J . Phys. Chem. 1993, 97, 2571-2575
257 1
Sequential Oscillations in Bromine Hydrolysis Controlled Oscillators in a Closed Reactor R. P. Rastogi’J and G. P. Misra Department of Chemistry, Banaras Hindu University, Varanasi-221005, India, and Medicinal Chemistry Division, Central Drug Research Institute, Lucknow-226001, India
Ishwar Das and Archana Sharma Department of Chemistry, Gorakhpur University, Gorakhpur-273009, India Received: May 26, 1992; In Final Form: November 24, 1992
Detailed results on birth, growth including sequential oscillations and decay for ascorbic acid-cyclohexanone-Ce4+-BrOj--H2S04 system in a batch reactor are reported. Oscillations are observed within a certain range of temperature and the system is monostable beyond the upper and lower limits of temperature. Results have been interpreted in terms of Field, Kiiriis and Noyes mechanism and bromine hydrolysis controlled model which has been modified for reactions in the closed reactor keeping in view the consecutive reactions associated with ascorbic acid. Numerical simulations indicate the existence of two types of oscillations as experimentally observed when the sequence ascorbic acid -oxalic acid -products is taken into account. There is an indication of time pause which becomes more evident when the sequence ascorbic acid intermediate (threonic acid) oxalic acid is taken into consideration. However, quantitative agreement is poor.
-
-
Introduction Bromine hydrolysis controlled (BHC) oscillator^^-^ are a typical class of bromate-driven oscillators which display variety of oscillatory b e h a ~ i o r .In~ earlier communications, such oscillators with mixed organic substrate have been reported by Rastogi et a1.s-8 and Sevcik and Adamcikova et aL9 Under certain circumstances complex oscillations are also observed. Rastogi and co-workers have investigated bifurcation phenomena in a Belousov-Zhabotinskii (BZ) system with oxalic acid cyclohexanone,8 glucose + acetone/cyclohexanone,IO and fructose + acetone/cyclohexanone. lo Sequential oscillations in BZ system with ascorbic acid acetone as mixed organic substrate have been reported]] recently. Similar dual frequency oscillationsI2 have also been reported by Srivastava, Mori and Hanazaki in case of chemical oscillators of the B-Z type in 0- and macetylphenols. However, a few overshoots have been observed in case of p-acetylphenol. Although preliminary results] I on bromate oscillators with ascorbic acid and cyclohexanone as mixed organic substrate have been reported, detailed studies on temporal oscillations in such systems have not been made. In this communication, we report studies on long-time behavior of such a system under different conditions in a batch reactor. It may be noted that “in closed system, species are consumed and products take their place, as such systems proceed to unique terminal states. There are no truly stationary states and indefinitely sustained oscillatory ones”. 3a The apparent nonequilibrium effects observed in such systems cannot be disentangled from genuine nonlinear behavior and hence make quantitative mechanistic interpretation difficult. This is also true for bifurcation features, since as pointed out by Erneux and Reiss,13bit is customary to assume that the bifurcation parameter X is independent of time. Model mathematical studies on delaying the transition of Hopf bifurcation induced by slow variation of the bifurcation parameter indicate that it is not possible to define uniquely the value of the bifurcation parameter at which the solution qualitatively changes from nearly steady to oscillatory. On the other hand, multistability and sustained oscillations are
+
+
’ Presently Emeritus Scientist at Central Drug Research Institute, Lucknow, India. 0022-3654/93/2091-2511 S04.00/0
possible in open system. However, the remarkable features of the closed systems are “how vivid a pictorial and mathematical representation is possible of the growth, number and duration of oscillations and how natural are their birth and death long after reaction has started and always long before reaction is overn.I3c Thus, batch reactors do provide a time trace of the system moving from instant of mixing of the reaction to ultimate equilibrium. It has to be noted that powerful methods of bifurcation analysis have been employed for investigating closed systems using the so-called “Pool Chemical a p p r o ~ i m a t i o n ” . ~ ~Using - ~ s these techniques, it has been shown that complex oscillations and even aperiodicity can exist as transient phenomena in closed chemical systems.I6 Experimentally also, chaos has been observed in closed systems.I7 We were motivated to investigate sequential oscillations in batch reactors with the above considerations in mind, although perhaps a parallel study of closed and open reactors would have been much more rewarding. The results have been analyzed in the context of the Field, Kiiriis and Noyes (FKN) mechanismI8 and bromine hydrolysis controlled (BHC) m0de1.l~ Attempt has been made to simulate the sequential oscillations. Complex oscillations similar to two oscillatory b i r h y t h m i ~ i t yand , ~ ~ sequential oscillat i o n ~are ~ ~observed , ~ ~ numerically. Experimental Section Materials and procedure were the same as reported earlier.” Bright platinum electrode in conjunction with calomel electrode were used for monitoring of redox potential with time in the batch reactor. Influence of concentration of ascorbic acid, cyclohexanone, Br03- and Ce4+on the nature of oscillations for BHC oscillator containing ascorbic acid + cyclohexanone have been studied. Concentration of ascorbic acid was varied in the range 0.014-0.079 M, cyclohexanone concentration was varied from 0.0159 to 0.0239 M, and the concentration of Br03- was varied in the range 0.0798-0.059 M. Experiments were also performed at various Ce4+concentrations. The concentration of Ce4was varied between 0.0003 and 0.025 M. Solutions of Br03and Ce4+in H2S04were added to mixed organic substrate solution 0 1993 American Chemical Society
2572
The Journal of Physical Chemistry, Vol. 97, No. I I 1993
Rastogi et al.
