First order reaction with no backreaction

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html96" HREF="node8.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html102" HREF="node8.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html104" HREF="node10.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node9.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="SECTION00031000000000000000">First order reaction with no backreaction</A></H2> <P> Using the Euler method which was derived in the lecture notes, a short program can be written to evaluate the concentration over a grid in time. Here a function <IMG WIDTH=31 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline193" SRC="img24.gif" > is first defined and then the Table function is used to generate the concentrations up to a time of 100. The time step size <I>h</I> is set to 1 so 100 steps are required. To make it easier to change the time step size and/or the total length of time, the number of timesteps is evaluated from the totaltime and stepsize. <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende4.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>Clear[k,h,nsteps,a,n,data]; k=0.01; totaltime=100; h=1; nsteps = Round[totaltime/h]; a[n_] := a[n] = a[n-1] - k*h*a[n-1] ; a[0] = 1; data = Table[{ n*h, a[n] }, { n, 0, nsteps }];</PRE></td></table> <P> Here the function <I>a</I>[<I>n</I>] is defined by the recursion relation obtained by the finite difference approximation of the first derivative in the rate law. The initial concentration of <I>A</I> is here chosen to be 1. The <B>Table</B> function evaluates the function and creates a list of the values. The generated list can be graphed using <B>ListPlot</B> <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send12.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>ListPlot[data, PlotJoined -> True ]</code></td></table> <P> <I>Q: Repeat the calculation with smaller and larger values for the step size, <I>h</I>, to find what range will give accurate results to, say, three significant figures. Compare the numerical calculations with the analytical solution by plotting them on the same graph.</I> <P> <HR><center> <table border=1> <td><A NAME="tex2html96" HREF="node8.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html102" HREF="node8.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html104" HREF="node10.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node9.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>