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<Text-field style="Heading 2" layout="Heading 2"><Font size="12">2.       Initial Value Problems</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font style="Text">Engineers develop mathematical models to describe processes of interest to them.  For example, the process of converting a reactant <Font mathsize="12" mathvariant="italic" mathbackground="[255,255,255]" italic="true" mathcolor="[0,0,0]" fontfamily="Times New Roman">A</Font> to a product <Font mathsize="12" mathvariant="italic" mathbackground="[255,255,255]" italic="true" mathcolor="[0,0,0]" fontfamily="Times New Roman">B </Font>in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition.  This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time.  In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs.</Font></Text-field>
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<Text-field style="Heading 2" size="12" layout="Heading 2"><Font size="12">2.1     Linear Ordinary Differential Equations</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font size="12">2.1.1   Introduction</Font></Text-field>
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<Text-field style="Text" layout="Normal">First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs).  Single first order equations can be integrated by classical methods (Rice and Do, 1995).  However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra.  In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations.</Text-field>
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<Text-field style="Heading 2" layout="Heading 2"><Font size="12">2.1.2   Homogeneous Linear ODEs</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider two linear ordinary differential equations:	 	</Text-field>
<Text-field style="Text" layout="Normal">                                                            <Image height="99" width="136" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlK0MhIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJWc4RiotJS5CT1VORFNfSEVJR0hURzYjJCIkISoqRiotJSlDSElMRFJFTkc2Ig==</Drawing-Root></Image>                                            (2.1)</Text-field>
<Text-field style="Text" layout="Normal">with the following initial conditions</Text-field>
<Text-field style="Text" layout="Normal">	 	                               <Image height="25" width="189" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlU0YhIiItJSlCT1VORFNfWUc2IyQiIz9GKi0lLUJPVU5EU19XSURUSEc2IyQiJSEqPUYqLSUuQk9VTkRTX0hFSUdIVEc2IyQiJF0jRiotJSlDSElMRFJFTkc2Ig==</Drawing-Root></Image>                                       (2.2)</Text-field>
<Text-field style="Text" layout="Normal">where y<Font subscript="true">1</Font> and y<Font subscript="true">2</Font> are the dependent variables; a<Font subscript="true">1</Font>, a<Font subscript="true">2</Font>, a<Font subscript="true">3</Font>, and a<Font subscript="true">4</Font> are constants.  This system of linear differential equations (equation(2.1)) can be written in matrix form as</Text-field>
<Text-field style="Text" layout="Normal">	 	                                            <Image height="42" width="69" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlZ0shIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJCFwRiotJS5CT1VORFNfSEVJR0hURzYjJCIkPyVGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>                                              (2.3)</Text-field>
<Text-field style="Text" layout="Normal">where the dependent variables are expressed  as a 2 x 1 matrix:</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">	 	                                             <Image height="52" width="70" zoomable="false">TUZOV3RLVWI8b2I8Uj1NRExDZE5iS0JHQ3I6UmtPeU51PVtDTlNET0U8O1pBblpARltCPltAZlo/RltBPltEPltDVlo9VlpFTltEXltEVls/Xlo+dlpAdlpFRltEPltBdjo+OmRWSmhrUGo6TFFjdG5jPmJgQXVgW1lJYz5HYk4/aVRpdU5Xc0dOcmpeW29YbkJYXFxKdnY8YHJHbmBaR1pwP3c/bnZiQVtmQHJCX3M6QV5ycWs6YWB2YWpFV11dRnQ9Zl5hdnB0Z2lpaXRzb2A6eWdXXm95QGtaXnlmRnZKQXlLZnFtPnFJXnB0R2o7T2N5aW1BQVs8Vm1UX1tbWF1JQXNLYXI+eGQ9Z19SbnBvUGp0T3haUWpdSXBqUHFnb29zb2FpbmxhaGhTSG5DT2U/R1plaGtkR2ZcXEheT15uS2FmalZwbG5obnFxblZ4dnZdaHlnUUZtTj9jc3hsQWdwVXB2bWFpSVBiUFZ2T05kcmhzTXF0UFFyUkZzbll4UVZgUmZoVVdda15jXFxPeEJnYFFmdXJvW2Q/dlVWeVZHbXhwXFxzcGFFPmtzWGVSYXg9R2JdYWVVZmNaYF9gZl9oX3Jpdnh2d1tIaVpScXZmTmdlV2hLXkE+WkNnYkhfYmhQYlpPSjE6XCJce1x9<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlK0whIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJCsoRiotJS5CT1VORFNfSEVJR0hURzYjJCIkPyZGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>                                             (2.4)</Text-field>
