Robert Griswold
and John F. Haughl Lebanon Valley College Annville, Pennsylvania
I
I
Analog Computer Simulation A n experiment in chemical kinetics
This paper describes analog computer simulation of an experiment in chemical kinetics. The particular experiment described is representative of a general type of experiment performed, with variations, over the past five years by students in the physical chemistry course a t Lebanon Valley College. The authors have previously described a similar experimenL2 The emphasis in this paper will be on the chemical aspects of the application of computer techniques to a specific problem in chemical kinetics. The design and function of analog computers have heeu described in detail elsewhere (1-3). I n addition, a number of papers dealing with the application of analog computers to problems of interest to chemists, have appeared in recent years (4-7). Analog computer simulation of the study of the kinetics of a complex chemical reaction provides one way whereby students can carry out an investigation with a degree of sophistication suitable for a course in physical chemistry a t the third year level. Such an approach is especially desirable in view of the considerable exposure to the study of elementary reaction kinetics students now receive, both in classroom and laboratory, during the first two years of college chemistry. Advantages of Analog Simulation
A major advantage of analog simulation arises from the fact that analog computers are capable of solving several differentialequations simultaneously. When a step-wise mechanism is postulated for a complex reaction, differential equations may be written describing the rate of change in the concentration of each reactant, intermediate, or product. Because the analog computer can simultaneously solve these equations, it is unnecessary to use the steady state assumption as is frequently done in approximate solutions. Another advantage lies in the fact that a large amount of interesting data can be obtained quite rapidly. Because of the nature of the computer, it is possible to follow the changes, with time, in the concentration of each reactant, intermediate, or product. Each run takes only a few minutes. Thus, using the analog method, a student can simulate a large number of experiments in a laboratory period of two or three hours. Data on concentration versus time for any or all reaction mixture components can be conveniently preserved in the form of chart recordings. Present address:' Department of Chemistry, Gordon college, Wenham. Massachusetts019R4. A similar paper on this subject was presen~edby the n u r l w r t hefore the 1)ivision of Clwniral K d ~ ~ m t iam t tlae 147th S a r i o n a l Meeting of t h p A m ~ r i c wC'l~en~~rnl Swirly, Pldadelphm, April 7 , 1Bfrl.
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Some advantages of the use of aualog as compared to digital computers might also be mentioned. In addition to the greater speed with which analog computers can perform the operations necessary for the experiment described here, there is the advantage of simplicity. A student can learn sufficient analog computer techniques with very little instruct,ion. Also, analog computers suitable for the type of experiment described here are considerably less expensive than digital computers. Summary of the Experiment
Student investigations into the theory of complex chemical reaction mechanisms can be designed to make good use of the advantages of the analog simulation approach. In the particular investigation described in this paper, there were four purposes to be achieved within the framework of one step-wise reaction mechanism postulated for one particular reaction. ?'lit. first, ~lud n~njor,purpw:e of the experiment was to studv the m:iuricr in which the overall snccific rate constant and the orders of reaction with respect to each reactant were dependent on the relative magnitudes of the specific rate constants for each step in the postulated mechanism. The magnitudes of the step-wise rate constants constituted the variable factor in the experiment since the same step-wise sequence of reactions was used by all students in all determinations. The experiment can be designed, by the instructor, so that with suitable choice of step-wise rate constants, a variety of overall rate constants and orders of reaction may be observed. Usually, two sets of step-wise rate constants were assigned t o each student. One of these sets was used by all students in the class, and the other set was different for each student. A second purpose of the experiment was a study of the conditions under which the steady state assumption may or may not be valid for the step-wise mechanism undrr considcr:ition. The ~ ~ t d conipurcr ng can be uscd in thc study of kinetic.; problem.: without making use of t,he steady state assumption. Graphs of concentration against time can he obtained for any postulated reaction intermediate. From such graphs, it can immediately be seen whether or not the concentration of the intermediate quickly reaches and maintains a constant, low concentration, as assumed in the steady state approach. A third purpose of the experiment is concerned with the relationship between order and molecularity. This purpose is related to the first, and is intended to focus the attention of the student on the fact that, although the molecularity of the overall reaction and of the individual steps is the same in all parts of the experiment, the overall order of the reaction may vary aa the relation
