific heat at constant pressure of the liquid" } ], "sympy": "Eq(h2, cp2)", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/pressure", "concept/saturated-vapour", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-63c03e0514", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "151", "location": "System in Different States of Aggregation", "latex": "h_{1} = (c_{p})_{2} + \\frac{dL}{d\\theta} - \\frac{L}{\\theta}", "name": null, "statement": "The specific heat of saturated vapour equals the liquid's specific heat at constant pressure plus the rate of change of latent heat with temperature minus the latent heat divided by the temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "h_1", "meaning": "specific heat of the saturated vapour" }, { "unit": null, "symbol": "(c_p)_2", "meaning": "specific heat at constant pressure of the liquid" }, { "unit": null, "symbol": "L", "meaning": "latent heat of vaporization" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(h1, cp2 + dL_dtheta - L/theta)", "physics": true, "states": [], "concepts": [ "concept/saturated-vapour", "concept/temperature", "quantity/latent-heat", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7e48c39cf6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "153", "location": "System in Different States of Aggregation", "latex": "\\left(\\frac{\\dd p}{\\dd v}\\right)_{2} = 0", "name": null, "statement": "One of the two conditions of the critical state: the pressure's derivative with respect to specific volume vanishes at the critical point.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(dp_dv_2, 0)", "physics": true, "states": [], "concepts": [ "concept/critical-point", "concept/partial-derivative", "concept/pressure", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-51eabeb049", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "153", "location": "System in Different States of Aggregation", "latex": "\\left(\\frac{\\dd^{2} p}{\\dd v^{2}}\\right)_{2} = 0", "name": null, "statement": "The second of the two conditions of the critical state: the second derivative of pressure with respect to specific volume vanishes at the critical point.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(d2p_dv2_2, 0)", "physics": true, "states": [], "concepts": [ "concept/critical-point", "concept/derivative", "concept/pressure", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-58bcda4bf2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "152", "location": "System in Different States of Aggregation", "latex": "p = p_{2} + \\left(\\frac{\\dd p}{\\dd v}\\right)_{2} (v - v_{2}) + \\tfrac{1}{2} \\left(\\frac{\\dd^{2} p}{\\dd v^{2}}\\right)_{2} (v - v_{2})^{2}\\Add{,}", "name": "Taylor expansion of pressure about the state 2", "statement": "For a small volume difference, Taylor's theorem expresses the pressure at any intermediate volume through its value and first two derivatives at state 2.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure at intermediate volume v" }, { "unit": null, "symbol": "p_2", "meaning": "pressure of state 2" }, { "unit": null, "symbol": "v", "meaning": "intermediate specific volume" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of state 2" } ], "sympy": "Eq(p, p_2 + dp_dv_2*(v - v_2) + Rational(1,2)*d2p_dv2_2*(v - v_2)**2)", "physics": false, "states": [ "theorem/taylor-expansion-of-pressure-about-the-state-2" ], "concepts": [ "concept/approximation", "concept/partial-derivative", "concept/pressure", "method/taylor-s-theorem", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e80287e960", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "153", "location": "System in Different States of Aggregation", "latex": "p_{1} = p_{2} = p_{3}\\Add{,}", "name": null, "statement": "In the three-state equilibrium the pressures of the vapour, liquid and solid are all equal.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p_1", "meaning": "pressure of the gaseous state" }, { "unit": null, "symbol": "p_2", "meaning": "pressure of the liquid state" }, { "unit": null, "symbol": "p_3", "meaning": "pressure of the solid state" } ], "sympy": "And(Eq(p_1, p_2), Eq(p_2, p_3))", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/state-of-aggregation", "concept/thermodynamic-equilibrium", "concept/triple-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1553950291", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "153", "location": "System in Different States of Aggregation", "latex": "\\phi_{1} - \\phi_{2} = \\frac{u_{1} - u_{2} + p_{1}(v_{1} - v_{2})}{\\theta}\\Add{,}", "name": null, "statement": "In three-state coexistence, the difference of specific entropies of the gaseous and liquid states equals the internal-energy difference plus pressure times volume difference, divided by the temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\phi_1", "meaning": "specific entropy of state 1" }, { "unit": null, "symbol": "\\phi_2", "meaning": "specific entropy of state 2" }, { "unit": null, "symbol": "u_1", "meaning": "specific internal energy of state 1" }, { "unit": null, "symbol": "u_2", "meaning": "specific internal energy of state 2" }, { "unit": null, "symbol": "p_1", "meaning": "common pressure" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of state 1" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of state 2" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(phi1 - phi2, (u1 - u2 + p1*(v1 - v2))/theta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/state-of-aggregation", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-05b493a4fc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "153", "location": "System in Different States of Aggregation", "latex": "\\phi_{2} - \\phi_{3} = \\frac{u_{2} - u_{3} + p_{1}(v_{2} - v_{3})}{\\theta}\\Add{.}", "name": null, "statement": "The corresponding relation between the liquid and solid states. Note: the book writes p_1 here where the pattern of the previous equation would give p_2; since all three pressures are equal (the line above) the statement is unaffected, but the subscript is as printed.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\phi_2", "meaning": "specific entropy of the liquid state" }, { "unit": null, "symbol": "\\phi_3", "meaning": "specific entropy of the solid state" }, { "unit": null, "symbol": "u_2", "meaning": "specific internal energy of the liquid" }, { "unit": null, "symbol": "u_3", "meaning": "specific internal energy of the solid" }, { "unit": null, "symbol": "p_1", "meaning": "common pressure (printed subscript 1)" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of the liquid" }, { "unit": null, "symbol": "v_3", "meaning": "specific volume of the solid" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" } ], "sympy": "Eq(phi2 - phi3, (u2 - u3 + p1*(v2 - v3))/theta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/state-of-aggregation", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c0e9d20b7c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "154", "location": "System in Different States of Aggregation", "latex": "M_{1} + (M_{2} + M_{3}) = M\\Add{,}", "name": null, "statement": "The three portion masses sum to the total mass of the substance.", "kind": "law", "symbols": [ { "unit": null, "symbol": "M_1", "meaning": "mass of the gaseous portion" }, { "unit": null, "symbol": "M_2", "meaning": "mass of the liquid portion" }, { "unit": null, "symbol": "M_3", "meaning": "mass of the solid portion" }, { "unit": null, "symbol": "M", "meaning": "total mass" } ], "sympy": "Eq(M1 + (M2 + M3), M)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "concept/thermodynamic-equilibrium", "quantity/mass" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e797c03eda", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "154", "location": "System in Different States of Aggregation", "latex": "M_{1} v_{1} + M_{2} v_{2} + M_{3} v_{3} = V\\Add{,}", "name": null, "statement": "The total volume is the sum of each portion's mass times its specific volume.", "kind": "law", "symbols": [ { "unit": null, "symbol": "M_1, M_2, M_3", "meaning": "masses of the three portions" }, { "unit": null, "symbol": "v_1, v_2, v_3", "meaning": "specific volumes of the three portions" }, { "unit": null, "symbol": "V", "meaning": "total volume" } ], "sympy": "Eq(M1*v1 + M2*v2 + M3*v3, V)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/mass", "quantity/specific-volume", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7537f40422", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "154", "location": "System in Different States of Aggregation", "latex": "M_{1} u_{1} + M_{2} u_{2} + M_{3} u_{3} = U\\Add{,}", "name": null, "statement": "The total energy is the sum of each portion's mass times its specific energy.", "kind": "law", "symbols": [ { "unit": null, "symbol": "M_1, M_2, M_3", "meaning": "masses of the three portions" }, { "unit": null, "symbol": "u_1, u_2, u_3", "meaning": "specific energies of the three portions" }, { "unit": null, "symbol": "U", "meaning": "total energy" } ], "sympy": "Eq(M1*u1 + M2*u2 + M3*u3, U)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/internal-energy", "quantity/mass", "quantity/specific-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9d5123db3f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "155", "location": "System in Different States of Aggregation", "latex": "p_{12} = p_{21}", "name": null, "statement": "The saturated vapour pressure is the same whether the vapour is referred to as in contact with the liquid or the liquid is referred to as in contact with the vapour.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "p_{12}", "meaning": "saturated pressure of vaporization (vapour in contact with liquid)" }, { "unit": null, "symbol": "p_{21}", "meaning": "saturated pressure of vaporization (liquid in contact with vapour)" } ], "sympy": "Eq(p_12, p_21)", "physics": true, "states": [], "concepts": [ "concept/corresponding-point", "concept/pressure", "concept/saturated-vapour", "concept/vaporization-curve" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-410dc688b1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "156", "location": "System in Different States of Aggregation", "latex": "\\frac{dp_{12}}{d\\theta} = \\frac{L_{12}}{\\theta (v_{1} - v_{2})}", "name": "Clapeyron equation (vaporization)", "statement": "The slope of the vaporization pressure curve at the fundamental point equals the latent heat divided by temperature times the volume difference of the two phases.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p_{12}", "meaning": "vaporization pressure" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature (fundamental state)" }, { "unit": null, "symbol": "L_{12}", "meaning": "heat of vaporization" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of vapour in the fundamental state" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of liquid in the fundamental state" } ], "sympy": "Eq(dp12_dtheta, L12/(theta*(v1 - v2)))", "physics": true, "states": [ "law/clapeyron-equation-vaporization" ], "concepts": [ "concept/pressure", "concept/temperature", "concept/triple-point", "concept/vaporization-curve", "quantity/latent-heat", "quantity/specific-volume", "theorem/clapeyron-equation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7e7c68c575", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "156", "location": "System in Different States of Aggregation", "latex": "\\frac{dp_{23}}{d\\theta} = \\frac{L_{23}}{\\theta (v_{2} - v_{3})}", "name": "Clapeyron equation (fusion)", "statement": "The slope of the fusion pressure curve at the fundamental point equals the latent heat of fusion divided by temperature times the volume difference of liquid and solid.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p_{23}", "meaning": "melting (fusion) pressure" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature (fundamental state)" }, { "unit": null, "symbol": "L_{23}", "meaning": "heat of fusion" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of the liquid in the fundamental state" }, { "unit": null, "symbol": "v_3", "meaning": "specific volume of the solid in the fundamental state" } ], "sympy": "Eq(dp23_dtheta, L23/(theta*(v2 - v3)))", "physics": true, "states": [ "law/clapeyron-equation-fusion" ], "concepts": [ "concept/fusion-curve", "concept/pressure", "concept/temperature", "quantity/latent-heat", "quantity/specific-volume", "theorem/clapeyron-equation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-af02b771c3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "156", "location": "System in Different States of Aggregation", "latex": "\\frac{dp_{31}}{d\\theta} = \\frac{L_{31}}{\\theta (v_{3} - v_{1})}", "name": "Clapeyron equation (sublimation)", "statement": "The slope of the sublimation pressure curve at the fundamental point equals the latent heat of sublimation divided by temperature times the volume difference of solid and vapour.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p_{31}", "meaning": "sublimation pressure" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature (fundamental state)" }, { "unit": null, "symbol": "L_{31}", "meaning": "heat of sublimation (as written, negative of L_13)" }, { "unit": null, "symbol": "v_3", "meaning": "specific volume of the solid in the fundamental state" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of the vapour in the fundamental state" } ], "sympy": "Eq(dp31_dtheta, L31/(theta*(v3 - v1)))", "physics": true, "states": [ "law/clapeyron-equation-sublimation" ], "concepts": [ "concept/pressure", "concept/sublimation-curve", "concept/temperature", "quantity/latent-heat", "quantity/specific-volume", "theorem/clapeyron-equation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-37da518888", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "157", "location": "System in Different States of Aggregation", "latex": "u = \\dfrac{U}{M}", "name": null, "statement": "The mean specific energy of the system is its total energy divided by its total mass.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u", "meaning": "mean specific energy of the system" }, { "unit": null, "symbol": "U", "meaning": "total energy" }, { "unit": null, "symbol": "M", "meaning": "total mass" } ], "sympy": "Eq(u, U/M)", "physics": true, "states": [], "concepts": [ "quantity/internal-energy", "quantity/mass", "quantity/specific-energy" ], "pages": [ "157", "171" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-207e4ca161", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "162", "location": "System in Different States of Aggregation", "latex": "\\Phi = M\\phi", "name": null, "statement": "For a single homogeneous state the total entropy is the total mass times the specific entropy.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\Phi", "meaning": "total entropy of the single-state system" }, { "unit": null, "symbol": "M", "meaning": "total mass" }, { "unit": null, "symbol": "\\phi", "meaning": "specific entropy" } ], "sympy": "Eq(Phi, M*phi)", "physics": true, "states": [], "concepts": [ "quantity/entropy", "quantity/mass" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9a264085fa", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "162", "location": "System in Different States of Aggregation", "latex": "\\Phi' = M\\phi' = M_{12} \\phi_{12} + M_{21} \\phi_{21}", "name": null, "statement": "For the vapour-liquid solution the total entropy is the sum of the entropies of the two portions, and its specific form is M times phi-prime.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\Phi'", "meaning": "total entropy of the two-state (vapour-liquid) solution" }, { "unit": null, "symbol": "\\phi'", "meaning": "specific entropy of the two-state solution" }, { "unit": null, "symbol": "M_{12}, M_{21}", "meaning": "masses of vapour and liquid portions" }, { "unit": null, "symbol": "\\phi_{12}, \\phi_{21}", "meaning": "specific entropies of the two portions" }, { "unit": null, "symbol": "M", "meaning": "total mass" } ], "sympy": "Eq(Phi_prime, M_12*phi_12 + M_21*phi_21)", "physics": true, "states": [], "concepts": [ "concept/saturated-vapour", "concept/state-of-aggregation", "quantity/entropy", "quantity/mass" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-38120d1693", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "162", "location": "System in Different States of Aggregation", "latex": "\\Phi'' = M\\phi'' = M_{1} \\phi_{1} + M_{2} \\phi_{2} + M_{3} \\phi_{3}\\Add{.}", "name": null, "statement": "For the three-state coexistence the total entropy is the sum of the entropies of the three portions.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\Phi''", "meaning": "total entropy of the three-state solution" }, { "unit": null, "symbol": "\\phi''", "meaning": "specific entropy of the three-state solution" }, { "unit": null, "symbol": "M_1, M_2, M_3", "meaning": "masses of the three portions" }, { "unit": null, "symbol": "\\phi_1, \\phi_2, \\phi_3", "meaning": "specific entropies of the three portions" } ], "sympy": "Eq(Phi_dd, M1*phi1 + M2*phi2 + M3*phi3)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/entropy", "quantity/mass" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-31c9571ea4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "163", "location": "System in Different States of Aggregation", "latex": "\\phi'' > \\phi' > \\phi", "name": null, "statement": "For any system with all partial masses positive, the specific entropy of the three-state solution exceeds that of the two-state solution, which exceeds that of the single-state solution (the claim is stated as shown for positive partial masses).", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\phi''", "meaning": "specific entropy, three-state solution" }, { "unit": null, "symbol": "\\phi'", "meaning": "specific entropy, two-state solution" }, { "unit": null, "symbol": "\\phi", "meaning": "specific entropy, single-state solution" } ], "sympy": "And(phi_dd > phi_p, phi_p > phi)", "physics": true, "states": [], "concepts": [ "concept/inequality", "concept/stability-of-equilibrium", "concept/state-of-aggregation", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cc53aa4623", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "159", "location": "System in Different States of Aggregation", "latex": "\\phi_{12} - \\phi_{21} = \\frac{u_{12} - u_{21} + p_{12} (v_{12} - v_{21})}{\\theta_{12}}", "name": null, "statement": "On the vaporization curve, the entropy difference of the vapour and liquid at corresponding points equals the internal-energy difference plus pressure times volume difference, divided by temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\phi_{12}, \\phi_{21}", "meaning": "specific entropies of vapour and liquid at corresponding points" }, { "unit": null, "symbol": "u_{12}, u_{21}", "meaning": "specific energies of vapour and liquid" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "v_{12}, v_{21}", "meaning": "specific volumes of vapour and liquid" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" } ], "sympy": "Eq(phi_12 - phi_21, (u_12 - u_21 + p_12*(v_12 - v_21))/theta_12)", "physics": true, "states": [], "concepts": [ "concept/corresponding-point", "concept/pressure", "concept/saturated-vapour", "concept/temperature", "concept/vaporization-curve", "quantity/entropy", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-44bd0fd06a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "160", "location": "System in Different States of Aggregation", "latex": "\\frac{du_{12}}{dv_{12}} = \\left(\\frac{\\dd u}{\\dd v}\\right)_{12} + \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{12} \\frac{d\\theta_{12}}{dv_{12}}", "name": null, "statement": "Chain rule: along the vaporization curve the slope du/dv equals the partial derivatives at constant temperature and constant volume, weighted by the change of saturation temperature with volume.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "u_{12}", "meaning": "specific energy of the saturated vapour" }, { "unit": null, "symbol": "v_{12}", "meaning": "specific volume of the saturated vapour" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/partial-derivative", "concept/vaporization-curve", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e82a962627", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "160", "location": "System in Different States of Aggregation", "latex": "\\frac{du_{12}}{dv_{12}} = \\theta_{12} \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{12} - p_{12} + (c_{v})_{12}\\, \\frac{d\\theta_{12}}{dv_{12}}", "name": null, "statement": "Along the vaporization-curve branch the slope du/dv is given by temperature times the pressure's thermal derivative minus pressure plus specific heat at constant volume times the saturation-temperature change with volume.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_{12}", "meaning": "specific energy of the saturated vapour" }, { "unit": null, "symbol": "v_{12}", "meaning": "specific volume of the saturated vapour" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "(c_v)_{12}", "meaning": "specific heat at constant volume along the saturation curve" } ], "sympy": "Eq(du12_dv12, theta_12*dp_dtheta_12 - p_12 + cv_12*dtheta12_dv12)", "physics": true, "states": [], "concepts": [ "concept/derivative", "concept/partial-derivative", "concept/pressure", "concept/temperature", "concept/vaporization-curve", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f9d0f5989c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "160", "location": "System in Different States of Aggregation", "latex": "\\frac{du_{21}}{dv_{21}} = \\theta_{21} \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{21} - p_{12} + (c_{v})_{21}\\, \\frac{d\\theta_{12}}{dv_{21}}", "name": null, "statement": "The corresponding relation for the liquid branch of the vaporization curve. Note: the book writes p_{12} and d\\theta_{12}/dv_{21} here; the text states p_{21}=p_{12} and \\theta_{21}=\\theta_{12}, so the printed subscripts are equivalent by those identities, but are reproduced exactly as printed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_{21}", "meaning": "specific energy of the liquid" }, { "unit": null, "symbol": "v_{21}", "meaning": "specific volume of the liquid" }, { "unit": null, "symbol": "\\theta_{21}", "meaning": "saturation temperature (liquid branch)" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure (printed subscript)" }, { "unit": null, "symbol": "(c_v)_{21}", "meaning": "specific heat at constant volume along the liquid branch" } ], "sympy": "Eq(du21_dv21, theta_21*dp_dtheta_21 - p_12 + cv_21*dtheta12_dv21)", "physics": true, "states": [], "concepts": [ "concept/corresponding-point", "concept/derivative", "concept/pressure", "concept/temperature", "concept/vaporization-curve", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f5f66c9c80", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "167", "location": "System in Different States of Aggregation", "latex": "\\frac{u_{12} - u_{21}}{v_{12} - v_{21}} = \\theta_{12}\\, \\frac{dp_{12}}{d\\theta_{12}} - p_{12}", "name": null, "statement": "The slope of the chord joining corresponding vapour and liquid points equals temperature times the pressure derivative along saturation minus the pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_{12}, u_{21}", "meaning": "specific energies of vapour and liquid" }, { "unit": null, "symbol": "v_{12}, v_{21}", "meaning": "specific volumes of vapour and liquid" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" } ], "sympy": "Eq((u_12 - u_21)/(v_12 - v_21), theta_12*dp12_dtheta12 - p_12)", "physics": true, "states": [], "concepts": [ "concept/corresponding-point", "concept/derivative", "concept/pressure", "concept/temperature", "concept/vaporization-curve", "quantity/specific-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d2969e0824", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "167", "location": "System in Different States of Aggregation", "latex": "\\frac{\\dd u}{\\dd v} = \\theta\\, \\frac{dp}{d\\theta} - p", "name": null, "statement": "At constant temperature the partial derivative of specific energy with respect to specific volume equals temperature times the thermal pressure derivative minus pressure.