<Text-field style="Text" layout="Normal">The 2 x 2 coefficient matrix <Font bold="true">A</Font> is written in this case as</Text-field>
<Text-field style="Text" layout="Normal">	</Text-field>
<Text-field style="Text" layout="Normal">	 	                                            <Image height="52" width="92" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlZ0shIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJD8qRiotJS5CT1VORFNfSEVJR0hURzYjJCIkPyZGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>                                           (2.5)</Text-field>
<Text-field style="Text" layout="Normal">and <Font bold="true">Y</Font><Font subscript="true">0</Font>  is the vector (2 x 1 matrix) of initial conditions:</Text-field>
<Text-field style="Text" layout="Normal">	 	                                              <Image height="52" width="73" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlU0whIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJEkoRiotJS5CT1VORFNfSEVJR0hURzYjJCIkPyZGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>                                            (2.6)</Text-field>
<Text-field style="Text" layout="Normal">The solution of  equation 95H(2.3) is given by:[2, 3] </Text-field>
<Text-field style="Text" layout="Normal">	 	                                              <Image height="52" width="73" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlU0whIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJEkoRiotJS5CT1VORFNfSEVJR0hURzYjJCIkPyZGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>                                            (2.7)</Text-field>
<Text-field style="Text" layout="Normal">where exp(<Font mathsize="12" fontweight="bold" mathvariant="bold" bold="true" mathbackground="[255,255,255]" mathcolor="[0,0,0]" fontfamily="Times New Roman">A</Font>t) is a n x n matrix and is called the matrix exponential or the exponential matrix of A (see Appendix A for a detailed derivation of equation (2.7)).  It is of interest to note that Professor Amundson introduced the exponential matrix to chemical engineers in 1966 on page 166 of his book.<Font superscript="true">[2]</Font>  At that time it was very difficult to obtain symbolically exp(<Font bold="true">A</Font>t).  We believe that the need to find the exponential matrix (exp(<Font bold="true">A</Font>t))  has discouraged chemical engineers from using the exponential matrix to solve coupled systems of ODEs.  Maple provides a command to find the exponential matrix as a function of t, the independent variable.  Maple can also find the exponential matrix as a function of parameters such as a<Font subscript="true">1</Font>, a<Font subscript="true">2</Font>, a<Font subscript="true">3</Font>, and a<Font subscript="true">4</Font> which we will illustrate by solving some classical problems in chemical engineering.  In general, for a linear system of  <Font mathsize="12" mathvariant="italic" mathbackground="[255,255,255]" italic="true" mathcolor="[0,0,0]" fontfamily="Times New Roman">n</Font>  simultaneous coupled first order differential equations, <Font mathsize="12" fontweight="bold" mathvariant="bold" bold="true" mathbackground="[255,255,255]" mathcolor="[0,0,0]" fontfamily="Times New Roman">Y</Font> is a n x 1  matrix, <Font mathsize="12" fontweight="bold" mathvariant="bold" bold="true" mathbackground="[255,255,255]" mathcolor="[0,0,0]" fontfamily="Times New Roman">A</Font> is a n x n matrix and, again, exp(<Font bold="true">A</Font>t) is a n x n matrix.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field style="Heading 2" layout="Heading 2"><Font size="12">2.1.3  First Order Irreversible Series Reactions</Font></Text-field>
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<Text-field style="Text" layout="Normal">Consider the first order series reactions<Font superscript="true">[4] <Image height="30" width="142" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlIVsjISIiLSUpQk9VTkRTX1lHNiMkIiNJRiotJS1CT1VORFNfV0lEVEhHNiMkIiU/OUYqLSUuQk9VTkRTX0hFSUdIVEc2IyQiJCskRiotJSlDSElMRFJFTkc2Ig==</Drawing-Root></Image></Font>.  The governing equations for this reaction scheme can be written as</Text-field>
<Text-field style="Text" layout="Normal">                                                                <Image height="86" width="142" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlZ0QhIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJT85RiotJS5CT1VORFNfSEVJR0hURzYjJCIkZylGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>                                         (2.8)</Text-field>