between the rate constants for the steps in t,he reaction is varied. A fourth and final purpose of the experiment is to introduce the student to a method of considerable value in research. For example, in studies on kinetics and mechanism of chemical reactions, the techniques used in the experiment described here could easily be applied to test the validity of a newly postulated step-wise mechanism. Reaction and Mechanism Investigated
As stated above, this experiment is set within the framework of one reaction, and one postulated step-wise mechanism. The reaction chosen was A+B+3C
(1)
The step-wise mechanism postulated was
I is a reaction intermediate. It may be noted that the above reaction has the same stoichiometric relationship between reactants and product as found for the reaction between nitrogen pentoxide and nitric oxide to give nitrogen dioxide. Also, the mechanism postulated above has the same form and stoichiometry as the mechanism for the nitrogen pentoxide-nitric oxide reaction postulated by Smith and Daniels (X),and further investigated by Nlills and Johnston (9). This latter mechanism was, in fact, the basis for the experiment originally described by the author^.^ The use of equations such as (1-4), using only letters to represent reactants, intermediates, and products, has the advantage of greater generality. The experiment need not be designed to conform with the experimental aspects of any known reaction. Thus, in the present case, the effect of varying the magnitudes of the step-wise rate constants can be studied independently of other effects such as, for example, variation in pressure, which has been found to cause variation in the order of the reaction of nitrogen pentoxide with nitric oxide (9).
scribed mathematically. The mathematical terms in rate equations are particularly adaptable to analog simulation. Concentrat,ions or pressures of reaction mixture components, or their simple multiples, may be represented by voltages. A variety of initial concentrations can be set for reactants, and even intermediates and products, if desired, by suitable choice of initial voltage conditions applied to integrating operational amplifiers. The input voltages to the integrators are related mathematically to the differential terms describing rates of change in concentration. Multiplication of concentrations by a constant may be accomplished by operational amplifiers, coefficient potentiometers, or both. Rate constants, or their simple multiples, are most convenient,lyrepresented by settings of coefficient potentiometers. Rlultiplication of two variable concentrations is performed by XY multipliers. The various components of the computer are represented symbolically in Figure 1. Operational amplifiers are represented by triangles, multipliers by labelled rectangles, and coefficient potentiometers by circles. The operational amplifiers numbered 1 4 are integrating amplifiers. The outputs of the integratprs, as indicated in Figure 1, are the concentrations of the reaction mixture components. Other concentration analogs of voltages, and various combinations of them, are indicated at various points in Figure 1, which is drawn in such a way that the operations performed by the various computer components can be followed around the circuit, and related back to eqns. (5-8). The program shown in Figure 1 requires an analog computer with at least twelve operational amplifiers.' As already stated, amplifiers 1-4 are integrators. The number of other amplifiers needed is determined to some extent by the nature of the XY multipliers used. The ones used are of the "quarter-square" variety, and re-
Programming the Computer
Figure 1 shows an analog computer program designed to simultaneously solve the differential equations which describe the rates of change in the concentrations of reactants A and B, the intermediate I, and the product C, shown in the eqns. (1-4). These differential equations, which are based on eqns. ( 2 4 ) , are
+ kdI)(C)
+d(A)/dl -d(C)/dt
=
-kdA) -kl(A)
-d(I)/dt
=
-kl(A)
-d(B)/dl
=
+ka (I)@)
=
+ kl (I)(C) - 2k3 (I)@)
+ kn (I)(C) + kr (I)(B)
(5) (6)
(7)
(8)
The signs used in eqns. (5-8) are written to give maximum economy ip the use of computer components. Thus, the same terms with the same signs are found in more than one equation. An analog computer program is designed so that certain properties of the computer or its components can be used as analogs of the properties of a system being de-
Figure 1. Analog computer program for the rimultaneovs solution of oqnr Ratios of feedback to input impedances ore unity, unless otherwise indicated.
(5-8).
