(************** 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[ 23522, 701]*) (*NotebookOutlinePosition[ 24187, 724]*) (* CellTagsIndexPosition[ 24143, 720]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Correction to the Debye-H\[UDoubleDot]ckel Limiting Law", "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 excellent book by ", StyleBox["S. I. Sandler, Chemical and Engineering Thermodynamics, 3rd \ Edition, Wiley 1999 (Illustration 9.1-8 page 654).", FontWeight->"Bold"] }], "Subsubtitle", Background->RGBColor[0.773449, 0.996109, 0.996109]], Cell[BoxData[ \(Off[General::"\"]\)], "Input"], Cell[CellGroupData[{ Cell["\<\ Activity coefficients of HCl in aqueous hydrochloric acid solutions versus \ HCl molality at 25\[Degree]C. \ \>", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(\(tbl1 = {{0.0005, 0.975}, {0.001, 0.965}, {0.005, 0.928}, {0.01, 0.904}, {0.05, 0.83}, {0.1, 0.796}, {0.5, 0.757}, {1.0, 0.809}, {3. , 1.316}, {5. , 2.38}, {8. , 5.90}, {10. , 10.44}, {12. , 17.25}, {14. , 27.3}, {16. , 42.4}};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(tbl2 = Transpose[{Map[Sqrt[#[\([1]\)]] &, \ tbl1], Map[Log[#[\([2]\)]] &, \ tbl1]}]\)], "Input"], Cell[BoxData[ \({{0.022360679774997897`, \(-0.025317807984289897`\)}, \ {0.03162277660168379`, \(-0.03562717764315116`\)}, {0.07071067811865475`, \ \(-0.07472354619593642`\)}, {0.1`, \(-0.10092591858996053`\)}, \ {0.22360679774997896`, \(-0.18632957819149348`\)}, {0.31622776601683794`, \ \(-0.22815609313775398`\)}, {0.7071067811865476`, \(-0.2783920255446883`\)}, \ {1.`, \(-0.2119563619236453`\)}, {1.7320508075688772`, 0.2745968329031255`}, {2.23606797749979`, 0.8671004876833833`}, {2.8284271247461903`, 1.7749523509116738`}, {3.1622776601683795`, 2.3456445824544927`}, {3.4641016151377544`, 2.847812143477369`}, {3.7416573867739413`, 3.3068867021909143`}, {4.`, 3.7471483622379123`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(plt1 = ListPlot[tbl2, PlotStyle \[Rule] \ {PointSize[0.02], RGBColor[1, 0, 0]}, DisplayFunction \[Rule] \ Identity]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Values of the parameters (\[Alpha], \[Beta] and a) appearing in the Debye-H\ \[UDoubleDot]ckel limiting law and its corrections. \ \>", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(\[Alpha] = 1.178; a = 4; \[Beta] = 0.3291;\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Corrections to the Debye-H\[UDoubleDot]ckel limiting law. We find \ \[Delta]=0.278995 and \[Beta]a=1.77879. \ \>", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[BoxData[ \(\(LogGamTheo = Map[\(-\[Alpha]\)\ \@#/\((1 + \[Beta]a\ \@#)\) + \[Delta]\ # &, \ \(Transpose[tbl1]\)[\([1]\)]];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol1 = FindMinimum[ Sum[\((tbl2[\([i, 2]\)] - LogGamTheo[\([i]\)])\)^2, {i, 1, 15}], {\[Delta], 0.5}, {\[Beta]a, 1.5}]\)], "Input"], Cell[BoxData[ \({0.058210275560314335`, {\[Delta] \[Rule] 0.2789953932699924`, \[Beta]a \[Rule] 1.7787895313686757`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ " \[Beta]a obtained using experimental data is equal to 1.77879. This value \ is close to 1.3164 obtained from Tabulated values of \[Beta] and for an \ average radius of hydration a=4 ", Cell[BoxData[ FormBox[ OverscriptBox["A", StyleBox["o", FontSize->9]], TraditionalForm]]], "." }], "Subsubtitle", Background->RGBColor[1, 1, 0.65098]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Beta]\ a\)], "Input"], Cell[BoxData[ \(1.3164`\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(plt2 = Plot[\(-\[Alpha]\)\ x/\((1 + \[Beta]a\ x)\) + \[Delta]\ x^2 /. sol1[\([2]\)], {x, 0, 4}, PlotStyle \[Rule] \ RGBColor[0, 0, 1], DisplayFunction \[Rule] \ Identity]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Debye-H\[UDoubleDot]ckel limiting law ", "Subsubtitle", Background->RGBColor[1, 1, 0.658824]], Cell[CellGroupData[{ Cell[BoxData[ \(plt3 = Plot[\(-\[Alpha]\)\ x, {x, 0, 4}, PlotStyle \[Rule] \ RGBColor[0, 1, 0], DisplayFunction \[Rule] \ Identity]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Plots of experimental data of activities versus HCl molality and the various \ theoretical models. 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