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<Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Text-field style="Text" layout="Normal">where k<Font subscript="true">1</Font> and k<Font subscript="true">2</Font> are rate constants and the initial conditions are C<Font subscript="true">A</Font>(0) = 1 mol/l, C<Font subscript="true">B</Font>(0) = 0, and  C<Font subscript="true">C</Font>(0) = 0.  The concentration of species C (C<Font subscript="true">C</Font>(t)) at any time is given by the material balance:</Text-field>
<Text-field style="Text" layout="Normal" alignment="centred">                                                            <Image height="33" width="222" zoomable="false">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<Drawing-Root>LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIlK0MhIiItJSlCT1VORFNfWUc2IyQiIiFGKi0lLUJPVU5EU19XSURUSEc2IyQiJT9BRiotJS5CT1VORFNfSEVJR0hURzYjJCIkSSRGKi0lKUNISUxEUkVORzYi</Drawing-Root></Image>                               (2.9)                      </Text-field>
<Text-field style="Text" layout="Normal">A Maple procedure for solving equations (2.8) subject to the initial conditions is presented below.  This procedure is useful for developing the correct coefficient matrix <Font bold="true">A</Font><Font encoding="UTF-8"> to avoid \342\200\234by hand\342\200\235 errors.</Font></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field style="Text" layout="Normal"><Font bold="true">Procedure</Font></Text-field>
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<Text-field style="List Item" layout="List Item" bullet="none">1.    Start the Maple program with a 'restart' command to clear all variables.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">2.    Call Maple's linear algebra package by using the 'with(linalg)' command.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">3.    Call Maple's plotting package by using the 'with(plots)' command.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">4.    Use Maple to enter the governing equations (equation (2.8)).</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">5.    Name the variable 'vars.'</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">6.    Name the right hand side of the equations eqs.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">7.    Next, use Maple to generate the coefficient matrix (<Font bold="true">A</Font>) using Maple's 'genmatrix' command.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">8.    Store the initial conditions for the dependent variables in the vector <Font bold="true">Y0</Font>.  The first row of <Font bold="true">Y0</Font> corresponds to the initial condition for the first dependent variable (C<Font subscript="true">A</Font>).  The second row of <Font bold="true">Y0 </Font>corresponds to the initial condition for the second variable (C<Font subscript="true">B</Font>).</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">9.    Find the matrix exponential of <Font bold="true">A</Font> (i.e., exp(<Font bold="true">A</Font>t)) as a function of the parameters (rate constants) and the independent variable (t) using the Maple command 'exponential(<Font bold="true">A</Font>,t)'.  Call this matrix mat.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">10.  Next, find the solution (sol) by multiplying mat by <Font bold="true">Y0</Font> using Maple's 'evalm' command.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">11.  The first row of the sol vector corresponds to C<Font subscript="true">A(t)</Font> and the second row corresponds to C<Font subscript="true">B(t)</Font>.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">12.  Note that the analytical solutions for C<Font subscript="true">A</Font> and C<Font subscript="true">B</Font> are obtained as functions of the parameters (rate constants) and the independent variable, t.  Lower case letters are used for these variables with k<Font subscript="true">1</Font> and k<Font subscript="true">2</Font> as parameters.</Text-field>
<Text-field style="List Item" layout="List Item" bullet="none">13.  For a given set of rate constants, plot the concentration profiles.</Text-field>
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<Text-field style="Heading 3" italic="false" layout="Heading 3"><Font italic="false">Example 2.1 Irreversible Series Reactions</Font></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(linalg):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
