Sequential first order reactions including backreactions

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html116" HREF="node10.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html120" HREF="node8.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html122" HREF="node12.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node11.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="SECTION00033000000000000000">Sequential first order reactions including backreactions</A></H2> <P> Your program can now be modified easily to include the backreactions. The reverse rate coefficients are denoted <I>km</I>1 and <I>km</I>2. <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende6.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>Clear[h,nsteps,k1,k2,km1,km2,a,b,c,data]; totaltime=100; h=0.5; nsteps = Round[totaltime/h]; k1=1.0 ; k2=0.1 ; km1=0.05; km2=0.1; a[0]=1; b[0]=0; c[0]=0; Do[ { a[n] = a[n-1] + h*( km1*b[n-1] - k1*a[n-1]), b[n] = b[n-1] + h*( k1*a[n-1] + km2*c[n-1] - k2*b[n-1] - km1*b[n-1]), c[n] = c[n-1] + h*( k2*b[n-1] - km2*c[n-1]) }, {n,1,nsteps,1}] //N; data = Table[{a[n], b[n], c[n]}, {n, 0, nsteps}]</PRE></td></table> <P> <I>Q: How much difference do the backreactions make? Investigate this questions for several sets of values of the forward and backward rate constants.</I> You might be tempted to get an analytical answer to this problem with <B>DSolve</B>. But think twice before you attempt that. The calculation takes over an hour and the output, even after invoking <B>Simplify</B>, takes about 250 lines. Hardly informative or useful. <P> <I>Q: Derive (by hand, or using Mahematica) a steady state approximation for this reaction, i.e. apply the assumption <P> <P> <IMG WIDTH=296 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath221" SRC="img25.gif" > <P> <P> to get a simplified rate law. </I> Hint: You can use the expression for <I>d</I>[<I>B</I>]/<I>dt</I> to isolate [<I>B</I>] and then eliminate [<I>B</I>] from the expression for <I>d</I>[<I>A</I>]/<I>dt</I>. <I>Q: What is the resulting rate law and what is its solution [<I>A</I>](<I>t</I>)? Carry out computations using the steady state approximation for differing values of the rate constants and compare results with your numerical calculations for eh exact reaction scheme by plotting the two on the same graph. Can you give a simple set of rules indicating the range of validity of the steady state approximation?</I> <P> <I>Q: A more powerful numerical approach (but one that is hidden from view.... which is not always good): Skim Chapter 9 in Bahder: Section 9.5 (pp585 et. seq.) discusses numerical solutions to sets of ODEs..... Looking at the infromation on MSE pp 586 set up and execute the solution of one of the previous sets of ODEs from this lab but now using the Mathematica function NDSolve[ ]. Compare your results with those using the Euler method. </I> <P> <HR><center> <table border=1> <td><A NAME="tex2html116" HREF="node10.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html120" HREF="node8.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html122" HREF="node12.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node11.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>