J. Phys. Chem. 1991, 95,6575-6580
6515
Hydrodynamic Turbulence and Diffusion-Controlled Reactlons. Simulation of the Effect of Stirring on the Osclllatlng Belousov-Zhabotlnsky Reaction with the Radicalator Model Zoltiin Noszticzius,**t Zsolt Bodnir,* Lis216 Garamszegi,t and Miria Wittmannt Institute of Physics, Technical University of Budapest, H- 1521 Budapest, Hungary, Department of Organic Chemical Technology, Technical University of Budapest, H- 1521 Budapest, Hungary, and Department of Physics and the Center for Nonlinear Dynamics, The University of Texas, Austin, Texas 78712 (Received: January 22, 1991; In Final Form: March 1 1 , 1991)
Different stirring effects of the Belousov-Zhabotinsky (BZ) reaction observed by previous authors and also reported here can be explained and modeled semiquantitatively by a diffusion-controlled radical-radical reaction step of the Radicalator. It is shown that hydrodynamic turbulence can accelerate the rate of a diffusion-controlled reaction considerably provided that both reactants of the diffusion-controlled step are present in low concentrations (less than 10" M). A stirring effect can indicate the presence of a diffusion-controlled step in the mechanism of other (non-BZ) reactions as well. Finally, it is found that uniform ultrasonic stirring can substitute for the less uniform mechanical stirring in the BZ reaction.
Introduction Experiments on chemical dynamic systems are performed either in closed "batch" or, more often, in open "continuously fed stirred tank" reactors (CSTRs). The latter have the advantage that a far from equilibrium situation can be maintained indefinitely. In both cases the reactors are stirred intensively and it is assumed that they are practically homogeneous. As long as there are no significant concentration differences within the reactor, the rate of stimng should not affect the chemical dynamics. While in most papers published on chemical dynamic systems the reactors are regarded to be "well-mixed", in recent years more and more authors observed stirring effects in CSTRs and even in batch reactors. Row, DeKepper, and B o h d e were the first to report' stirring effects in the C10; I- reaction in a CSTR. That study was continued and extended by Menzinger and co-worker~~-~ and by Luo and Epstein.6 A review paper on the (210,-+ I- reaction was published just recently? These observations were interpreted in the framework of different micromixing theories.*-" Thus, stimng effects in a CSTR were explained by an incomplete mixing of the reagent feed streams. It is more surprising that such effects were observed in batch reactors as well.1cz2 These observations were explained by statistical f l u c t u a t i ~ n sand ~ ~by ~ ~certain ~ absorption into or catalytic effects on the reactor wa11.18~z2 The aim of the present work is to study the effect of stirring on the oscillating Belousov-Zhabotinsky (BZ)23,24reaction in a batch reactor and to suggest a novel explanation based on the theories of diffusion-controlled reactions and hydrodynamic turbulence. It is important to remark that whenever the BZ reaction is studied in a reactor open to the atmosphere, atmospheric oxygen has a strong effect on the reaction. That oxygen effect is not the subject of the present study. A few years ago all stirring effects were thought to be caused by the atmospheric but k E i k and AdamEikovHI9 have shown that this is not the case. They found stirring effects even in an inert atmosphere. SevEik and AdamEikovH studied two different versions of the BZ reaction: one system with Ce catalyst and malonic acid substrate, which is the Yclassicalnsystem except for the temperature which was somewhat higher than the usual, 35 "C. (Bromate and aqueous sulfuric acid are not mentioned explicitly as they are "standard" components of the different BZ systems.) In their second experiment they applied Mn catalyst and oxalic acid Noszticzius et a1.'* found a stirring effect that was the same in CSTR and in batch. They applied Ce catalyst and oxalic acidacetone mixed substrate at 28 OC. Menzinger and J a n k o ~ s k i l ~ J ' j ~ were the first to report heterogeneities and stirring effects in a
+
Address correspondence to this author at the University of Texas. 'Institute of Physics, Technical University of Budapest. Department of Organic Chemical Technology, Technical University of Budapest.