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/ 577
quire as inputs plus and minus values of the two variables to be multiplied. Thus, operational amplifiers 5-7 serve simply as inverters. Amplifiers 8 and 9, which are used without input or feedback impedances, are also required by the design of the particular multipliers used. Amplifiers 10 and 11 are needed to partially modify the output of the multipliers, which, for the particular type used, is always -0.01 times the product of multiplication. Amplifier 12 is required by the magnitude and sign of the third term on the right in eqn. (6). It is possible that a smaller computer could he used for an experiment of this type, although the results may not be quite the same. Assuming use of the same kind of multipliers as described here, amplifiers 10 and 11 could be eliminated to accommodate the program to a smaller computer. However, the results obtainable would be severely limited unless an alternate method could be used to perform the functions of amplifiers 10 and 11. This limitation arises from the fact that the output of the multipliers must be multiplied by 100 to get the desired product of two concentrations to use as inputs to amplifiers 1-4. Amplifiers 10 and 11 each multiply by 10. Multiplication by another factor of 10 is accomplished by setting coeficient potentiometers 2 and 3 for 10 times the value required to represent stepwise rate constants k2 and ka. The requirement that some coefficientpotentiometers be set for 10 times the desired k value restricts the extent to which step-wise rate constants can differ from each other without using values that would give impractically low rates of simulated reaction. Elimination of amplifiers 10 and 11 would require that coefficient potentiometers be set to 100 times the ,value of a step-wise constant, making the restriction even more serious. For the sets of rate constants used in this experiment, amplifiers 10 and 11 are essential. Other modifications of the program shown in Figure 1 are also possible. For example, amplifiers 10 and 11 could be eliminated for adaptation to a smaller computer if the required tenfold multiplication were effected by a tenfold reduction of the input impedances on the integrating amplifiers, 1-4. Such a procedure would result in changing all of the signs shown in Figure 1, and in eqns. (5-8). Effecting a tenfold multiplication in this alternate fashion also makes it possible to increase the range of variability in the step-wise rate constants, if amplifiers 10 and 11 are kept in the program. However, the authors have found that such a degree of variability is not necessary to show the desired effects of change in orders of reaction resulting from variation of step-wise constants. The program shown in Figure 1 was used without the modification described in the preceding paragraph to preserve the advantage of permitting students to more easily relate Figure 1 to the functions of the computer components. Description of the Computer Used
The analog computer used by the authors was built from kits supplied by the Heath Company of Benton Harbor, Michigan. I t is one of the "ES" seriesa and has fifteen operational amplifiers. The "quarter-
' Currently available from Daystrom,
tmio, Canda.
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Journal of Chemical Education
Ltd., Cooksville, On-
square" type XY multipliers were from the Donner Division of Systron-Donner Corporation, of Concord, California. Experimental Procedure
Experimental procedure in this simulated kinetics experiment is based on the simultaneous solution of equs. (5-8) using an analog computer programed as shown in Figure 1. Solutions are made by integration and are generally obtained by recording the output voltages of the integrating amplifiers as functions of time, using a meter, an oscilloscope, or a chart recorder. The variation of these output voltages with time is analogous to the variation, with time, of the concentration of any component of a reaction mixture. The experiment was preceded by background preparation with lectures, assigned reading, and a prior experiment which required students to design, set up, and test analog programs based on the rate equations for simple first and second order reactions. Students were expected to be able to write differential equations such as eqns. (5-8) for any step-wise mechanism, and to relate the equations to a program such as given in Figure 1. However, they were not expected to be able to set up complex programs. Before starting the experiment, students were expected to have devised a definite plan for solving the assigned experimental problem. In line with the major purpose of the experiment as described above, the goal of the experimental procedures was to obtain data from which the overall rate constants and orders of reaction could be determined, using two different assigned sets of step-wise constants. The other purposes of the experiment could also be achieved by interpretation of these same data. Most students chose an initial rate method for runnine the exneriment. and data were collected bv simultaneonsly solving eqns. ( 5 8 ) as functions of time for a number of different initial concentrations. The initial rate approach is desirable for two reasons. The first of these arises from the fact that the computer used is most reliable when working with the relatively high voltages related to the initial portion of a concentration versus time curve. Another significant reason can be seen from the mechanism upon which the experiment is based. Only the initial rates can be expected to be free from the effects of intermediates and products. Since the computer was programed in advance by the instructor, procedures carried out by students consisted essentially of setting desired initial concentration analogs, setting coefficient potentiometers to represent assigned step-wise rate constants, pushing the "operate" switch, and collecting concentration versus time data, usually in the form of chart recordings. These procedures could be repeated as often as necessary to collect the data required by the students' previously planned approach.
