Eugene Hamori
Univers~tyof Delaware Newark, Delaware 1971I
Use of the Analog Computer in Teaching Relaxation Kinetics
During the last few years the subject concerning the study of fast reactions by relaxationkinetic techniques has become an integral part of many undergraduate courses in physical chemistry. The basic principles of this new approach have been discussed in THIS JOURNAL (1, 8 ) and in several other publications (8-9). Teaching the theory and experimentation of relaxation kinetics to students who do not have a sufficient grasp of the kinetic concept of chemical equilibrium is not an easy task, however. One of the many difficulties in treating this subject is the fact that the simple equilibria the teacher is forced to use when discussing relaxation kinetics are not suitable to demonstrate the full power and usefulness of this technique. The purpose of this paper is to present a supplementary method based on the utilization of the analog computer for teaching relaxation kinetics. The programs discussed below were based on our experience in teaching this new topic to undergraduate students a t the University of Delaware using the EAI TR-20 analog computer. The simulations can be used both in lectures and in the laboratory. Furthermore, these computer experiments can serve as starting points for a variety of student research projects. This paper assumes that the reader has a cursory knowledge of analog computation. Those who need some background are referred to the several good introductory articles which have appeared in THIS JOURNAL (1fM6). Although the proper scaling and the precise use of units are an indispensable part of any advanced analog computer operation, it has been our experience that students whose main interest is not analog computers (e.g., chemistry students) can be introduced to the field by problems and tasks which involve very limited skill in scaling. I n the examples discussed below scaling factors and units are not mentioned and the students (and teachers) are expected simply to accept that in these problems volts represent concentrations, (10V = unit conc.) and the units of first- and second-order rate constants are sec-' and (10V)-' sec-', respectively. Simulation of Relaxations in One-Step Equilibria
The electronic analog of the simplest chemical equilibrium
is represented in Figure 1 in the form of a circuit dia-
Figure 1. 1.1 Analog computer circuit diagrom, suitable for the simulation of relaxdion reodionr in the chemical equilibrium: A C. (b) Time-base generator circuit.
gram. If an analog computer is programmed (patched) according to this scheme, the time course of any reaction in this equilibrium, from a pre-set initial state to a final equilibrium state, can be displayed on an XY plotter or on an oscilloscope screen. When discussing this equilibrium with students, one can demonstrate on the computer the concentration changes of components A and C during a reaction. One can also show the effects of adjustments in the rate constants (kl and k,) and initial concentrations (Ao and Co) on the solutions displayed. Convenient settings of the potentiometers (P) for this display might be: k, = 0.25, k, = 0.25, SR = 0.15, A. = 0.80, Co/10 = 0.04. Function switches (FS) 1 and 2 should both be in Down (DN) positions. In a temperature-jump relaxation experiment, the equilibrium constant of the prevailing chemical equilibrium is changed by a fast temperature rise (1, 4, 6). This change initiates a relaxation reaction during which the components of the system assume their new equilibrium concentrations. On an analog computer programmed according to Figure 1, this event can be simulated as follows: Potentiometer klf is set to a value of 0.04 and the other potentiometers are kept a t the settings given above. The time-base generator (Fig. lb) is momentarily disconnected by placing FS2 in the UP position. Keeping FSl also in the UP position, the computer is put in Operate (OP) mode. After a few seconds the equilibrium, determined by K' = k11/k2,is established and the concentration of a selected component, e.g., that of A, can be observed as a steady value on the voltmeter of the computer or on the oscilloscope screen. If now FS1 is rapidly switched to the DN position, the rate constant kl' will be replaced by k, and the equilibrium constant K' by K = h/kz. Conse-
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quently, a change in A leading to the establishment of the new equilihrium concentration will be observable on the vo1tmcter.l The time course of this relaxation can be displayed on theoscilloscope or on a (fast) X Y plotter if the timebase generator is started a t the same time as the rate constant is changed. This can he accomplished by placing both functionswitchessimultaneously in the DN position at the start of the simulation. (If a double-pole device is available the switches can be activated by one lever.) To prepare for another simulation experiment FS1 and FS2 are again placed in the UP position and the computer is switched first to Reset (RE) and then to OP mode. If an X Y recorder is used, the students can replot the exponential curve recorded on