0022-3654/91/2095-6575$02.50/0
batch BZ system. They used ferroin catalyst and malonic acid substrate at 25 "C. In addition, they demonstrated heterogeneities and concentration fluctuations in their batch reactor using platinum microelectrodes.z2 In this work two different BZ systems are used to study the effect of stirring in a batch reactor. (All experiments were done in an inert atmosphere.) The first one is the so-called RBcz systemmJO-andsome of its modifications-which was successfully modeled by the Radicalator.m The RHcz system contains the usual components-Ce catalyst and malonic acid substrate-but the applied concentrations are somewhat exotic to ensure malonyl radical controlled oscillations. The second system is a classical one: Ce catalyst, malonic acid substrate, laboratory temperature (1) Roux, J. C.; DeKepper, P.; Boiinade, J. Phys. Lett. A 1983,97. 168. (2)Menzinger, M.; Boukalouch, M.; DeKepper, P.; Boissonade, J.; Roux, J. C.; Saadaoui, H. J. Phys. Chem. 1986, 90, 313. (3) Menzinger, M.; Giraudi, A. J. Phys. Chem. 1987, 91,4391. (4)Dutt, A. K.;Menzinger, M. J . Phys. Chem. 1990,94,4867. (5) Ochiai, E. 1.; Menzinger, M., preprint. (6)Luo, Y.; Epstein, I. R. J . Chem. Phys. 1986,85, 5733. (7) DeKepper, P.; Boissonade, J.; Epstein, I. R. J . Phys. Chem. 1990,94, 6525. (8)Horsthemke, W.; Hannon, L. J. Chem. Phys. 1984,81,4363. (9) Nicolis, G.;Frisch, H. Phys. Rev. A 1985,31, 439. (IO) Dewel, G.; Borclanans, P.; Walgraef, D. Phys. Rev. A 1985,31,1983. (11) Hannon, L.; Horsthemke, W. J . Chem. Phys. 1987,86,140. (12)Boissonade, J.; DeKepper, P. J . Chem. Phys. 1987,87,210. (13) F'uhl, A.; Nicolis, G. J . Chem. Phys. 1987,87, 1070. (14) Menzinger, M.; Jankowski, P. J. Phys. Chem. 1986, 90, 1217. (15) Ruoff, P.;Noyes, R. M. J . Phys. Chem. 1986,90,4700. (16)Menzinger, M.; Jankowski, P. J. Phys. Chem. 1986,90,6865. (17) Nagyp61, I.; Epstein. I. R. J . Phys. Chem. 1986,90,6285. (18) Noszticzius, Z.; Horsthemke, W.; McCormick, W. D.; Swinney, H. L. In Spatial Inhomogeneities and Transient Behaviour in Chemical Sysrems; Gray, P, Nicolis, G., Baras, F., Borckmans, P., Scotts, S. K.,Eds.;Manchester Universit Press: Manchester, U.K., 1990;p 647. (19) &vEik, P.; AdamEikov6, L. Chem. Phys. Leu. 1988, 146, 419. (20) Nagyp61, I.; Epstein, I. R. J . Chem. Phys. 1988, 89, 6925. (21)SevEik, P.; AdamEikov6, L. J. Chem. Phys. 1989,91, 1012. (22) Menzinger, M.; Jankowski, P. J . Phys. Chem. 1990,944123, (23) Zhabotinsky, A. M. Dokl. Akad. Nauk USSR 1%7. 157, 392. (24) Field, R. J., Burger, M.,Us.Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, 1985. (25) (a) VHradi, Z.9.;Beck, M. T. J. Chem.Soc., Chem. Commun. 1973, 30. (b) Beck, M. T.; VBradi. Z. B. Magy. Kem. Foly. 1973. 79, 46. (26) Roux, J. C.;Rossi, A. Compr. Rend. Acad. Sci. (Paris) 19f8. C 287,
151.