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<Text-field style="Text" layout="Normal">Enter the governing equations as follows:</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq[1]:=diff(C[A](t),t)=-k1*C[A](t);</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkiQ0c2IjYjSSJBR0YqNiNJInRHRipGLiwkKiZJI2sxR0YqIiIiRidGMiEiIg==</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eq[2]:=diff(C[B](t),t)=k1*C[A](t)-k2*C[B](t);</Text-field>
<Text-field style="Text" layout="Normal">Store the variables in 'vars:'</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkiQ0c2IjYjSSJCR0YqNiNJInRHRipGLiwmKiZJI2sxR0YqIiIiLSZGKTYjSSJBR0YqRi1GMkYyKiZJI2syR0YqRjJGJ0YyISIi</Equation></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">vars:=[C[A](t),C[B](t)];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NyQtJkkiQ0c2IjYjSSJBR0YmNiNJInRHRiYtJkYlNiNJIkJHRiZGKQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, store the right hand sides of equations 1 and 2 in 'eqs.'</Text-field>
</Input>
</Group>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">eqs:=[rhs(eq[1]),rhs(eq[2])];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NyQsJComSSNrMUc2IiIiIi0mSSJDR0YmNiNJIkFHRiY2I0kidEdGJkYnISIiLCZGJEYnKiZJI2syR0YmRictJkYqNiNJIkJHRiZGLUYnRi8=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Now use the 'genmatrix' Maple command to produce the coefficient matrix <Font bold="true">A.</Font></Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">A:=genmatrix(eqs,vars);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Specify the initial conditions.</Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Y0:=matrix(2,1,[1,0]);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Use Maple's 'exponential(A,t)' command to produce the exponential (<Font bold="true">A</Font>t) matrix.</Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">mat:=exponential(A,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">[encoded data, 4836 characters]</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Obtain the solution vector using Maple's 'evalm' command and matrix multiplication (&amp;*).</Text-field>
</Input>
</Group>
<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">sol:=evalm(mat&amp;*Y0);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The first row of 'sol' is the solution for the concentration of species A (C<Font subscript="true">A</Font>) and the second row is for the concentration of species B (C<Font subscript="true">B</Font>).</Text-field>
</Input>
</Group>
<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">ca:=sol[1,1];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComSSNrMUdGJyIiIkkidEdGJ0YsISIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cb:=sol[2,1];</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCQqKEkjazFHNiIiIiIsJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJkkjazJHRiVGJkkidEdGJUYmISIiRjItRik2IywkKiZGJEYmRjFGJkYyRiZGJiwmRjBGMkYkRiZGMkYy</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The concentration of species C is obtained from the material balance equation.</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cc:=1-ca-cb;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCgiIiJGIy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJkkjazFHRilGI0kidEdGKUYjISIiRi8qKEYtRiMsJi1GJTYjLCQqJkkjazJHRilGI0YuRiNGL0YvRiRGI0YjLCZGNkYvRi1GI0YvRiM=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Note that the concentrations of the species are obtained as functions of the rate constants (k<Font subscript="true">1</Font> and k<Font subscript="true">2</Font>) and the independent variable, t.  To illustrate the results, enter values for the rate constants (pars).<Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L98" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Text" executable="true" layout="Normal"><Font style="Maple Input">pars:={k1=2,k2=1};</Font></Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">PCQvSSNrMkc2IiIiIi9JI2sxR0YlIiIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L28" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, substitute these values for the rate constants into the concentrations store as C<Font subscript="true">A</Font>, B<Font subscript="true">B</Font>, and C<Font subscript="true">C</Font>.</Text-field>
</Input>
</Group>
