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$CellContext`xB Log[$CellContext`xB $CellContext`\[Gamma]B])}; Typeset`initDone$$ = True), SynchronousInitialization->True, UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.856457675423616*^9, 3.856457743024412*^9}, { 3.856457779228714*^9, 3.856457811084856*^9}, 3.8568662497600403`*^9, { 3.856875821748414*^9, 3.856875849597477*^9}, 3.856881370635153*^9, 3.8569427948919497`*^9, 3.857470900632456*^9, {3.857477124776979*^9, 3.857477142071396*^9}, {3.85747796082514*^9, 3.857477989671341*^9}, 3.857478066471649*^9, 3.857483547662328*^9}, CellLabel->"Out[4]=",ExpressionUUID->"67e7084f-75f9-47a8-b9ee-a68480dc78a8"] }, {2}]] }, Open ]], Cell[CellGroupData[{ Cell["Common Tangent Construction", "Section", CellChangeTimes->{{3.8568769516758966`*^9, 3.856876957319303*^9}},ExpressionUUID->"d463c09f-8e72-4701-b936-\ 94a195265394"], Cell[CellGroupData[{ Cell["\<\ When the Enthalpy of mixing is positive, in the regular and general solution \ models above, there\[CloseCurlyQuote]s a competition between molecular \ interactions and the configurational entropy at low temperatures, manifesting \ in the loss of convexity\ \>", "Item", CellChangeTimes->{{3.856876960454816*^9, 3.856877051468769*^9}, 3.856877182328566*^9, {3.856942825360029*^9, 3.856942830967259*^9}, { 3.857136435106707*^9, 3.857136517161417*^9}},ExpressionUUID->"8f5a63ad-71a6-4836-8f73-\ 1997ba9ee3e3"], Cell["\<\ As we\[CloseCurlyQuote]ve seen before, a loss of convexity in thermodynamic \ potentials suggests a phase transition\ \>", "Subitem", CellChangeTimes->{{3.856876960454816*^9, 3.856877078886392*^9}, 3.8569428491103907`*^9},ExpressionUUID->"15aee468-c3d8-48d2-8ff5-\ a0c790ed141d"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ In particular, this introduces a range of compositions where the free energy \ of a single homogeneous solution is higher than the weighted-average of the \ mechanical mixture of two solutions with the same average composition\ \>", "Item", CellChangeTimes->{{3.856876960454816*^9, 3.856877045376951*^9}, { 3.856877079904628*^9, 3.85687712288941*^9}},ExpressionUUID->"982d49c1-e597-4c1b-9788-\ 12c07bbf7325"], Cell["\<\ As such, the homogenous solution separates into two homogenous solutions with \ different compositions each to reach equilibrium\ \>", "Subitem", CellChangeTimes->{{3.856876960454816*^9, 3.856877045376951*^9}, { 3.856877079904628*^9, 3.85687713696253*^9}, {3.856877183890114*^9, 3.856877184261649*^9}},ExpressionUUID->"8975b716-77c8-4871-b0f2-\ 105e439b924e"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ At a single temperature, these limiting compositions can be computed with the \ common tangent construction\ \>", "Item", CellChangeTimes->{{3.856876960454816*^9, 3.856877036546154*^9}, 3.856877106429901*^9, 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Rational[207, 1000], -0.938659986429172}, { Rational[26, 125], -0.9397468563892248}, { Rational[209, 1000], -0.9408245857136683}, { Rational[21, 100], -0.9418932277966774}, { Rational[211, 1000], -0.9429528354759088}, { Rational[53, 250], -0.9440034610402311}, { Rational[213, 1000], -0.9450451562373086}, { Rational[107, 500], -0.9460779722810432}, { Rational[43, 200], -0.9471019598588747}, { Rational[27, 125], -0.9481171691389454}, { Rational[217, 1000], -0.9491236497771303}, { Rational[109, 