(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 65141, 2103]*) (*NotebookOutlinePosition[ 65953, 2132]*) (* CellTagsIndexPosition[ 65909, 2128]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["Differential Equations for Chemical Kinetics", FontSize->18]], "SectionFirst"], Cell[TextData[{ "A wide variety of chemical reactions can be modeled with coupled (often \ nonlinear) differential equations. These equations describe the time \ evolution of the concentrations of the various chemical species: reactants, \ intermediaries, catalysts, and products. Such problems are quite simple to \ set up and solve with ", StyleBox["Mathematica", FontSlant->"Italic"], ". The function ", StyleBox["NDSolve", FontFamily->"Courier"], " numerically integrates the differential equations that arise. The \ resulting concentrations can be plotted as a function of time and can also be \ used to accurately compute the expected concentration of the molecular \ species." }], "Text"], Cell[CellGroupData[{ Cell["Reaction with an intermediate", "Subsection"], Cell[TextData[{ "In the reaction ", Cell[BoxData[ \(TraditionalForm\`A + 2 B\[LongRightArrow]R + S\)]], ", the mechanism proceeds through an intermediate species ", Cell[BoxData[ \(TraditionalForm\`X\)]], "." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{GridBox[{ { RowBox[{"A", "+", RowBox[{"B", UnderoverscriptBox[GridBox[{ {"\[LongLeftArrow]"}, {"\[LongRightArrow]"} }, RowSpacings->-1], StyleBox["2", FontSize->9], StyleBox["1", FontSize->9]], "X"}]}]}, { RowBox[{"X", "+", RowBox[{"B", OverscriptBox["\[LongRightArrow]", StyleBox["3", FontSize->9]], "R"}], "+", "S"}]} }, ColumnAlignments->{Left}], StyleBox["}", SpanMaxSize->Infinity]}], "\[DoubleLongRightArrow]", "A"}], "+", \(2 B\[LongRightArrow]R\), "+", "S"}]], "NumberedEquation", AutoStyleOptions->{"UnmatchedBracketStyle"->None}], Cell["The rate equations corresponding to this reaction are", "Text"], Cell[BoxData[GridBox[{ {\(\[PartialD]\_t a[t] \[Equal] \ \(-k\_1\)\ a[t]\ b[t] + k\_2\ x[t]\)}, {\(\[PartialD]\_t b[t] \[Equal] \ \(-k\_1\)\ a[t]\ b[t] + k\_2\ x[t] - k\_3\ b[t]\ x[t]\)}, {\(\[PartialD]\_t x[t] \[Equal] k\_1\ a[t]\ b[t] - k\_2\ x[t] - k\_3\ b[t]\ x[t]\)} }, RowSpacings->2, ColumnAlignments->{"\[Equal]"}]], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`a[t]\)]], ", ", Cell[BoxData[ \(TraditionalForm\`b[t]\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`x[t]\)]], " represent the concentrations of the species ", Cell[BoxData[ \(TraditionalForm\`A\)]], ", ", Cell[BoxData[ \(TraditionalForm\`B\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`X\)]], " at a time ", Cell[BoxData[ \(TraditionalForm\`t\)]], " at a constant temperature. The parameters ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`k\_3\)]], ", the rate constants, are temperature dependent. If the temperature of the \ mixture of reactants is allowed to change, then an additional set of \ equations is needed to model the temperature variation of ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`k\_3\)]], " with time." }], "Text"], Cell[TextData[{ "Although these equations do not have an explicit analytic solution, they \ can be numerically integrated using the built in numerical differential \ equation solver, ", StyleBox["NDSolve", FontFamily->"Courier"], ". Here is the solution for a particular choice of the rate constants ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`k\_3\)]], ". In this example the initial concentrations of ", Cell[BoxData[ \(TraditionalForm\`A\)]], " and ", Cell[BoxData[ \(TraditionalForm\`B\)]], " are equal and that of the intermediary ", Cell[BoxData[ \(TraditionalForm\`X\)]], " is zero." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(k\_1 = 1/10;\)\), "\[IndentingNewLine]", \(\(k\_2 = 1/10;\)\), "\[IndentingNewLine]", \(\(k\_3 = 10;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(ndsolution = \[IndentingNewLine]\(NDSolve[{\[IndentingNewLine]\ \[PartialD]\_t a[t] \[Equal] \(-k\_1\)\ a[t]\ b[t] + k\_2\ x[t], \[PartialD]\_t b[ t] \[Equal] \(-k\_1\)\ a[t]\ b[t] + k\_2\ x[t] - k\_3\ b[t]\ x[t], \[PartialD]\_t x[t] \[Equal] k\_1\ a[t]\ b[t] - k\_2\ x[t] - k\_3\ b[t]\ x[t], \[IndentingNewLine]a[0] \[Equal] 1, \[IndentingNewLine]b[0] \[Equal] 1, \[IndentingNewLine]x[0] \[Equal] 0}, \[IndentingNewLine]{a, b, x}, {t, 0, 10}]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\)}], "Input", CellLabel->"In[1]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"a", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"b", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"x", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}}, "<>"]\), False, Editable->False]}]}], "}"}]], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[TextData[{ "The result is expressed in terms of a set of replacement rules that give \ the functions as ", StyleBox["InterpolatingFunction", FontFamily->"Courier"], " objects. The notation ", Cell[BoxData[ \(TraditionalForm\`InterpolatingFunction[{{0.`, 10. }}, "<>"]\)], FontFamily->"Courier"], ", for example, is a shorthand for the numerical information needed by the \ interpolation algorithm to reproduce the solution." }], "Text"], Cell[TextData[{ "Here is plot of the result. The concentration of ", Cell[BoxData[ \(TraditionalForm\`X\)]], " was multiplied by a factor of 50 so that it can be seen in the graph on \ the same scale as ", Cell[BoxData[ \(TraditionalForm\`A\)]], " and ", Cell[BoxData[ \(TraditionalForm\`B\)]], "). 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The initial concentrations of ", Cell[BoxData[ \(TraditionalForm\`A\)]], " and ", Cell[BoxData[ \(TraditionalForm\`B\)]], " are equal and the rate constants ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], "chosen as shown." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(k\_1 = 1\/10;\)\), "\n", \(\(k\_2 = 1\/20;\)\), "\n", \(ndsolution = \[IndentingNewLine]\(NDSolve[{\[IndentingNewLine]\ \[PartialD]\_t a[t] \[Equal] \(-k\_1\)\ a[t]\ x[ t], \[IndentingNewLine]\[PartialD]\_t b[ t] \[Equal] \(-k\_2\)\ b[t]\ y[ t], \[IndentingNewLine]\[PartialD]\_t x[ t] \[Equal] \(-k\_1\)\ a[t]\ x[t] + k\_2\ b[t]\ y[t], \[IndentingNewLine]\[PartialD]\_t y[ t] \[Equal] k\_1\ a[t]\ x[t] - k\_2\ b[t]\ y[t], \[IndentingNewLine]a[0] == 1, \[IndentingNewLine]b[0] == 1, \[IndentingNewLine]x[0] \[Equal] 1, \[IndentingNewLine]y[0] \[Equal] 0}, \[IndentingNewLine]{a, b, x, y}, {t, 0, 250}]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\)}], "Input", CellLabel->"In[6]:="], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"a", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 250.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"b", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 250.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"x", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 250.`}}, "<>"]\), False, Editable->False]}], ",", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 250.`}}, "<>"]\), False, Editable->False]}]}], "}"}]], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[{ "Here is plot of the result. The colors of the curves for ", Cell[BoxData[ \(TraditionalForm\`A\)]], ", ", Cell[BoxData[ \(TraditionalForm\`B\)]], ", ", Cell[BoxData[ \(TraditionalForm\`X\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`Y\)]], " are black, red, green, and blue respectively." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[Evaluate[{a[t], b[t], x[t], y[t]} /. ndsolution], {t, 0, 250}, PlotStyle \[Rule] {{AbsoluteThickness[2], RGBColor[0, \ 0, \ 0]}, {AbsoluteThickness[2], RGBColor[ .7, \ 0, \ 0]}, {AbsoluteThickness[2], RGBColor[0, \ .7, \ 0]}, {AbsoluteThickness[2], RGBColor[0, \ 0, \ .7]}}, Axes \[Rule] False, Frame \[Rule] True, PlotLabel -> StyleForm[ A\ StyleForm["\< B\>", FontColor \[Rule] RGBColor[ .7, 0, 0]]\ StyleForm["\< X\>", FontColor \[Rule] RGBColor[0, .7, 0]] StyleForm["\< Y\>", FontColor \[Rule] RGBColor[0, 0, .7]], FontFamily \[Rule] "\", FontWeight \[Rule] "\"]];\)\)], "Input", CellLabel->"In[9]:="], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica 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