~
>
T
E
-1
J
9 I-
Q
L
2
w
I-
O
a
X
0 0
w
a
TIME ( s a c ) TIME ( s e c )
Figure 1. Plot of redox potential versus time. System: cyclohexanone (9.5 x 10 ? M) + Br03- (7.98 X M) + Ce4+ (1.45 X M) + HzSOj ( I .5 M) + ascorbic acid concentration equal to (a) 7.9 X M, (b) 5.9 X IO M, (c) 5.0 X M, (d) 3.8 X lo-* M, (e) 2.8 X M, and (f) 1.4 X lo-* M. Temperature = 30.0 f 0.5 OC.
Figure 3. Plot of redox potential versus time. System: ascorbic acid (5.01 X IO-*M) +cyclohexanone (9.5 X M) + BrO3- (7.98 X M) + H2S04 (1.5 M) + Ce4+concentration equal to (a) 2.5 X IO-? M, (b) 9.5 x M, (c) 6.3 X IO-) M, (d) 3.2 X M, (e) 2.2 X IO-' M,(f) 1.4X IO--'M,(g)0.9X IO--'Mand(h)0.3X lO-'M. Temperature = 30.0 f 0.5 'C. 3.0 I
x
0 K
I
I TIME
(Sac)
Figure 2. Plot of redox potential versus time. System: ascorbic acid (5.01 x IO M) + BrO, (7.98 X IO-? M) + Ce4+ (1.45 X IO M) + HzSOl ( I .5 M) cyclohexanone concentration equal to (a) 2.39 X IO I M, (b) 1.92 X IO M, (c) 1.60 X IO I M, (d) 1.28 X 10 M, (e) 1.12 X IO I M, (f) 0.95 X IO I M, (g) 0.48 X 10 I M and (h) 0.16 X IO I M Temperature = 30.0 f 0.5 " C .
+
in 1.5 M H2S04 in the reactor. Typical traces for redox potential changes as a function of time are shown Figures 1-3. Results and Discussion
The experimental results confirm the occurrence of sequential oscillations as reported earlier.' I The results on oscillatory features are summarized in Table I. Two types of oscillations designated as type I and type I1 separated by a time pause are observed. Type I oscillations appear first and are generated by ascorbic acid/cyclohexanone" substrate in the BZ system. Type I1 oscillations appear later and are due to oxalic acid/cyclohexanone system. Onset time for type I1 oscillations (sum of life time of
2.2 1 30
I
I
I
I
3.1
3.2 3.3 3.4 1 IT x 1 0 j ~ Figure 4. Plot of log (time pause/s) versus inverse of temperature System: ascorbic acid (5.01 X IO-*M) +cyclohexanone (9.5 X M) + BrOl (7.98 X M) + Ce4+ (1.45 X M) + H$OI (1.5 M).