<Group labelreference="L29" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Ca:=subs(pars,ca);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComIiIjIiIiSSJ0R0YnRiwhIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L30" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cb:=subs(pars,cb);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCYqJiIiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRishIiJGJUYlKiZGJEYlLUYnNiMsJComRiRGJUYuRiVGL0YlRi8=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L31" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cc:=subs(pars,cc);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCgiIiJGIy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiI0YjSSJ0R0YpRiMhIiJGIyomRi1GIy1GJTYjLCRGLkYvRiNGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L32" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Plot the concentration profiles for these values of the rate constants as shown in Figure Exp. 2.1.1.</Text-field>
</Input>
</Group><Presentation-Block>
<Group labelreference="L135" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p1:=plot(Ca,t=0..10,thickness=3,color=blue,labels=[&quot;t(min)&quot;,&quot;Concentration (mols/l)&quot;],labeldirections=[horizontal, vertical],legend='C[A]'):</Text-field>
</Input>
</Group>
<Group labelreference="L133" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p2:=plot(Cb,t=0..10,linestyle=1,thickness=3,labels=[&quot;t(min)&quot;,&quot;Concentration (mols/l)&quot;],labeldirections=[horizontal, vertical],legend='C[B]'):</Text-field>
</Input>
</Group>
<Group labelreference="L134" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p3:=plot(Cc,t=0..10,linestyle=1,thickness=3,color=green,labels=[&quot;t(min)&quot;,&quot;Concentration (mols/l)&quot;],labeldirections=[horizontal, vertical],legend='C[C]'):</Text-field>
</Input>
</Group></Presentation-Block>
<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p4:=textplot([6,.9,k1=2.0*Unit(min^(-1))]):</Text-field>
</Input>
</Group>
<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p5:=textplot([6,.7,k2=1.0*Unit(min^(-1))]):</Text-field>
</Input>
</Group>
<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display([p1,p2,p3,p4,p5],title=&quot;Figure Exp. 2.1.1 Concentrations of A, B, and C as functions of time.&quot;);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="400" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="true">[encoded data, 28156 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L101" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L100" drawlabel="true">
<Input>
<Text-field style="Text" layout="Text Output">Suppose <Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L40" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">pars:={k1=1,k2=2};</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">PCQvSSNrMUc2IiIiIi9JI2syR0YlIiIj</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L157" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Ca:=subs(pars,ca);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJEkidEdGJyEiIg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L171" drawlabel="true">
<Input>
<Text-field style="Text" layout="Title" alignment="left">We get division by zero because we  have k<Font mathsize="12" mathvariant="normal" subscript="true" mathbackground="[255,255,255]" mathcolor="[0,0,0]" fontfamily="Times New Roman">1</Font><Font mathsize="12" mathvariant="normal" mathbackground="[255,255,255]" mathcolor="[0,0,0]" fontfamily="Times New Roman"> and k<Font subscript="true">2</Font> in the denominator of the expression for Cb.  To handle this problem, we can apply</Font><Equation executable="false" style="Text" input-equation="" display="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">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</Equation><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEqY29uc3RhbnRzRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSlhcHByb2FjaEYnRi9GMkY1LUkjbW5HRiQ2JFEiMUYnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzY3LUYsNiVRImlGJ0YvRjItRjY2LVEiLkYnRjJGOUY7Rj1GP0ZBRkNGRUZHRkotRiw2JVEiZUYnRi9GMkZlbi1GNjYtUSIsRidGMkY5L0Y8USV0cnVlRidGPUY/RkFGQ0ZFRkcvRktRLDAuMzMzMzMzM2VtRidGNS1GLDYlUSNhc0YnRi9GMkY1LUklbXN1YkdGJDYmLUYsNiVRImtGJ0YvRjItRiM2JEZPRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUY2Ni1RLSZSaWdodEFycm93O0YnRjJGOUY7L0Y+Rl9vRj9GQUZDRkUvRkhRLDAuMjc3Nzc3OGVtRicvRktGaHBGT0Y1LUY2Ni5RJGFuZEYnLyUlYm9sZEdGMUYyRjlGO0Y9Rj9GQUZDRkVGR0ZKRjVGYm9GNS1GZm82JkZoby1GIzYkLUZQNiRRIjJGJ0YyRjJGXXBGYHBGNUZjcEZPRjJGMkZlbkYy">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</Equation>  