Kinetics and Mechanism-A Games Approach Giinther Harsch Universitat Dortmund, Abteilung Chemie, 46 Dortmund 50, Federal Republic of Germany The basic questions in chemical kinetics are two: How do the concentrations of the reactants, intermediates, and products of a chemical reaction change quantitatively with time? Which events at the molecular level are responsible for these changes of concentrations? T h e first question is answered experimentally by measuring the concentrations a t various times. the temDerature usuallv being held constant throughout the &me of the reaction. ~ h second question is answered theoretically by postulating a sequence of elementary reactions that are in accordance with the stoichiometric and energetic requisites for the overall reaction and that explain the ohserved kinetic hehavior. However, the stoichiometric equation, which summarizes the elementary reactions, itself con-tains very poor information about the t w e s and sequences of these reactions. For example, A = H, whiii it sl~ecific~lly excludes mechnnisms such a6 A 2F3, could stnnd for mnnv other mechanisms, a few of which are shown in Figure 1. The standard procedure for elucidating the mechanism of an experimentally studied chemical reaction is to postulate mechanisms, to write down the rate laws for all mechanisms and substances (in Fig. 1 this has been done only for A), to integrate the rate laws in order to get the time laws and to comnare these with the exoerimental data. If it turns out that the experimental concentration versus time curves are in a r cordance with one of the theoretical time laws (for nronerlv . . . . chosen values of the rate constants k), the mechanism in auestion is said to he ~ossihle:otherwise it is definitivelv
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Chemical kinetics can never provide sufficient conditions for a definitive positive proof of a unique mechanism. This is so because the number of intermediates (the concentrations of which may he too low to he detected) is quite unlimited. Thus, the proof of a mechanism is a difficult job. students' prohlems in understanding how mechanisms are elucidated hy chemical kinetics arise from the fact that mechanisms (which tell individual molecular events) and experimental data (which tell the kinetic hehavior of macroscopic samples containing a huge numher of molecules) are correlated only in a coded way by means of a mathematical construct (rate laws and time laws). If the mathematical competences of the students are poor, students have no chance of bridging the gap between the submicroscopic level of mechanisms and the macroscopic level of concentrations versus time curves. Even if thev are well acauainted with integral calculus they often make use of i t q i i t e formally and do not realize that these deterministic eauations are the consequences of statistical laws governing molecular random events. Statistical Games for Simulallng Chemlcal Klnellcs I propose an approach to chemical kinetics and mechanism using statistical games, the prototype of which was originally developed by the well-known physicists Paul and Tatiana Ehrenfest ( I ) . The Ehrenfests used it for illustrating Boltzmann's H-Theorem. Although this context seems to he far removed from chemical kinetics a t first glance, the Ehrenfests' model has the same mathematical structure as the simple racemization reaction A s B of optical antipodes. This
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Figure 1. MeIhodoicgiCal scheme wrrelating t k macroscopic kinetic bahaviw of a reaction whose stoichiomeiricequation is A = B, with some micrwwpic mechanisms compatible with the fixed staichiomehy.
Figure 2. Mental reorganlzatian of Ehrenlests' twocontainer model in order to show ils close congenialitywith a monomolecuiar racemiration between A a B of optical antipades. is shown in Figure 2. The left part represents the Ehrenfesw' model: There are two n~ntainersA and H holdina n numtwred notps (lottery tickets). Random numhers 2 in [he range 1 5 z 5 n are produced, once every I0 s. The ticket whose number is generated immediately jumps from one container into the other. 'l'his is of course a prototype for demonstrating mass action and rquilihrium, hut it is interesting to note that neither the Ehrenfests nor Kohlrausch and Schrndinger (2).who analyzed the mathematical structurr of this model 19 years later, mentioned the analogy to chemical kinetics. The only hint for interpreting the Ehrenfest model in a context other than Roltzmann's H-'l'heurem is given by Kohlrauich and Schrodinger at the end of page 309, where their calculations arrived at a differental equation that they recognized as heing eauivalent to Smol~~chowski'swell-known euuation that governs the diffusion of a Brownian particle under the influence of an elastic force pulling it into an equilibrium position. The analo~yto chemical kinetics is better seen if we rearrange Ehrenfests' model as shown in Figurt 2: Instead of numbered tickets we take equal-sized balls without numbers. Instead of two containers A and B we only use one (reaction vessel) and store the information A or B ("state" of the original tickets) as colors (A =black, B =white) of the balls.