this instrument on a semilogarithmic plot and establish the relaxation time (7) for the observed relaxation process. Since for this equilibrium 7-1 = k~ k2, (3) from the value of r and from the ratio of equilibrium concentrations (C,/A. = k&), the individual rate constants of the reaction k, and k* can he determined. In a laboratory experiment, the settings of potentiometers k, and k2 may he concealed, and the students can he asked to determine the "unknown" values of these rate constants from the r and K values they establish experimentally on the computer. They should remember that in this reaction r is independent of the equilibrium concentrations of the reactants. Although very simple and easy to simulate, this equilibrium is not a convenient example for a more general discussion of the theory of relaxation kinetics. For instance, it is an important requirement in relaxation studies that the equilibrium under investigation be subjected only to small perturbations (3). In the equilibrium discussed above, however, this is not a restriction and perturbations of any size will lead to simple firsborder relaxations. For this reason, even the original time course of the reaction from any initial to equilibrium concentrations is usable for obtaining r. (This original reaction can be considered as a relaxation process which is initiated not by a perturbation of K but by a hypothetical large "concentration jump" of the components from the equilibrium to the initial concentrations.) The simulation of a relaxation reaction in the equilibrium
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switching FS1 rapidly from its UP position to D S position when the computer is in OP mode. If the timebase generator (Fig. lb) is started simultaneously d h this change, the relaxation curve of any of the components (A, B, or C ) can be displayed on the oscilloscope or on the X Y plotter. In spite of the sizable change in k,, the perturbation of the equilibrium is sufficiently small and the relaxation curves observed will approach simple exponential lines whose time constant r can be determined from half-life measurements or from the replotting of the curves on semilogarithmic plots. In this equilihrium the relationship between the relaxation time observed and the rate constants is given by the equation where the subscripts e designate equilihrium conditions (3). If several relaxation curves are recorded which were generated with various initial concentrations (Ao,Bo,and Co),from the differentequilibrium values of components A and B, a plot of l/r versus (A, Be) can be constructed. I t follows from eqn. (3) that the slope and the intercept of the straight line obtained yield k, and k2, respectively. In the analog program shown in Figure 2a, the initial concentrations of the components can be adjusted conveniently by changing the setting of P 12. The values of A, and B,, on the other hand, can be either read from the concentration curves displayed on the oscilloscope, or can be obtained in a separate step by placing the computer in OP mode and measuring the voltages of integrator 1 and amplifier 9 with the voltmeter of the computer. During a demonstration it can be emphasized to the students, at this point, that the determination of the rate constants in this equilibrium is rather complicated using classical kinetic techniques. Relaxation methods (e.g., temperature jump), on the other hand, will easily yield these rate constants as could be seen in the analog computer simulation. In the following section a method will be discussed which allows a very convenient measurement of the r and A, B. values during the simulation of the relaxation reactions in eqr. (2). If the computer is placed in Repetitive Operation (REP OP) mode it will display in rapid succession on the oscilloscope screen the time
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will be considered next. The analog circuit to be constructed is shown in Figure 2a. For a simple demonstration of the time course of the reaction, the following settings of the potentiometers might be used: kl = 0.30, lcr = 0.15, kl' = 0.10, A0 = O.60,Bo = 0.20, Co/10 = 0.05 and T = 1.00. The relaxation reaction due to a sudden change of the equilihrium constant can be simulated in this circuit by 'If it is desirable to simulate s. more realistic temperaturejump experiment in which typically both rate constants change, the circuit of Figure 1 can be easily modified to include another function switch and potentiometer (kt'). This addition, bowever, would not change the relaxation behavior of the system since r (see below) is determined solely by the final rate constants (k,, k3), and it is uneffected by the size of perturbation which is controlled by the initial rate constante (k,' and, if used, ' 1 . This argument also applies to the simulation of more complex relaxations discussed in following parts of this paper. 40
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Figure 2. (a1 Analog progrmm for the rimulotion of relaxation reactions in tho equilibrium: A 8 C. Ib) Circuit diagram for the geherotion See text for d~tailr. of functions 10lA A,) or 1012A A. - A,).