type I oscillations and time pause) has been plotted in Figure 4 against reciprocal of temperature (K). A straight line isobtained. The oscillationsoccur within a rangeof ascorbic acid concentration and lower and upper limits are observed. Onset time, time period, life time and number of cycles decrease with increase in ascorbic acid concentration for type I oscillations while the reverse trend was observed for type I1 oscillations (Table I). The oscillatory characteristics at various cyclohexanone concentrations in the range 0.0159-0.2394 M are recorded in Figure 2. Onset time and life time for type I oscillations increase with decrease in cyclohexanone concentration whereas no regular trend could be observed for type I1 oscillations. The system exhibits complex periodic behaviour at low cyclohexanone concentration for type I1 oscillations similar to that observed in other systems by Rastogi and Srivastava.Io Since the time pause is related with time
Bromine Hydrolysis Controlled Oscillators
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2573
TABLE I: Dependence of Onset Time, Lifetime, and Time Period for Type I and Type I1 Oscillations on Concentrations of Ascorbic Acid and Br03type I oscillation type I1 oscillations onset time (s)
time period (s)
lifetime (s)
* *
* *
* *
* *
*
14
20 88
2 7
664 386
1154 746
26.4 17.6
962 340
50 28
*
0.059 0.064 0.069 0.079
*
*
*
290 274 212
12 12 14
12 20 88
*
*
*
*
Br03- Concentration/ M'
*
* *
* *
* *
* *
* *
2 7
184 386
478 746
18.8 17.6
450 340
33 28
+
TABLE 11: Values of Critical Limits of Concentration of Specific Reactants and Temperatures type I 1 oscillations
lower limit
upper lower limit limit ascorbic acidi' concentration 0.059M 0.028M 0.059M 0.014M cyclohexanone'' concentration 0.1436M 0.0798M 0.2235M 0.0159M 0.069M f 0.064M BrOI ' concentration f Ce4+#'concentration 0.0032M 0.0009M f 0.0003M 10.0 60.0 5.0 temperature' (k0.5 "C) 60.0 upper limit
+
+
Cyclohexanone (9.5 X 10 M) BrO3- (7.98 X M) Ce4+ (1.45 x IO 3 M) HzS04(1.5 M) ascorbic acid. Temperature = 30.0 f 0.5 OC. Ascorbic acid (5.01 X M) + B r 0 3 (7.98 X M) Ce4+(1.45 X 10-3 M) H2S04 (1.5 M) +cyclohexanone. Temperature = 30.0 f 0.5 "C. Ascorbic acid (5.01 X M) +cyclohexanone (9.5 X I 0 ? M ) C e J f ( I . 4 5 X IO 'M)+H#.04(1.5M)+BrO3. Temperature = 30.0 f 0.5 OC. d Ascorbic acid (5.01 X IO-? M) +cyclohexanone (9.5 X IO M) BrOj (7.98 X M) + H2S04 (1.5 M) + Ce4+. Temperature = 30.0 f 0.5 OC. Ascorbic acid (5.01 X 10 M) + cyclohexanone (9.5 X IO M) + BrO3 (7.98 X M) + Ce4+ (1.45 X 10 M) H2S04 (1.5 M ) . / N o upper limit was found in the concentration range studied.
+
+
*
1
[
type I oscillations
*
*
*
Asterisks signify that no oscillations are observed. System: cyclohexanone (9.5 X IO-* M) + BrO3- (7.98 X Temperature = 30.0 f 0.5 OC. System: ascorbic acid (5.01 X M) cyclohexanone (9.5 HzSOj (1.5 M). Temperature = 30.0 f 0.5 OC.
+ HzSOj (1.5 M).
