(* 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[ 319888, 6592] NotebookOptionsPosition[ 303458, 6330] NotebookOutlinePosition[ 305979, 6378] CellTagsIndexPosition[ 305898, 6373] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lecture 05 - 02/14/2022", "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}, {3.853751852803602*^9, 3.853751857475161*^9}},ExpressionUUID->"f5d9d907-2b54-43cd-897c-\ 99ae57a61323"], Cell[CellGroupData[{ Cell["Convexity of Thermodynamic Potentials", "Chapter", 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we\[CloseCurlyQuote]ll jump the gun a \ bit and introduce some ", StyleBox["postulates ", FontWeight->"Bold"], "of Thermodynamic potentials, to allow us to continue our discussion on the \ Van der Waals gas" }], "Item", CellChangeTimes->{{3.853752273920609*^9, 3.853752278100286*^9}, { 3.853752381756166*^9, 3.8537524678776283`*^9}, {3.853752512846182*^9, 3.853752579003854*^9}, {3.8537526806256*^9, 3.85375275619162*^9}},ExpressionUUID->"b62cfdac-1250-4fde-8293-\ ca9d57add2c2"], Cell[TextData[{ StyleBox["Note: ", FontWeight->"Bold"], "This approach can be termed \[OpenCurlyQuote]phenomenological \ thermodynamics\[CloseCurlyQuote] and follows the treatment of Callen - nicely \ summarized by this short paper: https://arxiv.org/pdf/1404.5273.pdf" }], "Subitem", CellChangeTimes->{{3.853752273920609*^9, 3.853752278100286*^9}, { 3.853752381756166*^9, 3.8537524678776283`*^9}, {3.853752512846182*^9, 3.853752579003854*^9}, {3.8537526806256*^9, 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"Subsubitem", CellChangeTimes->{{3.853754483854239*^9, 3.8537545487581*^9}, { 3.853754582878977*^9, 3.853754614629553*^9}, {3.853754805556758*^9, 3.853754809526805*^9}},ExpressionUUID->"4ef71e33-b93b-4ff4-8646-\ 8148a79ef700"], Cell[TextData[{ "Volume, ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["V", "TI"], TraditionalForm], "errors" -> {}, "input" -> "V", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "6f525e86-2654-4e18-9f48-5ee3b9657b86"] }], "Subsubitem", CellChangeTimes->{{3.853754483854239*^9, 3.8537545487581*^9}, { 3.853754582878977*^9, 3.853754620876851*^9}},ExpressionUUID->"6363187e-7747-469b-931b-\ 37d628184fe0"], Cell[TextData[{ "Mole numbers, ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"{", SubscriptBox[ StyleBox["n", "TI"], StyleBox["j", "TI"]], "}"}], TraditionalForm], "errors" -> {}, "input" -> "\\{n_j\\}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 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energy with the product of this intensive variable \ and its ", StyleBox["extensive conjugate ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["X", "TI"], TraditionalForm], "errors" -> {}, "input" -> "X", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "8c16be2c-f91e-4607-8b9c-b610eb2a2eae"], StyleBox["\[LineSeparator]", FontWeight->"Bold"], Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[CapitalPhi]", "\[LongEqual]", StyleBox["U", "TI"], "-", StyleBox["Y", "TI"], StyleBox["X", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> "\\mathcal{\\Phi} = \\mathcal{U} - Y X", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "5c04b87a-a280-472d-8440-958a139f9451"] }], "Subsubitem", CellChangeTimes->{{3.853754903769964*^9, 3.853754995974317*^9}, { 3.853755029526492*^9, 3.853755034870706*^9}, {3.853755372187297*^9, 3.853755376083564*^9}, {3.8537561188843403`*^9, 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