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e
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course of the reaction from the pre-set initial conditions to the equilibrium state. If it is assured that the initial concentrations (Ao,Bo,and Co)are relatively close to the equilibrium values of the respective components, the displayed functions will be first-order relaxation curves generated by recurring small concentration jumps from equilibrium values to initial concentrations. Either the time function generator shown in Figure l b or the built-in time sweep of the analog computer can be used to display these curves on the oscilloscope screen. The observation of the relaxation curves, however, will immediately indicate an obvious difficulty: Since the concentrations of components do not change much during the relaxation process, one observes only very slightly curved exponential lines which are not suitable for quantitative measurements. To overcome this difficulty, it is desirable therefore to display an amplified form of the relaxation curve such as 10(A - A ,), for instance. Figure 2b (without the connections marked with dotted lines) is a circuit diagram for the generation of this amplified function. Note that A, is not supplied to the system as a constant value, but the computer records (stores) that information by a "trackand-hold" circuit which monitors the time course of the concentration change of A during every run of the REP OP cycle, in the following manner: Since integrator 5 is in a cross-switched state,l in the OP part of the cycle of the computer, it wil! let signal A pass through without change, but during the R E part of the cycle it will store (hold) the last value of A (e.g., A,). Integrator 6, in the RE mode, will use this signal as an initial condition and in the OP mode will display it unchanged. Thus the output of this integrator will always be a constantly "updated" value of A,. This arrangement makes possible the slow variation of A. and B, dun'ng the display and consequently allows the simultaneous observation of the changes occurring in the relaxation curve. Using the circuits of Figure 2 in conjunction with a time base generator (Fig. lb) it is possible to plot a series of relaxation curves corresponding to various values of A, and B, on the XY plotter in the following way: (a) adjustment of P 12 in the RE mode, (b) switching to OP mode and reading A, and B. on the voltmeter, (c) switching to RE mode, and (d) switching to OP mode and plotting. The evaluation of the curves recorded on the XY plotter yields k, and kz in the manner discussed above. If the connections designated by the dotted lines are made in the circuit of Figure 2b, the function 10(2A A. - A,) can be displayed on the oscilloscope screen. This function changes from 10(Ao- A ,) to - 10(ApA ,) during the relaxation process and it is zero a t time t, = rln2 (see Appendix). Thus, one can directly read from the oscilloscope display the "cross-over time" of the curve which is simply related to therelaxation time of the reaction (Fig. 3). Using this function, the measurement of data for the construction of the 1/r versus A, B, plots can be very conveniently carried out in the following manner: (a) the computer is run in REP OP mode displaying lO(2A - A. - A,) and the crossover time t, is recorded, (b) the computer is placed in a single-run OP mode and A, and B, are measured on the voltmeter, (c) a new set of initial conditions is produced by changing P 12, and the result again is observed in R E P OP mode and t, is recorded, etc. Using the
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Figure 3.
Time plot of
- Ao - A d .
he function 10i2A
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BI plot constructad from doto obtained in Figwe 4. l/r versus A* the malog computer simulation of the reaction, A B 7 3 C.
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initial conditions: AO = 0.50, Bo = 0.50, Co/lO = 0.25, kl = 0.20, k2 = 0.10 and T = 0.10, we obtained the 1/r versus A, B, plot shown in-Figure 4 in this manner. In spite of the large errors, which are due to the difficulties associated with the reading of t, on our small oscilloscope, the data follow the linear relationship expected. The line drawn represents eqn. (3) with values of lcl and k2 given above. (In a laboratory experiment the students can be asked to determine the unknown kl and kz values of this equilibrium by this procedure.)
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Simulation of Relaxations in a Complex Equilibrium
Following a classical kinetic approach it is extremely difficult to determine the individual rate constants in the equilibrium kl
B
k-I
k.,
A+B*C*D
(4)
The problem involved here is mainly due to the complex mathematical equation which has to be used for the evaluation of the observed ratesa These difficulties are in sharp contrast with the ease of solution of these type of multiple equilibria by relaxation-kinetic and/or analog-computer techniques. Figure 5a is the circuit diagram of the electronic analog of equilibrium (4). It can be used to generate the time course of any process in this equilibrium. Relaxation reactions can be simulated in two ways in this circuit. (One procedure is analogous to a temperature-jump, and the other, to a concentration-jump 2 That is, when the computer is in RE mode the integrator is in OP mode, and when the computer is in OP made the integrator is in RE mode. a h sn example, see Ref. (10) for the exact mathematical solution of the equilibrium: A e B Ft C.
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Figure 5. (01 Circuit diagram for the simulation of relaxation reactions in the equilibrium: A B C $0. lb) Logarithmic time base generator circuit.