500], -0.9501214509239375}, { Rational[219, 1000], -0.9511106212312799}, { Rational[11, 50], -0.9520912088591255}, { Rational[221, 1000], -0.9530632614820193}, { Rational[111, 500], -0.9540268262954941}, { Rational[223, 1000], -0.9549819500223599}, { Rational[28, 125], -0.9559286789188802}, { Rational[9, 40], -0.9568670587808387}, { Rational[113, 500], -0.9577971349494943}, { Rational[227, 1000], -0.9587189523174349}, { Rational[57, 250], -0.9596325553343205}, { Rational[229, 1000], 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250], -0.9792030378278107}, { Rational[253, 1000], -0.9799245391908793}, { Rational[127, 500], -0.9806388430734339}, { Rational[51, 200], -0.9813459838129448}, { Rational[32, 125], -0.982045995430446}, { Rational[257, 1000], -0.9827389116340556}, { Rational[129, 500], -0.9834247658224362}, { Rational[259, 1000], -0.9841035910882073}, { Rational[13, 50], -0.9847754202212976}, { Rational[261, 1000], -0.9854402857122465}, { Rational[131, 500], -0.9860982197554569}, { Rational[263, 1000], -0.9867492542523937}, { Rational[33, 125], -0.9873934208147349}, { Rational[53, 200], -0.9880307507674724}, { Rational[133, 500], -0.9886612751519651}, { Rational[267, 1000], -0.9892850247289476}, { Rational[67, 250], -0.9899020299814875}, { Rational[269, 1000], -0.9905123211179043}, { Rational[27, 100], -0.9911159280746386}, { Rational[271, 1000], -0.9917128805190772}, { Rational[34, 125], -0.9923032078523425}, { Rational[273, 1000], -0.9928869392120303}, { Rational[137, 500], -0.9934641034749144}, { Rational[11, 40], -0.9940347292596061}, { Rational[69, 250], -0.994598844929174}, { Rational[277, 1000], -0.9951564785937284}, { Rational[139, 500], -0.9957076581129626}, { Rational[279, 1000], -0.996252411098661}, { Rational[7, 25], -0.9967907649171657}, { Rational[281, 1000], -0.9973227466918102}, { Rational[141, 500], -0.9978483833053159}, { Rational[283, 1000], -0.9983677014021552}, { Rational[71, 250], -0.9988807273908756}, { Rational[57, 200], -0.9993874874463967}, { Rational[143, 500], -0.999888007512267}, { Rational[287, 1000], -1.0003823133028935`}, { Rational[36, 125], -1.0008704303057354`}, { Rational[289, 1000], -1.0013523837834692`}, { Rational[29, 100], -1.0018281987761202`}, { Rational[291, 1000], -1.002297900103163}, { Rational[73, 250], -1.0027615123655966`}, { Rational[293, 1000], -1.003219059947984}, { Rational[147, 500], -1.0036705670204673`}, { Rational[59, 200], -1.004116057540752}, { Rational[37, 125], -1.0045555552560628`}, { Rational[297, 1000], 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Rational[171, 500], -1.018664060821649}, { Rational[343, 1000], -1.018845198780417}, { Rational[43, 125], -1.0190212686602695`}, { Rational[69, 200], -1.019192285812896}, { Rational[173, 500], -1.0193582654497708`}, { Rational[347, 1000], -1.0195192226431362`}, { Rational[87, 250], -1.0196751723269677`}, { Rational[349, 1000], -1.019826129297933}, { Rational[7, 20], -1.019972108216336}, { Rational[351, 1000], -1.0201131236070464`}, { Rational[44, 125], -1.0202491898604198`}, { Rational[353, 1000], -1.0203803212332052`}, { Rational[177, 500], -1.020506531849442}, { Rational[71, 200], -1.0206278357013419`}, { Rational[89, 250], -1.020744246650164}, { Rational[357, 1000], -1.020855778427074}, { Rational[179, 500], -1.0209624446339967`}, { Rational[359, 1000], -1.0210642587444525`}, { Rational[9, 