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "u", "meaning": "specific energy" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": "Eq(du_dv, theta*dp_dtheta - p)", "physics": true, "states": [], "concepts": [ "concept/partial-derivative", "concept/pressure", "concept/temperature", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cbf43437fe", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "167", "location": "System in Different States of Aggregation", "latex": "\\frac{dp_{12}}{d\\theta_{12}} = \\frac{\\dd p}{\\dd \\theta} + \\frac{\\dd p}{\\dd v} · \\frac{dv_{12}}{d\\theta_{12}}", "name": null, "statement": "Total derivative of the saturation pressure with respect to saturation temperature: the thermal derivative plus the volume derivative times the volume change along saturation.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v_{12}", "meaning": "specific volume of saturated vapour" } ], "sympy": "Eq(dp12_dtheta12, dp_dtheta + dp_dv*dv12_dtheta12)", "physics": true, "states": [], "concepts": [ "concept/derivative", "concept/partial-derivative", "concept/pressure", "concept/saturated-vapour", "concept/temperature", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cc4f77f7f5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "164", "location": "System in Different States of Aggregation", "latex": "p_{12} (v - v_{12}) + (u - u_{12}) - \\theta_{12} (\\phi - \\phi_{12}) = 0", "name": null, "statement": "The plane in (v, u, phi) space through a corresponding pair of points on the vaporization curve; it contains both points and the line joining them.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "v, u, \\phi", "meaning": "variable rectangular coordinates of the plane" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "v_{12}, u_{12}, \\phi_{12}", "meaning": "coordinates of the saturated vapour point" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" } ], "sympy": "Eq(p_12*(v - v_12) + (u - u_12) - theta_12*(phi - phi_12), 0)", "physics": false, "states": [], "concepts": [ "concept/corresponding-point", "concept/developable-surface", "concept/saturated-vapour", "concept/surface" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8eab142a41", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "164", "location": "System in Different States of Aggregation", "latex": "v = \\frac{\\lambda v_{12} + \\mu v_{21}}{\\lambda + \\mu}", "name": null, "statement": "Points on the straight line joining corresponding points lie at weighted averages of the two corresponding specific volumes, with positive weights lambda and mu.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "v", "meaning": "specific volume of a point on the joining line" }, { "unit": null, "symbol": "v_{12}, v_{21}", "meaning": "specific volumes of the corresponding points" }, { "unit": null, "symbol": "\\lambda, \\mu", "meaning": "arbitrary positive weights" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/corresponding-point", "concept/developable-surface", "concept/line", "concept/ruled-surface" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-020abfefa5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "165", "location": "System in Different States of Aggregation", "latex": "\\delta \\phi' = \\frac{\\delta u + p_{12}\\, \\delta v}{\\theta_{12}}", "name": null, "statement": "The variation of the two-state specific entropy equals the variation of specific energy plus pressure times variation of specific volume, divided by the saturation temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\phi'", "meaning": "specific entropy of the two-state solution" }, { "unit": null, "symbol": "u", "meaning": "mean specific energy" }, { "unit": null, "symbol": "v", "meaning": "mean specific volume" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" } ], "sympy": "Eq(delta_phi_p, (delta_u + p_12*delta_v)/theta_12)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/saturated-vapour", "concept/temperature", "law/first-law-of-thermodynamics", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bb15d0ce16", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "165", "location": "System in Different States of Aggregation", "latex": "\\delta (\\phi' - \\phi) = \\left(\\frac{1}{\\theta_{12}} - \\frac{1}{\\theta}\\right) \\delta u + \\left(\\frac{p_{12}}{\\theta_{12}} - \\frac{p}{\\theta}\\right) \\delta v", "name": null, "statement": "The variation of the difference between the two entropy surfaces is a linear form in the variations of u and v, which vanishes along the curve of contact.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\phi'", "meaning": "specific entropy, two-state solution" }, { "unit": null, "symbol": "\\phi", "meaning": "specific entropy, single-state solution" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature of the state" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure of the state" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" } ], "sympy": "Eq(delta_phi_p - delta_phi, (1/theta_12 - 1/theta)*delta_u + (p_12/theta_12 - p/theta)*delta_v)", "physics": true, "states": [], "concepts": [ "concept/derivative", "concept/pressure", "concept/surface", "concept/temperature", "concept/thermodynamic-equilibrium", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e840de8f24", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "166", "location": "System in Different States of Aggregation", "latex": "\\theta\\, \\delta^{2} (\\phi' - \\phi) = (\\delta \\theta - \\delta \\theta_{12})\\, \\delta \\phi + (\\delta p_{12} - \\delta p)\\, \\delta v", "name": null, "statement": "At points of contact the second variation of the entropy difference, scaled by temperature, is expressed through the variations of temperature, saturation temperature, pressure and entropy.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" }, { "unit": null, "symbol": "\\phi", "meaning": "specific entropy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "p_{12}", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/derivative", "concept/pressure", "concept/stability-of-equilibrium", "concept/surface", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-eb59fef41f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "166", "location": "System in Different States of Aggregation", "latex": "\\delta \\phi = \\frac{c_{v}}{\\theta}\\, \\delta \\theta + \\frac{\\dd p}{\\dd \\theta}\\, \\delta v", "name": null, "statement": "The variation of specific entropy in terms of the variations of temperature and specific volume, with specific heat at constant volume (the book cites Eq. 81 for this).", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "specific entropy" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(delta_phi, cv/theta*delta_theta + dp_dtheta*delta_v)", "physics": true, "states": [], "concepts": [ "concept/derivative", "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/specific-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-aa13c73477", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "166", "location": "System in Different States of Aggregation", "latex": "\\delta p = \\frac{\\dd p}{\\dd \\theta}\\, \\delta \\theta + \\frac{\\dd p}{\\dd v}\\, \\delta v", "name": null, "statement": "The variation of pressure is the sum of its thermal and volume derivatives times the variations of temperature and specific volume.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume" } ], "sympy": "Eq(delta_p, dp_dtheta*delta_theta + dp_dv*delta_v)", "physics": true, "states": [], "concepts": [ "concept/partial-derivative", "concept/pressure", "concept/temperature", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7216db55c2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "167", "location": "System in Different States of Aggregation", "latex": "\\delta \\theta_{12} = \\frac{c_{v}\\, \\delta \\theta - \\theta\\, \\dfrac{\\dd p}{\\dd v} · \\dfrac{dv_{12}}{d\\theta_{12}}\\, \\delta v}{c_{v} - \\theta\\, \\dfrac{\\dd p}{\\dd v} \\left(\\dfrac{dv_{12}}{d\\theta_{12}}\\right)^{2}}", "name": null, "statement": "The variation of saturation temperature expressed through the variations of temperature and specific volume of the state.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature of the state" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume of the state" }, { "unit": null, "symbol": "v_{12}", "meaning": "specific volume of the saturated vapour" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/derivative", "concept/partial-derivative", "concept/saturated-vapour", "concept/temperature", "quantity/specific-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c35c28f7dc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "167", "location": "System in Different States of Aggregation", "latex": "\\delta^{2} (\\phi' - \\phi) = -\\frac{\\dd p}{\\dd v} · \\frac{c_{v}}{\\theta} · \\frac{\\left(\\dfrac{dv_{12}}{d\\theta_{12}}\\, \\delta \\theta - \\delta v\\right)^{2}}{c_{v} - \\theta\\, \\dfrac{\\dd p}{\\dd v} \\left(\\dfrac{dv_{12}}{d\\theta_{12}}\\right)^{2}}", "name": null, "statement": "The second variation of the entropy difference is essentially positive, since c_v is positive and dp/dv is negative for stable states, so the two-state surface rises above the single-state surface, establishing stability of the two-state solution within its region.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\phi'", "meaning": "specific entropy, two-state solution" }, { "unit": null, "symbol": "\\phi", "meaning": "specific entropy, single-state solution" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume (positive)" }, { "unit": null, "symbol": "\\theta", "meaning": "temperature" }, { "unit": null, "symbol": "\\theta_{12}", "meaning": "saturation temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/higher-order-derivative", "concept/partial-derivative", "concept/pressure", "concept/stability-of-equilibrium", "quantity/entropy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a853c4143d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "168", "location": "System in Different States of Aggregation", "latex": "v &= \\frac{\\lambda v_{1} + \\mu v_{2} + \\nu v_{3}}{\\lambda + \\mu + \\nu}", "name": null, "statement": "The specific volume of a point on the plane spanned by the three corner points is the weighted average of the corner volumes v1, v2, v3, with weights lambda, mu, nu.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "v", "meaning": "volume per unit mass (v = V/M)" }, { "unit": null, "symbol": "v_1, v_2, v_3", "meaning": "volumes per unit mass at the three corner points of the fundamental triangle" }, { "unit": null, "symbol": "lambda, mu, nu", "meaning": "positive weights of the three corner points" } ], "sympy": "Eq(v, (lambda_*v1 + mu*v2 + nu*v3)/(lambda_ + mu + nu))", "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/associated-triangles", "concept/plane-triangle" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-096537c457", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "168", "location": "System in Different States of Aggregation", "latex": "u &= \\frac{\\lambda u_{1} + \\mu u_{2} + \\nu u_{3}}{\\lambda + \\mu + \\nu}", "name": null, "statement": "The specific energy of a point on the plane spanned by the three corner points is the weighted average of the corner energies u1, u2, u3, with weights lambda, mu, nu.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "u", "meaning": "internal energy per unit mass (u = U/M)" }, { "unit": null, "symbol": "u_1, u_2, u_3", "meaning": "internal energies per unit mass at the three corner points" }, { "unit": null, "symbol": "lambda, mu, nu", "meaning": "positive weights of the three corner points" } ], "sympy": "Eq(u, (lambda_*u1 + mu*u2 + nu*u3)/(lambda_ + mu + nu))", "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/associated-triangles", "concept/plane-triangle", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-216ee73835", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "169", "location": "System in Different States of Aggregation", "latex": "M\\, \\delta\\phi'' = \\phi_{1}\\, \\delta M_{1} + \\phi_{2}\\, \\delta M_{2} + \\phi_{3}\\, \\delta M_{3}", "name": null, "statement": "The variation of the total entropy of the mixed system equals the sum of the entropies of the three portions weighted by the variations of their masses.", "kind": "result", "symbols": [ { "unit": null, "symbol": "M", "meaning": "total mass of the system" }, { "unit": null, "symbol": "phi''", "meaning": "mean specific entropy of the mixed state" }, { "unit": null, "symbol": "phi_1, phi_2, phi_3", "meaning": "specific entropies of the three portions (solid, liquid, gaseous)" }, { "unit": "mass", "symbol": "M_1, M_2, M_3", "meaning": "masses of the three portions" } ], "sympy": "Eq(M*dphi2, phi1*dM1 + phi2*dM2 + phi3*dM3)", "physics": true, "states": [], "concepts": [ "concept/associated-triangles", "concept/state-of-aggregation", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d51c825f14", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "169", "location": "System in Different States of Aggregation", "latex": "\\delta M_{1} + \\delta M_{2} + \\delta M_{3} &= 0\\Add{,}", "name": null, "statement": "The variations of the three portion masses sum to zero, so the total mass is unchanged.", "kind": "law", "symbols": [ { "unit": "mass", "symbol": "M_1, M_2, M_3", "meaning": "masses of the three portions" } ], "sympy": "Eq(dM1 + dM2 + dM3, 0)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/mass" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e1c8b41c66", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "169", "location": "System in Different States of Aggregation", "latex": "v_{1}\\, \\delta M_{1} + v_{2}\\, \\delta M_{2} + v_{3}\\, \\delta M_{3} &= M\\, \\delta v\\Add{,}", "name": null, "statement": "The variation of the total volume equals the sum of the volume variations of the three portions.", "kind": "law", "symbols": [ { "unit": null, "symbol": "v_1, v_2, v_3", "meaning": "specific volumes of the three portions" }, { "unit": null, "symbol": "M", "meaning": "total mass of the system" }, { "unit": null, "symbol": "v", "meaning": "specific volume of the whole system" } ], "sympy": "Eq(v1*dM1 + v2*dM2 + v3*dM3, M*dv)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3e68783718", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "169", "location": "System in Different States of Aggregation", "latex": "u_{1}\\, \\delta M_{1} + u_{2}\\, \\delta M_{2} + u_{3}\\, \\delta M_{3} &= M\\, \\delta u\\Add{.}", "name": null, "statement": "The variation of the total internal energy equals the sum of the energy variations of the three portions.", "kind": "law", "symbols": [ { "unit": null, "symbol": "u_1, u_2, u_3", "meaning": "specific internal energies of the three portions" }, { "unit": null, "symbol": "M", "meaning": "total mass of the system" }, { "unit": null, "symbol": "u", "meaning": "specific internal energy of the whole system" } ], "sympy": "Eq(u1*dM1 + u2*dM2 + u3*dM3, M*du)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-29a3789d2a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "169", "location": "System in Different States of Aggregation", "latex": "\\delta \\phi'' = \\frac{\\delta u + p_{1}\\, \\delta v}{\\theta_{1}}", "name": null, "statement": "The variation of the mean specific entropy of the mixed state equals the variation of specific energy plus p1 times the variation of specific volume, divided by the temperature theta_1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi''", "meaning": "mean specific entropy of the mixed state" }, { "unit": null, "symbol": "u", "meaning": "specific internal energy" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "p_1", "meaning": "pressure of the first state (an absolute constant here)" }, { "unit": "degrees", "symbol": "theta_1", "meaning": "temperature of the first state (an absolute constant here)" } ], "sympy": "Eq(dphi2, (du + p1*dv)/theta1)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/tangent", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3e97d4f3ce", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "170", "location": "System in Different States of Aggregation", "latex": "\\theta_{1}^{2}\\, \\delta^{2} (\\phi'' - \\phi')\n = \\left[\\delta u - \\left(\\theta_{1}\\, \\frac{dp_{12}}{d\\theta_{12}} - p_{1}\\right) \\delta v\\right] \\delta \\theta_{12}.", "name": null, "statement": "The second variation of the entropy difference along the sheet (12) is proportional to the variation of temperature theta_12 times a linear combination of the variations of u and v; its sign decides stability.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi''", "meaning": "mean specific entropy of the mixed state" }, { "unit": null, "symbol": "phi'", "meaning": "specific entropy of the developable surface sheet (12)" }, { "unit": "degrees", "symbol": "theta_12", "meaning": "common temperature of the two-phase sheet (12)" }, { "unit": null, "symbol": "p_12", "meaning": "common pressure of the two-phase sheet (12) as a function of theta_12" }, { "unit": null, "symbol": "theta_1, p_1", "meaning": "temperature and pressure of the first state" }, { "unit": null, "symbol": "u, v", "meaning": "specific energy and specific volume of the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/developable-surface", "concept/point-of-tangency", "concept/pressure", "concept/stability-of-equilibrium", "concept/temperature", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d912d81fbb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "170", "location": "System in Different States of Aggregation", "latex": "\\frac{M_{12}\\, \\delta v_{12} + M_{21}\\, \\delta v_{21} - M\\, \\delta v}{v_{12} - v_{21}} = \\frac{M_{12}\\, \\delta u_{12} + M_{21}\\, \\delta u_{21} - M\\, \\delta u}{u_{12} - u_{21}}", "name": null, "statement": "Eliminating the masses M_12 and M_21 gives a relation between the variations of volume and energy for the two-phase states along the coexistence line.", "kind": "result", "symbols": [ { "unit": "mass", "symbol": "M_12, M_21", "meaning": "masses of the two coexisting states on the sheet (12)" }, { "unit": null, "symbol": "v_12, v_21", "meaning": "specific volumes of the two coexisting states" }, { "unit": null, "symbol": "u_12, u_21", "meaning": "specific energies of the two coexisting states" }, { "unit": null, "symbol": "M", "meaning": "total mass of the system" }, { "unit": null, "symbol": "v, u", "meaning": "specific volume and specific energy of the whole system" } ], "sympy": "Eq((M12*dv12 + M21*dv21 - M*dv)/(v12 - v21), (M12*du12 + M21*du21 - M*du)/(u12 - u21))", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "concept/thermodynamic-equilibrium", "method/elimination", "quantity/internal-energy", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bbbff32189", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "133", "location": "System in Different States of Aggregation", "latex": "M_{1} u_{1} + M_{2} u_{2} + M_{3} u_{3} = U", "name": null, "statement": "The energies of the portions add up to the given energy of the system.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "u_1, u_2, u_3", "meaning": "specific energies of the portions" }, { "unit": null, "symbol": "U", "meaning": "given energy of the system" } ], "sympy": "Eq(M_1*u_1 + M_2*u_2 + M_3*u_3, U)", "physics": true, "states": [], "concepts": [ "concept/external-conditions-of-equilibrium", "quantity/internal-energy", "quantity/specific-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-42f682eb81", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "133", "location": "System in Different States of Aggregation", "latex": "M_{1} + M_{2} + M_{3} = M", "name": null, "statement": "The masses of the solid, liquid and gaseous portions add up to the total mass of the system.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "M_1, M_2, M_3", "meaning": "masses of the three portions of the system" }, { "unit": null, "symbol": "M", "meaning": "total mass of the system" } ], "sympy": "Eq(M_1 + M_2 + M_3, M)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/mass" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a0564c7b66", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "133", "location": "System in Different States of Aggregation", "latex": "M_{1} v_{1} + M_{2} v_{2} + M_{3} v_{3} = V", "name": null, "statement": "The volumes of the portions, each mass times its specific volume, add up to the given volume of the system.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "v_1, v_2, v_3", "meaning": "specific volumes of the portions" }, { "unit": null, "symbol": "V", "meaning": "given volume of the system" } ], "sympy": "Eq(M_1*v_1 + M_2*v_2 + M_3*v_3, V)", "physics": true, "states": [], "concepts": [ "concept/external-conditions-of-equilibrium", "concept/state-of-aggregation", "quantity/specific-volume", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8aa7e87c09", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "134", "location": "System in Different States of Aggregation", "latex": "\\Phi = M_{1} \\phi_{1} + M_{2} \\phi_{2} + M_{3} \\phi_{3}", "name": null, "statement": "The entropy of the system is the sum of the masses times the specific entropies of the portions.