(27)Treindl, L.;FHbiln, P. Collecr. Czech. Chem. Commun. 1980. 45, 1168. (28) Farage, J. V.; Janjic, D. Chimia 1980, 34, 342. (29)Patonay, G.; Noszticzius, Z. Reacr. Kinfr. Catal. Lett. 1981, 17,187. (30) (a) Fbrsterling, H. D.; MurBnyi, S.;Noszticzius, Z. J . Phys. Chem. 1990.94.2915. (b) Fhterling, H. D.; MurHnyi, S.;Noszticzius, Z. Reacr. Kiner. Catal. Letr. 1990,42,217. (31)RBcz, K. Ph.D. Thesis, L. Ebtvbs University, Budapest. Hungary, 1984.
0 1991 American Chemical Society
6576 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 TABLE I: Initial C o ” ~~
~
Noszticzius et al.
for Different Ex~eri~~ents
~
init concn, mol/L svstem Ricz modified Ricz I modified Ricz 11 classical
0.004
IMAl 0.6 0.6
0.0075 0.05
0.6 0.1
INaBrO-1 0.015
salt bridge
0
Teflon stirrer
iquid Level
U/
-
1cm
Figure 1. (a) Schematic drawing of the apparatus. (b) Visual picture of the Teflon stirrer (magnified, not at scale).
(23 “C), and usual the concentration range. Finally, the experimentally observed stirring effects are simulated by the Radicalator model by assuming that hydrodynamic turbulence accelerates the diffusion-limited step of the Radicalator. In general it is an important conclusion that stirring effect in a batch reactor can indicate the presence of some diffusion-limited step in the mechanism of the reaction in question. Experimental Section
Chemicals. With the exception of malonic acid, all chemicals, namely, H#04 (96% Merck), NaBr03, Ca(N03)’, and Ce(S04), (Fluka), were of reagent grade and were used without further purification. Malonic acid (Reachim USSR, purum) was recrystallized three times: first from acetontchloroform, then from ethyl acetate, and finally again from acetontchloroform following the method of Noszticzius et alau The main contaminant of the Reachim malonic acid was chloride. It was possible to remove chloride by titrating solutions of unpurified malonic acid with silver nitrate in the presence of a silver wire eleclrode. Then the solution was filtered to remove the AgCl precipitate. Malonic acid stock solutions prepared this way gave results identical with those solutions prepared from three-times recrystallized samples. Apparatus. The main parts of our apparatus are depicted schematically in Figure la. The overall redox potential was measured with a bright platinum electrode with a surface area of 2 cm2. The reference electrode (not shown in the picture) was Ag/AgCl in 0.1 M KCl connected to the reactor via a “salt” bridge filled with 0.05 M sulfuric acid. The frequency of the stirrer (Heidolph RZR 50L equipped with turbo propeller) was controlled between 0 and 2000 rotations per minute (rpm). The Teflon stirrer is shown in Figure Ib. The propeller draws gas bubbles into the liquid above 300 rpm due to its special construction. To avoid any oxygen effect an inert atmosphere of high-purity nitrogen was applied. Inlet and outlet for a high-purity nitrogen gas stream and injection ports for bromate and cerium solutions are not shown in the schematic drawing of the apparatus. Procedure. In all experiments the final volume of the mixedreactant solutions was 36.5 cm3. First a solution of malonic and sulfuric acid (35 cm3) was pipeted into the reactor. Then the system was closed and flushed with high-purity nitrogen to remove any oxygen from the reactor and the solution. To facilitate the oxygen removal from the liquid phase, a rapid stirring of 1300 rpm was applied simultaneously for 6 min. Then the stirring rate was set to its final value and an oxygen-free sodium bromate solution (1 cm3) was injected into the reactor from a plastic syringe. Finally, after a waiting period of 3 min, an oxygen-free cerium solution (0.5 cm’) was injected from another syringe to start the reaction. That waiting period is necessary because right
1ce3+1
ICe4+l
