Reversible first order reaction:

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html68" HREF="node5.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html74" HREF="node2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html76" HREF="node7.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node6.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="SECTION00024000000000000000">Reversible first order reaction:</A></H2> <P> So far we have completely neglected back reactions. Thus the validity of the solutions obtained so far is limited to ``initial" rates which are those seen experimentally just following the mixing of reactants. Now consider the elementary reactions <P> <P> <IMG WIDTH=280 HEIGHT=11 ALIGN=BOTTOM ALT="displaymath147" SRC="img10.gif" > <P> <P> with ``forward'' rate <P> constant <IMG WIDTH=55 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline149" SRC="img11.gif" > and <P> <P> <IMG WIDTH=280 HEIGHT=11 ALIGN=BOTTOM ALT="displaymath151" SRC="img12.gif" > <P> <P> with ``backward'' rate <P> constant <IMG WIDTH=51 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline153" SRC="img13.gif" > . In the simple case of a first order reaction in both directions, the solution valid for all times is easy to obtain. <P> The rate law is <P> <P> <IMG WIDTH=359 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath155" SRC="img14.gif" > <P> <P> where <I>kf</I> is the rate constant for the forward reaction and <I>kb</I> for the back reaction. From mass balance, we have <P> <P> <IMG WIDTH=347 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath161" SRC="img15.gif" > <P> <P> which can be used to eliminate [<I>C</I>](<I>t</I>). The solution can then be found by typing <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende1.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>{[Clear[a,a0,c0,kb,kf]; DSolve[{a'[t] == -kf*a[t] + kb*(c0 + a0 - a[t]), a[0]==a0}, a[t], t]}</PRE></td></table> <P> <I>Q: Determine the equilibrium concentration of <I>A</I> from the mathematical expression for the function. Plot and verify your answer by comparing with the exact alegbric results for the equilibrium consant (in terms of the rate constants) and the law of mass action. </I> <P> <HR><center> <table border=1> <td><A NAME="tex2html68" HREF="node5.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html74" HREF="node2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html76" HREF="node7.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node6.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>