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experiment.) In the first method, an equilibrium is established initially in the circuit by keeping the computer in OP mode for several seconds while FS1 is in the UP position. Then, FS1 is rapidly switched from rate ' to a slightly different rate constant k, constant k ~ (UP) (DN). The ensuing reestablishment of the equilibrium can be displayed by starting a linear time sweep (e.g., Fig. lb) simultaneously with the rate constant change. The other method involves the setting of the initial concentrations of the components (Ao, B,, C,, and Do) slightly off from their equilibrium values and switching the computer from RE to OP mode. The time base generator must be left on (i.e., Fig. l b FS2 DN) all the time during this operation. This latter method is more convenient than the first one since it can be adopted to a REP OP display. Using this second procedure we have chosen the following settings of the potentiometers to simulate a relaxation reaction in equilibrium (4): A. = 0.32, Bo/lO = 0.029, Co = 0.099, DJ10 = 0.007, kl = k, = 0.900,Scz = Ic_z = 0.030, P13 = 0.902 and PI7 = 0.190. Figure 6 shows a sketch of the double relaxation process we observed in this manner. A fast and a slow process, indicating the presence of two equilibria, are clearly discernible. Notice, that in the circuit shown in Figure 5a provision has been made for the amplified display of the relaxation curve of component C similarly to the circuit of Figure 2b. In this program, however, the constant sampling of C, can be bypassed by the use of P13 and FS2. (By balancing amplifier 19 in the equilibrium state of the computer with P13, a constant C, value can be supplied to the circuit.) This mode of operation provides a very stable display which is a desirable feature for quantitative measurements. The use of a logarithmic time sweep for the display of 42
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relaxation curves has been suggested (3) but we are not aware of the application of this particular method in any reported experimental studies. This may be due to the fact that the most commonly used oscilloscopes can be equipped with a logarithmic time function generator only at considerable expense. Since many analog computers contain logarithmic diode function generators, logarithmic sweep rates can be easily applied when using these instruments. If this display method is used, the composite exponential relaxation curves (e.g., Fig. 6) are transformed into multiple-stage sigmoidal curves whose "vertical" inflection points correspond to the relaxation times involved in the observed process. As an example, Figure 7 shows a hypothetical relaxation process displayed as a function of logarithm of time which is characterized by 3 relaxation times ( T I , 7 2 , and T ~ ) . The main attractiveness of this procedure lies not in the quantitative measurement of relaxation timesthe corresponding inflection points occur exactly at the logarithms of relaxation times only if these are sufficiently separated (17)-but in their easy qualitative identification. Also, shifts in relaxation times can be very easily detected on these sigmoidal curves. Figure 5b is the circuit diagram we used to generate a logarithmic time base for the display of the relaxation curves of eqnilibrium (4). To expand the cycle span of our log function generator we used it in combination uith a quarter-square multiplier which was patched as a square-root extractor. In this manner the output of the log function generator was changed from 5 log (time) const. to 2.5 log (time) const. and this made possible the representation of a considerable time span on the oscilloscope screen. Using the rapid switching method (see above) the two stage sigmoidal curves characterizing the relaxation process of equilibrium (4) can be diaplayed on the oscilloscope screen or on the XY plotter (not shown). It
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Figure 6. A reloxolion curve displaying a fast and slow exponential process in the equilibrium: A B F? C 0.
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~ time curve for a reloxalion process Figure 7. Concentration v e r log rhoracterired b y three relaxation timer.
advanced group of undergraduate students) the students can be asked to determine the "unknown" rate constants of equilibrium (4) using these circuits on the analog computer. The procedure they might use is the following. By changing the total concentration of the reactants, various sets of the 71, r ~and , A. B, values can be obtained. It follows from eqn. (5) that a plotting of l / n versus A , B, will yield 1c1 and Ll. These constants, in turn, can be utilized in calculating the denominator in eqn. (6). Then, plotting l/r2 versus [l k-l/kl(A, Be)]-' will yield ic, and k-% from the respective slope and intercept of the straight lines obtained.
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Summary The rimulotion of a relaxation procear in the equilibrium A B ~3C D. The ordinote repre5ents 1 OOIC - C,) and the obrcisro the logarithm d the time elapsed after the perturbation. The orrows indicate the calculated relaxation timer of the equilibrium. (The characteri3tic horizontal flattening of the line ot the left ride of the curve could b e observed a t different setting., but the displays obtained in this manner were somewhat vnrtable and were not witable for photographic recording.) Figure 8.