WSOd
2 x 10-3
3 3 3 1
I 0-3
1.5 x 10-3 I 0-3
Figure 2a,b 2c,d 3a 3b
after the addition of bromate the potential of the platinum electrode rises rapidly. No absolute electrode potentials were measured; only relative potential changes of the platinum electrode were recorded with a potentiometric recorder (Radelkis OH-814). All experiments were performed at 23 f 1 O C . When ultrasonic stirring was applied (Tesla UC 002 BM 1, 20-kHz working frequency), an intensive thermostating was arranged to avoid a warming up due to the ultrasound power. Numerical Metbods. Numerical integration of the kinetic differential equations was performed with the program DIFFGL,’, which is based on Gear’s methodP using an IBM/AT compatible personal computer equipped with a mathematical coprocessor. ReSdtS First we studied the effect of stirring on the RBcz system and on a modified RBcz system. The initial concentrations used in different experiments are collected in Table I. The results can be seen in Figure 2. Both systems are sensitive to stirring, and in both cases it is the amplitude of the oscillations that is affected. Increasing the rate of stirring results in a larger amplitude. In the case of the original RBcz system (Figure 2a,b), however, there was a problem when the rate of stirring was low (45 rpm, Figure 2a). Namely, 45 rpm was not high enough to homogenize the system within few seconds. It was necessary to insert a period at 300 rpm right before and after the injection of cerium. We had to choose that low stirring rate (45 rpm) to see a characteristic difference between the low and high rates. Thus, we modified somewhat the concentrations of the original RBcz system to get rid of such problems. The modified RBcz system (Figure 2c,d) showed a characteristic stirring sensitivity at higher stirring rates. When the experimental results of Figure 2 are interpreted in terms of nonlinear dynamics, the observations are compatible with a supercritical Hopf b i f ~ r c a t i o n ’ ~where , ~ ~ stirring rate is the bifurcation parameter. Parts a and b of Figure 2 can represent a scenario right after a supercritical Hopf bifurcation where the amplitude of the oscillations grows rapidly with the increasing parameter value. Figure 2, Parts c and d of on the other hand, can represent a scenario before a supercritical Hopf bifurcation. In that case, there are only damped oscillations but the damping decreases with increasing parameter value as the system nears its bifurcation point. It is a common drawback of the previous systems, however, that only a few oscillations can be observed in these experiments. There is not enough time to study the effect of different stirring rates within one experiment. Thus we modified the RBcz system again to produce more oscillations. The result is depicted in Figure 3a. Here the stirring effect can be observed within one experiment. The result is similar to the previous cases; the amplitude of the oscillations increases with increasing stirring rate. We compared this behavior with that of a YclassicalnBZ system. The result is ~~~~~~~~
(32) Wrsterling, H. D.; Kuhn, H. Praxis der Physikalische Chemir, VCH Verlagsgesellschaft: Weinheim, Germany, 1985. (33) Gear,C. W. Numerical Initial Value Problems in Ordinary Dvferenriul Equations; Prentice-Hall: Englewood Cliffs, NJ, 1972; Chapter 1 I . (34) Noszticzius, 2.;McCormick, W. D.; Swinney, H. L. J. Phys. Chem. 1987, 91, 5129. (35) Hassard, B. D.; Kazarinoff, N . D.; Wan, Y. H. Theory and Applications of Hopf Bijurcation;Cambridge University Press: Cambridge, U.K., 1981. (36) (a) Noszticzius, 2.;Stirling, P.; Wittmann, M. J. Phys. Chem. 1985, 89, 4914. (b) Noszticzius. 2.;Wittmann, M.; Stirling, P. J . Chem. Phys. 1987, 86, 1922. (37) Robertson, E. B.; Dunford, H. B. J . Am. Chem. Soc. 1964,86,5080.