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "phi_1, phi_2, phi_3", "meaning": "specific entropies of the portions" } ], "sympy": "Eq(Phi, M_1*phi_1 + M_2*phi_2 + M_3*phi_3)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-144dcc2370", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "134", "location": "System in Different States of Aggregation", "latex": "\\delta \\phi = \\frac{\\delta u + p\\, \\delta v}{\\theta}", "name": null, "statement": "For an infinitesimal change of state, the change in specific entropy equals the change in specific energy plus pressure times change in specific volume, divided by the temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "specific entropy" }, { "unit": null, "symbol": "u", "meaning": "specific energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": "Eq(delta_phi, (delta_u + p*delta_v)/theta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/specific-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-da7931c349", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "135", "location": "System in Different States of Aggregation", "latex": "\\theta_{1} = \\theta_{2} = \\theta_{3} (= \\theta)", "name": null, "statement": "In equilibrium the temperatures of the three portions are all equal to a common temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta_1, theta_2, theta_3", "meaning": "temperatures of the three portions" }, { "unit": null, "symbol": "theta", "meaning": "common temperature of the system" } ], "sympy": "And(Eq(theta_1, theta_2), Eq(theta_2, theta_3))", "physics": true, "states": [], "concepts": [ "concept/temperature", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-33db367916", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "135", "location": "System in Different States of Aggregation", "latex": "p_{1} = p_{2} = p_{3}", "name": null, "statement": "In equilibrium the pressures of the three portions are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p_1, p_2, p_3", "meaning": "pressures of the three portions" } ], "sympy": "And(Eq(p_1, p_2), Eq(p_2, p_3))", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c239bcbfbc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "135", "location": "System in Different States of Aggregation", "latex": "\\phi_{1} - \\phi_{2} = \\frac{(u_{1} - u_{2}) + p_{1}(v_{1} - v_{2})}{\\theta}", "name": null, "statement": "In equilibrium between the first and second portions, their difference in specific entropy is fixed by their energies, pressure and volumes at the common temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi_1, phi_2", "meaning": "specific entropies of portions 1 and 2" }, { "unit": null, "symbol": "u_1, u_2", "meaning": "specific energies of portions 1 and 2" }, { "unit": null, "symbol": "p_1", "meaning": "common pressure" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of portions 1 and 2" }, { "unit": null, "symbol": "theta", "meaning": "common temperature" } ], "sympy": "Eq(phi_1 - phi_2, ((u_1 - u_2) + p_1*(v_1 - v_2))/theta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/thermodynamic-equilibrium", "quantity/entropy", "quantity/specific-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9de0df905f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "135", "location": "System in Different States of Aggregation", "latex": "\\phi_{2} - \\phi_{3} = \\frac{(u_{2} - u_{3}) + p_{2}(v_{2} - v_{3})}{\\theta}", "name": null, "statement": "In equilibrium between the second and third portions, their difference in specific entropy is fixed by their energies, pressure and volumes at the common temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi_2, phi_3", "meaning": "specific entropies of portions 2 and 3" }, { "unit": null, "symbol": "u_2, u_3", "meaning": "specific energies of portions 2 and 3" }, { "unit": null, "symbol": "p_2", "meaning": "common pressure" }, { "unit": null, "symbol": "v_2, v_3", "meaning": "specific volumes of portions 2 and 3" }, { "unit": null, "symbol": "theta", "meaning": "common temperature" } ], "sympy": "Eq(phi_2 - phi_3, ((u_2 - u_3) + p_2*(v_2 - v_3))/theta)", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/thermodynamic-equilibrium", "quantity/entropy", "quantity/specific-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-46502a4bf4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "137", "location": "System in Different States of Aggregation", "latex": "\\theta\\, \\delta^{2} \\Phi = -\\tsum M_{1} \\left(\\frac{(c_{v})_{1}}{\\theta}\\, \\delta \\theta_{1}^{2} - \\left(\\frac{\\dd p_{1}}{\\dd v}\\right)_{\\theta} \\delta v_{1}^{2}\\right)", "name": null, "statement": "The second variation of the entropy, multiplied by the temperature, is given in independent variables; it is negative, and the entropy is a maximum, when specific heat is positive and the isothermal change of pressure with volume is negative.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "entropy of the system" }, { "unit": null, "symbol": "M_1", "meaning": "mass of a portion" }, { "unit": null, "symbol": "(c_v)_1", "meaning": "specific heat at constant volume of a portion" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "p_1", "meaning": "pressure of a portion" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of a portion" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/stability-of-equilibrium", "concept/temperature", "quantity/entropy", "quantity/specific-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3ea07f64af", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "136", "location": "System in Different States of Aggregation", "latex": "\\int_{v_{2}}^{v_{1}} p\\, dv = p_{1} (v_{1} - v_{2})", "name": null, "statement": "For two states of aggregation in contact, the integral of pressure along the isotherm between the two specific volumes equals the constant pressure times the difference of volumes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure along the isotherm" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of the vapour and the liquid" }, { "unit": null, "symbol": "p_1", "meaning": "pressure of coexistence" } ], "sympy": "Eq(Integral(p, (v, v_2, v_1)), p_1*(v_1 - v_2))", "physics": true, "states": [], "concepts": [ "concept/definite-integral", "concept/pressure", "concept/saturated-vapour", "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-903d77ccfb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "136", "location": "System in Different States of Aggregation", "latex": "\\int_{v_{3}}^{v_{2}} p\\, dv = p_{2}(v_{2} - v_{3})", "name": null, "statement": "The corresponding integral condition holds between the second and third portions in contact.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure along the isotherm" }, { "unit": null, "symbol": "v_2, v_3", "meaning": "specific volumes of portions 2 and 3" }, { "unit": null, "symbol": "p_2", "meaning": "pressure of coexistence" } ], "sympy": "Eq(Integral(p, (v, v_3, v_2)), p_2*(v_2 - v_3))", "physics": true, "states": [], "concepts": [ "concept/definite-integral", "concept/pressure", "concept/state-of-aggregation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a670e7d455", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "135", "location": "System in Different States of Aggregation", "latex": "\\phi_{1} - \\phi_{2} = \\frac{u_{1} - u_{2}}{\\theta} + \\frac{1}{\\theta} \\int_{v_{2}}^{v_{1}} p\\, dv", "name": null, "statement": "The entropy difference between two portions is obtained by integrating along an isotherm: the energy difference over temperature plus the integral of pressure over volume divided by temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi_1, phi_2", "meaning": "specific entropies of the two portions" }, { "unit": null, "symbol": "u_1, u_2", "meaning": "specific energies" }, { "unit": null, "symbol": "theta", "meaning": "temperature of the isotherm" }, { "unit": null, "symbol": "p", "meaning": "pressure along the isotherm" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes" } ], "sympy": "Eq(phi_1 - phi_2, (u_1 - u_2)/theta + Integral(p, (v, v_2, v_1))/theta)", "physics": true, "states": [], "concepts": [ "concept/definite-integral", "concept/isothermal-curve", "concept/pressure", "quantity/entropy", "quantity/specific-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b88a6d8656", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "140", "location": "System in Different States of Aggregation", "latex": "\\frac{R\\theta}{v_{1} - a} - \\frac{c}{\\theta (v_{1} + b)^{2}} = \\frac{R\\theta}{v_{2} - a} - \\frac{c}{\\theta (v_{2} + b)^{2}}", "name": null, "statement": "Under Clausius' equation of state, the saturated vapour and the liquid in contact have equal pressure at the same temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "absolute gas constant" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "a, b, c", "meaning": "constants of Clausius' equation for the substance" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of the saturated vapour" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of the liquid" } ], "sympy": "Eq(R*theta/(v_1 - a) - c/(theta*(v_1 + b)**2), R*theta/(v_2 - a) - c/(theta*(v_2 + b)**2))", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/clausius-equation", "concept/pressure", "concept/saturated-vapour" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0f0b2f3608", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "140", "location": "System in Different States of Aggregation", "latex": "R\\theta \\log \\frac{v_{1} - a}{v_{2} - a} - \\frac{c}{\\theta} \\left(\\frac{1}{v_{2} + b} - \\frac{1}{v_{1} + b}\\right) = (v_{1} - v_{2}) \\left(\\frac{R\\theta}{v_{1} - a} - \\frac{c}{\\theta (v_{1} + b)^{2}}\\right)", "name": null, "statement": "Under Clausius' equation of state, the equal-area condition on the isotherm (the integral of pressure between the two volumes equals the coexistence pressure times the volume difference) takes this explicit form.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "absolute gas constant" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "a, b, c", "meaning": "constants of Clausius' equation for the substance" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of the saturated vapour" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of the liquid" } ], "sympy": "Eq(R*theta*log((v_1 - a)/(v_2 - a)) - c/theta*(1/(v_2 + b) - 1/(v_1 + b)), (v_1 - v_2)*(R*theta/(v_1 - a) - c/(theta*(v_1 + b)**2)))", "physics": true, "states": [], "concepts": [ "concept/clausius-equation", "concept/definite-integral", "concept/pressure", "concept/saturated-vapour" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1d2aa1d4ab", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "140", "location": "System in Different States of Aggregation", "latex": "u - \\theta \\phi = f", "name": "free energy per unit mass", "statement": "The free energy per unit mass is defined as specific energy minus temperature times specific entropy.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u", "meaning": "specific energy" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "phi", "meaning": "specific entropy" }, { "unit": null, "symbol": "f", "meaning": "free energy per unit mass" } ], "sympy": "Eq(f, u - theta*phi)", "physics": true, "states": [ "quantity/free-energy" ], "concepts": [ "concept/temperature", "quantity/entropy", "quantity/specific-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-21276ad018", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "141", "location": "System in Different States of Aggregation", "latex": "f_{2} - f_{1} = p_{1} (v_{1} - v_{2})", "name": null, "statement": "In equilibrium between vapour and liquid, the difference of free energies per unit mass equals the coexistence pressure times the volume difference.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f_1, f_2", "meaning": "free energies per unit mass of vapour and liquid" }, { "unit": null, "symbol": "p_1", "meaning": "coexistence pressure" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of vapour and liquid" } ], "sympy": "Eq(f_2 - f_1, p_1*(v_1 - v_2))", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/saturated-vapour", "quantity/free-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-92ebe5a75d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "141", "location": "System in Different States of Aggregation", "latex": "\\left(\\frac{\\dd f}{\\dd \\theta}\\right)_{v} = -\\phi", "name": null, "statement": "At constant specific volume, the rate of change of free energy with temperature equals minus the specific entropy.", "kind": "law", "symbols": [ { "unit": null, "symbol": "f", "meaning": "free energy per unit mass" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "v", "meaning": "specific volume (held constant)" }, { "unit": null, "symbol": "phi", "meaning": "specific entropy" } ], "sympy": "Eq(Derivative(f, theta), -phi)", "physics": true, "states": [], "concepts": [ "concept/partial-derivative", "concept/temperature", "quantity/entropy", "quantity/free-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b731185761", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "141", "location": "System in Different States of Aggregation", "latex": "\\left(\\frac{\\dd f}{\\dd v}\\right)_{\\theta} = -p", "name": null, "statement": "At constant temperature, the rate of change of free energy with specific volume equals minus the pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "f", "meaning": "free energy per unit mass" }, { "unit": null, "symbol": "v", "meaning": "specific volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature (held constant)" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": "Eq(Derivative(f, v), -p)", "physics": true, "states": [], "concepts": [ "concept/partial-derivative", "concept/pressure", "quantity/free-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b4323c460a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "142", "location": "System in Different States of Aggregation", "latex": "(u_{1} - u_{2}) + p_{1} (v_{1} - v_{2}) = \\theta (v_{1} - v_{2})\\, \\frac{dp_{1}}{d\\theta}", "name": "Clapeyron equation", "statement": "The latent heat of vaporization equals the temperature times the volume change times the slope of the vapour-pressure curve (Clapeyron's relation).", "kind": "law", "symbols": [ { "unit": null, "symbol": "u_1, u_2", "meaning": "specific energies of saturated vapour and liquid" }, { "unit": null, "symbol": "p_1", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of saturated vapour and liquid" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": "Eq((u_1 - u_2) + p_1*(v_1 - v_2), theta*(v_1 - v_2)*Derivative(p_1, theta))", "physics": true, "states": [ "theorem/clapeyron-equation" ], "concepts": [ "concept/pressure", "concept/saturated-vapour", "concept/temperature", "quantity/latent-heat", "quantity/specific-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b600679148", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "142", "location": "System in Different States of Aggregation", "latex": "\\phi_{1} - \\phi_{2} = (v_{1} - v_{2})\\, \\frac{dp_{1}}{d\\theta}", "name": null, "statement": "The difference of specific entropies of vapour and liquid equals their volume difference times the slope of the vapour-pressure curve.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi_1, phi_2", "meaning": "specific entropies of vapour and liquid" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of vapour and liquid" }, { "unit": null, "symbol": "p_1", "meaning": "saturated vapour pressure" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": "Eq(phi_1 - phi_2, (v_1 - v_2)*Derivative(p_1, theta))", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/saturated-vapour", "concept/temperature", "quantity/entropy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a80d3df803", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "142", "location": "System in Different States of Aggregation", "latex": "L = u_{1} - u_{2} + p_{1}(v_{1} - v_{2})", "name": "heat of vaporization", "statement": "The latent heat of vaporization is the change of energy plus the external work done against the saturated vapour pressure.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat of vaporization of unit mass" }, { "unit": null, "symbol": "u_1, u_2", "meaning": "specific energies of vapour and liquid" }, { "unit": null, "symbol": "p_1", "meaning": "constant pressure of saturated vapour" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of vapour and liquid" } ], "sympy": "Eq(L, u_1 - u_2 + p_1*(v_1 - v_2))", "physics": true, "states": [ "quantity/latent-heat" ], "concepts": [ "concept/pressure", "concept/work", "quantity/internal-energy", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1e8aa2a618", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "142", "location": "System in Different States of Aggregation", "latex": "W = -p_{1}(v_{1} - v_{2})", "name": null, "statement": "The external work performed during vaporization at constant pressure equals minus the pressure times the volume increase.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "W", "meaning": "external work per unit mass" }, { "unit": null, "symbol": "p_1", "meaning": "constant pressure of saturated vapour" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of vapour and liquid" } ], "sympy": "Eq(W, -p_1*(v_1 - v_2))", "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/work", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-61f8624ddc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "142", "location": "System in Different States of Aggregation", "latex": "L = \\theta (v_{1} - v_{2})", "name": "Clapeyron equation", "statement": "The heat of vaporization equals the absolute temperature times the difference of specific volumes of vapour and liquid; this relation was deduced by Clapeyron from Carnot's theory and rigorously proved by Clausius.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat of vaporization of unit mass" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "v_1, v_2", "meaning": "specific volumes of saturated vapour and liquid" } ], "sympy": "Eq(L, theta*(v_1 - v_2))", "physics": true, "states": [ "theorem/clapeyron-equation" ], "concepts": [ "concept/saturated-vapour", "concept/temperature", "quantity/latent-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5e15281954", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "145", "location": "System in Different States of Aggregation", "latex": "u_{1} = c_{v} \\theta + \\const", "name": null, "statement": "For a perfect gas the specific energy is a linear function of temperature with the specific heat at constant volume as slope.", "kind": "law", "symbols": [ { "unit": null, "symbol": "u_1", "meaning": "specific energy of the vapour" }, { "unit": null, "symbol": "c_v", "meaning": "specific heat at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": "Eq(u_1, c_v*theta + const)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "quantity/specific-energy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d5d509585f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "146", "location": "System in Different States of Aggregation", "latex": "\\frac{d\\theta}{dp_{1}} = \\frac{\\theta (v_{1} - v_{2})}{L}", "name": null, "statement": "The change of melting point (or boiling point) with pressure is the temperature times the volume change divided by the latent heat.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "melting temperature" }, { "unit": null, "symbol": "p_1", "meaning": "melting pressure" }, { "unit": null, "symbol": "v_1", "meaning": "specific volume of the liquid" }, { "unit": null, "symbol": "v_2", "meaning": "specific volume of the solid" }, { "unit": null, "symbol": "L", "meaning": "latent heat of fusion" } ], "sympy": "Eq(Derivative(theta, p_1), theta*(v_1 - v_2)/L)", "physics": true, "states": [], "concepts": [ "concept/melting-point", "concept/pressure", "concept/temperature", "quantity/latent-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-088141d2af", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-in-different-states-of-aggregation", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "147", "location": "System in Different States of Aggregation", "latex": "\\frac{L}{\\theta} = \\phi_{1} - \\phi_{2}", "name": null, "statement": "The heat of phase change divided by the temperature equals the difference of specific entropies of the two states.", "kind": "result", "symbols": [ { "unit": null, "symbol": "L", "meaning": "latent heat" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "phi_1, phi_2", "meaning": "specific entropies of the two states" } ], "sympy": "Eq(L/theta, phi_1 - phi_2)", "physics": true, "states": [], "concepts": [ "concept/state-of-aggregation", "concept/temperature", "quantity/entropy", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cd704b4aa5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "191", "location": "System of any Number of Independent Constituents", "latex": "\\lambda = \\frac{R}{m} \\theta^{2} · \\frac{d \\log \\dfrac{p}{p_{0}}}{d\\theta}", "name": "Kirchhoff's formula", "statement": "The heat evolved when salt sufficient for saturation dissolves in one gram of pure water equals R/m times θ² times the temperature derivative of log(p/p₀).", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\lambda", "meaning": "heat of solution of the salt in saturating a solution (heat given out when salt dissolves in unit mass of water)" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of water vapour" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure of the saturated solution" }, { "unit": null, "symbol": "p_0", "meaning": "vapour pressure of pure water at temperature θ" } ], "sympy": "Eq(lambda_, R/m*theta**2*Derivative(log(p/p_0), theta))", "physics": true, "states": [ "theorem/kirchhoff-s-formula" ], "concepts": [ "concept/perfect-gas", "concept/saturated-vapour", "concept/saturation-point", "concept/solution", "law/first-law-of-thermodynamics", "quantity/absolute-temperature", "quantity/heat-effect", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6dcded4acb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "192", "location": "System of any Number of Independent Constituents", "latex": "c' = \\frac{M_{2}'}{M_{1}'}", "name": null, "statement": "The concentration of the second constituent in the first phase is the ratio of its mass to the mass of the first constituent in that phase.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c'", "meaning": "concentration of the second constituent in the first phase" }, { "unit": null, "symbol": "M_1'", "meaning": "mass of the first constituent in the first phase" }, { "unit": null, "symbol": "M_2'", "meaning": "mass of the second constituent in the first phase" } ], "sympy": "Eq(c_p, M2p/M1p)", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/independent-constituent", "concept/phase" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c31d64130f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "192", "location": "System of any Number of Independent Constituents", "latex": "M_{1}'\\, \\frac{\\dd^{2} \\Psi'}{\\dd M_{1}'\\, \\dd M_{2}'} = \\varphi'", "name": null, "statement": "The quantity φ' is defined as M₁' times the mixed second derivative of Ψ' with respect to M₁' and M₂'.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\varphi'", "meaning": "characteristic quantity of the first phase, depending on its nature, θ, p and c' only" }, { "unit": null, "symbol": "\\Psi'", "meaning": "thermodynamic potential of the first phase" }, { "unit": null, "symbol": "M_1'", "meaning": "mass of the first constituent in the first phase" }, { "unit": null, "symbol": "M_2'", "meaning": "mass of the second constituent in the first phase" } ], "sympy": "Eq(phi_p, M1p*Derivative(Psi_p, M1p, M2p))", "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/partial-derivative", "concept/phase" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b11f7960a4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "193", "location": "System of any Number of Independent Constituents", "latex": "\\frac{\\dd^{2} \\Psi'}{\\dd M_{1}'^{2}} = -\\frac{M_{2}'}{M_{1}'^{2}} · \\varphi'", "name": null, "statement": "The second derivative of Ψ' with respect to M₁' equals minus M₂'/M₁'² times φ'.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Psi'", "meaning": "thermodynamic potential of the first phase" }, { "unit": null, "symbol": "\\varphi'", "meaning": "characteristic quantity of the first phase" }, { "unit": null, "symbol": "M_1'", "meaning": "mass of the first constituent in the first phase" }, { "unit": null, "symbol": "M_2'", "meaning": "mass of the second constituent in the first phase" } ], "sympy": "Eq(Derivative(Psi_p, (M1p, 2)), -M2p/M1p**2*phi_p)", "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/homogeneous-function", "concept/partial-derivative" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6922eedbc9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "193", "location": "System of any Number of Independent Constituents", "latex": "\\delta^{2} \\Psi < 0", "name": null, "statement": "For stable equilibrium at constant temperature and pressure the second variation of Ψ is negative.