First, the 'limit' command is applied with respect to the first parameter, <Equation executable="false" style="Text" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiYtRiw2JVEia0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjFGJ0Y4RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUkjbW9HRiQ2LVEtJlJpZ2h0QXJyb3c7RidGOC8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmZuLUY+NiRRIzEuRidGOEY4</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L144" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cb:=subs(pars,cb);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiMiIiJJInRHRihGLSEiIkYvLUYkNiMsJEYuRi9GLQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Next, the 'limit' command is applied with respect to the second parameter, for <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> also:              </Text-field>
</Input>
</Group>
<Group labelreference="L167" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cc:=subs(pars,cc);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCgiIiJGIy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiI0YjSSJ0R0YpRiMhIiJGIyomRi1GIy1GJTYjLCRGLkYvRiNGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L190" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cb:=limit(cb,k1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCQqJiwmLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComSSNrMkdGKiIiIkkidEdGKkYvISIiRi8tRiY2IywkRjBGMUYxRi8sJkYuRi9GL0YxRjFGMQ==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cb:=limit(Cb,k2=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGKCEiIkYqIiIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L179" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Similarly, limits can be applied for Cc also:</Text-field>
</Input>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cc:=limit(Cc,k1=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCgiIiJGIy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiI0YjSSJ0R0YpRiMhIiJGIyomRi1GIy1GJTYjLCRGLkYvRiNGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L108" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Cc:=limit(Cc,k2=1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCgiIiJGIy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiI0YjSSJ0R0YpRiMhIiJGIyomRi1GIy1GJTYjLCRGLkYvRiNGLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L51" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">We can use these results to plot the concentration profiles for the case when k<Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L183" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p1:=plot(Ca,t=0..10,thickness=3,color=blue,labels=[&quot;t(min)&quot;,&quot;Concentration (mol/l)&quot;],labeldirections=[horizontal, vertical],legend='C[B]'):</Text-field>
</Input>
</Group>
<Group labelreference="L181" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p2:=plot(Cb,t=0..10,linestyle=1,thickness=3,color=red,labels=[&quot;t(min)&quot;,&quot;Concentration (mol/l)&quot;],labeldirections=[horizontal, vertical],legend='C[B]'):</Text-field>
</Input>
</Group>
<Group labelreference="L184" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p3:=plot(Cc,t=0..10,linestyle=1,thickness=3,color=green,labels=[&quot;t(min)&quot;,&quot;Concentration (mol/l)&quot;],labeldirections=[horizontal, vertical],legend='C[C]'):</Text-field>
</Input>
</Group>
<Group labelreference="L182" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p4:=textplot([6,.8,(k1=1.0*Unit(min^(-1)))]):</Text-field>
</Input>
</Group>
<Group labelreference="L185" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">p5:=textplot([6,.6,(k2=1.0*Unit(min^(-1)))]):</Text-field>
</Input>
</Group>
<Group labelreference="L186" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display([p1,p2,p3,p4,p5],title=&quot;Figure Exp. 2.1.2 Concentrations of A, B, and C as functions of time.&quot;);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="400" type="two-dimensional" width="400" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="true">[encoded data, 25692 characters]</Plot></Text-field>
</Output>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">We can solve for the time at which B is at its maximum by differentiating Cb with respect to t:</Text-field>
</Input>
</Group>