Hydrodynamic Turbulence and Diffusion-Controlled Reactions The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6577 E
2
:-
.b
U u
u
loo
L 100
200
300
400
500
time/s
100
200
300
(00
500
-
100.
timds
100
200
400
300
time/s
500
Figure 2. Stirring effects in the RBcz system (a and b) and in a modified Rgcz system (c and d). Potentiometric t r a m of a platinum electrode. Observe the increasing amplitude of the oscillations with the increasing stirring rate. See text for concentrations.
shown in Figure 3b. In this case it is not the amplitude but the period of the oscillations that is increased with increasing rate of stirring. A similar behavior was observed by Menzinger and JankowskiZZin a ferroin-catalyzed BZ system. It is interesting to observe that while it was the time period of the different R5cz iystems that was independent of the stirring rate within the experimental error, it is now the amplitude of the oscillations of the classical BZ system that is practically independent of the stirring rate. A further interesting feature is the irregular time period at the high stirring rate. When the amount of the liquid used in the experiment is increased to 50 mL, hydrodynamic fluctuation decreases and the time period becomes more regular (experiment not shown). This result suggests again some connections between the rate of the reaction and hydrodynamic turbulence, a connection already suspected by Sorensen.)* Finally, we studied the classical BZ system of Figure 3b without mechanical stirring. We observed no oscillations without any form of stimng but we found that mechanical stirring can be substituted by ultrasound. In this case the effect of ultrasonic stirring with 20 kHz was qualitatively equivalent to the effect of mechanical stirring with 45 rpm; the time period and the amplitude of the observed oscillations were roughly the same in both cases.
Discussion Rdkahtor MoQl The Radicalator model used here (see Table I1 for the mechanism and for the rate constants) was created by Fijrsterling, Murhyi, and N o ~ z t i c z i u sto ~ ~explain the nonbromide-controlled oscillations observed by them in the Racz system. The main feature of their mechanism is a negative feedback loop where the control intermediate is malonyl radical instead of the more usual bromide ion. (While bromide does play a certain role in the original Radicalator, that role is negligible. For example, omitting all the bromide-containing steps (Rl-R3) from the full mechanism makes no visible difference in the oscillatory behavior.) The control reaction (R10) is a radical-radical
MA'
+ Br02'
-
P
(R10) reaction between malonyl (MA') and bromine dioxide (Br02') radicals. According to measurements by Fbrsterling and Nosz~~
(38) Sorenm, P. G. In Kinetics of Physicochemical Oscillations; 1979; Vol. 1. p 41 (Preprints, Technische Hochschule Aachen, Germany 1979).
1
100
'"1
200
300
400
-
600 timds
500
a "classica1"BZ s y s t e m 1000
2000
-
3000 t i m d s
Figure 3. Stirring effect (a) in the modified Ricz system I1 and (b) in a classical BZ system. See text for initial concentrations. Observe the different behavior of the two systems.
t i c z i ~ s , (R10) )~ is a very fast diffusion-controlled reactiona and they calculated that klo is around 5 X lo9M-' 6'in a 1 M sulfuric acid medium. In the case of a diffusion-controlledreaction, the ratedetermining step is the diffusion of the reactants toward each other bemuse the probability of reaction at each encounter of the reactants approaches unity. The encounter rate of two species (39) Fbrsterling, H. D.; Noszticzius, 2.J , Phys. Chem. 1989, 93, 2740. (40) (a) Noyes, R. M. frog. React. Kinet. 1%1,1, 129. (b) Clark, I. D.; Wayne, R. P. In Chemical Kinetics; Bamford, C. H., Tipper, C. F. H., Eds.; Elsevier: Amsterdam 1969; Vol. 2, pp 305-310. (c) Ovchinnikov, A. A.; Timashev, S.F.; Belyy, A. A. Kinetics of Diffusion Controlled Chemical Processes; Nova Science Publishers: Commack, NY, 1989.
6578 The Journal of Physical Chemistry, Vol. 95, No. 17. 1991
Noszticzius et al.
