Second order reactions: II

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html58" HREF="node4.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html64" HREF="node2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html66" HREF="node6.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node5.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="SECTION00023000000000000000">Second order reactions: II</A></H2> <P> For an elementary bimolecular reaction involving two species, <P> <P> <IMG WIDTH=327 HEIGHT=15 ALIGN=BOTTOM ALT="displaymath131" SRC="img6.gif" > <P> <P> the rate law may be formulated in two (entirely equivalent) ways: <P> <P> <IMG WIDTH=311 HEIGHT=35 ALIGN=BOTTOM ALT="displaymath133" SRC="img7.gif" > <P> <P> or <P> <P> <IMG WIDTH=311 HEIGHT=35 ALIGN=BOTTOM ALT="displaymath135" SRC="img8.gif" > <P> <P> We only have one differential equation and can from that only obtain one function, not both <I>A</I> and <I>B</I> independently. The additional constraint we need is the mass balance. Assuming the stoichiometric coefficients are <I>a</I> = <I>b</I>= 1, we have <P> <P> <IMG WIDTH=330 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath143" SRC="img9.gif" > <P> <P> and can use that to eliminate <I>B</I>[<I>t</I>] from the differential equation. Applying DSolve to the result <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send6.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>ans=DSolve[{a'[t] == -k*a[t]*(b0-a0 +a[t]), a[0]==a0}, a[t], t]</code></td></table> <P> Mathematica provides the answer in the form of a replacement rule. To get the result in a format that can be fed into Mathematica for evaluation, you must evaluate the rule. At the same time, it is convenient to make <B>a[t]</B> into a function. Both of these tasks can be accomplished with the command <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send7.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>a[t_]=a[t]/.ans</code></td></table> <P> which tells Mathematica to create the function <B>a[t]</B> using the rule specified by <B>ans</B>. If this is not self-evident please review the syntax of such ``patterns", ``function definitions" and ``replacement rules" in sections 1.6 and 1.7 of MSE, and in the introductory exercises! <P> <I>Q: Plot the concentration as a function of time.</I> <P> <I>Q: Do this part over for a reaction in which species A combines with two other A's, giving third order kinetics for the destruction of A. First state the rate law, and then use Mathematica to solve and plot data for a typical case. </I> <P> <HR><center> <table border=1> <td><A NAME="tex2html58" HREF="node4.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html64" HREF="node2.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html66" HREF="node6.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node5.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>