(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 58462, 1711]*) (*NotebookOutlinePosition[ 59126, 1734]*) (* CellTagsIndexPosition[ 59082, 1730]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Activity Coefficients in Solvated Mixtures:\nChemical and \ Physical Interactions", "Subsubtitle", FontSize->18, FontVariations->{"CompatibilityType"->0}], StyleBox["\n", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Author's Data", FontSize->14, FontWeight->"Bold"], StyleBox[": Housam Binous", FontSize->14], StyleBox["\n", TextAlignment->Center, FontFamily->"MS Shell Dlg", FontSize->8.5, Background->RGBColor[0.605478, 0.996109, 0.605478]], StyleBox["Department of Chemical Engineering\nNational Institute of Applied \ Sciences and Technology\nTunis, TUNISIA\nEmail: binoushousam@yahoo.com ", FontSize->14, FontWeight->"Plain"] }], "Title", TextAlignment->Center, Background->RGBColor[0.605478, 0.996109, 0.605478]], Cell[TextData[{ StyleBox["Acknowledgement :", FontWeight->"Bold"], "\nThis problem was presented in the paper by H. G. Harris and J. M. \ Prausnitz, Thermodynamics of Solutions with Physical and Chemical \ Interactions, Solubility of Acetylene in Organic Solvents, I&EC Fundamentals, \ Vol 8, NO. 2, May 1969, pp 180-188", StyleBox[".", FontWeight->"Bold"] }], "Subsubtitle", Background->RGBColor[0.773449, 0.996109, 0.996109]], Cell[BoxData[ \(Off[General::"\"]\)], "Input"], Cell[TextData[{ "We consider the reaction of formation of complex AB:\n\t\t\t", Cell[BoxData[ \(TraditionalForm\`A + B = AB\)]], "\n(component 2 is component A)" }], "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[CellGroupData[{ Cell["\<\ Activities of components A, B and AB as a fuction of \"true\" mole fractions \ and \"true\" activity coefficients.\ \>", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[CellGroupData[{ Cell[BoxData[ \(a\_A = \(\[Gamma]\_A\) z\_A\)], "Input"], Cell[BoxData[ \(z\_A\ \[Gamma]\_A\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a\_B = \(\[Gamma]\_B\) z\_B\)], "Input"], Cell[BoxData[ \(z\_B\ \[Gamma]\_B\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a\_AB = \(\[Gamma]\_AB\) z\_AB\)], "Input"], Cell[BoxData[ \(z\_AB\ \[Gamma]\_AB\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Agebraic rearrangment to get the extend of complex formation, \[Xi].\ \>", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[CellGroupData[{ Cell[BoxData[ \(sol = Solve[{z\_B \[Equal] \ \((x\_1 - \[Xi])\)/\((1 - \[Xi])\), z\_A \[Equal] \ \((x\_2 - \[Xi])\)/\((1 - \[Xi])\), z\_AB \[Equal] \ \[Xi]/\((1 - \[Xi])\), K\[Gamma] \[Equal] \ \[Gamma]\_AB/\((\[Gamma]\_A\ \[Gamma]\_B)\), K \[Equal] \ a\_AB/\((a\_A\ a\_B)\)}, {\[Xi]}, {z\_A, z\_B, z\_AB, \[Gamma]\_AB, \[Gamma]\_A\ , \[Gamma]\_B}] // Simplify\)], "Input"], Cell[BoxData[ \({{\[Xi] \[Rule] \(K\[Gamma] + K\ x\_1 + K\ x\_2 - \@\(\(-4\)\ K\ \((K + \ K\[Gamma])\)\ x\_1\ x\_2 + \((K\[Gamma] + K\ x\_1 + K\ x\_2)\)\^2\)\)\/\(2\ \ \((K + K\[Gamma])\)\)}, {\[Xi] \[Rule] \(K\[Gamma] + K\ x\_1 + K\ x\_2 + \@\(\ \(-4\)\ K\ \((K + K\[Gamma])\)\ x\_1\ x\_2 + \((K\[Gamma] + K\ x\_1 + K\ \ x\_2)\)\^2\)\)\/\(2\ \((K + K\[Gamma])\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Xi] = \(sol[\([1, 