TABLE 11: Radicalator Model and Its Parameters as They Were Used in Numerical Simulationsa Br- + HOBr + H+ F? Br, + H,O k , = 8 X lo9 M-, s-I Br- + HBr02 + H+ s 2HOBrBr- + BrO< + 2H+ s HOBr + HBrO, 2HBr0, ,a HOBr + BrO< + H+ HBr02 + Br03- + H+ s Br2O4 + H20 Br2O4 ,a 2Br0; Ce3++ Br0,' + H+ s Ce4++ HBr02 Ce4++ MA s Ce3++ MA' + H+ HOBr + MA BrMA + H20 2MA' + H20 MA + TA MA' + BrO; P, + P, 2Br03- + 2H+ HBr02 + HBr04 init concn, M chemical component BrOC H+ MA Ce3+ Ce4+ HBr0, Br204 RZcz system 0.015 4.00 0.6 1 x 10-3 7 x 10-7 2 x lo-' classical system 0.050 1.29 0.1 2 x 10-3 7 x 10-7 2 x 10-7 All experimentally determined rate constants were taken from the 1iterature;'O no parameter fitting was attempted. In the calculations, bromate, hydrogen ion, and malonic acid concentrationswere regarded to be constant ("pool of chemicals" approximation). Hydrogen ion concentrations were calculated according to Robertson and D ~ n f o r d .Initial ~ ~ concentrations not shown in the table were zero.
-
-4
in a nonstirred solution can be calculated,40 on the basis of certain assumptions, by using Fick's laws of diffusion. Now an important problem should be addressed in this respect: is hydrodynamic turbulence able to affect the rate of a diffusion-controlled reaction? Furthermore, if the answer is yes, what is the magnitude of such an effect? These problems are discussed in the next paragraph. Diffusion-Controlled Reactions and Hydrodynamic Turbulence. It is rather obvious that convection and stirring accelerates the component transport on a macroscopic level very effectively. Molecular diffusion on the other hand is a slow process on the macroscale; in the absence of any convection the homogenization of a reactor with linear dimensions of few centimeters would take several days. The average distance separating two reactive radicals, however, is not on the macroscale. For example, in a lo4 M solution there is one particle in a cube whose edge is 1 pm long. Assuming that the concentration of both reactive intermediates is lo4 M, the reaction partners are separated by a distance of 0.5 pm or less. At this small distance the molecular diffusion is considerably fast. The time constant T of a diffusional relaxation process on a distance d is given4I by eq 1, where Do
-
-
= d2/r2D0 (1) is the molecular diffusion coeffcint, which is usually cm2 s-I as an order of magnitude estimate. The above formula gives a 25-ps time constant for a 0.5-pm distance. Thus, the real question is whether the turbulent transport is able to compete with molecular diffusion on such a submicrometer length scale or not. According to the theory of turbulent diffusion this seems possible. The turbulent diffusion coefficient is by expression 2, DWRB= v ( d / d d 2 (2) where Y is the kinematic viscosity (- 10-2 cm2 s-l), d is the distance in question (an average distance between the reactive radicals in this case), and dKis a characteristic length scale determined by hydrodynamic condition^:^^^^^ T
dK = ( v / w ) ~ / ~ X - I / ~ (3) In (3) w is the angular velocity of a stirrer of radius A. For example, in our experiments the radius of the stirrer was 1.25 cm and the maximum frequency of stirring was 2000 min-' (w = 209 d);thus dK = 5 pm and &UEB = lo4 cm2 s-I for d = 0.5 pm. The effective diffusion Coefficient can be e~timated'~ as a sum of the molecular and turbulent diffusion coefficients: = Do + DTURB (4) Regarding our numerical example, the effective diffusion coefficient can be 1 order of magnitude higher than the molecular one. In the case of a diffusion-controlled reaction, an increase (41) Crank, J. The Mathematics of Dl//usion: Clarendon Press: Oxford, U.K., 1975. (42) Aronovitz, J. A.; Nelson, D. R. Phys. Rev. A 1984, 29, 2012.