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "\\Psi", "meaning": "total thermodynamic potential of the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/maximum", "concept/stability-of-equilibrium", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a508b10cf9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "194", "location": "System of any Number of Independent Constituents", "latex": "\\frac{L_{1}}{\\theta^{2}}\\, d\\theta - \\frac{s_{1}}{\\theta}\\, dp - \\varphi'\\, dc' + \\varphi''\\, dc'' = 0", "name": null, "statement": "For the first constituent passing between the two phases, the displacement of equilibrium links dθ, dp, dc' and dc''.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L_1", "meaning": "heat absorbed per unit mass of the first constituent passing from the first to the second phase" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "s_1", "meaning": "change of volume per unit mass of the first constituent passing between phases" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "c'", "meaning": "concentration of the second constituent in the first phase" }, { "unit": null, "symbol": "c''", "meaning": "concentration of the second constituent in the second phase" }, { "unit": null, "symbol": "\\varphi'", "meaning": "characteristic quantity of the first phase" }, { "unit": null, "symbol": "\\varphi''", "meaning": "characteristic quantity of the second phase" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/differential", "concept/displacement", "concept/mechanical-equilibrium", "concept/phase-rule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9d17b9e0d6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "195", "location": "System of any Number of Independent Constituents", "latex": "\\frac{L_{2}}{\\theta^{2}}\\, d\\theta - \\frac{s_{2}}{\\theta}\\, dp - \\varphi'\\, \\frac{dc'}{c'} + \\varphi''\\, \\frac{dc''}{c''} = 0", "name": null, "statement": "For the second constituent passing into the second phase, the displacement of equilibrium links dθ, dp, dc' and dc''.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L_2", "meaning": "heat absorbed per unit mass of the second constituent passing to the second phase" }, { "unit": null, "symbol": "s_2", "meaning": "change of volume per unit mass of the second constituent passing between phases" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "c'", "meaning": "concentration of the second constituent in the first phase" }, { "unit": null, "symbol": "c''", "meaning": "concentration of the second constituent in the second phase" }, { "unit": null, "symbol": "\\varphi'", "meaning": "characteristic quantity of the first phase" }, { "unit": null, "symbol": "\\varphi''", "meaning": "characteristic quantity of the second phase" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/differential", "concept/displacement", "concept/mechanical-equilibrium", "concept/phase-rule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-70db3afef3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "195", "location": "System of any Number of Independent Constituents", "latex": "dp = \\frac{\\left(\\dfrac{c''}{c'} - 1\\right) \\theta \\varphi'\\, dc'}{s_{1} + c'' s_{2}}", "name": null, "statement": "Along an isotherm the change of vapour pressure with concentration of the liquid is given by this expression.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "vapour pressure" }, { "unit": null, "symbol": "c'", "meaning": "concentration of the second constituent in the liquid" }, { "unit": null, "symbol": "c''", "meaning": "concentration of the second constituent in the vapour" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "\\varphi'", "meaning": "characteristic quantity of the liquid phase" }, { "unit": null, "symbol": "s_1", "meaning": "change of volume per unit mass of the first constituent passing between phases" }, { "unit": null, "symbol": "s_2", "meaning": "change of volume per unit mass of the second constituent passing between phases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/isothermal-curve", "concept/partial-derivative", "concept/pressure", "concept/saturated-vapour" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-80d55a3b17", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "195", "location": "System of any Number of Independent Constituents", "latex": "dc'' = \\frac{\\left(\\dfrac{1}{s_{1}} + \\dfrac{1}{c's_{2}}\\right)}{\\left(\\dfrac{1}{s_{1}} + \\dfrac{1}{c'' s_{2}}\\right)} · \\frac{\\varphi'}{\\varphi''}\\, dc'", "name": null, "statement": "Along an isotherm the change of the vapour concentration is proportional to the change of the liquid concentration, so both concentrations change in the same sense.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c'", "meaning": "concentration of the second constituent in the liquid" }, { "unit": null, "symbol": "c''", "meaning": "concentration of the second constituent in the vapour" }, { "unit": null, "symbol": "s_1", "meaning": "change of volume per unit mass of the first constituent passing between phases" }, { "unit": null, "symbol": "s_2", "meaning": "change of volume per unit mass of the second constituent passing between phases" }, { "unit": null, "symbol": "\\varphi'", "meaning": "characteristic quantity of the liquid phase" }, { "unit": null, "symbol": "\\varphi''", "meaning": "characteristic quantity of the vapour phase" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/common-ratio", "concept/concentration", "concept/isotherm" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a419dde12a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "194", "location": "System of any Number of Independent Constituents", "latex": "c'' = 0", "name": null, "statement": "The second constituent occurs only in the first phase, so its concentration in the second phase is zero.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "c''", "meaning": "concentration of the dissolved substance in the second phase (pure solvent)" } ], "sympy": "Eq(c_pp, 0)", "physics": true, "states": [], "concepts": [ "concept/dissolved-substance", "concept/solution", "concept/solvent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a73da99b73", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "196", "location": "System of any Number of Independent Constituents", "latex": "\\frac{L}{\\theta^{2}}\\, d\\theta - \\frac{s}{\\theta}\\, dp - \\varphi\\, dc = 0", "name": null, "statement": "For a solution in contact with the pure solvent vapour, the displacement of equilibrium links dθ, dp and dc.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat of vaporization of the solution, per unit mass of solvent" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "s", "meaning": "change of volume per unit mass of the solvent passing into the vapour" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/differential", "concept/mechanical-equilibrium", "concept/solution", "concept/solvent", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4598c83ef4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "197", "location": "System of any Number of Independent Constituents", "latex": "\\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{c} = \\frac{L}{\\theta · s}", "name": null, "statement": "At constant concentration the rate of change of vapour pressure with temperature equals L divided by θ times s.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution, held constant" }, { "unit": null, "symbol": "L", "meaning": "heat of vaporization of the solution" }, { "unit": null, "symbol": "s", "meaning": "change of volume per unit mass of the solvent passing into the vapour" } ], "sympy": "Eq(Derivative(p, theta), L/(theta*s))", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/partial-derivative", "concept/temperature", "quantity/latent-heat", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d94e45e2ac", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "197", "location": "System of any Number of Independent Constituents", "latex": "s = v = \\frac{R}{m} · \\frac{\\theta}{p}", "name": null, "statement": "Assuming Boyle's and Gay-Lussac's laws for the vapour, its specific volume equals R/m times θ/p.", "kind": "law", "symbols": [ { "unit": null, "symbol": "s", "meaning": "change of volume per unit mass of the solvent passing into the vapour" }, { "unit": null, "symbol": "v", "meaning": "specific volume of the vapour" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the vapour" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure" } ], "sympy": "Eq(s, v, R/m*theta/p)", "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/saturated-vapour", "concept/temperature", "law/boyle-s-law", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c74fd04a22", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "198", "location": "System of any Number of Independent Constituents", "latex": "\\Delta = \\frac{R}{m} \\theta^{2} \\left(\\frac{\\dd \\log \\dfrac{p}{p_{0}}}{\\dd \\theta}\\right)_{c}", "name": "Kirchhoff's formula", "statement": "The heat of dilution equals R/m times θ² times the temperature derivative of log(p/p₀) at constant concentration.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\Delta", "meaning": "heat of dilution of the solution (heat given out on adding unit mass of solvent to a large quantity of the solution)" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the vapour" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "p_0", "meaning": "vapour pressure of the pure solvent at temperature θ" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution, held constant" } ], "sympy": "Eq(Delta, R/m*theta**2*Derivative(log(p/p_0), theta))", "physics": true, "states": [ "theorem/kirchhoff-s-formula" ], "concepts": [ "concept/partial-derivative", "concept/solution", "person/gustav-robert-kirchhoff", "quantity/heat-effect", "quantity/heat-of-dilution", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f64240e7c6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "198", "location": "System of any Number of Independent Constituents", "latex": "\\left(\\frac{\\dd p}{\\dd c}\\right)_{\\theta} = -\\frac{\\theta\\varphi}{s}", "name": null, "statement": "At constant temperature the vapour pressure of a solution falls as its concentration rises.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution (positive)" }, { "unit": null, "symbol": "s", "meaning": "change of volume per unit mass of the solvent passing into the vapour" } ], "sympy": "Eq(Derivative(p, c), -theta*phi/s)", "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/concentration", "concept/lowering-of-vapour-pressure", "concept/partial-derivative", "concept/temperature", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-aac5b85949", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "199", "location": "System of any Number of Independent Constituents", "latex": "\\frac{p - p_{0}}{p} = \\frac{cm\\varphi}{R}", "name": "Wüllner's law", "statement": "The relative decrease of the vapour pressure is proportional to the concentration of the solution.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "p_0", "meaning": "vapour pressure of pure solvent at the same temperature" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the solvent vapour" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": "Eq((p - p_0)/p, c*m*phi/R)", "physics": true, "states": [ "law/w-llner-s-law" ], "concepts": [ "concept/characteristic", "concept/concentration", "concept/lowering-of-vapour-pressure", "concept/proportion", "theorem/lowering-of-vapour-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2d6bda1396", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "200", "location": "System of any Number of Independent Constituents", "latex": "\\left(\\frac{\\dd \\theta}{\\dd c}\\right)_{p} = \\frac{\\theta^{2} \\varphi}{L}", "name": null, "statement": "At constant pressure the boiling point rises with the concentration of the solution.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature (boiling point)" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution (positive)" }, { "unit": null, "symbol": "L", "meaning": "heat of vaporization of the solution" } ], "sympy": "Eq(Derivative(theta, c), theta**2*phi/L)", "physics": true, "states": [], "concepts": [ "concept/boiling-point-elevation", "concept/characteristic", "concept/concentration", "concept/partial-derivative", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-96aea18a0c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "200", "location": "System of any Number of Independent Constituents", "latex": "\\theta - \\theta_{0} = \\frac{c\\theta^{2} \\varphi}{L}", "name": null, "statement": "For dilute solutions the elevation of the boiling point is proportional to the concentration.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "boiling point of the solution" }, { "unit": null, "symbol": "\\theta_0", "meaning": "boiling point of the pure solvent" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution" }, { "unit": null, "symbol": "L", "meaning": "heat of vaporization of the solution" } ], "sympy": "Eq(theta - theta_0, c*theta**2*phi/L)", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/boiling-point-elevation", "concept/characteristic", "concept/concentration" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e1820622d2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "201", "location": "System of any Number of Independent Constituents", "latex": "\\left(\\frac{\\dd \\theta'}{\\dd c}\\right)_{p} = -\\frac{\\theta^{2} \\varphi}{L'}", "name": null, "statement": "At constant pressure the freezing or saturation point falls as the concentration of the solution rises, when L' is the heat of solidification.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\theta'", "meaning": "freezing point or saturation point of the solution" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution" }, { "unit": null, "symbol": "L'", "meaning": "heat of solidification of the solution (heat of precipitation of the salt)" } ], "sympy": "Eq(Derivative(theta_prime, c), -theta**2*phi/Lp)", "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/concentration", "concept/freezing-point-depression", "concept/heat-of-precipitation", "concept/heat-of-solidification", "concept/partial-derivative", "concept/saturation-point" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-803d5eaa28", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "202", "location": "System of any Number of Independent Constituents", "latex": "\\theta_{0}' - \\theta' = \\frac{c\\theta^{2} \\varphi}{L'}", "name": null, "statement": "For dilute solutions the lowering of the freezing point is proportional to the concentration.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\theta_0'", "meaning": "freezing point of the pure solvent" }, { "unit": null, "symbol": "\\theta'", "meaning": "freezing point of the solution" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution" }, { "unit": null, "symbol": "L'", "meaning": "heat of solidification of the solution" } ], "sympy": "Eq(theta_0p - theta_prime, c*theta**2*phi/Lp)", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/characteristic", "concept/concentration", "concept/freezing-point-depression" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1fb2ee7621", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "204", "location": "System of any Number of Independent Constituents", "latex": "\\frac{L}{\\theta^{2}}\\, d\\theta - \\frac{s'}{\\theta}\\, dp' - \\frac{s''}{\\theta}\\, dp'' - \\varphi\\, dc = 0", "name": null, "statement": "For a solution separated from pure solvent by a semipermeable membrane, the displacement of equilibrium links dθ, dp', dp'' and dc.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat of removal of the solvent from the solution" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "s'", "meaning": "change of volume of the solution during removal of unit mass of solvent (negative)" }, { "unit": null, "symbol": "s''", "meaning": "change of volume of the pure solvent during removal of unit mass (positive)" }, { "unit": null, "symbol": "p'", "meaning": "pressure in the solution" }, { "unit": null, "symbol": "p''", "meaning": "pressure in the pure solvent" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/differential", "concept/heat-of-removal", "concept/mechanical-equilibrium", "concept/semipermeable-membrane" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-596f9caf74", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "204", "location": "System of any Number of Independent Constituents", "latex": "\\left(\\frac{\\dd P}{\\dd c}\\right)_{\\theta} = -\\frac{\\theta\\varphi}{s'}", "name": null, "statement": "At constant temperature the osmotic pressure increases with the concentration of the solution, since s' is negative.", "kind": "law", "symbols": [ { "unit": null, "symbol": "P", "meaning": "osmotic pressure, the difference p' − p'' of the pressures in the two phases" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution (positive)" }, { "unit": null, "symbol": "s'", "meaning": "change of volume of the solution during removal of unit mass of solvent (negative)" } ], "sympy": "Eq(Derivative(P, c), -theta*phi/s_p)", "physics": true, "states": [], "concepts": [ "concept/characteristic", "concept/concentration", "concept/partial-derivative", "concept/temperature", "quantity/osmotic-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-15b391c81d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "205", "location": "System of any Number of Independent Constituents", "latex": "P = \\frac{c\\theta\\varphi}{v}", "name": null, "statement": "For small concentrations the osmotic pressure equals c times θ times φ divided by the specific volume v of the solution.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "P", "meaning": "osmotic pressure of the solution" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" }, { "unit": null, "symbol": "\\theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "\\varphi", "meaning": "characteristic quantity of the solution" }, { "unit": null, "symbol": "v", "meaning": "specific volume of the solution" } ], "sympy": "Eq(P, c*theta*phi/v)", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/characteristic", "quantity/osmotic-pressure", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e80ab95979", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "175", "location": "System of any Number of Independent Constituents", "latex": "\\Psi = \\Phi - \\frac{U + pV}{\\theta}", "name": null, "statement": "The function Psi is defined as the entropy minus the sum of energy and pressure-volume term, divided by the absolute temperature.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Psi", "meaning": "the function Psi of the system, defined in this chapter" }, { "unit": null, "symbol": "Phi", "meaning": "entropy" }, { "unit": null, "symbol": "U", "meaning": "energy of the system" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": "Eq(Psi, Phi - (U + p*V)/theta)", "physics": true, "states": [], "concepts": [ "concept/energy", "concept/function", "concept/pressure", "concept/quantity-psi-function", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/volume" ], "pages": [ "175", "207" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "planck-treatise-on-thermodynamics-1903/ch-gaseous-system" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e4e744706e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "175", "location": "System of any Number of Independent Constituents", "latex": "\\delta \\Psi = 0", "name": null, "statement": "The condition of thermodynamic equilibrium at constant temperature and pressure is that Psi does not change under any variation compatible with the given conditions.", "kind": "law", "symbols": [ { "unit": null, "symbol": "delta", "meaning": "variation compatible with the given conditions (not a change of equilibrium state)" }, { "unit": null, "symbol": "Psi", "meaning": "the function Psi of the system" } ], "sympy": "Eq(delta*Psi, 0)", "physics": true, "states": [], "concepts": [ "concept/characteristic-function", "concept/equilibrium-condition", "concept/quantity-psi-function", "concept/small-variation", "concept/thermodynamic-equilibrium", "concept/variation-of-sign", "quantity/psi-function" ], "pages": [ "175", "215", "230" ], "chapters": [ "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5482c8baa1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "175", "location": "System of any Number of Independent Constituents", "latex": "\\Psi = \\Psi' + \\Psi'' + \\dots +\\Psi^{\\beta}", "name": null, "statement": "The function Psi of the whole system is the sum of the contributions of its beta phases.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "Psi", "meaning": "the function Psi of the whole system" }, { "unit": null, "symbol": "Psi'", "meaning": "the Psi function of the first phase" }, { "unit": null, "symbol": "beta", "meaning": "number of phases in the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/phase", "concept/quantity-psi-function", "concept/sum" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f1e6a049fd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "175", "location": "System of any Number of Independent Constituents", "latex": "\\Psi' = \\Phi' - \\frac{U' + pV'}{\\theta}", "name": null, "statement": "For a single phase, Psi' is defined as its entropy minus its energy plus pressure-volume term, divided by the temperature.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Psi'", "meaning": "the Psi function of the first phase" }, { "unit": null, "symbol": "Phi'", "meaning": "entropy of the first phase" }, { "unit": null, "symbol": "U'", "meaning": "energy of the first phase" }, { "unit": null, "symbol": "V'", "meaning": "volume of the first phase" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/phase", "concept/pressure", "concept/temperature", "quantity/entropy", "quantity/internal-energy", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-89590a0259", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "176", "location": "System of any Number of Independent Constituents", "latex": "\\Psi' = \\frac{\\dd \\Psi'}{\\dd M_{1}'}\\, M_{1}' + \\frac{\\dd \\Psi'}{\\dd M_{2}'}\\, M_{2}' + \\dots + \\frac{\\dd \\Psi'}{\\dd M_{\\alpha}'}\\, M_{\\alpha}'.", "name": "Eulerian equation", "statement": "Because Psi' is homogeneous of the first degree in the masses, it equals the sum of each mass times its partial derivative with respect to that mass.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "Psi'", "meaning": "the Psi function of the first phase" }, { "unit": null, "symbol": "M_1', M_2', ..., M_alpha'", "meaning": "masses of the independent constituents in the first phase" }, { "unit": null, "symbol": "alpha", "meaning": "number of independent constituents" } ], "sympy": null, "physics": false, "states": [ "theorem/euler-s-theorem-for-homogeneous-functions" ], "concepts": [ "concept/function", "concept/homogeneous-function", "concept/independent-constituent", "concept/theorem-euler-s-theorem-for-homogeneous-functions", "method/differentiation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3d7f12c290", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "177", "location": "System of any Number of Independent Constituents", "latex": "\\frac{\\dd \\Psi'}{\\dd M_{1}'} = \\frac{\\dd \\Psi''}{\\dd M_{1}''} = \\dots = \\frac{\\dd \\Psi^{\\beta}}{\\dd M_{1}^{\\beta}}", "name": null, "statement": "In equilibrium, the partial derivative of Psi with respect to the mass of the first constituent takes the same value in every phase.