(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 1947604, 38067] NotebookOptionsPosition[ 1924522, 37647] NotebookOutlinePosition[ 1925065, 37666] CellTagsIndexPosition[ 1925022, 37663] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lecture 17 - 04/04/2022", "Subtitle", CellChangeTimes->{{3.8525512993398438`*^9, 3.8525513206118402`*^9}, { 3.852652054138073*^9, 3.8526520591301193`*^9}, {3.853194369726288*^9, 3.8531943739664793`*^9}, 3.8531971130005827`*^9, {3.853361889945813*^9, 3.853361893353859*^9}, {3.854455749568426*^9, 3.8544557604776297`*^9}, { 3.8545621484661007`*^9, 3.8545621501697083`*^9}, {3.85499650158072*^9, 3.854996504060568*^9}, {3.855176893175762*^9, 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{"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, "Droplines" -> { "freeformCursorMode" -> True, "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 1}, {0., 0.9999999948979591}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{{3.858002541891447*^9, 3.8580025525996923`*^9}, 3.8580089428903313`*^9, 3.858082172333852*^9}, CellLabel->"Out[39]=",ExpressionUUID->"b01154be-f8fe-4fcf-8233-04de3ec86cfb"] }, Open ]] }, Open ]] }, Open ]], Cell["\<\ As we\[CloseCurlyQuote]ve seen before, atoms on the surface have a higher \ average energy\ \>", "Item", 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StyleBox["i", "TI"], StyleBox["int", FontSlant -> "Plain"]]}], ")"}]}], TraditionalForm], "errors" -> {}, "input" -> "dE = T \\left(dS^{\\alpha}+dS^{\\beta}+dS^{\\mathrm{int}} \\right) - P \ \\left(d V^{\\alpha} + d V^{\\beta} \\right) + \\gamma dA + \\sum_i \\mu_i \ \\left(dn_i^{\\alpha}+dn_i^{\\beta}+dn_i^{\\mathrm{int}} \\right)", "state" -> "Boxes"|>, "TeXAssistantTemplate"], "\[IndentingNewLine]", "\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["d", "TI"], StyleBox["E", "TI"], "\[LongEqual]", StyleBox["T", "TI"], StyleBox["d", "TI"], StyleBox["S", "TI"], "-", StyleBox["P", "TI"], StyleBox["d", "TI"], StyleBox["V", "TI"], "+", "\[Gamma]", StyleBox["d", "TI"], StyleBox["A", "TI"], "+", UnderscriptBox["\[Sum]", StyleBox["i", "TI"], LimitsPositioning -> True], SubscriptBox["\[Mu]", StyleBox["i", "TI"]], StyleBox["d", "TI"], SubscriptBox[ StyleBox["n", "TI"], StyleBox["i", "TI"]]}], TraditionalForm], "errors" -> {}, "input" -> "dE = T dS - P dV + \\gamma dA + \\sum_i \\mu_i dn_i", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.858011579298419*^9, 3.858011581333877*^9}, { 3.8580116260506487`*^9, 3.858011643921938*^9}, 3.8580117499680853`*^9, { 3.858012004637471*^9, 3.858012027393211*^9}, 3.85801210766879*^9},ExpressionUUID->"ef28ac83-8a49-4b1e-8b6a-\ f9531e8ee426"], Cell["\<\ Finally, since all the differentials above are extensive, we can integrate \ the expression to give\ \>", "Item", CellChangeTimes->{{3.858012041609556*^9, 3.858012065713231*^9}},ExpressionUUID->"e8ca7445-18eb-43d3-a5cb-\ 1c9655b7e036"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["E", "TI"], "\[LongEqual]", StyleBox["T", "TI"], StyleBox["S", "TI"], "-", StyleBox["P", "TI"], StyleBox["V", "TI"], "+", "\[Gamma]", StyleBox["A", "TI"], "+", UnderscriptBox["\[Sum]", StyleBox["i", "TI"], LimitsPositioning -> True], SubscriptBox["\[Mu]", StyleBox["i", "TI"]], SubscriptBox[ StyleBox["n", "TI"], 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