+
no. of cycles
* *
*
parameter
lifetime (s)
time period (s)
*
*14
+
onset time (s)
*
* 212 272
time pause (s)
Ascorbic Acid Concentration/Mh
*
0.014 0.028 0.038 0.05 0.059
*
no. of cycles
+
1
+
+
M)
M)
Values of the critical limits of concentration of the specific reactants and the critical limits of temperature are summarized in Table 11. The results are internally consistent. It has to be noted that the lower limit of Br03- concentration for the two types of oscillations is approximately the same. Field and BoydI9haveshown that bromine hydrolysiscontrolled (BHC) model is successful in simulating oscillations in BZ system with oxalic acid + acetone as organic substrate. Quite recently, Rastogi and Misra26 have shown that critical limits of acetone concentration and also of Br0,- concentration based on numerical computation agree quite well with experimentally observed values. However, in order to model the observed sequential oscillations and in view of the evidence" of the decomposition of ascorbic acid into oxalic acid in the reaction system, we have modified the mechanism of BHC model as follows. We will call this model as M1:
Br0,-
Br0,-
ki
+ Br- + 2H+
HBrO, required for the conversion of ascorbic acid to produce the requisite amount of oxalic acid to generate type I1 oscillations, which in turn may be expected to be related to the rate constant, hence the plot of logarithm of time pause against reciprocal of temperature in degree would be a straight line. This expectation is justified by Figure 4. Oscillations at various Br03-concentrations have been studied in the concentration range 0.059-0.095 M. Lower limit of Br03concentration for both type I and I1 oscillations is observed. It has been observed that ceric ion concentration also influences the oscillatory characteristics (Figure 3). Experiments were performed in the range 0.0003-0.0250 M of Ce4+concentration. Lower and upper limits of ceric ion concentration for type I oscillations were observed at 0.0003 and 0.0032 M, respectively. However, only lower limit for type I1 oscillations is observed at Ce4+ concentration equal to 0.003 M. Upper limit of Ce4+ concentration for type I1 oscillations could not be observed in the concentration range studied. Onset time, life time and number of type I oscillations increase and time period decreases with increase in Ce4+concentration whereas no regular trend could be noticed for type I1 oscillations. Oscillatory behaviour in the reaction system has been studied at different temperatures (5.0-60.0 f 0.5 "C). Lower and upper limits of temperatures were found at 5.0 and 60.0 "C respectively at which we do not get any oscillations of either type. At 10.0 "C only type I oscillations disappear. Further, it has been observed that onset time, time period, lifetime, and number of oscillations decrease with increase in temperature for type I and I1 oscillations.
M) + Ce4+ (1.45 X M) Ce4+ (1.45 X
X
HBrO,
-
+ HOBr
k2
+ Br- + Ht
2HOBr
ki
+ HBrO, + H+
2Br0,'
+ H,O
kl
Ce3++ BrO,'
-
k4
+ H+ + Ce4++ HBrO,
k5
2HBr0,
Br0,-
+ HOBr + Ht
k6
+ Ce4+ oxalic acid + Ce3+ ascorbic acid + HOBr oxalic acid + Brka HOBr + Br- + H+ Br, + H,O k ascorbic acid
ki
F=
-8
koa
+ H+ + enol + H+
cyclohexanone Br,
+ enol + H+
oxalic acid
-
k-9,
k9b
-
+ Ce4+
oxalic acid
Br-
kio
+ other products
Ce3++ inert products
+ HOBr
kii
Br-
The above mechanism has justification since accumulation of oxalic acid at the start of type I1 oscillations has been confirmed by TLC studies."
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
2514
TABLE 111: Value of Parameters Used in Comwtations’ Darameter
O1
value
Darameter
value
~ ~ - 3 s ’ 3 X IO6 M-z s I 42 M-2 s-I 4.2 x 107 M - IS-I 8 X IO4 M-2 s-I 8.9 x 103 M-1 S-I 3 x 103 M - IS-I
kn k-n
8 X IO9 M-2 S-I I IO s-I 2.33 x 10-4 M - 1 S-I 1.0 x 103 M - 1 S K I 5.0 x 109 M - 1 S-I 27.5 M-l S-I 150 M-I s-I
~
ki k2
k3 k-3 k4
k. 4 ks ksb
k9, k-94 k9b kio ki I
Rastogi et al.
-7
a
kl-ks, ks, and k n are due to Field and Fosterling.28 kg,, k-s,, and are due to Dubois et aL30 klo and k l l are due to Field and Boyd.I9
-* 1
a
I
I
c . Ascorbic a c i d d , Oxolic a c i d
-10
c
Ascorbic acid
d
O x a k acid
-4
-10
‘
I
0
300
600
I
I
900
1200
I
1500
1600
2lOO
T i m e (Sec)
Figure 6. Plot of computed values of (a) log [Ce4+], (b) log [Br-1, (c) log [ascorbic acid], (d) log [oxalic acid], and (e) log [threonic acid] versus time basedon model M2. [Total cerium] = I .5 X lP3M, [ascorbic M, [Br03-] = 8.0 X lo-* M, and [cyclohexanone] = acid] = 5.0 X 9.0 X L L ~ M. k6’ = 5.0 X lo2 M-I S-I, kj‘ = 1.0 X IO4 M-’ s-I , k 12 = 10 M-I s-l, and k13 = IO2 M-I s-I, [H+] = 2.0 M .