-
Treatment of Data
The treatment of data described here is based on the initial rate method. This method has been found to be most suitable for this simulated kinetics study, and will be the only one discussed in this paper. Data on initial rates were obtained by striking tangents a t zero time on the concentration versus time
graphs obtained from a chart recorder. Graphs for either reactant, or for the product, could be used for initial rate determination except in cases where very rapid initial changes made it difficult t o determine the slope accurately. Equations suitable for the treatment of initial rate data are easily derived. A general expression for the overall rate of reaction, R, may be written R
=
k(A).(By
(9
where k is the overall specific rate constant, and a and b are the orders of reaction with respect to each of the reactants in eqn. (1). For the initial rate, Ro,and init,ial concentrations, (A)o and (B)o, eqn. (9), upon taking logarithms, becomes log R, = log k
+ a log (Ah + b log (B)o
(10)
~nitialrate data may be treated to obtain the overall rate constant and the orders of reaction with respect to each reactant by the application of eqns. (11-14), which are derived from eqn. (10). For example, for data from runs in which the initial concentration of A was varied and the initial concentration of I3 held oonstant
+ constant
(11)
+ b lag (B)o
(12)
log Ro = a log (A),
where constant
=
log k
Similarly, when the initial concentration of B was varied, and the initial concentration of A held oonstant
+ constant
(13)
+ a log (A)o
(14)
log Ro = b log (B)o
where constant
=
log k
Equations (11) and (13) are linear equations, and the orders of reaction, a and b, can be readily determined from the slopes of appropriate graphs of the logarithms of initial rates versus the logarithms of initial concentrations. The overall rate constant, k , can be calculated from the intercept of such a graph, using eqn. (12) or (14). The units used in the treatment of data were "machine" units. I n keeping with the analog principle, "machine" concentrations were represented by voltages, as stated earlier, and "machine" time units were arbitrary, often being represented simply by the units of recorder chart paper. These "machine" units could easily be converted to molar concentrations or gas pressures, and real time units, if desired.
TIME Figvre 2. Representotire curves of concentrotions of all readion components versus time, bored 12-Al on the flrrt set of step-wire rote constant. shown in Teble 1, on4 12-8) on the second set of conrtonts.
first set was used by all members of the class, and the second set is typical of constants chosen to give orders of reaction and overall rate constants different than would be obtained with the first set. Representative concentration versus time curves for reactants, intermediates, and products are shown in Figure 2. The differencesbetween Figures 2-A and 2-B arise from the use of the two different sets of step-wise rate constants shown in Table 1. Some typical results of treating initial rate data taken from concentration versus time curves, such as those shown in Figure 2, are presented in Figure 3 and Table 2. When logarithms of initial rates are plotted against logarithms of initial concentration in accord with eqns. (11) and (13), the graphs shown in Figure 3 are obtained for each of the two sets of step-wise constants. The orders of reaction determined, and the overall rate constants calculated using eqns. (12) and (I&),are summarized in Table 2 for both sets of step-wise rate con-
Results of the Experiment
Some typical student results will now be presented. These results were obtained by treating initial rate data as described in the preceding section. These data resulted from carrying out the experiment using the two sets of step-wise rate constants shown in Table 1. The
LOGARITHM OF INITIAL CONCENTRATION Figvre 3. Logarithms of initial rates verrvr logarithms of initial concentrotions of reactants, bored (3-A) on the flrst set of step-wise rote conrtantr shown in Table I , and 13-81 on the second set d conrtonh.