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is best, however, to perform the simulation as a recurring concentration jump on the REP OP mode of the computer and display the curves as permanent figures on the oscilloscope. To accomplish this, we used the following settings of the potentiometers on the diagram shown in Figure 5b: 1,:0.398, lb:0.131 and 1,:0.001. li was adjusted to provide the minimum allowed voltage to the log function generator. Figure S is a direct photograph of the oscilloscope display obtained in the REP OP simulation of a concentration jump using the circuits of Figures 5 a and b (the potentiometer settings were those given above). The characteristic horizontal flattening of the line a t the left side of the graph (see Fig. 7) could be observed on the oscilloscope screen at a different setting of the time base generator, but due to the persistent flickering of this part of the curve the display was not suitable for photographic recording. The two vertical inflection points characterizing rl and r2 are clearly observable, however, on the photograph. If in equilibrium (4) the relaxation times are sufficiently different they can be expressed (18) as
I n this paper we presented the analog-computer simulation of relaxation reactions in three chemical equilibria of increasing complexity. Since the perturbation and display methods used were also of various degree of difficulty, it is hoped that the simulations discussed here can find uses in a broad spectrum of teaching applications. It appears that the logical extension of this particular treatment of relaxationkinetic problems would be the design and use of hybrid (analog/digital) computer programs. The author is indebted to Mr. John W. Chang for his technical assistance and to Dr. Conrad N. Trumbore for his helpful comments on the manuscript. Special thanks are due to Dr. David E. Lamb for his inspiring introduction to analog computation. Appendix Due to the small perturbation employed the function A A, is a simple exponential function which at t = m becomes zero. The half of the total change of this function being (Ao A,)& the quantity A - A , - (Ao - A,)/2 represents the ereess over the half change which is zero exactly at the half life of the relsxabion process: 1, = 4 , 2 = In 2/k = r in 2. The same applies for the twenty-fold value of this quantity: 20[A A , - (Ao - A,)/2] = 10(ZA - A. - A , ) .
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(1) FIN" -~~ (2) SWINEHART, J . H., J. CXEM.EDUC.; 44. 524 (i967). (3) E I ~ E N M., , A N D DEMAEIER,L. in "Teohniquea of Organic Chemiatry" (Ed21078: FAIEB.S. L., LEWIS, E. 8.. A N D W E ~ O E R G EA,). R, Interscienoe Publishers (division of John Wiley 61 Sons, Ino.), New York. 1963, Vol. VIII, Part 11. Chapter XVIII. (4) FRENCII, T. C.. HAMUER, G. C., TAXAHISAI. M . T., AGBERTY. R. A.. Eooens, F.. K U ~ T I W K., A N D DE MAEYER, L. C. M. in "Methods in Enrymology:' Academic Press, New York. 1969, Vol. XVI (Editor: ~
~
and
I n the computer simulation we observed that A, = 3.14 V and B, = 2.87 V. Using the k1 = k, = 0.900 and kz = Ic-% = 0.030 values set on the potentiometers, we calculated from eqns. (5) and (6) 0.69 and 24.2 sec, respectively, for the two relaxation times r1 and T ~ . On Figure 8 arrows indicate the positions of these predicted relaxation times. It can be seen that the agreement with the actual inflection points is very satisfactory. I n a laboratory situation (e.g., a special project to an
K R ~ R H E CG. X ,C., HAMORI, E., DAV J . Amer. Chcm. Soc., 88, 246 (19661. H A M O ~E.. I . A N D SCXERAOA. H. A,. J. Phm. Chen.. 71, 4147 (1967). OSUURN: J. o.,J. CHEM.EDUC., 38, 492 (i9611. C a ~ n w M. . L., J. C ~ MEouc., . 43, 579 (19661. T ~ n n u r rF. , D.. J. CXEM.EDUC.44. 64 (1967); ermts, 44, 486 (1967). Gmswofio, R., AND H ~ u a " ,J. F., J. C n e ~ Eonc., . 45, 576 (1968). Lonor. N. G., J. CXEM.EDOC.,46, 861 (1969). O ~ ~ n iR. a ,W., J. CneM. Eoac., 47, 651 (19701. Lownr, T.M., A N D JOHN,W. T..J. Cham. Sac., 97,2634 (19101. LEE. YANN.. Di.sem.ti~n to be submitted, University of Delaware. ~ ~ o n I.. n ,AND HAMMEB. G. C.. Chemical Kinetics," McGraw Hill Book Company, New York, 1966, p. 141.
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