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Psi'", "meaning": "the Psi function of the first phase" }, { "unit": null, "symbol": "M_1'", "meaning": "mass of the first constituent in the first phase" }, { "unit": null, "symbol": "beta", "meaning": "number of phases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/independent-constituent", "concept/phase", "concept/thermodynamic-equilibrium", "method/differentiation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-607065685d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "177", "location": "System of any Number of Independent Constituents", "latex": "M_{\\alpha} = M_{\\alpha}' + M_{\\alpha}'' + \\dots + M_{\\alpha}^{\\beta}", "name": null, "statement": "The total mass of each independent constituent is the sum of its masses in all the phases.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "M_alpha", "meaning": "total mass of the alpha-th independent constituent" }, { "unit": null, "symbol": "M_alpha^beta", "meaning": "mass of that constituent in the beta-th phase" }, { "unit": null, "symbol": "alpha", "meaning": "number of independent constituents" }, { "unit": null, "symbol": "beta", "meaning": "number of phases" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/independent-constituent", "concept/phase", "concept/sum" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9fa770a29d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "178", "location": "System of any Number of Independent Constituents", "latex": "\\alpha\\beta + 2", "name": null, "statement": "The state of the beta phases depends on alpha times beta masses plus temperature and pressure, which are alpha beta + 2 variables.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "number of independent constituents" }, { "unit": null, "symbol": "beta", "meaning": "number of phases" } ], "sympy": "alpha*beta + 2", "physics": false, "states": [], "concepts": [ "concept/phase-rule", "concept/real-number", "concept/variable" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5f3e4dda36", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "179", "location": "System of any Number of Independent Constituents", "latex": "\\bigl[(\\alpha - 1)\\beta + 2\\bigr] - \\bigl[\\alpha (\\beta - 1)\\bigr] = \\alpha - \\beta + 2", "name": null, "statement": "After the internal conditions are satisfied, alpha minus beta plus two internal variables remain undetermined.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "number of independent constituents" }, { "unit": null, "symbol": "beta", "meaning": "number of phases" } ], "sympy": "Eq((alpha - 1)*beta + 2 - alpha*(beta - 1), alpha - beta + 2)", "physics": true, "states": [], "concepts": [ "concept/internal-conditions-of-equilibrium", "concept/phase-rule", "concept/variable" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-60973405e7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "179", "location": "System of any Number of Independent Constituents", "latex": "\\beta \\leq \\alpha + 2", "name": "phase rule", "statement": "The number of phases cannot exceed the number of independent constituents by more than two.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "beta", "meaning": "number of phases" }, { "unit": null, "symbol": "alpha", "meaning": "number of independent constituents" } ], "sympy": "beta <= alpha + 2", "physics": true, "states": [ "concept/phase-rule" ], "concepts": [ "concept/independent-constituent", "concept/phase" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7f591321af", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "184", "location": "System of any Number of Independent Constituents", "latex": "\\delta\\, d\\Psi = 0", "name": null, "statement": "The condition of equilibrium for a neighbouring equilibrium state requires the variation of the infinitesimal change dPsi to vanish.", "kind": "law", "symbols": [ { "unit": null, "symbol": "delta", "meaning": "variation of masses consistent with the given external conditions" }, { "unit": null, "symbol": "d", "meaning": "change from one equilibrium state to a slightly different one" }, { "unit": null, "symbol": "Psi", "meaning": "the function Psi of the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/differential", "concept/thermodynamic-equilibrium", "concept/variation-of-sign" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-58c67cad38", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "184", "location": "System of any Number of Independent Constituents", "latex": "\\frac{\\dd \\Psi}{\\dd \\theta} = \\frac{U + pV}{\\theta^{2}}", "name": null, "statement": "The partial derivative of Psi with respect to temperature equals energy plus pressure-volume term divided by the square of the temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Psi", "meaning": "the function Psi of the system" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "U", "meaning": "energy of the system" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/temperature", "method/differentiation", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b4360c8e14", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "184", "location": "System of any Number of Independent Constituents", "latex": "\\frac{\\dd \\Psi}{\\dd p} = -\\frac{V}{\\theta}", "name": null, "statement": "The partial derivative of Psi with respect to pressure equals minus the volume divided by the temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Psi", "meaning": "the function Psi of the system" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/temperature", "method/differentiation", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4a988168aa", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "183", "location": "System of any Number of Independent Constituents", "latex": "d\\Phi' = \\frac{dU' + p\\, dV'}{\\theta}", "name": null, "statement": "For an infinitely small change of a phase without change of its masses, the change in entropy is the heat term divided by temperature.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Phi'", "meaning": "entropy of the first phase" }, { "unit": null, "symbol": "U'", "meaning": "energy of the first phase" }, { "unit": null, "symbol": "V'", "meaning": "volume of the first phase" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/differential", "concept/heat", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-56c9e829d2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "176", "location": "System of any Number of Independent Constituents", "latex": "\\Delta \\Psi = \\eps\\Psi'", "name": null, "statement": "When all masses are increased in the ratio 1 + epsilon to 1, Psi increases by epsilon times Psi'.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Delta Psi", "meaning": "change of Psi under proportional increase of all masses" }, { "unit": null, "symbol": "eps", "meaning": "very small number by which the masses are increased" }, { "unit": null, "symbol": "Psi'", "meaning": "the Psi function of the first phase" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/homogeneous-function", "concept/small-variation" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b5a63bc1dc", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "187", "location": "System of any Number of Independent Constituents", "latex": "\\frac{dp}{d\\theta} = \\frac{Q}{\\theta\\, \\delta V}", "name": null, "statement": "The rate of change of the equilibrium pressure with temperature equals the heat absorbed divided by the temperature and by the change of volume.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "equilibrium pressure" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "Q", "meaning": "heat absorbed by the system during the virtual change" }, { "unit": null, "symbol": "delta V", "meaning": "change of volume of the system in the virtual change" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/mechanical-equilibrium", "concept/pressure", "concept/rate-of-change", "concept/temperature", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-dbf91c6b02", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "187", "location": "System of any Number of Independent Constituents", "latex": "L = \\theta\\, \\frac{dp}{d\\theta} (v'' - v')", "name": null, "statement": "The heat of vaporization per unit mass equals temperature times the slope of the vapour pressure curve times the difference of the specific volumes of vapour and liquid.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat of vaporization per unit mass" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure" }, { "unit": null, "symbol": "v'", "meaning": "specific volume of the liquid" }, { "unit": null, "symbol": "v''", "meaning": "specific volume of the vapour" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/pressure", "concept/temperature", "quantity/latent-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5bb5d6a375", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "188", "location": "System of any Number of Independent Constituents", "latex": "Q = \\theta · \\frac{dp}{d\\theta} · \\delta V", "name": null, "statement": "For a salt solution in three phases at constant temperature, pressure and concentration, the heat absorbed equals temperature times the pressure slope times the volume change.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed by the system" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure" }, { "unit": null, "symbol": "delta V", "meaning": "change of volume of the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/pressure", "concept/temperature", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7461956fd9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "189", "location": "System of any Number of Independent Constituents", "latex": "\\delta V = \\bigl[(v'' + cv''') - (1 + c) v'\\bigr]\\, \\delta M_{1}''", "name": null, "statement": "The volume change on evaporating water from a salt solution while precipitating salt equals the specific-volume combination times the mass of vapour formed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "delta V", "meaning": "increase of the total volume of the system" }, { "unit": null, "symbol": "v'", "meaning": "specific volume of the solution" }, { "unit": null, "symbol": "v''", "meaning": "specific volume of the water vapour" }, { "unit": null, "symbol": "v'''", "meaning": "specific volume of the solid salt" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution, mass of salt per mass of water" }, { "unit": "mass", "symbol": "delta M_1''", "meaning": "mass of water vapour formed" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/variation-of-sign", "quantity/specific-volume", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c80a64fa97", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "189", "location": "System of any Number of Independent Constituents", "latex": "L = \\theta\\, \\frac{dp}{d\\theta} \\bigl(v'' + cv''' - (1 + c)v'\\bigr)", "name": null, "statement": "The heat needed to evaporate water from a solution and precipitate salt equals temperature times the pressure slope times the specific-volume combination.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat required to evaporate unit mass of water from the solution and precipitate the corresponding salt" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure" }, { "unit": null, "symbol": "v'", "meaning": "specific volume of the solution" }, { "unit": null, "symbol": "v''", "meaning": "specific volume of the vapour" }, { "unit": null, "symbol": "v'''", "meaning": "specific volume of the solid salt" }, { "unit": null, "symbol": "c", "meaning": "concentration of the solution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "quantity/heat-effect", "quantity/latent-heat", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-012b14d2bf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "189", "location": "System of any Number of Independent Constituents", "latex": "v'' = \\frac{R}{m} · \\frac{\\theta}{p}", "name": "perfect gas equation for specific volume", "statement": "The specific volume of the vapour, treated as a perfect gas, equals the gas constant divided by the molecular weight, times temperature over pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "v''", "meaning": "specific volume of the vapour" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the vapour" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure of the vapour" } ], "sympy": "Eq(v2, (R/m)*(theta/p))", "physics": true, "states": [ "law/perfect-gas-equation-for-specific-volume" ], "concepts": [ "concept/perfect-gas", "concept/pressure", "concept/temperature", "quantity/absolute-gas-constant", "quantity/molecular-weight", "quantity/specific-volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9f2feaa4bb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-system-of-any-number-of-independent-constituents", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "189", "location": "System of any Number of Independent Constituents", "latex": "L = \\frac{R}{m} \\theta^{2} · \\frac{d \\log p}{d\\theta}", "name": null, "statement": "With the vapour treated as a perfect gas and the liquid volume neglected, the heat of vaporization equals gas constant over molecular weight times the square of temperature times the derivative of log pressure with respect to temperature.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat of vaporization of the solution per unit mass of water" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "m", "meaning": "molecular weight of the vapour" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/logarithm", "concept/perfect-gas", "quantity/absolute-gas-constant", "quantity/latent-heat", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b1eb743090", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "208", "location": "Gaseous System", "latex": "V = \\frac{R\\theta}{p} (n_{1} + n_{2} + \\dots) = \\frac{R\\theta}{p} \\tsum n_{1}", "name": null, "statement": "The volume of a mixture of perfect gases equals R times temperature over pressure times the total number of molecules.", "kind": "law", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the mixture" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of kind 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/gas-mixture", "concept/perfect-gas", "concept/pressure", "concept/temperature", "law/boyle-s-law", "law/law-of-combining-volumes", "quantity/number-of-molecules", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-ec089e71f7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "208", "location": "Gaseous System", "latex": "U_{1} = \\tsum n_{1} (c_{v_{1}}\\theta + h_{1})", "name": null, "statement": "The energy of a perfect gas depends only on temperature and equals the number of molecules times (molecular heat at constant volume times temperature plus a constant).", "kind": "law", "symbols": [ { "unit": null, "symbol": "U_1", "meaning": "energy of the n_1 molecules of the gas" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of the gas" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of the gas at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "h_1", "meaning": "constant for the gas" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "concept/temperature", "quantity/atomic-heat", "quantity/internal-energy", "quantity/specific-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-694651c1cd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "208", "location": "Gaseous System", "latex": "U = \\tsum n_{1} (c_{v_{1}}\\theta + h_{1})", "name": null, "statement": "The total energy of a mixture of perfect gases is the sum of the energies of its constituents.", "kind": "law", "symbols": [ { "unit": null, "symbol": "U", "meaning": "total energy of the mixture" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of kind 1" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "h_1", "meaning": "constant for gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/energy", "concept/gas-mixture", "concept/temperature", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f5250d52e3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "209", "location": "Gaseous System", "latex": "\\Phi = \\tsum n_{1} \\left(c_{v_{1}} \\log \\theta + R \\log \\frac{\\theta}{p}\\right) + C", "name": null, "statement": "The entropy of a perfect gas mixture is a sum over constituents plus a constant of integration that depends only on composition.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "entropy of the mixture" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of kind 1" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "C", "meaning": "constant of integration, depending on composition" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/constant-of-integration", "concept/gas-mixture", "concept/perfect-gas", "concept/pressure", "concept/temperature", "quantity/atomic-heat", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6af7933b49", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "212", "location": "Gaseous System", "latex": "n (c_{v} \\log \\theta + R \\log \\frac{\\theta}{p} + k)", "name": null, "statement": "The entropy of a perfect gas with n molecules is n times (c_v log temperature + R log(temperature/pressure) + k).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of molecules, M/m" }, { "unit": null, "symbol": "c_v", "meaning": "molecular heat at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "k", "meaning": "constant including log(R/m)" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/perfect-gas", "quantity/atomic-heat", "quantity/entropy", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9ec19c7703", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "213", "location": "Gaseous System", "latex": "n = \\dfrac{M}{m}", "name": null, "statement": "The number of molecules equals the mass divided by the molecular weight.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of molecules" }, { "unit": null, "symbol": "M", "meaning": "mass" }, { "unit": null, "symbol": "m", "meaning": "molecular weight" } ], "sympy": "Eq(n, M/m)", "physics": true, "states": [], "concepts": [ "quantity/mass", "quantity/molecular-weight", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f9704b4c3c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "213", "location": "Gaseous System", "latex": "\\Phi = \\tsum n_{1} (c_{v_{1}} \\log \\theta + R \\log \\frac{\\theta}{p_{1}} + k_{1})", "name": "Gibbs's proposition", "statement": "The entropy of a gas mixture is the sum of the entropies each gas would have alone at the same temperature in the total volume, using each gas's partial pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "entropy of the mixture" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of gas 1" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "p_1", "meaning": "partial pressure of gas 1" }, { "unit": null, "symbol": "k_1", "meaning": "constant of gas 1" } ], "sympy": null, "physics": true, "states": [ "law/gibbs-s-proposition" ], "concepts": [ "concept/gas-mixture", "concept/perfect-gas", "person/gibbs", "quantity/entropy", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-659a2ae639", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "213", "location": "Gaseous System", "latex": "\\tsum p_{1} = p", "name": "Dalton's law", "statement": "The pressure of a gas mixture is the sum of the partial pressures of its constituents.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p_1", "meaning": "partial pressure of gas 1" }, { "unit": null, "symbol": "p", "meaning": "total pressure of the mixture" } ], "sympy": null, "physics": true, "states": [ "law/dalton-s-law" ], "concepts": [ "concept/dalton", "concept/gas-mixture", "concept/pressure", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4b9c9201a5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "213", "location": "Gaseous System", "latex": "c_{1} = \\frac{n_{1}}{n_{1} + n_{2} + \\dots}", "name": null, "statement": "The concentration of a gas in a mixture is the number of its molecules divided by the total number of molecules.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1 in the mixture" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of gas 1" }, { "unit": null, "symbol": "n_2", "meaning": "number of molecules of gas 2" } ], "sympy": "Eq(c1, n1/(n1 + n2))", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/gas-mixture", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-165068cebd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "213", "location": "Gaseous System", "latex": "p_{1} = c_{1} p", "name": null, "statement": "The partial pressure of a gas equals its concentration times the total pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p_1", "meaning": "partial pressure of gas 1" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "p", "meaning": "total pressure" } ], "sympy": "Eq(p1, c1*p)", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/pressure", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-42966d88eb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "213", "location": "Gaseous System", "latex": "\\Phi = \\tsum n_{1} (c_{v_{1}} \\log \\theta + R \\log \\frac{\\theta}{pc_{1}} + k_{1})", "name": null, "statement": "The entropy of a gas mixture as a function of temperature, pressure and numbers of molecules, with concentrations included.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "entropy of the mixture" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of gas 1" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "p", "meaning": "total pressure" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "k_1", "meaning": "constant of gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/gas-mixture", "concept/perfect-gas", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-838ae84f1d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "214", "location": "Gaseous System", "latex": "C = \\tsum n_{1} (k_{1} - R \\log c_{1})", "name": null, "statement": "The constant of integration of the mixture entropy equals the sum over constituents of n_1 times (k_1 minus R log c_1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "constant of integration of the mixture entropy" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of gas 1" }, { "unit": null, "symbol": "k_1", "meaning": "constant of gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/constant-of-integration", "concept/gas-mixture", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3d9f7e917f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "214", "location": "Gaseous System", "latex": "-n_{1} R \\log c_{1} - n_{2} R \\log c_{2}", "name": null, "statement": "The entropy change on diffusion of two gases at constant temperature and pressure is minus R times n log c summed over the gases, which is positive.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n_1", "meaning": "number of molecules of gas 1" }, { "unit": null, "symbol": "n_2", "meaning": "number of molecules of gas 2" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "c_2", "meaning": "concentration of gas 2" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/diffusion", "concept/increase-of-entropy", "concept/irreversible-process", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b87721437a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "215", "location": "Gaseous System", "latex": "\\Psi = \\tsum n_{1} (\\varphi_{1} - R \\log c_{1})", "name": null, "statement": "The characteristic function of a gas mixture is the sum over constituents of n_1 times (phi_1 minus R log c_1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "Psi", "meaning": "characteristic function Psi" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of gas 1" }, { "unit": null, "symbol": "varphi_1", "meaning": "function of temperature and pressure only" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic-function", "concept/concentration", "concept/gas-mixture", "quantity/psi-function" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-60cb399748", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "215", "location": "Gaseous System", "latex": "c_{v_{1}} \\log \\theta - \\frac{h_{1}}{\\theta} + R \\log \\frac{\\theta}{p} + k_{1} - c_{v_{1}} - R = \\varphi_{1}", "name": null, "statement": "Defines phi_1 as a function of temperature and pressure only, independent of the number of molecules.