-10
1
0
\ 300
600
900
1200
1500
I
1800
Time I Sec I
Figure 5. Plot of computed values of (a) log [Ce4+], (b) log [Br-), (c) log [ascorbic acid], and (d) log [oxalic acid] versus time based on model M, [ascorbic acid] = 5.0 X M, M1. [Total cerium] = 1.5 X M. k6 = 5.0 [Br03-] = 8.0 X IO-*M and [cyclohexanone] = 9.0 X X lo6 M-I S-I and k j = 1.0 X IO4 M-1SKI, [H+] = 2.0 M. For the purpose M, whereas of computation, the initial value of [Br-J was taken to be M. Results are the concentration of oxalic acid was taken as M. unaffected if initial [Br-] was taken to be
The above mechanism leads to a set of differential equations containing eleven variables. Numerical solution of the differential equations wereobtained with H C L “PC/AT”computer and using subroutine27STIFF3 and a set of rate constants largely due to Field and FBrsterling.28 Values of k6 and kj have been assigned by us. The rate constants value are given in Table 111. Experiments on relative rates of oxidation of ascorbic acid and oxalic acid indicated that ascorbic acid is oxidized at a faster rate. Hence, a higher value than that for oxalic acid was assigned to k6 and k7. The contention is further justified since life time of type I oscillations is always much shorter than type I1 oscillations. Results are plotted in Figure 5. Oscillations in sequence of two different frequencies are observed corresponding to type I and type I1 oscillations. However, there is no pause. In order to check in what way type I and type I1 are related with ascorbic acid concentration and oxalic acid concentration, the concentration of these species was also computed as a function
of time. The results are plotted in Figure 5c,d. The results are consistent with Figure 5a.b. Further, Figure 5 shows that type I1 oscillations start when the oxalic acid concentration attains the maximum value and at this instant the type I1 oscillations start. The nature of oscillations are slightly affected when H+ concentration is changed in the numerical computation. When H+concentration is reduced, the number of cycles in type I1 oscillations are increased whereas the character of type I oscillations practically remains unchanged. Thus, when H+ concentration is 1 M, the number of cycles is nine whereas for H+ concentration is 1.5 M, the same is equal to three. The mechanism of this reaction can be understood broadly in terms of the FKN mechanism.Is Type I oscillations occur on account of oscillator containing ascorbic acid and cyclohexanone. These stop when the lower limit of [ascorbic acid] is reached. During this period and also during pause, ascorbic acid is broken down into oxalic acid. Again when sufficient amount of oxalic acid has accumulated, oscillations are initiated by the second oscillator containing oxalic acid and cyclohexanone. Numerical results described above do not clearly demonstrate the occurrence of time pause. Further whereas the observed ratio of cycles of type I1 oscillations to that for type I is much larger but numerical simulation leads to contrary result. Hence, it was thought to further modify the mechanism. As pointed out in an earlier paper,” oxidation of ascorbic acid proceeds as follows:
-
-
IO1
ascorbic acid threonic acid oxalic acid Keeping this in view, the mechanism was modified by replacing reactions 6 and 7 by following set of reactions 6‘ and 7’ and adding the set of two reactions 12 and 13: ascorbic acid
+ Ce4+
kb‘
threonic acid
+ Ce”
(6’)
Bromine Hydrolysis Controlled Oscillators
-
The Journal of Physical Chemistry, Vol. 97, No. 1 1 , 1993 2575
ki’
+ HOBr threonic acid + Brk12 threonic acid + Ce4+ oxalic acid + Ce” k13 threonic acid + HOBr oxalic acid + Br-
ascorbic acid
-
(7’) (1 2) ( 1 3)
This mechanism is designated as M2 and leads to a set of differential equations containing 12 variables (for batch reactor). The values of the rate constants were similar to that used earlier except those for steps 6’,12, and 13. Computed results are plotted in Figure 6. Oscillations of two types are clearly evident. Time pause is also indicated but not strongly. Computations have been made for different values of k I 2and kI3. At a fixed kI2as we decrease klj,a stage comes when oscillationsof type I1 disappear. Similar behavior is obtained when there is a decrease in kl2. These results confirm the earlier cotention that oscillations of type I1 start when sufficient amount of oxalic acid is produced since oxalic acid is produced by both the steps 12 and 13. Although in the above case, chemical evidence1I and numerical computations both suggest that decomposition of one organic substrate into another organic substrate generates type I and type I1 oscillations for mixed substrate having acetone/cyclohexanone, there are cases where time-pause can not be explained easily such as in oxalic acid-acetone mixed substrate system reported by Wittman et Acknowledgment. Thanks are due to UGC, New Delhi, for supporting the investigation under special assistance program of Chemistry Department, Gorakhpur University. A S . is indebted to UGC for the award of a Senior Research Fellowship. G.P.M. is thankful to the CSIR, New Delhi, for the award of Senior Research Fellowship. References and Notes ( I ) Noszticzius, Z.; BMiss, J. J . Am. Chem. SOC.1979, 101, 3177.