25], -1.0211612341043899`}, { Rational[361, 1000], -1.021253383932998}, { Rational[181, 500], -1.0213407213235177`}, { Rational[363, 1000], -1.021423259244036}, { Rational[91, 250], 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Rational[387, 1000], -1.021995922505044}, { Rational[97, 250], -1.0219623373736377`}, { Rational[389, 1000], -1.021924248392589}, { Rational[39, 100], -1.0218816654170713`}, { Rational[391, 1000], -1.0218345981956385`}, { Rational[49, 125], -1.021783056370777}, { Rational[393, 1000], -1.0217270494794306`}, { Rational[197, 500], -1.0216665869535357`}, { Rational[79, 200], -1.021601678120539}, { Rational[99, 250], -1.0215323322039125`}, { Rational[397, 1000], -1.0214585583236582`}, { Rational[199, 500], -1.0213803654968083`}, { Rational[399, 1000], -1.02129776263792}, { Rational[2, 5], -1.0212107585595582`}, { Rational[401, 1000], -1.0211193619727768`}, { Rational[201, 500], -1.0210235814875896`}, { Rational[403, 1000], -1.0209234256134376`}, { Rational[101, 250], -1.0208189027596482`}, { Rational[81, 200], -1.0207100212358846`}, { Rational[203, 500], -1.0205967892525991`}, { Rational[407, 1000], -1.0204792149214659`}, { Rational[51, 125], -1.0203573062558182`}, { Rational[409, 1000], 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Rational[54, 125], -1.0161503464544883`}, { Rational[433, 1000], -1.015922404277362}, { Rational[217, 500], -1.0156903028738995`}, { Rational[87, 200], -1.01545404774467}, { Rational[109, 250], -1.0152136443017072`}, { Rational[437, 1000], -1.0149690978687849`}, { Rational[219, 500], -1.0147204136816819`}, { Rational[439, 1000], -1.014467596888451}, { Rational[11, 25], -1.0142106525496715`}, { Rational[441, 1000], -1.0139495856387104`}, { Rational[221, 500], -1.0136844010419659`}, { Rational[443, 1000], -1.0134151035591161`}, { Rational[111, 250], -1.0131416979033583`}, { Rational[89, 200], -1.0128641887016414`}, { Rational[223, 500], -1.0125825804949014`}, { Rational[447, 1000], -1.0122968777382846`}, { Rational[56, 125], -1.0120070848013687`}, { Rational[449, 1000], -1.011713205968383}, { Rational[9, 20], -1.0114152454384184`}, { Rational[451, 1000], -1.0111132073256357`}, { Rational[113, 250], -1.0108070956594704`}, { Rational[453, 1000], -1.01049691438483}, { Rational[227, 500], 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Rational[153, 250], -0.9787053226388565}, { Rational[613, 1000], -0.9784633949216411}, { Rational[307, 500], -0.9782189533924466}, { Rational[123, 200], -0.977971987981213}, { Rational[77, 125], -0.977722488509594}, { Rational[617, 1000], -0.9774704446903808}, { Rational[309, 500], -0.9772158461269245}, { Rational[619, 1000], -0.9769586823125481}, { Rational[31, 50], -0.976698942629948}, { Rational[621, 1000], -0.9764366163505914}, { Rational[311, 500], -0.9761716926341042}, { Rational[623, 1000], -0.9759041605276484}, { Rational[78, 125], -0.9756340089652957}, { Rational[5, 8], -0.9753612267673888}, { Rational[313, 500], -0.9750858026398967}, { Rational[627, 1000], -0.9748077251737584}, { Rational[157, 250], -0.9745269828442218}, { Rational[629, 1000], -0.9742435640101706}, { Rational[63, 100], -0.9739574569134449}, { Rational[631, 1000], -0.9736686496781498}, { Rational[79, 125], -0.9733771303099576}, { Rational[633, 1000], -0.9730828866953984}, { Rational[317, 500], 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