<Group labelreference="L58" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Eqmax:=diff(cb,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCQqKEkjazFHNiIiIiIsJiomSSNrMkdGJUYmLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJComRilGJkkidEdGJUYmISIiRiZGJiomRiRGJi1GKzYjLCQqJkYkRiZGMkYmRjNGJkYzRiYsJkYpRjNGJEYmRjNGMw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L59" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"><Font style="Text">Next solve 'eqmax' to find the time at which Cb is at a maximum in terms of the rate constants </Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L60" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">tmax:=solve(Eqmax,t);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCQqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkjazJHRikiIiJJI2sxR0YpISIiRi0sJkYsRi9GLkYtRi9GLw==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L61" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"><Font style="Text">Substitute this value for time into the concentration equation for B(cb(t)) to find the maximum value of Cb as a function of the rate constants </Font><Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
</Input>
</Group>
<Group labelreference="L62" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cbmax:=subs(t=tmax,cb);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">LCQqKEkjazFHNiIiIiIsJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKihJI2syR0YlRiYtSSNsbkdGKjYjKiZGL0YmRiQhIiJGJiwmRi9GNEYkRiZGNEY0LUYpNiMqKEYkRiZGMEYmRjVGNEYmRiZGNUY0RjQ=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L63" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The equation for 'cbmax' can be simplified further by using Maple's 'simplify' command.</Text-field>
</Input>
</Group>
<Group labelreference="L64" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">cbmax:=simplify(cbmax);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">KihJI2sxRzYiIiIiLCYpKiZJI2syR0YkRiVGIyEiIiomRilGJSwmRilGKkYjRiVGKkYlKUYoKiZGI0YlRixGKkYqRiVGLEYq</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L65" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"><Font style="Text">A maximum for the concentration of B(cbmax) for the special case of </Font><Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field>
</Input>
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<Group labelreference="L66" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="2D Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="limit(cbmax, k2 = k1);" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZILUkjbWlHRiQ2J1EmbGltaXRGJ0YvLyUnaXRhbGljR0Y5RjJGNS1JKG1mZW5jZWRHRiQ2Ji1GIzYsLUZMNidRJmNibWF4RidGL0ZPRjJGNS1GLDYvUSIsRidGL0YyRjVGNy9GO0YxRjxGPkZARkJGREZGL0ZKUSwwLjMzMzMzMzNlbUYnLUZMNidRI2syRidGL0ZPRjJGNS1GLDYvUSI9RidGL0YyRjVGN0Y6RjxGPkZARkJGRC9GR1EsMC4yNzc3Nzc4ZW1GJy9GSkZgby1GTDYnUSNrMUYnRi9GT0YyRjVGLy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0Y5RjJGNUYvRjJGNS1GLDYvUSI7RidGL0YyRjVGN0ZmbkY8Rj5GQEZCRkRGRkZhb0YvRmVvRmhvRjJGNQ==">QyQtSSZsaW1pdEc2IjYkSSZjYm1heEdGJS9JI2syR0YlSSNrMUdGJSIiIg==</Equation></Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMhIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L67" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The time needed for the concentration of B to reach a maximum, for the case when k<Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiZGKy1GIzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0YrRjg=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiZGKy1GIzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0YrRjg=</Equation>= k<Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiZGKy1GIzYkLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0YrRjg=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiZGKy1GIzYkLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0YrRjg=</Equation> can be found as</Text-field>
</Input>
</Group>
<Group labelreference="L68" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">limit(tmax,k2=k1);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUkmbWZyYWNHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiUtSSNtaUdGJDYlUSNrMUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRidGLy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRi8lKWJldmVsbGVkR1EmZmFsc2VGJw==">KiRJI2sxRzYiISIi</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L86" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Note that when k<Font subscript="true">1 </Font>= k<Font subscript="true">2</Font>, 'cbmax' is independent of the rate constants, the time taken to reach this maximum value e<Font superscript="true">-1</Font> is inversely proportional to the rate constant k<Font subscript="true">1</Font>.</Text-field>
</Input>
</Group>
</Worksheet>