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stants. Two values of the overall rate constant are obtained for each set of step-wise constants, one by eqn. (12) and the other by eqn. (14). Table 2 shows good agreement between the two determinations of k for each set of step-wise constants. The two sets of results presented in Table 2 show the achievement of the principal aims of the experiment. As can be seen, when the experiment is performed using the first set of step-wise rate constants, the reaction shown in eqn. (1) was found to be first-order with respect t o reactant A, and approximately zero-order with respect to reactant B. Using the second set of constants, the reaction was found to be approximately firstorder with respect t o each reactant. I n both cases presented, the same step-wise mechanism has been used, thus leaving the molecularity of each step unchanged. However, by changing the step-wise rate constants, different orders of reaction, as well as differentvalues of the overall rate constant have been observed. Some interesting conclusions regarding the applicability of the steady state approximation can be made by examination of the concentration versus time curves for the intermediate, I, in Figure 2. In Figure 2-A, the concentration of the intermediate is seen to rapidly reach a low, practically constant concentration. Such a low, constant concentration of intermediate is not observed in Figure 2-B. Thus, for the step-wise mechanism shown in eqns. ( 2 4 ) , the steady state assumption would be valid for the first set of step-wise rate constants shown in Table 1,but not for the second set. I n the case, described in the preceding par&graph, where the steady state assumption was supported by experimental observation, a rate law derived on t h e basis of the steady state assumption should be the same as the rate law determined experimentally. Agreement between derived and observed rate laws may be easily demonstrated. Applying the steady state assumption to eqns. ( 2 4 ) , the rate law derived is
Equation (15) predicts first-order with respect to reactant A in the initial part of the reaction where the concentration of the product C is negligible. As shown in Table 2, the predicted first-order behavior was found when the experiment was performed using the first set of step-wise rate constants. The importance of working with initial rates in this experiment is seen in the deviations from first-order behavior observed after the initial part of the concentration versus time curves in Figure 2-A. Careful examination shows that successive small fractional lifetimes become progressively longer when, for strict first-order, they should be constant. Such deviation is predicted by eqn. (15), and may be attributed to the increase in concentration of the product C. Since the results illustrated in Figure 2-B do not show the conditions necessary for the steady state assumption, the rate law derived from the steady state assumption would not be expected to agree with the experimental rate law. As can be seen, eqn. (15) fails completely to predict those results presented in Table 2, which are based on the use of the second set of step-wise constants and the initial rate method. Comparison of the predictions of eqn. (15) with the results shown in Table 2 for both sets of step-wise rate constants, clearly demonstrates the limitations of the steady state approximation, as applied to the study of the kinetics and mechanism of complex reactions. In this experiment, even though the same sequence of steps for the reaction mechanism was always used, the validity of the steady state assumption was found to be dependent on the magnitudes of the step-wise rate constants. Literature Cited
(1) KORN,G. A., AND KORN,T . M., "Electronic Analog Com~uters,"( Z r d &.I, McGraw-Hill Book Co., Inc., New York, 1956. (2) KORN,G. A,, AND KORN,T. M., "Electronic Andolog and Hybrid Computers," McGraw-Hill Book Co., Inc., New York, 1964. (3) SMITH,G. W., AND WOOD,R. C., "Principles of Analog Computation," McGraw-Hill Book Ca., Inc., New York, 1959. (4) OSBIJRN,J. O., J. CHEM.EDUC.,38,492-495 (1961). (5) WHEELER, R. C. H., "Computer Application Series," Donner Scientific Division, Systron-Donner Corporation, Concord, California, 1962. (1966). M. L., J. CHEM.EDUC.43, '.(6) CORRIN, (?) TABBUTT, FREDERICK D., J. CHEM. .DUC., 44,64 (1967). (8) SMITH, J. H., A N D DANIELS,F., J . Am. Chem. Sac., 69, 1735 (1947). (9) MILLS,R. L., A ~ JOHNSTON, D H. S., J . Am. Chem. Sac., 73, 938 (1951).
$
Fifth Caribbean Chemical Conference The Fifth Caribbean Chemical Conferencewill be held in the Chemistry Department of the University of the West Indies in Barbados from January 6, 1969 until January 11, 1969. Plenary Lectures will be given by distinguished British, Canadian, and American chemkts. Time will be given in the program for the presentation of reseilrch papers (15 min each plus discussion). The program will include aseminar on "Chemical Education in the Caribbean." The registration fee is $12.00. All inquiries should be sent to Dr. R. C. Russell, University of the West Indies, P.O. Box 64, Bridgetown, Barbados.
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