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Volume 61 Number 12 December 1984
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Figve 3. Kinetics of Ehrentests'twwontainer model. Tlm points were obtained by me physicists Kohlrausch and Schmdiw (1926).The cuves were calculated using eqn. (6)in Figure 1.
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Fioure 5. Kinetics of A B (basicversion). . The .ooints reoresent the averaae res~lts01 twe ~1UdBnls.The curves were calculated usmg eqn. ( I ) m Fog~re1. Note the constant half-toler, 2 = 100 lame ~nits.
systems, etc.). In all cases that I have studied in detail (3) the results of the games were in full accordance with the known analvtical solutions of the kinetic rate laws. he following six examplesdemunstrate the simplicity and validitv of the rames'a~oroach,which it is hooed will redure the problems in teachingand learning chemical kinetics since i t uses as little mathematics as possible.
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Six Examples ,m
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Figure 4. Kinetics of A B for the A-balls. Ail tries were recwded. The curve with k = 11140 is related to the basis set of rules. The curve with k = 11280 requires a mcdified "reaction rule" for drawn A-balls, reducing the probability of reaction by a factorof 0.5.
Instead of generating random numbers we put our hand into the container and draw single halls blindly and successively. Every selected hall is identified visually as an A-hall or a B-hall. Instead of jumping from one container to another, every drawn hall is replaced by a hall of the opposite color taken from stnck and mixed with the others in the container before the next ball is drawn. In effect, the ball "changes color." Instead of one jump per 10 s (absolute time counting) we specify that each try-independent of its result and its real period and frequeney-will represent 1 unit of time (relative time counting). T h e last rule is permissible because in chemical systems (under constant ambient conditions) the number of collisions hetween the particles per second is huge and constant, say, for example 1030 s-'. Therefore, we can always find a minute time unit (here 10-3''s) within which just 1collision occurs (on the average). If we identify 1collision = 1activation event (reality) with 1activation event (model) = 1try, we see a t once that the time rule is iustified bv kinetic eas theorv. The model time ( n u m k r of tiies) is a reiative timewhich is"propnnional t the real time. orovidpd that the total number of balls dues not change d&ing the reaction (which is equivalent to constant volume conditions). It is also clear that the mathematical structure of the Ehrenfest model has not been changed by this mental rearrangement. Therefore, we need not play this game in order to get data but can just take the original data generated by Kohlrausch and Schrdinger according to the Ehrenfests'lotto rules. We see (Fig. 3) that their results (points) are in accordance with the integrals of the kinetic rate laws for an equilibrium reaction A B (see eqn. (6) in Fig. 1). With n h = 100, n b = 0, and k l = k2 = 1/100, we obtain the theoretical curves and recognize a t once the obvious meaning of the rate constants k~ and kz: they measure the chance for any individual ball to be drawn in any one try-that is, the activation probability of an individual molecule per unit of time. This model may be further developed in order to cover more complex kinetics (including oscillations, instabilities in open
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Example 1: The Monomolecular Reaction A B Put 140 (black) balls A into an urn. Pull out successively and blindly single halls and determine after each try which sort of hall you have drawn. If you have drawn an A-hall, change its color to a (white)B-hall and mix this product particle back into the container. If you have drawn a B-ball, mix this useless activated particle hack into the container without changing its color. Each try (independent from its result) is counted as one unit of time. 'lhe result of this game isshow in Fiww4 (lower curve). Every single try w a rerordrd. The ahwe rules apply for k = I, 140. This value of the rate constant may Ire diminished to k = I/;! X 11140 = 112811hy specifying a new "re&ion rule" far A: If you have drawn an A-hali, flin a eoin: if it is "heads." mix A back in without chaneine .. its color. othrru.i.iechange it intoa B-ldl as urual. 'The kinetic rurve of A now falls moreslmvlv than in the former case. but the type uianalytical function = f u ) is therame in both rases. In order to show this it is conven:ent to average the resulls of, say, five students !Fig. 51 so that statistical fluctuations nearly disappear. I t is clearl~setn that the half.life is constant thn~~chout theruurseof the reaction and is equal to 100 units of time. It ;I not necessary for the eeneral intended areuments to identifv this criterion with the cxpmential iunmun. Hwrvrr, ifstudents are fnmillnr with i f they may ~alrulatrt h ratecm.