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "varphi_1", "meaning": "function of temperature and pressure for gas 1" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "h_1", "meaning": "constant of gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "k_1", "meaning": "constant of gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/function", "concept/pressure", "concept/temperature", "quantity/atomic-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8c9fa502a4", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "215", "location": "Gaseous System", "latex": "\\tsum (\\varphi_{1} - R \\log c_{1})\\, \\delta n_{1} + \\tsum n_{1}\\, \\delta(\\varphi_{1} - R \\log c_{1}) = 0", "name": null, "statement": "The variation of Psi under a chemical change vanishes, expanded into terms in phi, concentrations and molecule numbers.", "kind": "law", "symbols": [ { "unit": null, "symbol": "delta", "meaning": "variation" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of gas 1" }, { "unit": null, "symbol": "varphi_1", "meaning": "function of temperature and pressure for gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/mechanical-equilibrium", "concept/variation-of-sign" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6ce771b120", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "216", "location": "Gaseous System", "latex": "\\delta n_{1} : \\delta n_{2} : \\dots = \\nu_{1} : \\nu_{2} : \\dots", "name": null, "statement": "The simultaneous changes in molecule numbers of a reaction are in the ratio of the integers nu.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "delta n_1", "meaning": "change in number of molecules of kind 1" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer, positive or negative, for kind 1 in the reaction" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/variation-of-sign", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d434911f3d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "216", "location": "Gaseous System", "latex": "\\tsum (\\varphi_{1} - R \\log c_{1}) \\nu_{1} = 0", "name": null, "statement": "The equilibrium condition of a chemical reaction in the gas mixture.", "kind": "law", "symbols": [ { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1 in the reaction" }, { "unit": null, "symbol": "varphi_1", "meaning": "function of temperature and pressure for gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-89dea9bc90", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "216", "location": "Gaseous System", "latex": "\\nu_{1} \\log c_{1} + \\nu_{2} \\log c_{2} + \\dots = \\frac{\\nu_{1} \\varphi_{1} + \\nu_{2} \\varphi_{2} + \\dots}{R}", "name": null, "statement": "At equilibrium the weighted sum of log concentrations equals the weighted sum of phi functions divided by R.", "kind": "law", "symbols": [ { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1 in the reaction" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "varphi_1", "meaning": "function of temperature and pressure for gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/logarithm", "concept/mechanical-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-cff68b7335", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "215", "location": "Gaseous System", "latex": "c_{1} + c_{2} + \\dots = 1", "name": null, "statement": "The concentrations of all kinds of molecules in a mixture sum to one.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "c_2", "meaning": "concentration of gas 2" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/gas-mixture" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7077c547d3", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "216", "location": "Gaseous System", "latex": "\\frac{\\tsum \\nu_{1} (k_{1} - c_{v_{1}} - R)}{R} = \\log a", "name": null, "statement": "Defines the constant a as the exponential of the sum of nu times (k_1 minus c_v1 minus R), divided by R.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant of the equilibrium condition" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1 in the reaction" }, { "unit": null, "symbol": "k_1", "meaning": "constant of gas 1" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/constant", "concept/logarithm" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e869c35663", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "216", "location": "Gaseous System", "latex": "\\frac{\\tsum \\nu_{1} h_{1}}{R} = b", "name": null, "statement": "Defines the constant b from the constants h_1 of the reacting gases.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "b", "meaning": "constant tied to the heat effect of the reaction" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1 in the reaction" }, { "unit": null, "symbol": "h_1", "meaning": "constant of gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/constant", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-110d8ba3fd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "216", "location": "Gaseous System", "latex": "\\frac{\\tsum \\nu_{1} c_{v_{1}}}{R} = c", "name": null, "statement": "Defines the constant c from the molecular heats of the reacting gases.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c", "meaning": "constant from molecular heats of the reaction" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1 in the reaction" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/constant", "quantity/atomic-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8d705882b5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "217", "location": "Gaseous System", "latex": "\\nu_{1} \\log c_{1} + \\nu_{2} \\log c_{2} + \\dots = \\log a + (\\nu_{1} + \\nu_{2} + \\dots) \\log \\frac{\\theta}{p} - \\frac{b}{\\theta} + c \\log \\theta", "name": null, "statement": "The equilibrium condition written in logarithms of concentrations, temperature and pressure.", "kind": "law", "symbols": [ { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "a", "meaning": "equilibrium constant" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect" }, { "unit": null, "symbol": "c", "meaning": "constant from molecular heats" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/logarithm", "concept/mechanical-equilibrium", "concept/pressure", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a9c58ed43f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "217", "location": "Gaseous System", "latex": "\\prod c_{1}^{\\nu_{1}} = a\\left(\\frac{\\theta}{p}\\right)^{\\tsum \\nu_{1}} e^{-\\efrac{b}{\\theta}} \\theta^{c}", "name": null, "statement": "The equilibrium condition as a product of concentrations powered by nu equals a constant times temperature over pressure to the total nu, times an exponential in b over temperature, times temperature to c.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1" }, { "unit": null, "symbol": "a", "meaning": "equilibrium constant" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect" }, { "unit": null, "symbol": "c", "meaning": "constant from molecular heats" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/mechanical-equilibrium", "concept/pressure", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b10a5af58f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "217", "location": "Gaseous System", "latex": "\\prod c_{1}^{\\nu_{1}} = a e^{-\\efrac{b}{\\theta}} \\left(\\frac{\\theta}{p}\\right)^{\\tsum \\nu_{1}}", "name": null, "statement": "With c = 0 (constant atomic heats in reactions) the equilibrium condition of a gaseous reaction reduces to this form.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of gas 1" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1" }, { "unit": null, "symbol": "a", "meaning": "equilibrium constant" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/mechanical-equilibrium", "concept/pressure", "concept/temperature", "quantity/atomic-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5430b27408", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "217", "location": "Gaseous System", "latex": "\\prod c_{1}^{\\nu_{1}} = ae^{-\\efrac{b}{\\theta}} \\left(\\frac{\\theta}{p}\\right)^{\\tsum \\nu_{1}}", "name": null, "statement": "General equilibrium condition for any gaseous chemical change: the product of concentrations raised to the nu's equals a times exponential of minus b over temperature times (temperature over pressure) to the total nu.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of kind 1" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1 in the reaction" }, { "unit": null, "symbol": "a", "meaning": "equilibrium constant" }, { "unit": null, "symbol": "b", "meaning": "constant related to the heat effect" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/law-of-mass-action", "concept/mechanical-equilibrium", "concept/pressure", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-de0938ee8b", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "217", "location": "Gaseous System", "latex": "Q = \\delta U + p\\, \\delta V", "name": null, "statement": "The heat received equals the change of energy plus pressure times the change of volume (first law).", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed" }, { "unit": null, "symbol": "U", "meaning": "energy" }, { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "V", "meaning": "volume" }, { "unit": null, "symbol": "delta", "meaning": "variation" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/energy", "concept/heat", "concept/pressure", "concept/work", "law/first-law-of-thermodynamics", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0bfd1e9695", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "217", "location": "Gaseous System", "latex": "Q = \\tsum (c_{v_{1}} \\theta + h_{1} + R\\theta)\\, \\delta n_{1}", "name": null, "statement": "The heat absorbed at constant temperature and pressure in an infinitesimal reaction.", "kind": "law", "symbols": [ { "unit": null, "symbol": "Q", "meaning": "heat absorbed" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "h_1", "meaning": "constant of gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "delta n_1", "meaning": "change in number of molecules of kind 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/heat", "law/first-law-of-thermodynamics", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7619cd1f67", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "218", "location": "Gaseous System", "latex": "L = \\tsum (c_{v_{1}} \\theta + h_{1} + R\\theta) \\nu_{1}", "name": null, "statement": "The heat absorbed in a finite reaction at constant temperature and pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat absorbed in the reaction" }, { "unit": null, "symbol": "c_{v_1}", "meaning": "molecular heat of gas 1 at constant volume" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "h_1", "meaning": "constant of gas 1" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/heat", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-803c6efb62", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "218", "location": "Gaseous System", "latex": "L = Rb + R\\theta \\tsum \\nu_{1}", "name": null, "statement": "The heat absorbed in a reaction equals R b plus R times temperature times the total nu.", "kind": "result", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat absorbed in the reaction" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1" } ], "sympy": "Eq(L, R*b + R*theta*Sum(nu1, ...))", "physics": true, "states": [], "concepts": [ "concept/heat", "concept/work", "quantity/heat-effect", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-92428de629", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "218", "location": "Gaseous System", "latex": "L = 1.97 (b + \\theta \\tsum \\nu_{1})", "name": null, "statement": "The heat absorbed in a reaction in calories, with R expressed as 1.97 cal per degree.", "kind": "result", "symbols": [ { "unit": "cal", "symbol": "L", "meaning": "heat absorbed in the reaction" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect" }, { "unit": null, "symbol": "theta", "meaning": "temperature" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3b05497970", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "219", "location": "Gaseous System", "latex": "L = 1.97 \\{b + (\\nu_{1} + \\nu_{2} + \\dots) \\theta\\}", "name": null, "statement": "The heat absorbed at constant temperature and pressure written with the sum of nu's explicit.", "kind": "result", "symbols": [ { "unit": "cal", "symbol": "L", "meaning": "heat absorbed in the reaction" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect" }, { "unit": null, "symbol": "nu_1", "meaning": "simple integer for kind 1" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2b7b2536cd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "219", "location": "Gaseous System", "latex": "c_{1}^{-2} c_{2}^{1} c_{3}^{1} = ae^{-\\efrac{b}{\\theta}}", "name": null, "statement": "Equilibrium condition for the dissociation of hydriodic acid (2 HI giving H2 and I2).", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of HI" }, { "unit": null, "symbol": "c_2", "meaning": "concentration of H2" }, { "unit": null, "symbol": "c_3", "meaning": "concentration of I2" }, { "unit": null, "symbol": "a", "meaning": "equilibrium constant" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/dissociation", "concept/hydriodic-acid", "concept/mechanical-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fc8f50bb1a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "220", "location": "Gaseous System", "latex": "a = 0.120", "name": null, "statement": "The constant a for hydriodic acid dissociation, from Bodenstein's measurements, has the value 0.120.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "equilibrium constant for hydriodic acid dissociation" } ], "sympy": "Eq(a, 0.120)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/constant", "concept/dissociation", "concept/hydriodic-acid" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-dfec285d1e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "220", "location": "Gaseous System", "latex": "b = 1300", "name": null, "statement": "The constant b for hydriodic acid dissociation has the value 1300.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "constant related to heat effect, for hydriodic acid dissociation" } ], "sympy": "Eq(b, 1300)", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/dissociation", "concept/hydriodic-acid", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0fb43ebc55", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "221", "location": "Gaseous System", "latex": "L = 1.97 (14690 + \\theta) = 28900 + 1.97\\theta", "name": null, "statement": "The heat of dissociation of a molecule of iodine as a function of temperature, in calories.", "kind": "result", "symbols": [ { "unit": "cal", "symbol": "L", "meaning": "heat of dissociation of a molecule of iodine" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dissociation-of-iodine-vapour", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-90b3d8b451", "chapter": "planck-treatise-on-thermodynamics-1903/ch-gaseous-system", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "219", "location": "Gaseous System", "latex": "\\frac{c_{2}c_{3}}{c_{1}^{2}} = \\frac{n_{2}n_{3}}{n_{1}^{2}} = ae^{-\\efrac{b}{\\theta}}", "name": null, "statement": "Equilibrium of the hydriodic acid reaction within the graded dissociation system, with iodine dissociation included.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of HI" }, { "unit": null, "symbol": "c_2", "meaning": "concentration of H2" }, { "unit": null, "symbol": "c_3", "meaning": "concentration of I2" }, { "unit": null, "symbol": "a", "meaning": "equilibrium constant for hydriodic acid" }, { "unit": null, "symbol": "b", "meaning": "constant related to heat effect for hydriodic acid" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/dissociation", "concept/graded-dissociation", "concept/hydriodic-acid", "concept/mechanical-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-88bce78dd2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "225", "location": "Dilute Solutions", "latex": "U &= n_{0} u_{0} + n_{1} u_{1} + n_{2} u_{2} + \\dots\\Add{,}", "name": null, "statement": "The internal energy of a dilute solution is the sum over molecule kinds of each kind's number times its per-molecule energy, which is linear in the numbers.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy of the solution" }, { "unit": null, "symbol": "n_0, n_1, n_2", "meaning": "numbers of molecules of the solvent and of each dissolved substance" }, { "unit": null, "symbol": "u_0, u_1, u_2", "meaning": "energy per molecule of each kind, depending only on temperature, pressure and the nature of the molecules" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/dissolved-substance", "concept/linear-function", "concept/solvent", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-aa1712a81e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "225", "location": "Dilute Solutions", "latex": "V &= n_{0} v_{0} + n_{1} v_{1} + n_{2} v_{2} + \\dots\\Add{.}", "name": null, "statement": "The volume of a dilute solution is the sum over molecule kinds of each kind's number times its per-molecule volume.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the solution" }, { "unit": null, "symbol": "n_0, n_1, n_2", "meaning": "numbers of molecules of the solvent and of each dissolved substance" }, { "unit": null, "symbol": "v_0, v_1, v_2", "meaning": "volume per molecule of each kind" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/dissolved-substance", "concept/linear-function", "concept/solvent", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-704f940bd9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "224", "location": "Dilute Solutions", "latex": "\\frac{U}{n_{0}} = u_{0} + u_{1}\\, \\frac{n_{1}}{n_{0}} + u_{2}\\, \\frac{n_{2}}{n_{0}} + \\dots\\Add{,}", "name": null, "statement": "The energy per solvent molecule is a linear function of the ratios of dissolved to solvent molecule numbers.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "U", "meaning": "internal energy of the solution" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" }, { "unit": null, "symbol": "u_0, u_1, u_2", "meaning": "constants depending on temperature, pressure and nature of the molecules" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/common-ratio", "concept/dilute-solution", "concept/dissolved-substance", "concept/linear-function", "concept/solvent", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3eb8a73715", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "225", "location": "Dilute Solutions", "latex": "\\frac{U}{n_{0}} = u_{0} + u_{1}\\, \\frac{n_{1}}{n_{0}} + \\dots + u_{11} \\left(\\frac{n_{1}}{n_{0}}\\right)^{2} + 2u_{12}\\, \\frac{n_{1}}{n_{0}} · \\frac{n_{2}}{n_{0}} + u_{22} \\left(\\frac{n_{2}}{n_{0}}\\right)^{2} + \\dots\\Add{.}", "name": null, "statement": "A more accurate expansion of energy per solvent molecule including quadratic terms for the dissolved substances' mutual interactions.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "u_11, u_12, u_22", "meaning": "coefficients for interactions among dissolved molecules" }, { "unit": null, "symbol": "n_0, n_1, n_2", "meaning": "molecule numbers of solvent and dissolved substances" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/coefficient", "concept/dilute-solution", "concept/function-of-several-variables", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-07d0833d6a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "225", "location": "Dilute Solutions", "latex": "V' = (n_{0} + 1) v_{0} + n_{1} v_{1} + n_{2} v_{2} + \\dots", "name": null, "statement": "After adding one solvent molecule, the solution volume becomes the old sum with n_0 increased by one.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V'", "meaning": "volume after dilution by one solvent molecule" }, { "unit": null, "symbol": "v_0", "meaning": "volume of one solvent molecule" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/small-variation", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-aa055d1009", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "225", "location": "Dilute Solutions", "latex": "U' = (n_{0} + 1) u_{0} + n_{1} u_{1} + n_{2} u_{2} + \\dots\\Add{.}", "name": null, "statement": "After adding one solvent molecule, the energy becomes the old sum with n_0 increased by one.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "U'", "meaning": "energy after dilution by one solvent molecule" }, { "unit": null, "symbol": "u_0", "meaning": "energy of one solvent molecule" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/small-variation", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-66b766e18d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "225", "location": "Dilute Solutions", "latex": "U' - (U + u_{0}) + p \\bigl\\{V' - (V + v_{0})\\bigr\\}", "name": null, "statement": "The heat absorbed on adding one solvent molecule at constant temperature and pressure, by the first law; it vanishes for a dilute solution.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure" }, { "unit": null, "symbol": "U, V", "meaning": "energy and volume before dilution" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/heat", "concept/pressure", "law/first-law-of-thermodynamics" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-555d0a7cf1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "226", "location": "Dilute Solutions", "latex": "d\\phi_{0} = \\frac{du_{0} + p\\, dv_{0}}{\\theta}", "name": null, "statement": "The function phi_0 of temperature and pressure has differential equal to the solvent's energy plus pressure-volume differential over temperature.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi_0", "meaning": "function of temperature and pressure for the solvent" }, { "unit": null, "symbol": "u_0, v_0", "meaning": "energy and volume per solvent molecule" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/exact-differential", "concept/pressure", "concept/temperature", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8515481e69", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "227", "location": "Dilute Solutions", "latex": "\\Phi = n_{0} \\phi_{0} + n_{1} \\phi_{1} + n_{2} \\phi_{2} + \\dots + C,", "name": null, "statement": "The entropy of a dilute solution is the sum of molecule numbers times their phi functions plus an integration constant depending only on the molecule numbers.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "integration constant, a function of the molecule numbers only" }, { "unit": null, "symbol": "phi_0, phi_1, phi_2", "meaning": "functions of temperature and pressure for each molecule kind" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/constant", "concept/dilute-solution", "method/integration", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bec2ec2bc9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "228", "location": "Dilute Solutions", "latex": "C = n_{0} (k_{0} - R \\log c_{0}) + n_{1} (k_{1} - R \\log c_{1}) + \\dots\\Add{.