(2) (a) Noszticzius, Z.; Stirling, P.; Wittmann, M. J . Phys. Chem. 1985, 89, 4914. (b) Noszticzius, Z.; Wittmann, M.;Stirling, P. J . Chem. Phys. 1987.86, 1922. (3) Bar-Eli, K.; Noyes. R. M. J. Chem. Phys. 1987, 86, 1927. (4) Field, R. J., Burger, M., Eds. Oscillations and Travelling Waves in Chemical Systems; New York, 1985.
( 5 ) Rastogi, R. P.;Yadav, R. D.;Singh,S.;Sharma,A. IndianJ. Chem. 1985, 24A, 43. (6) Rastogi. R. P.; Verma, M. K. Indian. J. Chem. 1983, 22A, 830. (7) Rastogi, R. P.; Verma, M. K. Indian. J . Chem. 1983, 22A, 917. (8) Rastogi, R. P.; Misra, G. P. J . Phys. Chem. 1987, 91, 3007. (9) Sevcik, P.; Adamcikova, L. J . Phys. Chem. 1985, 89, 5178. (101 Rastoni. R. P.:Srivastava.Sannita Chem. Phvs. Lett. 1989.164.173. ( I I j RastogilR. P.;Das, Ishwar;Sh&ma,ArchanaJ. Chem.Soc:,Faiaday Trans. I 1989, 85, 201 1. (12) Srivastava, P. K.; Mori, Y.;Hanazaki, I. J . Phys. Chem. 1991, 95, 1636. (13) (a) Gray, P. (p 34), (b) Erneux, T.; Reiss, E. L. (p 267), (c) Gray, P. (p 52) In Spatial Inhomogeneities and Transient Behaviours in Chemical Kinetics; Gray, P., Nicolis, G., Baras, F., Borckmans, P., Scott, S. K., Eds.; Manchester University Press: Manchester, 1990. (14) Gray, P.; Scott, S. K. Ber. Bunsen-Ges Phys. Chem. 1986, 90,985. (15) Merkin, J. H.; Needham, D. J.; Scott, S.K. Proc. R. SOC.London, Ser. A 1986, 406. 299. (16) Scott, S. K.; Peng, B.; Tomlin, A. S.; Showalter, K. J . Chem. Phys. 1991, 94, 1134. (17) Rastogi, R. P.; Srivastava, P. K., in ref 13, p 653. (18) Field, R. J.; Koros, E.; Noyes, R. M. J . Amer. Chem. SOC.1972,94, 8649. (19) Field, R. J.; Boyd, P. M. J . Phys. Chem. 1985, 89, 3707. (20) Beck, M. T.; Varadi, Z. Reacf. Kinet. Catal. Lett. 1977, 6, 275. (21) Cooke, D. 0. Int. J . Chem. Kinet. 1982, 14, 1047. (22) Salter, L. F.; Sheppard, J. G.Int. J . Chem. Kinef. 1982, 14, 815. (23) Decroly, 0.; Goldbeter, A. Proc. Natl. Acad. Sei. U.S.A. 1982, 79, 6917. (24) Kaner, R. J.; Epstein, 1. R. J . Am. Chem. SOC.1987, 141, 241. (25) Heilwel, J.; Henchman, M. J.; Epstein, 1. R. J. Am. Chem. SOC.1979, 101, 3698. (26) Rastogi, R. P.; Misra, G. P. Chem. Phys. Lett. 1990, 174, 617. (27) Villadsen, J.; Michelsen, M. L. Solution of Differential Equation models by Polynomial Approximation; Prentice-Hall: Old Tappan, N J, 1978. (28) Field, R. J.; Fbrsterling, H. D. J . Phys. Chem. 1986, 90. 5400. (29) Wittman, M.; Stirling, P.; Bcdiss, P. Chem. Phys. Left. 1987, 141, 241. (30) Dubois, J. E.; El-Alaoni, M.; Toublee, J. J . Am. Chem. SOC.1981, 103, 5393.
