(* 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[ 474950, 8953] NotebookOptionsPosition[ 461390, 8713] NotebookOutlinePosition[ 463355, 8755] CellTagsIndexPosition[ 463235, 8749] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lecture 04 - 02/09/2022", "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}},ExpressionUUID->"3f5ed32e-d36b-4831-8e70-\ 07bc5563378e"], Cell[CellGroupData[{ Cell["Liquid - Gas Phase Transitions", "Chapter", CellChangeTimes->{{3.852551340964005*^9, 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3.853362989512271*^9}},ExpressionUUID->"b65a7ca5-e7e4-4f34-b0b9-\ aab04acc7f7b"] }, Open ]], Cell[TextData[{ "One of these relations, called the ", ButtonBox["isothermal bulk modulus", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Bulk_modulus"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Bulk_modulus"], " , relates the change in the pressure of material following a change in its \ volume at fixed temperature" }], "Item", CellChangeTimes->{{3.853362836297984*^9, 3.853362928519536*^9}, { 3.853362972263578*^9, 3.853362973911716*^9}, {3.853363025647715*^9, 3.853363089432975*^9}},ExpressionUUID->"a5e7db8b-7261-483a-9fb1-\ d8fcabea0c10"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ StyleBox["K", "TI"], StyleBox["T", "TI"]], "\[LongEqual]", "-", StyleBox["V", "TI"], SubscriptBox[ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", StyleBox["P", "TI"]}], RowBox[{"\[PartialD]", StyleBox["V", "TI"]}]], "\[VerticalSeparator]"}], 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The reasoning goes as follows:\ \>", "Item", CellChangeTimes->{{3.853362836297984*^9, 3.853362855094796*^9}, { 3.853363097634355*^9, 3.853363097635551*^9}, 3.853363237619362*^9},ExpressionUUID->"8f941ce5-f963-4512-b086-\ 4711a065fa86"], Cell["\<\ If the bulk modulus becomes negative, then a decrease in volume would be \ resisted by decreasing pressure and the volume would continue to shrink until \ it goes to zero (or the bulk modulus becomes positive). \ \>", "Subitem", CellChangeTimes->{{3.853362836297984*^9, 3.853362855094796*^9}, { 3.853363100817333*^9, 3.853363106568837*^9}, {3.853363230339158*^9, 3.8533632362427263`*^9}},ExpressionUUID->"5a2ef94d-b89c-4f63-8b3b-\ 3a10427405d4"], Cell["\<\ Thus a negative isothermal bulk modules would produce spontaneous volume \ changes \[LongDash] i.e. a phase transition.\ \>", "Subitem", CellChangeTimes->{{3.853362836297984*^9, 3.853362855094796*^9}, { 3.853363100817333*^9, 3.853363106568837*^9}, {3.853363230339158*^9, 3.8533632563352833`*^9}},ExpressionUUID->"4ea8d466-2c98-47c4-a6f3-\ 656b88106f41"] }, Open ]], Cell["Mathematically, we can express this stability condition as ", "Item", CellChangeTimes->{{3.853362836297984*^9, 3.853362855094796*^9}, { 3.853363100817333*^9, 3.853363106568837*^9}, {3.853363230339158*^9, 3.853363266931617*^9}},ExpressionUUID->"fcde1524-e546-472d-af70-\ ab1bf8a6d5a1"], Cell[BoxData[ RowBox[{"\t", TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ RowBox[{ RowBox[{"\[Delta]", StyleBox["V", "TI"], "\[Delta]", StyleBox["P", "TI"]}], "\[VerticalSeparator]"}], StyleBox["T", "TI"]], "<", "0"}], TraditionalForm], "errors" -> {}, "input" -> "\\left.\\delta V \\delta P \\right|_T < 0", "state" -> "Boxes"|>, "TeXAssistantTemplate"]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.7735647893570647`*^9, 3.7735647902824917`*^9}, { 3.7735660158612933`*^9, 3.773566016706119*^9}, {3.7735676244674263`*^9, 3.773567625284314*^9}, {3.7735678890224657`*^9, 3.773567891494249*^9}, { 3.7735683430528593`*^9, 3.7735683437834396`*^9}, 3.773724600013646*^9, { 3.853362665796761*^9, 3.853362666140524*^9}, {3.853363295175158*^9, 3.853363320973236*^9}}, FontSize->18, CellTags-> "eq:van-der-waals",ExpressionUUID->"9e476b17-4fa2-40b4-84f7-7a9741d03bb8"], Cell["\<\ which highlights that in order for the system to be stable, pressure must \ decrease with increasing volume. \ \>", "Subitem", CellChangeTimes->{{3.8533633290565023`*^9, 3.853363335675214*^9}},ExpressionUUID->"21c1851b-9465-433e-b8fd-\ e53770ac754c"], Cell[TextData[{ "For a gas obeying the van der Waals equation of state, there exists a \ critical temperature, ", Cell[BoxData[ FormBox[ SubscriptBox["T", "c"], TraditionalForm]],ExpressionUUID-> "02e1b065-240b-423a-aa64-3779753ae7c1"], ", above which eq. (5) is always satisfied \[Dash] highlighting the stable \ region." }], "Subitem", CellChangeTimes->{{3.8533633290565023`*^9, 3.8533633423083363`*^9}},ExpressionUUID->"7db7400d-3c27-4ed9-be2e-\ 8b9ccaa512ef"], Cell[CellGroupData[{ Cell[TextData[{ "In order to find ", Cell[BoxData[ FormBox[ SubscriptBox["T", "c"], TraditionalForm]],ExpressionUUID-> "d58357ad-b018-438f-8047-2eddaf67a486"], ", the conditions for a negative slope in pressure versus molar volume need \ to be worked out. As such, we rearrange (3) for pressure:" }], "Item", CellChangeTimes->{{3.8533633760750513`*^9, 3.853363397725234*^9}},ExpressionUUID->"5d7216bc-3859-4404-abee-\ b4c48fabe4e1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"vdWEquationOfStatePressure", "[", RowBox[{"a_", ",", "b_", ",", "r_"}], "]"}], "[", RowBox[{"molarVolume_", ",", "temperature_"}], "]"}], "=", RowBox[{"pressure", "/.", RowBox[{"First", "[", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"vdWEquationOfState", "[", RowBox[{"a", ",", "b", ",", "r"}], "]"}], "[", RowBox[{"pressure", ",", "molarVolume", ",", "temperature"}], "]"}], ",", "pressure"}], "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.85336347686228*^9, 3.853363491791638*^9}}, CellLabel->"In[10]:=",ExpressionUUID->"b3f2e389-dc6e-4b10-b934-7af9efd34e47"], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{ RowBox[{"-", "a"}], " ", "b"}], "+", RowBox[{"a", " ", "molarVolume"}], "-", RowBox[{ SuperscriptBox["molarVolume", "2"], " ", "r", " ", "temperature"}]}], RowBox[{ RowBox[{"(", RowBox[{"b", "-", "molarVolume"}], ")"}], " ", SuperscriptBox["molarVolume", "2"]}]]], "Output", CellChangeTimes->{3.853363494370277*^9}, CellLabel->"Out[10]=",ExpressionUUID->"1d1362f1-4aa2-4d87-8610-4f43589f7f84"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We require that the molar volume exhibits an inflection point\ \>", "Item", CellChangeTimes->{{3.853363523968565*^9, 3.853363557106702*^9}},ExpressionUUID->"f7c41806-b144-4299-8563-\ ce059e7d78d4"], Cell[CellGroupData[{ Cell["\<\ This amounts to the first and second derivatives (w.r.t. molar volume) vanish\ \ \>", "Subitem", CellChangeTimes->{{3.853363523968565*^9, 3.853363582359413*^9}},ExpressionUUID->"fb9f34d3-1c0b-4998-9401-\ 012d0bda610b"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"extremumEquation", "=", RowBox[{"Simplify", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Derivative", "[", RowBox[{"1", ",", "0"}], "]"}], "[", RowBox[{"vdWEquationOfStatePressure", "[", RowBox[{"a", ",", "b", ",", "r"}], "]"}], "]"}], "[", RowBox[{"vCritical", ",", "tCritical"}], "]"}], "\[Equal]", "0"}], "\[IndentingNewLine]", "]"}]}]], "Input", CellChangeTimes->{{3.853363601464505*^9, 3.853363604816352*^9}}, CellLabel->"In[11]:=",ExpressionUUID->"b460a484-c8c0-4c98-934c-14cf6fe032fa"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"2", " ", "a", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"b", "-", "vCritical"}], ")"}], "2"]}], "-", RowBox[{"r", " ", "tCritical", " ", SuperscriptBox["vCritical", "3"]}]}], RowBox[{ RowBox[{"(", RowBox[{"b", "-", "vCritical"}], ")"}], " ", "vCritical"}]], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.853363608897586*^9}, CellLabel->"Out[11]=",ExpressionUUID->"a231a323-4fc3-4837-a3c8-bb053ef20eb8"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"inflectionEquation", "=", RowBox[{"Simplify", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Derivative", "[", RowBox[{"2", ",", "0"}], "]"}], "[", RowBox[{"vdWEquationOfStatePressure", "[", RowBox[{"a", ",", "b", ",", "r"}], "]"}], "]"}], "[", RowBox[{"vCritical", ",", "tCritical"}], "]"}], "\[Equal]", "0"}], "\[IndentingNewLine]", "]"}]}]], "Input", CellChangeTimes->{{3.853363617079802*^9, 3.853363621714244*^9}}, CellLabel->"In[12]:=",ExpressionUUID->"53609d8e-8ecb-4c07-8b2b-71c1d071a73f"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"3", " ", "a", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"b", "-", "vCritical"}], ")"}], "3"]}], "+", RowBox[{"r", " ", "tCritical", " ", SuperscriptBox["vCritical", "4"]}]}], RowBox[{ RowBox[{"(", RowBox[{"b", "-", "vCritical"}], ")"}], " ", "vCritical"}]], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.853363627836646*^9}, CellLabel->"Out[12]=",ExpressionUUID->"0a0587a1-6414-4511-9cb1-288bff4b11c3"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Coding comment: ", FontWeight->"Bold"], "The derivatives above were evaluated using the ", StyleBox["Derivative", FontSlant->"Italic"], " operator. This is referred to as functional differentiation, and we often \ find it to be the most natural choice in functional programming (e.g. when \ manipulating mathematical expressions).\n\nAn alternative is to use partial \ derivatives notation, which instead acts on expressions and is given by the \ ", StyleBox["D", FontSlant->"Italic"], " function in Mathematica. The examples below are provided in hope of \ demystifying the differences and similarities between the two." }], "Item", CellChangeTimes->{{3.8533636321926003`*^9, 3.853363644554522*^9}},ExpressionUUID->"04cec547-f3ef-4ac9-ad5d-\ 5d48ca00495c"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"myFunction", "[", "univariateArgument_", "]"}], ":=", RowBox[{"4", " ", SuperscriptBox["univariateArgument", "3"]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"functionalDerivative", "=", RowBox[{ RowBox[{"Derivative", "[", "1", "]"}], "[", "myFunction", "]"}]}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"functionalDerivative", "[", "x", "]"}], "\[IndentingNewLine]", RowBox[{"functionalDerivative", "[", "3", "]"}]}], "Input", CellChangeTimes->{{3.85336367166615*^9, 3.853363727425283*^9}, { 3.853364027293235*^9, 3.8533640337974367`*^9}}, CellLabel->"In[22]:=",ExpressionUUID->"e0a72177-11dc-4449-9958-7a83a44cceff"], Cell[BoxData[ RowBox[{ RowBox[{"12", " ", SuperscriptBox["#1", 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Cell[CellGroupData[{ Cell["\<\ In this form, the equation can be solved for any of the critical variables in \ terms of the other two.\ \>", "Item", CellChangeTimes->{{3.853364096314439*^9, 3.85336409978657*^9}},ExpressionUUID->"98c39f85-57e2-4195-9233-\ ef33629986f3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"criticalVolume", "=", RowBox[{"vCritical", "/.", RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"vdWEquationOfStateCritical", "[", "r", "]"}], "[", RowBox[{"pCritical", ",", "vCritical", ",", "tCritical"}], "]"}], ",", " ", "vCritical"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.853364110334344*^9, 3.8533641399026337`*^9}, { 3.8533647501903343`*^9, 3.853364751965526*^9}}, CellLabel->"In[44]:=",ExpressionUUID->"516bbe1f-50bd-444e-b774-8e2d9a7575f8"], Cell[BoxData[ FractionBox[ RowBox[{"3", " ", "r", " ", "tCritical"}], RowBox[{"8", " ", "pCritical"}]]], "Output", CellChangeTimes->{{3.853364115803385*^9, 3.853364140405221*^9}, 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