}", "name": null, "statement": "The integration constant C is fixed by matching the solution to the ideal-gas mixture, giving a concentration-dependent expression.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k_0, k_1, k_2", "meaning": "constants for each molecule kind" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_0, c_1", "meaning": "concentrations (mole fractions) of each kind" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/constant", "concept/dissolved-substance", "concept/perfect-gas", "concept/solvent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d0d4153600", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "228", "location": "Dilute Solutions", "latex": "c_{0} = \\frac{n_{0}}{n_{0} + n_{1} + n_{2} + \\dots}", "name": null, "statement": "The concentration of the solvent is its molecule number divided by the total number of molecules.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_0", "meaning": "concentration of the solvent" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/solvent", "quantity/number-of-molecules" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2dde7430cf", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "228", "location": "Dilute Solutions", "latex": "\\Phi = n_{0} (\\phi_{0} + k_{0} - R \\log c_{0}) + n_{1} (\\phi_{1} + k_{1} - R \\log c_{1}) + \\dots\\Add{.}", "name": null, "statement": "The entropy of a dilute solution written in terms of concentrations.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_0, c_1", "meaning": "concentrations of solvent and dissolved substances" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/dilute-solution", "quantity/entropy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-42e84bd6ed", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "228", "location": "Dilute Solutions", "latex": "\\phi_{0} + k_{0} - \\frac{u_{0} + pv_{0}}{\\theta} &= \\varphi_{0}\\Add{,}", "name": null, "statement": "Defines phi_0 (here written varphi_0) as a function of temperature and pressure only.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "varphi_0", "meaning": "function of temperature and pressure for the solvent" }, { "unit": null, "symbol": "k_0", "meaning": "constant for the solvent" } ], "sympy": "Eq(varphi0, phi0 + k0 - (u0 + p*v0)/theta)", "physics": true, "states": [], "concepts": [ "concept/function", "concept/pressure", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-152e316d3d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "229", "location": "Dilute Solutions", "latex": "\\Psi = n_{0} (\\varphi_{0} - R \\log c_{0}) &+ n_{1} (\\varphi_{1} - R \\log c_{1}) \\\\ &+ n_{2} (\\varphi_{2} - R \\log c_{2}) + \\dots\\Add{.}", "name": null, "statement": "The Psi function of a dilute solution: the sum over molecule kinds of number times (phi minus R log concentration). This determines the thermodynamic properties of a dilute solution.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Psi", "meaning": "Psi function of the whole system" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "c_0, c_1, c_2", "meaning": "concentrations of each molecule kind" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/dilute-solution", "concept/dissolved-substance", "concept/solvent", "quantity/psi-function" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1fb2418747", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "230", "location": "Dilute Solutions", "latex": "\\tsum \\nu_{0} \\log c_{0} + \\nu_{1} \\log c_{1} + \\nu_{2} \\log c_{2} + \\dots &= \\frac{1}{R} \\tsum \\nu_{0} \\varphi_{0} + \\nu_{1} \\varphi_{1} + \\dots \\\\ &= \\log K.", "name": null, "statement": "Equilibrium condition for a chemical change: the weighted sum of log concentrations equals log K, a constant independent of molecule numbers.", "kind": "law", "symbols": [ { "unit": null, "symbol": "nu_0, nu_1, nu_2", "meaning": "integer ratios of molecule changes in the reaction" }, { "unit": null, "symbol": "K", "meaning": "equilibrium constant, independent of molecule numbers" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/constant", "concept/logarithm", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6e430e92a8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "231", "location": "Dilute Solutions", "latex": "\\frac{\\dd \\log K}{\\dd \\theta} = \\frac{L}{R\\theta^{2}}", "name": null, "statement": "The temperature dependence of the equilibrium constant is set by the heat absorbed by the reaction.", "kind": "law", "symbols": [ { "unit": null, "symbol": "K", "meaning": "equilibrium constant" }, { "unit": null, "symbol": "L", "meaning": "heat absorbed in the change at constant temperature and pressure" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "theta", "meaning": "temperature" } ], "sympy": "Eq(Derivative(log(K), theta), L/(R*theta**2))", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/heat", "concept/logarithm", "concept/temperature", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d58bc2b0d5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "232", "location": "Dilute Solutions", "latex": "\\frac{\\dd \\log K}{\\dd p} = -\\frac{s}{R\\theta}\\Add{.}", "name": null, "statement": "The pressure dependence of the equilibrium constant is set by the volume change of the reaction.", "kind": "law", "symbols": [ { "unit": null, "symbol": "s", "meaning": "increase of volume of the system in the change" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": "Eq(Derivative(log(K), p), -s/(R*theta))", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/logarithm", "concept/pressure", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-fbeff07c23", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "231", "location": "Dilute Solutions", "latex": "s = \\tsum \\nu_{0} v_{0} + \\nu_{1} v_{1} + \\nu_{2} v_{2} + \\dots", "name": null, "statement": "The volume increase of a reaction is the weighted sum of molecular volumes.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s", "meaning": "volume increase of the system" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f3ffcd9fdd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "231", "location": "Dilute Solutions", "latex": "L = \\tsum (\\nu_{0} u_{0} + \\nu_{1} u_{1} + \\dots) + p(\\nu_{0} v_{0} + \\nu_{1} v_{1} + \\dots);", "name": null, "statement": "The heat absorbed in the change equals the energy change plus pressure times the volume change, by the first law.", "kind": "law", "symbols": [ { "unit": null, "symbol": "L", "meaning": "heat absorbed at constant temperature and pressure" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/heat", "concept/pressure", "law/first-law-of-thermodynamics", "quantity/internal-energy" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bc97b30821", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "232", "location": "Dilute Solutions", "latex": "\\log K = \\log a - \\frac{b}{\\theta} + (\\nu_{1} + \\nu_{2} + \\dots) \\log \\frac{\\theta}{p}.", "name": null, "statement": "Explicit form of log K for a reaction with constants a and b, which recovers the earlier special equations.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a, b", "meaning": "constants of the reaction" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/constant", "concept/logarithm", "concept/pressure", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8b39b434b7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "235", "location": "Dilute Solutions", "latex": "2\\, \\frac{\\dd \\log c_{1}}{\\dd \\theta} = \\frac{1}{R} · \\frac{L}{\\theta^{2}}.", "name": null, "statement": "For the water dissociation H2O to H+ and OH-, the temperature derivative of log c1 relates to the heat of dissociation.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of H+ ions (equal to OH- concentration)" }, { "unit": "calorie", "symbol": "L", "meaning": "heat necessary to dissociate one H2O molecule" } ], "sympy": "Eq(2*Derivative(log(c1), theta), L/(R*theta**2))", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/dissociation", "concept/ion", "concept/logarithm", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d3c907de54", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "235", "location": "Dilute Solutions", "latex": "-\\log c_{0} + \\log c_{1} + \\log c_{2} = K", "name": null, "statement": "Equilibrium condition for the dissociation of water into H+ and OH- ions.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_0", "meaning": "concentration of H2O" }, { "unit": null, "symbol": "c_1, c_2", "meaning": "concentrations of H+ and OH- ions" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/dissociation", "concept/ion", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-ba98912c9e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "235", "location": "Dilute Solutions", "latex": "L = \\frac{4045000}{\\theta}", "name": null, "statement": "Thomsen's measured heat of neutralization, used as the heat of dissociation of water, as a function of mean temperature (in calories).", "kind": "formula", "symbols": [ { "unit": "calorie", "symbol": "L", "meaning": "heat of dissociation of one H2O molecule" }, { "unit": null, "symbol": "theta", "meaning": "mean temperature" } ], "sympy": "Eq(L, 4045000/theta)", "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/heat-effect", "unit/calorie" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c3b9a17d33", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "236", "location": "Dilute Solutions", "latex": "c_{1} = C e^{-\\efrac{513000}{\\theta^{2}}}", "name": null, "statement": "Integrated relation giving the dissociation concentration of water as a function of temperature with an integration constant C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "integration constant, determined from the dissociation at 18 degrees C" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" } ], "sympy": "Eq(c1, C*exp(-513000/theta**2))", "physics": true, "states": [], "concepts": [ "concept/constant", "concept/dissociation", "concept/ion", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-81f0f338f9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "236", "location": "Dilute Solutions", "latex": "c_{1} = 6.1 e^{-\\efrac{513000}{\\theta^{2}}} × 10^{-7}", "name": null, "statement": "Degree of dissociation of water at any temperature, with the constant fixed by the measured value at 18 degrees C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "degree of dissociation of water (concentration of H+)" }, { "unit": null, "symbol": "theta", "meaning": "absolute temperature" } ], "sympy": "Eq(c1, 6.1*exp(-513000/theta**2)*10**(-7))", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/dissociation", "concept/ion", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8dd62d49ba", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "237", "location": "Dilute Solutions", "latex": "\\frac{c_{2}^{2}}{c_{1}} = K\\Add{.}", "name": null, "statement": "Equilibrium of the dissociation of a binary electrolyte: the square of the ion concentration over the undissociated concentration is constant (Ostwald's dilution law).", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of undissociated molecules" }, { "unit": null, "symbol": "c_2", "meaning": "concentration of each ion" }, { "unit": null, "symbol": "K", "meaning": "dissociation constant" } ], "sympy": "Eq(c2**2/c1, K)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/dissociation", "concept/electrolyte", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a22c88c2ff", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "238", "location": "Dilute Solutions", "latex": "K = \\dfrac{\\lambda_{v}}{\\lambda_{\\infty} (\\lambda_{\\infty} - \\lambda_{v})^{v}}", "name": null, "statement": "Ostwald's law of dilution of binary electrolytes in terms of molecular conductivities (from the footnote).", "kind": "law", "symbols": [ { "unit": null, "symbol": "lambda_v", "meaning": "molecular conductivity at dilution v" }, { "unit": null, "symbol": "lambda_infinity", "meaning": "molecular conductivity at infinite dilution" }, { "unit": null, "symbol": "v", "meaning": "molecular volume of the electrolyte" } ], "sympy": "Eq(K, lambda_v/(lambda_inf*(lambda_inf - lambda_v)**v))", "physics": true, "states": [], "concepts": [ "concept/dilution-law", "concept/dissociation", "concept/electrolyte", "quantity/conductivity" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a8e3eccac9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "237", "location": "Dilute Solutions", "latex": "c_{1} + c_{2} = c", "name": null, "statement": "The total concentration of undissociated and dissociated acid molecules is given.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "c", "meaning": "total concentration of the acid" } ], "sympy": "Eq(c1 + c2, c)", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/electrolyte" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-227c2d381f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "239", "location": "Dilute Solutions", "latex": "\\frac{c_{2} c_{3}}{c_{1}} = K", "name": null, "statement": "First dissociation equilibrium of sulphuric acid, H2SO4 into H+ and HSO4-.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of H2SO4 molecules" }, { "unit": null, "symbol": "c_2", "meaning": "concentration of H+ ions" }, { "unit": null, "symbol": "c_3", "meaning": "concentration of HSO4- ions" }, { "unit": null, "symbol": "K", "meaning": "equilibrium constant of the first dissociation" } ], "sympy": "Eq(c2*c3/c1, K)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/dissociation", "concept/electrolyte", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-bb32c0c6a6", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "239", "location": "Dilute Solutions", "latex": "\\frac{c_{2} c_{4}}{c_{3}} = K'", "name": null, "statement": "Second dissociation equilibrium of the HSO4- ion into H+ and SO4 ions.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_4", "meaning": "concentration of SO4 ions" }, { "unit": null, "symbol": "K'", "meaning": "equilibrium constant of the second dissociation" } ], "sympy": "Eq(c2*c4/c3, Kp)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/dissociation", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-471f45b43e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "239", "location": "Dilute Solutions", "latex": "2c_{4} + c_{3} = c_{2}", "name": null, "statement": "Conservation of SO4 radicals and hydrogen atoms: the ion count condition for two independent constituents.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "c_2, c_3, c_4", "meaning": "concentrations of H+, HSO4- and SO4 ions" } ], "sympy": "Eq(2*c4 + c3, c2)", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/independent-constituent", "concept/ion", "concept/phase-rule" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1188b1d409", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "239", "location": "Dilute Solutions", "latex": "c_{1} + c_{3} + c_{4} = c", "name": null, "statement": "The quantity of sulphuric acid in the solution is given, summing its undissociated and dissociated forms.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "c", "meaning": "given total concentration of sulphuric acid" } ], "sympy": "Eq(c1 + c3 + c4, c)", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/electrolyte", "concept/solution" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-469d1a024a", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "231", "location": "Dilute Solutions", "latex": "\\frac{\\dd \\varphi_{0}}{\\dd \\theta} = \\frac{u_{0} + pv_{0}}{\\theta^{2}};\\quad", "name": null, "statement": "Temperature derivative of the solvent's phi function (stated with its pressure-derivative companion).", "kind": "result", "symbols": [ { "unit": null, "symbol": "varphi_0", "meaning": "function of temperature and pressure for the solvent" }, { "unit": null, "symbol": "u_0, v_0", "meaning": "energy and volume per solvent molecule" } ], "sympy": "Eq(Derivative(varphi0, theta), (u0 + p*v0)/theta**2)", "physics": true, "states": [], "concepts": [ "concept/exact-differential", "concept/function", "concept/pressure", "concept/temperature" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-eab16f482f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "231", "location": "Dilute Solutions", "latex": "\\frac{\\dd \\varphi_{0}}{\\dd p} = -\\frac{v_{0}}{\\theta}", "name": null, "statement": "Pressure derivative of the solvent's phi function is minus the molecular volume over temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v_0", "meaning": "volume per solvent molecule" } ], "sympy": "Eq(Derivative(varphi0, p), -v0/theta)", "physics": true, "states": [], "concepts": [ "concept/function", "concept/pressure", "concept/temperature", "quantity/volume" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b424247de9", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "241", "location": "Dilute Solutions", "latex": "-\\log c_{1} = \\log K", "name": null, "statement": "At fixed temperature and pressure, the equilibrium condition fixes the concentration of the dissolved gas in the solution.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of the dissolved gas (CO2) in the solution" }, { "unit": null, "symbol": "K", "meaning": "equilibrium constant of the reaction, fixed by temperature and pressure" } ], "sympy": "Eq(-log(c_1), log(K))", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/thermodynamic-equilibrium", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-653455d2d2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "240", "location": "Dilute Solutions", "latex": "c_{1} = \\frac{n_{1}}{n_{0} + n_{1}}", "name": null, "statement": "The concentration of the dissolved gas in the liquid is its number of molecules divided by all molecules in the liquid phase.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of the dissolved gas in the liquid" }, { "unit": null, "symbol": "n_1", "meaning": "number of molecules of the dissolved gas" }, { "unit": null, "symbol": "n_0", "meaning": "number of molecules of the solvent (water)" } ], "sympy": "Eq(c_1, n_1/(n_0 + n_1))", "physics": false, "states": [], "concepts": [ "concept/concentration", "concept/dissolved-substance", "concept/solvent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-529ec179ac", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "241", "location": "Dilute Solutions", "latex": "\\frac{\\dd \\log c_{1}}{\\dd p} = \\frac{1}{R} · \\frac{s}{\\theta}", "name": null, "statement": "The change of the logarithm of the dissolved-gas concentration with pressure, at constant temperature.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of the dissolved gas" }, { "unit": null, "symbol": "p", "meaning": "pressure of the free gas" }, { "unit": null, "symbol": "R", "meaning": "gas constant in the book's units (R = 1.97 when heat is in calories)" }, { "unit": null, "symbol": "s", "meaning": "increase of volume of the system on evaporation of one gram molecule" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": "Eq(Derivative(log(c_1), p), s/(R*theta))", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/pressure", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d1172cfcb8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "241", "location": "Dilute Solutions", "latex": "\\frac{\\dd \\log c_{1}}{\\dd \\theta} = -\\frac{1}{R} · \\frac{L}{\\Erratum{\\theta_{2}}{\\theta^{2}}}", "name": null, "statement": "The change of the logarithm of the dissolved-gas concentration with temperature. The book marks the denominator as an erratum: printed as theta-2, corrected to theta-squared; the sympy form is withheld because the erratum is not resolved in the source text.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of the dissolved gas" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" }, { "unit": "calorie", "symbol": "L", "meaning": "heat absorbed during isothermal-isopiestic evaporation of one gram molecule of CO2" }, { "unit": null, "symbol": "R", "meaning": "gas constant in the book's units" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/temperature", "quantity/heat-effect", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-0ea8992f66", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "242", "location": "Dilute Solutions", "latex": "c_{1} = Cp", "name": "Henry's law", "statement": "The concentration of the dissolved gas is proportional to the pressure of the free gas on the solution.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of the dissolved gas" }, { "unit": null, "symbol": "C", "meaning": "factor measuring the solubility of the gas; depends on temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure of the free gas over the solution" } ], "sympy": "Eq(c_1, C*p)", "physics": true, "states": [ "law/henry-s-law" ], "concepts": [ "concept/concentration", "concept/pressure", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a29ccb20eb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "242", "location": "Dilute Solutions", "latex": "L = -\\frac{R \\theta^{2}}{C} · \\frac{\\dd C}{\\dd \\theta}", "name": null, "statement": "The heat effect of absorption of the gas from the solution can be calculated from the temperature variation of the solubility factor C.", "kind": "formula", "symbols": [ { "unit": "calorie", "symbol": "L", "meaning": "heat absorbed when one gram molecule of the gas evaporates from the solution" }, { "unit": null, "symbol": "C", "meaning": "solubility factor of Henry's law" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant in the book's units" } ], "sympy": "Eq(L, -R*theta**2/C*Derivative(C, theta))", "physics": true, "states": [], "concepts": [ "concept/temperature", "quantity/heat-effect", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-a40093dac2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "241", "location": "Dilute Solutions", "latex": "s = \\frac{R\\theta}{p}", "name": null, "statement": "The volume of one gram molecule of a perfect gas at temperature theta and pressure p, taken from the gas equation (16).