1, 2]\)] /. x\_1 + x\_2 \[Rule] 1\) /. K\ x\_1 + K\ x\_2 \[Rule] \ K // Simplify\)], "Input"], Cell[BoxData[ \(\(K + K\[Gamma] - \@\(\(-4\)\ K\ \((K + K\[Gamma])\)\ x\_1\ x\_2 + \((K\ \[Gamma] + K\ x\_1 + K\ x\_2)\)\^2\)\)\/\(2\ \((K + K\[Gamma])\)\)\)], \ "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\"True\" activity coefficients using van Laar equations", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Gamma]\_A = Exp[v\_A/\((R\ T)\) \((\[Alpha]\_\(A - B\)\ \[CapitalPhi]\_B^2 + \ \[Alpha]\_\(A - AB\)\ \[CapitalPhi]\_AB^2 + \((\[Alpha]\_\(A - B\) + \[Alpha]\ \_\(A - AB\) - \[Alpha]\_\(B - AB\))\)\ \[CapitalPhi]\_B\ \ \[CapitalPhi]\_AB)\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(v\_A\ \((\[Alpha]\_\(A - AB\)\ \ \[CapitalPhi]\_AB\%2 + \((\[Alpha]\_\(A - AB\) + \[Alpha]\_\(A - B\) - \ \[Alpha]\_\(\(-AB\) + B\))\)\ \[CapitalPhi]\_AB\ \[CapitalPhi]\_B + \ \[Alpha]\_\(A - B\)\ \[CapitalPhi]\_B\%2)\)\)\/\(R\ T\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Gamma]\_B = Exp[v\_B/\((R\ T)\) \((\[Alpha]\_\(A - B\)\ \[CapitalPhi]\_A^2 + \ \[Alpha]\_\(B - AB\)\ \[CapitalPhi]\_AB^2 + \((\[Alpha]\_\(A - B\) + \[Alpha]\ \_\(B - AB\) - \[Alpha]\_\(A - AB\))\)\ \[CapitalPhi]\_A\ \ \[CapitalPhi]\_AB)\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(v\_B\ \((\[Alpha]\_\(A - B\)\ \[CapitalPhi]\_A\%2 \ + \((\(-\[Alpha]\_\(A - AB\)\) + \[Alpha]\_\(A - B\) + \[Alpha]\_\(\(-AB\) + \ B\))\)\ \[CapitalPhi]\_A\ \[CapitalPhi]\_AB + \[Alpha]\_\(\(-AB\) + B\)\ \ \[CapitalPhi]\_AB\%2)\)\)\/\(R\ T\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Gamma]\_AB = Exp[\((v\_B + 0.75\ v\_A)\)/\((R\ T)\) \((\[Alpha]\_\(A - AB\)\ \ \[CapitalPhi]\_A^2 + \[Alpha]\_\(B - AB\)\ \[CapitalPhi]\_B^2 + \((\[Alpha]\_\ \(A - AB\) + \[Alpha]\_\(B - AB\) - \[Alpha]\_\(A - B\))\)\ \[CapitalPhi]\_A\ \ \[CapitalPhi]\_B)\)]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\(\((0.75`\ v\_A + v\_B)\)\ \((\[Alpha]\_\(A - AB\)\ \ \[CapitalPhi]\_A\%2 + \((\[Alpha]\_\(A - AB\) - \[Alpha]\_\(A - B\) + \ \[Alpha]\_\(\(-AB\) + B\))\)\ \[CapitalPhi]\_A\ \[CapitalPhi]\_B + \[Alpha]\_\ \(\(-AB\) + B\)\ \[CapitalPhi]\_B\%2)\)\)\/\(R\ T\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPhi]\_B = z\_B\ v\_B/\((z\_A\ v\_A + z\_B\ v\_B + z\_AB\ v\_AB)\)\)], "Input"], Cell[BoxData[ \(\(v\_B\ z\_B\)\/\(v\_A\ z\_A + v\_AB\ z\_AB + v\_B\ z\_B\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPhi]\_A = z\_A\ v\_A/\((z\_A\ v\_A + z\_B\ v\_B + z\_AB\ v\_AB)\)\)], "Input"], Cell[BoxData[ \(\(v\_A\ z\_A\)\/\(v\_A\ z\_A + v\_AB\ z\_AB + v\_B\ z\_B\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPhi]\_AB = z\_AB\ v\_AB/\((z\_A\ v\_A + z\_B\ v\_B + z\_AB\ v\_AB)\)\)], "Input"], Cell[BoxData[ \(\(v\_AB\ z\_AB\)\/\(v\_A\ z\_A + v\_AB\ z\_AB + v\_B\ z\_B\)\)], \ "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "\"True\" mole fractions of each species in terms of the extend of \ reaction, \[Xi], and the \"apparent\" mole fractions, ", Cell[BoxData[ \(TraditionalForm\`x\_1\ and\ x\_2\)]], "." }], "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(z\_B = \((x\_1 - \[Xi])\)/\((1 - \[Xi])\); z\_A = \ \((x\_2 - \[Xi])\)/\((1 - \[Xi])\); z\_AB = \[Xi]/\((1 - \[Xi])\);\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Van Laar parameters for physical interactions", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(\(\[Alpha]\_\(A - B\) = \(\(\((\[Delta]\_A - \[Delta]\_B)\)^2 /. \ \[Delta]\_A \[Rule] \ \[Delta]\_B - \@\[Alpha]\) /. \[Alpha] \[Rule] \ R\ T/v\_B\) /. \[Delta]\_B \[Rule] \ 6.5;\)\)], "Input"], Cell[BoxData[ \(\(\[Delta]\_AB = \(\(\((\((\[Delta]\_A^2\ v\_A + \[Delta]\_B^2\ v\_B)\)/ v\_AB)\)^0.5 /. \[Delta]\_A \[Rule] \ \[Delta]\_B - \@\ \[Alpha]\) /. \[Alpha] \[Rule] \ R\ T/v\_B\) /. \[Delta]\_B \[Rule] \ 6.5;\)\)], "Input"], Cell[BoxData[ \(\(\[Alpha]\_\(A - AB\) = \(\(\((\[Delta]\_A - \[Delta]\_AB)\)^2 /. \ \[Delta]\_A \[Rule] \ \[Delta]\_B - \@\[Alpha]\) /. \[Alpha] \[Rule] \ R\ T/v\_B\) /. \[Delta]\_B \[Rule] \ 6.5 // Simplify;\)\)], "Input"], Cell[BoxData[ \(\(\[Alpha]\_\(B - AB\) = \(\(\((\[Delta]\_B - \[Delta]\_AB)\)^2 /. \ \[Delta]\_A \[Rule] \ \[Delta]\_B - \@\[Alpha]\) /. \[Alpha] \[Rule] \ R\ T/v\_B\) /. \[Delta]\_B \[Rule] \ 6.5;\)\)], "Input"], Cell[BoxData[ \(\(\[Delta]\_A = \ \(\[Delta]\_B - \@\[Alpha] /. \[Delta]\_B \[Rule] \ 6.5\) /. \[Alpha] \[Rule] \ R\ T/v\_B;\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\"Apparent\" activity coefficientsof component 2", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(\(x\_1 = 1 - x\_2;\)\)], "Input"], Cell[BoxData[ \(\(\[Gamma]\_2 = z\_A\ \[Gamma]\_A/x\_2;\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Molar volumes, gas constant and temperature", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(v\_A = 100; v\_B = 100; v\_AB = v\_B + 0.75\ v\_A;\)], "Input"], Cell[BoxData[ \(\(R = 1.987;\)\)], "Input"], Cell[BoxData[ \(\(T = 298.15;\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Computing \"apparent\" activity coefficient versus mole fraction for different values of the equilibrium constant K=5, 10 and 0.2.\ \>", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(\(K = 5;\)\)], "Input"], Cell[BoxData[ \(For[i = 0, i < \ 101, {x\_2 = i\ 0.01 + 10^\(-5\), sol = FindRoot[ K\[Gamma] == \[Gamma]\_AB/\((\[Gamma]\_A\ \[Gamma]\_B)\), {K\ \[Gamma], 0.1}], Gam5[i] = \[Gamma]\_2 /. sol, \(i++\)}]\)], "Input"], Cell[BoxData[ \(\(tbl5 = Table[{i\ 0.01, Gam5[i]}, {i, 0, 100}];\)\)], "Input"], Cell[BoxData[ \(\(K = 10;\)\)], "Input"], Cell[BoxData[ \(For[i = 0, i < \ 101, {x\_2 = i\ 0.01 + 10^\(-5\), sol = FindRoot[ K\[Gamma] == \[Gamma]\_AB/\((\[Gamma]\_A\ \[Gamma]\_B)\), {K\ \[Gamma], 0.1}], Gam10[i] = \[Gamma]\_2 /. sol, \(i++\)}]\)], "Input"], Cell[BoxData[ \(\(tbl10 = Table[{i\ 0.01, Gam10[i]}, {i, 0, 100}];\)\)], "Input"], Cell[BoxData[ \(\(K = 0.2;\)\)], "Input"], Cell[BoxData[ \(For[i = 0, i < \ 101, {x\_2 = i\ 0.01 + 10^\(-5\), sol = FindRoot[ K\[Gamma] == \[Gamma]\_AB/\((\[Gamma]\_A\ \[Gamma]\_B)\), {K\ \[Gamma], 0.1}], Gam02[i] = \[Gamma]\_2 /. sol, \(i++\)}]\)], "Input"], Cell[BoxData[ \(\(tbl02 = Table[{i\ 0.01, Gam02[i]}, {i, 0, 100}];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Plotting the \"apparent\" activity coefficient versus mole fraction for different values of the equilibrium constant K=5, 10 and 0.2. 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