Jay Martin Anderson Bryn Mawr College Bryn Mawr. Pennsylvania 19010
The scientific method demands a comparison between the result of experiment and the theory or model contrived for the physical situation under ohsewation. Simulation provides one method of comparison between the theory, model, or mechanism for a chemical reaction and the time-hy-time measurements of concentrations of species undergoing that reaction. Outside the usual scientific methodology of ohsevation and hypothesis formation, simulation can provide useful insights. Simulation may afford a glimpse of the consequences when the experiment may he too long, too difficult, or too risky, as in the fate of fluorocarbons or nitrogen oxides in the stratosphere. Simulation has proven to he very useful in systems in which there is both a physical (or chemical or biological) component and a societal component, as in environmental policy analysis. In this paper we shall examine one tool for system simulation, some chemical examples to which the tool can he applied, and some generalizations from these examples to nonchemical systems.
Computer Simulation in Chemical Kinetics For example, one would represent the reaction step reactant -product by the flow diagram
When the mechanism for this reaction ster, is estahlished (or hypothesized! the dependence of the rate upon cuncentrutionsof reactants is shown with broken lines. For examde, a reaction step first-order in reactant would be shown h$
Simulation Using System Dynamics
The one tool which will he discussed and used in this paper is System Dynamics, a technique for the description and analysis of general systems developed by Professor J. W. Forrester a t M.I.T. ( I , 2). Svstem Dvnamics deniands that our model include rates of tiansformkion and levels of accumulation of material, the de~endenceof each rate on levels (concentrations, partial pr&ures), and parameters quantify the model (rate constants). System Dynamics lends itself naturally to flowcharting and digital computation; the DYNAMO language (3) has grown up to accommodate the technique. I t is helpful to use the flow-charting notation of DYNAMO, if not the computer language itself. This notation is shown in Figure 1. We see there that a box and the letter L represent a level, the accumulation or depletion of material, and the result of integrating differential equations. The valve symhol and the letter R represent a rate, and a heavy line the flow of material. Dashed lines indicate functional dependence ( R depends on L, R is a function of L ) , or, more broadly, information links. A bar and the letter C indicate a constant, and a circle and the symbol A indicate an auxiliary function helpful in defining a rate.
This diagram corresponds to writing
A flow diagram alone cannot easily illustrate reaction orders, howeve;. The same diagram would he used to represent a reaction step second-order in reactant, save possibly for namine the rate constant K2 rather than K1. Bv adiustinethe -~~ relative magnitude of k.b and k b , the student A n seethe conseouences of a ranid first sten or a r a ~ i second d steD. .. and begin develop theLideaof theAsteady-state. The ~edaeoeical issue to he faced is whether simulation. in . replacing the calculus in chemistry, does a disservice to the student or a service. I contend that it is a service, for it gives him chemical answers a t once while providing an incent&e to d e v e l o ~his mathematical Drowess for the rigorous analysis of thisand similar prohle&s. The wide variety of output devices available to computer users admit numerous possibilities for display of results. In our "recipe" ( 4 ) , we rely primarily on plots made by a line printer (as does DYNAMO), hut have also used CalComp graphics in a hatch-mode environment and the CalCompcompatible TSP plotter in a time-sharing environment. Figure 3 was produced on the TSP device. u
w
The lodine Clock Reaction
As an example of a more complex chemical reaction, we consider the familiar iodine clock reaction. eauations and ahbre\.inted flow diagram for which are ahownin Figure 4. This fieure shows onlv the flow of atoms and not the rnerhanism orthe reaction. A diagram with a simplified mechanism is constructed below. A discussion of this reaction as a lecture-demonstration has frequently appeared in this Journal Presented at the 169th National Meeting, American Chemical Society, April 7-11, 1975,Philadelphia, Pennsylvania. Volume 53,Number 9. September 1976 / 561
3H,S03
A
Figwe 2. Con-utive and flow diagram.
..--..* KBC
,
j_.-.l
first&
10---3504'+ 3
a
I - + 3 FhO
C
B '_..--.
-
slow
+
I
reactions: mechanisms, differential q m t i o n ~ .
Figure 4. action.
Mechanism and abbreviated flow diagram for me iodine clock r e
Figure 5.
Simplified flow diagram for the icdine clock reaction
Figure 3. Simulation of the model shown in Figure 2. The cancentrations (arbitrary scale) of A (reactant). B (intermediate), and C (prcduct) as a function of time (arbitrary scale).
( 5 ) , and a brief discussion of the kinetics has been given by Yost and Russell (6). Our concern here is with modeling and simulation, and not with precise values of rate constants. Consequently, we can make use of the experimental finding that the second (iodine-producing) and third (iodine-consuming)steps are rapid in order to simplify the model. Were this simplification not possible, the time interval of integration would he much smaller, the computer time required much larger, hut little new information would he availahle. We are relying on a very crude numerical method for carrying out the integration, but methods for accommodating fast steps of reaction mechanisms without simplification are available (7,s)and have been applied to several examples. The simplification we use is t o suggest that, when sulfite (sulfurous acid) falls to a very low value, reaction 3 stops altoeether. and reaction 2 takes d a c e in an instant. The result is ;hat a pulse of iodine atoms are transferred from the iodate and iodide surer to the iodine state all at once. The resulting flow diagram is shown in Figure 5. Here the rates of flow are shown with the avvn,vriatc stiochiometric coefficients.There are only two independent rates: RED, the slow rate of reduction of iodate to iodide by sulfite (step 1on Figure 4), and IZP, the pulse of iodine atoms transferred when the "clock" goes off. The CLIP function compares the current level of sulfite to a critical minimum level of sulfite to trigger the pulse. A simulation of this reaction is shown in Figure 6 in the form of an annotated printer-plot of the concentrations of lo3-, I-, 12, and S o ? versus time. I t should he clear that the utility of this model and simulation lies primarily in aiding a student to conceptualize the multistep reaction process, and only secondarily in establishing quantitative parameters descriptive of the scheme. 562 / Journal of Chemical Education
TIME Figxe 6. Simulation of the &I M. o < [sos2-I 4x 20 s.