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "volume of one gram molecule of the gas (increase of volume of the system)" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "p", "meaning": "pressure" } ], "sympy": "Eq(s, R*theta/p)", "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/perfect-gas", "concept/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b696c284b2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "241", "location": "Dilute Solutions", "latex": "\\frac{\\dd \\log c_{1}}{\\dd p} = \\frac{1}{p}", "name": null, "statement": "Substituting the perfect-gas volume into the pressure derivative gives the logarithmic derivative of the concentration as one over the pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of the dissolved gas" }, { "unit": null, "symbol": "p", "meaning": "pressure of the free gas" } ], "sympy": "Eq(Derivative(log(c_1), p), 1/p)", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b3d9237aee", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "243", "location": "Dilute Solutions", "latex": "\\log c_{1} = \\frac{L}{R\\theta} + \\const", "name": null, "statement": "If the heat effect L is independent of temperature, integrating the temperature relation gives the logarithm of the concentration as a constant plus L over R theta.", "kind": "result", "symbols": [ { "unit": "calorie", "symbol": "L", "meaning": "heat of vaporization of the dissolved substance, taken as constant" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/concentration", "quantity/heat-effect" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-2b36ca3272", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "243", "location": "Dilute Solutions", "latex": "L = -R\\theta^{2} \\frac{\\dd \\log c_{1}}{\\dd \\theta}", "name": null, "statement": "The heat effect of precipitating one gram molecule of a salt from a saturated solution is obtained from the temperature variation of the solubility (van't Hoff's use on succinic acid).", "kind": "formula", "symbols": [ { "unit": "calorie", "symbol": "L", "meaning": "heat effect of precipitation of one gram molecule of the solid" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of the undissolved-salt molecules in the solution" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": "Eq(L, -R*theta**2*Derivative(log(c_1), theta))", "physics": true, "states": [], "concepts": [ "concept/saturation-point", "quantity/heat-effect", "quantity/solubility" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b1aecace6d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "244", "location": "Dilute Solutions", "latex": "\\frac{c_{2}^{2}}{c_{1}} = K'", "name": null, "statement": "In a saturated solution of a dissociating salt, the dissociated and undissociated molecule concentrations satisfy a constant equilibrium relation at given temperature and pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_2", "meaning": "concentration of the dissociated ions of the salt" }, { "unit": null, "symbol": "c_1", "meaning": "concentration of the undissociated salt molecules" }, { "unit": null, "symbol": "K'", "meaning": "equilibrium constant of the dissociation reaction" } ], "sympy": "Eq(c_2**2/c_1, Kp)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/dissociation", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8487af050f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "246", "location": "Dilute Solutions", "latex": "\\frac{n_{1} + n_{2} + n_{3} + \\dots}{n_{0}} = \\log K", "name": null, "statement": "For a solvent passing to the vapour phase, the ratio of dissolved molecules to solvent molecules equals log K, which is therefore a small quantity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" }, { "unit": null, "symbol": "n_1, n_2, n_3", "meaning": "numbers of the dissolved molecules of each kind" }, { "unit": null, "symbol": "K", "meaning": "equilibrium constant of the transformation" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dissolved-substance", "concept/logarithm", "concept/solvent", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-27421b0824", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "246", "location": "Dilute Solutions", "latex": "\\frac{n_{1} + n_{2} + n_{3} + \\dots}{n_{0}} = \\frac{1}{R} \\left(\\frac{m_{0}}{m_{0}'}\\, \\varphi_{0}' - \\varphi_{0}\\right)", "name": null, "statement": "The ratio of dissolved molecules to solvent molecules is given by the difference of the solvent's potential functions in the two phases, scaled by the molecular weights.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m_0", "meaning": "molecular weight of the solvent in the liquid" }, { "unit": null, "symbol": "m_0'", "meaning": "molecular weight of the solvent in the vapour" }, { "unit": null, "symbol": "φ_0, φ_0'", "meaning": "potential functions of the solvent in the liquid and vapour phases (defined earlier in the book)" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/phase", "concept/solvent", "concept/thermodynamic-equilibrium", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-663aed8c52", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "247", "location": "Dilute Solutions", "latex": "\\theta - \\theta_{0} = \\frac{R\\theta^{2}}{n_{0} L} (n_{1} + n_{2} + n_{3} + \\dots)", "name": null, "statement": "The elevation of the boiling point of a dilute solution follows from the number of dissolved molecules, the temperature, and the heat of vaporization.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "boiling temperature of the solution" }, { "unit": null, "symbol": "θ_0", "meaning": "boiling temperature of the pure solvent at the same pressure" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" }, { "unit": "calorie", "symbol": "L", "meaning": "heat of vaporization of one gram molecule of the solvent" }, { "unit": null, "symbol": "R", "meaning": "gas constant (1.97 in calories)" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/boiling-point-elevation", "concept/dissolved-substance", "concept/solvent", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-e65bb543ed", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "248", "location": "Dilute Solutions", "latex": "\\theta - \\theta_{0} = \\frac{c \\theta^{2} \\varphi}{L}", "name": null, "statement": "The elevation of the boiling point from the general theory, in terms of the mass ratio of dissolved substance to solvent and the quantity phi.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "ratio of the mass of the dissolved non-volatile substance to the mass of the solvent" }, { "unit": null, "symbol": "φ", "meaning": "quantity defined in the general theory (eq. 165)" }, { "unit": null, "symbol": "L", "meaning": "heat of vaporization per unit mass of the solvent" }, { "unit": null, "symbol": "θ", "meaning": "absolute boiling temperature" } ], "sympy": "Eq(theta - theta_0, c*theta**2*phi/L)", "physics": true, "states": [], "concepts": [ "concept/boiling-point-elevation", "concept/concentration", "quantity/latent-heat" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-6187a23fea", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "249", "location": "Dilute Solutions", "latex": "\\varphi = \\frac{R(n_{1} + n_{2} + \\dots)}{n_{1} m_{1} + n_{2} m_{2} + \\dots}", "name": null, "statement": "The two theories agree only if phi takes this molecular value, which relates it to the number of dissolved molecules and their molecular weights.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "quantity from the general theory (eq. 165)" }, { "unit": null, "symbol": "n_1, n_2", "meaning": "numbers of dissolved molecules of each kind" }, { "unit": null, "symbol": "m_1, m_2", "meaning": "molecular weights of the dissolved kinds" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/dissolved-substance", "concept/solution", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-9abd1c5b2f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "249", "location": "Dilute Solutions", "latex": "c\\varphi = \\frac{R(n_{1} + n_{2} + n_{3} + \\dots)}{n_{0} m_{0}}", "name": null, "statement": "For dilute solutions the product of the mass ratio and phi reduces to a quantity set by the number of dissolved molecules and the solvent's mass.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "mass ratio of dissolved substance to solvent" }, { "unit": null, "symbol": "φ", "meaning": "quantity from the general theory" }, { "unit": null, "symbol": "n_0 m_0", "meaning": "mass of the solvent" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/dissolved-substance", "concept/solvent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-3ad5198e96", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "249", "location": "Dilute Solutions", "latex": "p_{0} - p = \\frac{R\\theta}{n_{0}s} (n_{1} + n_{2} + n_{3} + \\dots)", "name": null, "statement": "For dilute solutions, the lowering of the vapour pressure equals a quantity proportional to the total number of dissolved molecules.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p_0", "meaning": "vapour pressure of the solvent without dissolved molecules at the given temperature" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "s", "meaning": "change of volume of the system on evaporation of one gram molecule of solvent" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/lowering-of-vapour-pressure", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-60c9a9c2e8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "250", "location": "Dilute Solutions", "latex": "p_{0} - p = \\frac{m_{0}'p (n_{1} + n_{2} + \\dots)}{n_{0} m_{0}}", "name": null, "statement": "If the solvent vapour is a perfect gas and the solution's volume is negligible, the lowering of the vapour pressure is given in terms of the molecular weights and the number of dissolved molecules.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m_0", "meaning": "molecular weight of the solvent in the liquid" }, { "unit": null, "symbol": "m_0'", "meaning": "molecular weight of the solvent in the vapour" }, { "unit": null, "symbol": "p_0", "meaning": "vapour pressure of pure solvent" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/lowering-of-vapour-pressure", "concept/perfect-gas", "quantity/molecular-weight", "quantity/pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1358fa44b2", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "250", "location": "Dilute Solutions", "latex": "\\frac{p_{0} - p}{p} = (n_{1} + n_{2} + n_{3} + \\dots)\\, \\frac{m_{0}'}{n_{0} m_{0}}", "name": null, "statement": "The relative lowering of the vapour pressure of a dilute solution is set by the number of dissolved molecules; the book notes the common form, which holds only when the solvent's molecular weight is the same in liquid and vapour (m_0 = m_0').", "kind": "result", "symbols": [ { "unit": null, "symbol": "p_0", "meaning": "vapour pressure of pure solvent" }, { "unit": null, "symbol": "p", "meaning": "vapour pressure of the solution" }, { "unit": null, "symbol": "m_0", "meaning": "molecular weight of the solvent in the liquid" }, { "unit": null, "symbol": "m_0'", "meaning": "molecular weight of the solvent in the vapour" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/lowering-of-vapour-pressure", "concept/solvent", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-88383fd275", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "250", "location": "Dilute Solutions", "latex": "\\theta_{0}' - \\theta' = \\frac{R \\theta^{2}}{n_{0} L'} (n_{1} + n_{2} + n_{3} + \\dots)", "name": null, "statement": "The depression of the freezing point of a dilute solution is proportional to the number of dissolved molecules.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "θ_0'", "meaning": "freezing temperature of the pure solvent at the given pressure" }, { "unit": null, "symbol": "θ'", "meaning": "freezing temperature of the solution" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" }, { "unit": "calorie", "symbol": "L'", "meaning": "heat of solidification of one gram molecule of the solvent" }, { "unit": null, "symbol": "R", "meaning": "gas constant (1.97 in calories)" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/freezing-point-depression", "concept/heat-of-solidification" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-ac490ba14c", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "251", "location": "Dilute Solutions", "latex": "P = \\frac{R\\theta}{n_{0} m_{0} v} (n_{1} + n_{2} + n_{3} + \\dots)", "name": null, "statement": "The osmotic pressure of a dilute solution is set by the number of dissolved molecules and the temperature.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "P", "meaning": "osmotic pressure of the solution" }, { "unit": null, "symbol": "v", "meaning": "specific volume of the solution" }, { "unit": null, "symbol": "n_0 m_0", "meaning": "mass of the solvent" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/dilute-solution", "concept/temperature", "quantity/osmotic-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-7fa2ed48eb", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "251", "location": "Dilute Solutions", "latex": "P = \\frac{R\\theta}{V} (n_{1} + n_{2} + n_{3} + \\dots)", "name": null, "statement": "The osmotic pressure has the same form as the characteristic equation of a mixture of perfect gases, with the dissolved molecules as the gas.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "osmotic pressure" }, { "unit": null, "symbol": "V", "meaning": "volume of the solution (approximately n_0 m_0 v)" }, { "unit": null, "symbol": "θ", "meaning": "absolute temperature" }, { "unit": null, "symbol": "R", "meaning": "gas constant" }, { "unit": null, "symbol": "n_1, n_2, n_3", "meaning": "numbers of dissolved molecules of each kind" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/characteristic-equation", "concept/dilute-solution", "concept/perfect-gas", "quantity/osmotic-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8f2990e83e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "248", "location": "Dilute Solutions", "latex": "c = \\frac{n_{1} m_{1} + n_{2} m_{2} + \\dots}{n_{0} m_{0}}", "name": null, "statement": "The mass ratio of dissolved substance to solvent is expressed in molecule numbers and molecular weights.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c", "meaning": "ratio of the mass of the dissolved substance to the mass of the solvent" }, { "unit": null, "symbol": "n_1, n_2", "meaning": "numbers of dissolved molecules of each kind" }, { "unit": null, "symbol": "m_1, m_2", "meaning": "molecular weights of dissolved kinds" }, { "unit": null, "symbol": "n_0, m_0", "meaning": "number and molecular weight of solvent molecules" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/concentration", "concept/dissolved-substance", "quantity/molecular-weight" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b1930515a0", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "254", "location": "Dilute Solutions", "latex": "c_{1} + c_{2} + \\dots + \\log c_{0}' = \\log K", "name": null, "statement": "For a solvent evaporating from a liquid solution, the concentrations of dissolved molecules in the liquid and the solvent's concentration in the vapour are related by the equilibrium constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_1, c_2", "meaning": "concentrations of dissolved molecules in the liquid" }, { "unit": null, "symbol": "c_0'", "meaning": "concentration of solvent molecules in the vapour" }, { "unit": null, "symbol": "K", "meaning": "equilibrium constant of the evaporation" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/phase", "concept/solvent", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-f6c74d34b7", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "241", "location": "Dilute Solutions", "latex": "\\log c_{0}' = \\log K", "name": null, "statement": "When the solvent's vapour molecules far outnumber others, the solvent's vapour concentration does not depend on the composition of the solution, so the partial pressure of the solvent equals that of the pure solvent.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_0'", "meaning": "concentration of solvent molecules in the vapour" }, { "unit": null, "symbol": "K", "meaning": "equilibrium constant of the evaporation" } ], "sympy": "Eq(log(c0p), log(K))", "physics": true, "states": [], "concepts": [ "concept/concentration", "concept/solvent", "quantity/partial-pressure" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-5b71f79266", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "255", "location": "Dilute Solutions", "latex": "(c_{1} + c_{2} + \\dots) - (c_{1}' + c_{2}' + \\dots) = \\log K", "name": null, "statement": "When the dissolved substance also passes into the vapour, the boiling-point elevation and vapour-pressure lowering depend on the difference of the concentrations in liquid and vapour, not on the liquid concentrations alone.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c_1, c_2", "meaning": "concentrations of dissolved molecules in the liquid" }, { "unit": null, "symbol": "c_1', c_2'", "meaning": "concentrations of the same molecules in the vapour" }, { "unit": null, "symbol": "K", "meaning": "equilibrium constant" } ], "sympy": null, "physics": true, "states": [], "concepts": [ "concept/boiling-point-elevation", "concept/concentration", "concept/lowering-of-vapour-pressure", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-58a511a6f5", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "253", "location": "Dilute Solutions", "latex": "\\frac{c_{1}'}{c_{1}} = K", "name": "Nernst's law of distribution", "statement": "For each kind of molecule with the same molecular weight in both phases, the ratio of its concentrations in the two phases is a constant independent of other molecules present.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c_1", "meaning": "concentration of the molecule in the liquid" }, { "unit": null, "symbol": "c_1'", "meaning": "concentration of the same molecule in the vapour" }, { "unit": null, "symbol": "K", "meaning": "constant ratio of distribution" } ], "sympy": "Eq(c1p/c_1, K)", "physics": true, "states": [ "law/distribution-law" ], "concepts": [ "concept/concentration", "concept/phase", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-8c7a5015f8", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "256", "location": "Dilute Solutions", "latex": "\\frac{n_{2}^{2}}{n_{1} n_{0}} = K", "name": null, "statement": "For the dissociation of a weak electrolyte in a dilute solution, the ion count squared over the undissociated count and solvent count is a constant at fixed temperature and pressure.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n_2", "meaning": "number of ions of the electrolyte in solution" }, { "unit": null, "symbol": "n_1", "meaning": "number of undissociated molecules of the electrolyte" }, { "unit": null, "symbol": "n_0", "meaning": "number of solvent molecules" }, { "unit": null, "symbol": "K", "meaning": "dissociation equilibrium constant" } ], "sympy": "Eq(n_2**2/(n_1*n_0), K)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/dissociation", "concept/electrolyte", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c67c59701e", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "256", "location": "Dilute Solutions", "latex": "\\frac{n_{2}'^{2}}{n_{1}' n_{0}'} = K'", "name": null, "statement": "The same dissociation equilibrium holds for the second solution, with its own counts.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n_2'", "meaning": "number of ions in the second solution" }, { "unit": null, "symbol": "n_1'", "meaning": "number of undissociated molecules in the second solution" }, { "unit": null, "symbol": "n_0'", "meaning": "number of solvent molecules in the second solution" }, { "unit": null, "symbol": "K'", "meaning": "dissociation equilibrium constant" } ], "sympy": "Eq(n2p**2/(n1p*n0p), Kp)", "physics": true, "states": [], "concepts": [ "concept/dissociation", "concept/electrolyte", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-4275fd758f", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "256", "location": "Dilute Solutions", "latex": "\\bar{n}_{0} = n_{0} + n_{0}'", "name": null, "statement": "After mixing, the number of water molecules is the sum of the water molecules of the two solutions.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "n̄_0", "meaning": "number of water molecules after mixing" }, { "unit": null, "symbol": "n_0, n_0'", "meaning": "water molecule numbers of the two solutions before mixing" } ], "sympy": "Eq(nbar_0, n_0 + n0p)", "physics": false, "states": [], "concepts": [ "concept/conservation-of-molecules", "concept/solvent" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-d1e0de4f58", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "256", "location": "Dilute Solutions", "latex": "\\bar{n}_{2} + \\bar{n}_{4} = n_{1}' + n_{2}'", "name": null, "statement": "After mixing, the sodium atoms are conserved: their number equals that of the sodium-bearing molecules and ions of the second solution.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "n̄_2, n̄_4", "meaning": "numbers of sodium acetate molecules and Na+ ions after mixing" }, { "unit": null, "symbol": "n_1', n_2'", "meaning": "numbers of sodium-bearing molecules and ions in the second solution" } ], "sympy": "Eq(nbar_2 + nbar_4, n1p + n2p)", "physics": false, "states": [], "concepts": [ "concept/conservation-of-molecules", "concept/dissociation", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-c8d1708e97", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "256", "location": "Dilute Solutions", "latex": "\\bar{n}_{1} + \\bar{n}_{3} = n_{1} + n_{2}", "name": null, "statement": "After mixing, the hydrogen atoms are conserved across acetic acid and H+ ions.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "n̄_1, n̄_3", "meaning": "numbers of acetic acid molecules and H+ ions after mixing" }, { "unit": null, "symbol": "n_1, n_2", "meaning": "numbers of acetic acid molecules and H+ ions in the first solution" } ], "sympy": "Eq(nbar_1 + nbar_3, n_1 + n_2)", "physics": false, "states": [], "concepts": [ "concept/conservation-of-molecules", "concept/electrolyte", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-1f7d92a63d", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "256", "location": "Dilute Solutions", "latex": "\\bar{n}_{3} + \\bar{n}_{4} = \\bar{n}_{5}", "name": null, "statement": "Electrical neutrality: the number of positive ions equals the number of negative ions in the mixed solution.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "n̄_3, n̄_4", "meaning": "numbers of H+ and Na+ ions after mixing" }, { "unit": null, "symbol": "n̄_5", "meaning": "number of acetate (negative) ions after mixing" } ], "sympy": "Eq(nbar_3 + nbar_4, nbar_5)", "physics": true, "states": [], "concepts": [ "concept/conservation-of-molecules", "concept/electrolyte", "concept/ion" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-67359c98a1", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "257", "location": "Dilute Solutions", "latex": "\\frac{\\bar{c}_{3} \\bar{c}_{5}}{\\bar{c}_{1}} = K", "name": null, "statement": "At equilibrium, the dissociation of acetic acid in the mixed solution satisfies a constant ratio of ion to undissociated concentrations.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c̄_1", "meaning": "concentration of acetic acid molecules after mixing" }, { "unit": null, "symbol": "c̄_3", "meaning": "concentration of H+ ions after mixing" }, { "unit": null, "symbol": "c̄_5", "meaning": "concentration of acetate ions after mixing" }, { "unit": null, "symbol": "K", "meaning": "dissociation equilibrium constant of acetic acid, the same as in eq. (241)" } ], "sympy": "Eq(cbar_3*cbar_5/cbar_1, K)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/dissociation", "concept/thermodynamic-equilibrium" ] }, { "id": "planck-treatise-on-thermodynamics-1903/eq-b59bb3a5bd", "chapter": "planck-treatise-on-thermodynamics-1903/ch-dilute-solutions", "book": "planck-treatise-on-thermodynamics-1903", "edition": "Longmans, Green, and Co., 1903, translated by Alexander Ogg (edition to be confirmed from the copy)", "page": "257", "location": "Dilute Solutions", "latex": "\\frac{\\bar{c}_{4} \\bar{c}_{5}}{\\bar{c}_{2}} = K'", "name": null, "statement": "At equilibrium, the dissociation of sodium acetate in the mixed solution satisfies a constant ratio of ion to undissociated concentrations.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c̄_2", "meaning": "concentration of sodium acetate molecules after mixing" }, { "unit": null, "symbol": "c̄_4", "meaning": "concentration of Na+ ions after mixing" }, { "unit": null, "symbol": "c̄_5", "meaning": "concentration of acetate ions after mixing" }, { "unit": null, "symbol": "K'", "meaning": "dissociation equilibrium constant of sodium acetate, the same as in eq. (242)" } ], "sympy": "Eq(cbar_4*cbar_5/cbar_2, Kp)", "physics": true, "states": [], "concepts": [ "concept/chemical-reaction", "concept/concentration", "concept/dissociation", "concept/thermodynamic-equilibrium" ] } ], "exercise_sets": [], "problems": [], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }