(* 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[ 245694, 5444] NotebookOptionsPosition[ 226969, 5125] NotebookOutlinePosition[ 229795, 5178] CellTagsIndexPosition[ 229714, 5173] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["3.029 Spring 2022\[LineSeparator]Lecture 14 - 03/16/2022", "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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3.856374915535125*^9, 3.8563753956491117`*^9}, FontSize->18, CellTags-> "eq:van-der-waals",ExpressionUUID->"d75d2ef5-3406-4c92-88d6-14b428f28fdc"], Cell[CellGroupData[{ Cell["Quick note on (imo confusing) notation", "Item", CellChangeTimes->{{3.856374960486924*^9, 3.856374976166004*^9}},ExpressionUUID->"08ef85bf-c1e6-4413-ac9a-\ cd24c24acaa5"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Molar quantities ", FontSlant->"Italic"], "are intensive properties obtained by normalizing an extensive property by \ the number of moles" }], "Subitem", CellChangeTimes->{{3.856374960486924*^9, 3.856375011284904*^9}, { 3.8563751896512423`*^9, 3.856375204948391*^9}},ExpressionUUID->"08c6a37d-e43b-48b8-9524-\ 6b239006f427"], Cell[TextData[{ "They are typically represented with an ", StyleBox["underbar", FontWeight->"Bold"] }], "Subsubitem", CellChangeTimes->{{3.856374960486924*^9, 3.8563750287113543`*^9}},ExpressionUUID->"09134213-7d6b-437c-ab9a-\ 22ec510f57bb"], Cell[TextData[{ "E.g. the molar volume of pure Ni would be ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ UnderscriptBox[ StyleBox["V", "TI"], "_"], StyleBox["Ni", FontSlant -> "Plain"]], "\[LongEqual]", FractionBox[ StyleBox["V", "TI"], StyleBox["N", "TI"]], "\[LongEqual]", "7", SuperscriptBox[ StyleBox["cm", FontSlant -> "Plain"], "3"], "/", StyleBox["mol", FontSlant -> "Plain"]}], TraditionalForm], "errors" -> {}, "input" -> "\\underline{V}_{\\mathrm{Ni}}=\\frac{V}{N} = 7 \ \\mathrm{cm}^3/\\mathrm{mol}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "5b48339c-b327-4354-94e6-3645af112c4f"] }], "Subsubitem", CellChangeTimes->{{3.856374960486924*^9, 3.856375046154765*^9}, { 3.856375148344421*^9, 3.8563751579595613`*^9}},ExpressionUUID->"35d4b155-5faf-4853-87f3-\ 6091c968b035"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Partial molar quantities ", FontSlant->"Italic"], "are intensive properties obtained by taking a partial derivative of an \ extensive property, wrt to the number of moles of a component" }], "Subitem", CellChangeTimes->{{3.856374960486924*^9, 3.856375046154765*^9}, { 3.85637514958722*^9, 3.8563751783403*^9}, {3.856375210972471*^9, 3.856375269690261*^9}},ExpressionUUID->"0d417084-482c-4616-bbfe-\ fcb8a803d0a8"], Cell[TextData[{ "They are typically represented with an ", StyleBox["overbar", FontWeight->"Bold"] }], "Subsubitem", CellChangeTimes->{{3.856374960486924*^9, 3.856375046154765*^9}, { 3.85637514958722*^9, 3.8563751783403*^9}, {3.856375210972471*^9, 3.856375303241503*^9}},ExpressionUUID->"58b521d5-c1b1-4931-8cb5-\ 695979ab447a"], Cell[TextData[{ "E.g. as we saw above, the partial molar volume is given by ", Cell[BoxData[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ SubscriptBox[ OverscriptBox[ StyleBox["V", "TI"], "_"], StyleBox["j", "TI"]], "\[LongEqual]", SubscriptBox[ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", StyleBox["V", "TI"]}], RowBox[{"\[PartialD]", SubscriptBox[ 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3.856383756987796*^9, 3.8563838473234253`*^9}},ExpressionUUID->"7dd119b9-368f-46ac-9a51-\ 492703bfde81"], Cell[TextData[{ "since the two atoms are indistinguishable there are ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["N", "TI"], RowBox[{"(", RowBox[{ StyleBox["N", "TI"], "-", "1"}], ")"}], "/", "2"}], TraditionalForm], "errors" -> {}, "input" -> "N\\left(N-1\\right)/2", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "af906e25-0ed6-42ee-98c9-be9e007168af"], " ways to placing these first two atoms" }], "Subsubitem", CellChangeTimes->{{3.856383261421822*^9, 3.856383299130522*^9}, { 3.856383756987796*^9, 3.8563838869896927`*^9}},ExpressionUUID->"09400039-20d0-45b9-95d4-\ f8c14a1e7fe7"], Cell[TextData[{ "similarly, the first three atoms can be placed ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ StyleBox["N", "TI"], RowBox[{"(", RowBox[{ StyleBox["N", "TI"], "-", "1"}], ")"}], RowBox[{"(", RowBox[{ 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