Sequential reactions without backreaction

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html78" HREF="node6.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html82" HREF="node2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html84" HREF="node8.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node7.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><H2><A NAME="SECTION00025000000000000000">Sequential reactions without backreaction</A></H2> <P> Again, for the moment, ignoring back reactions, consider the system of (elementary) transformations: <P> <P> <IMG WIDTH=280 HEIGHT=11 ALIGN=BOTTOM ALT="displaymath167" SRC="img16.gif" > <P> <P> rate constant <IMG WIDTH=53 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline169" SRC="img17.gif" > <P> <P> <IMG WIDTH=281 HEIGHT=11 ALIGN=BOTTOM ALT="displaymath171" SRC="img18.gif" > <P> <P> rate constant <IMG WIDTH=53 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline173" SRC="img19.gif" > . An ``intermediate" <I>B</I> is created and then consumed during the course of the overall transformation of <I>A</I> into <I>C</I>. This mechanism leads to the rate equations: <P> <P> <IMG WIDTH=526 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath181" SRC="img20.gif" > <P> <P> This is a set of three coupled differential equations for the three functions <IMG WIDTH=92 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline183" SRC="img21.gif" > and [<I>C</I>](<I>t</I>) . DSolve can handle this. It is convenient to group the three equations and the three initial conditions into one lump <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende2.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>Clear[a,b,c,t,k1,k2]; equations = { a'[t] == -k1*a[t], b'[t] == k1*a[t] - k2*b[t], c'[t] == k2*b[t], a[0] == 1.0, b[0] == 0, c[0] == 0 }; answer=DSolve[equations, {a[t], b[t], c[t]}, t]</PRE></td></table> <P> Again, it is convenient to get the solution in a format that can be fed directly into Mathematica for evaluation <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende3.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>a[t_]=a[t]/.answer; b[t_]=b[t]/.answer; c[t_]=c[t]/.answer;</PRE></td></table> <P> <I>Plot the three functions for a few different values of the rate constants. Try for example: <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send8.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>k1=1.0;</code></td></table> <P> and <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send9.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>k2=0.1;</code></td></table> <P> (note that <IMG WIDTH=13 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline187" SRC="img22.gif" > must not be equal to <IMG WIDTH=13 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline189" SRC="img23.gif" > in the solution given by Mathematica).</I> You can construct a plot with all three concentrations together on the same plot by typing <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send10.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>plABC = Plot[{a[t],b[t],c[t]}, {t,0,50}, PlotRange -> {0,1}]</code></td></table> <P> Here a plot object is created and stored under the name `plABC'. The plot can be displayed, printed and manipulated later by referring to that name. To display the plot, type <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send11.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>Show[plABC]</code></td></table> <P> <I>Q: How does the concentration of the intermediate <I>B</I> depend on the ratio of the rate constants? </I> <P> <HR><center> <table border=1> <td><A NAME="tex2html78" HREF="node6.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html82" HREF="node2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html84" HREF="node8.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node7.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I>Hannes Jonsson<BR> Modified by Thomas L. Marchioro II<BR> and the <A NAME="tex2html1" HREF="/UCES/">Undergraduate Computational Engineering and Science project</A> </I> </ADDRESS> </BODY>