of the effective diffusion coefficient will result in a proportional increase of the rate constant of that reactionam In other words, the rate constant of the radical-radical reaction can be increased with stirring by a factor of 10 if the radical concentration is lo4 M or less. At higher radical concentrations, however, the effect of stirring should decrease considerably. For example, a similar calculation assuming a lo4 M radical concentration gives hRB = lod cmz S-I, which is 100 times smaller than in the previous case. (Still a 10% effect compared to Do, however.) Simulation of Stirring Effects with the Radicalator Model. As we have seen in the previous paragraph, increasing the rate of stirring will increase the rate constant of a diffusion-controlled reaction. Thus we made numerical experiments by changing the rate constant kloof the radical-radical reaction in our model. A new notation was introduced
kR k 1 , / ( 5 X 10' M-l S-I 1 (5) to show the relative rate of (R10) compared to the original value in Table 11. For the original Radi~alator,'~ kR = 1. The results of such numerical experiments with the RBcz system are shown in Figure 4. When the experimental results of Figure 2a and b are compared with the simulations of Figure 4, there is a qualitative similarity. On a quantitative level, however, we can observe certain deviations. In the simulations there is a supercritical Hopf bifurcation a t kR = 0.38. In the experiments we were not able to observe that Hopf bifurcation but we think it should occur around 0.9 or 0.8, that is much nearer to kR = 1 than in the simulations. This conclusion can be explained qualitatively as follows. The decreased amplitude in Figure 2a compared to Figure 2b suggests that the bifurcation point is not very far away. Moreover, stirring cannot accelerate the rate of the radical-radical reaction considerably if one of the radicals is always in a relatively high concentration. In the case of sinusoidal oscillations, the BrOZ*radical concentration is always above lo-' M (see Figure 4, kR = 0.4). Thus-regarding our considerations in the previous paragraphstirring can cause only a 20-30'56 increase of the rate constant but not more. Consequently there is roughly a 2-fold discrepancy between the experimental and theoretical position of the Hopf bifurcation. A part of the discrepancy is due to the higher viscosity of the 3 M sulfuric acid as the diffusion coefficient is inversely proportional to the viscosity of the solution. The viscosity ratio of 3 and 1 M sulfuric acid solutions43 is 1.44. The relatively high (0.6 M) concentration of malonic acid should increase the viscosity of the solution even further. In the end, it is interesting to remark that the time period of the sinusoidal oscillations in the simulations and in the experiments agree surprisingly well; it is 19 s in both cases. (43) CRC Handbook of Chemistry and Physics; CRC Press: Cleveland, OH, 1975; p D-262.
Hydrodynamic Turbulence and Diffusion-Controlled Reactions The Journal of Physical Chemistry, Vol. 95,No. f7, I991 6579
L
, 50
1
6]
MA.
i
100
50
1
6/
100
100
-
150t/s
I
150 t / r
kR=0.40
MA*
1
50
r
150 t / s
k,=0.361
Y i
-
50
100
I 1
1
-
150 t/r
Figure 5. Computer simulationsof the effect of stirring in the classical BZ system. For the explanation of kR, see the caption of the previous
I
figure.
50
100
-
150 t / s
’
Figure 4. Computer simulations of the effect of stirring in the Ricz system with the Radicalator model. Calculated bromine dioxide (BrO,’) and malonyl radical (MA’)concentrations as a function of time. kR,the relative rate constant of the BrO,’ + MA’reaction. (See expression 9, increases with increasing stirring rate. The different k~ values representing different stirring rates are displayed together with the diagrams. Other parameters used in the computer simulations are given in Table 11.
Simulations of the modified Ricz system I of Figure 2c and d gave qualitatively correct results that were damped oscillations (not shown here). The damping decreased with the increasing kR in agreement with the experiments. Simulations of the modified Ricz system I1 of Figure 3a gave sustained oscillations (not shown here), but these oscillations were again of the relaxation type, not sinusoidal. In other words, the real system was nearer to the Hopf bifurcation than its model simulation, just as in the case of the original Rgcz system. Finally, we simulated stirring effects in the “classical” BZ system of Figure 3b with the Radicalator. The results are shown in Figure 5. Again, there is a good qualitative agreement between the simulations and the experiments. This is somewhat surprising, because the Radicalator does not calculate with bromomalonic acid, which is an important intermediate-’ in the later stages of the BZ reaction. (The longer time period of the simulations, 130 s versus the experimental 70 s, is probably due to the omission of the bromomalonic acid reactions.) It is clear from the figures that to simulate the experimentally observed stirring effects kR should be increased by a factor of 5 . This is much larger than in the case of the sinusoidal oscillations. Is it realistic to assume that stirring can increase kR more in the case of relaxation-type oscillations? We think that the answer is yes and we suggest the following explanation. In the case of relaxation oscillations there are relatively long time intervals during which the concentration of the more concentrated radical-in this
case the malonyl radical-is 10-8 M or below. As a consequence, it is possible that stirring can increase the rate of the radicalradical reaction by 1 order of magnitude. It is worth noting that in the model simulations the time period of the classical system is more sensitive to a change in kR than that of the Ricz system, as in the real experiments. In the classical system, increasing kR by a factor of 5 increases the time period by a factor of 2. In the Ricz system a 10-fold increase of kR is needed to produce a similar effect. (See Figures 5 and 4 for comparison.) Stirring Effects and Other Rate Constants. In the Radicalator there are two diffusion-controlled reactions; (R10) the bromine dioxide radical-malonyl radical reaction, the effect of which was analyzed previously, and ( R l ) the bromide-hypobromous acid reaction. In Figures 4 and 5 to simulate stirring we changed klo exclusively and omitted any change in k l . Such an omission can be justified by the relative insensitivity of the system toward variations in k l . According to our calculations, increasing or decreasing kl by 2 orders of magnitude had virtually no effect on the dynamics of the system. This is not surprising because, as was mentioned, neglecting all reactions of the Radicalator containing bromide ion leaves the dynamics practically unchanged. There are two more radical-radical reactions in the Radicalator, (-R5”),the dimerization of Br02’ radicals, and (R9), the disproportionation of malonyl radicals. These reactions are fast, but somewhat slower than diffusion controlled. (See Table I1 for numerical data.) While most probably the rate-determining step in these cases is the chemical reaction alone, a small contribution from diffusion cannot be excluded. Thus, we performed model calculations by varying k+ and k9. Increasing any of these rate constants gradually by 1 order of magnitude, we first observed sinusoidal oscillations and then a supercritical Hopi bifurcation and damped oscillations. Such behavior, however, is just contrary to the experimental one. Therefore, these reactions cannot be responsible for the observed stirring effects.
Conclusions (i) Stirring effects observed in two variants of the BelousovZhabotinsky reaction studied here can be explained, at least semiquantitatively, by the Radicalator model. This strongly suggests that diffusion-controlled radical-radical reactions can play a fundamental role in different versions of the BZ reaction. This does not mean, of course, that other reactions of the mechanism are unimportant. For example, in the case of bromide-controlled BZ systems, (R2), the bromous acid-bromide ion reaction should have a central role. (R2), however, cannot explain stirring effects because its rate constanta is 100 times smaller than that of a diffusion-controlled reaction. At any rate, the chief ~~
(44)
Stuk, L.; Roberts, J.; McCormick, W.D.;Noszticzius, Z . J . fhys.
Chem. 1990, 94,6734. (45) Farsterling, H.D.;Murinyi, S.Z . Nofurforsch., in press. (46) Gyergyi, L.;Rmp,S.L.; Field, R. J. J. fhys. Chem. 1991,95,3159. (47) Cyargyi, L.; Turinyi, T.; Field, R.J. J . fhys. Chem. 1990,94,7162.
(48) (a) Noszticzius, Z.; Noszticzius, E.; Schelly, Z . A. J . fhys. Chem. 1983, 87, 510. (b) Tyson, J. J. In ref 24, Chapter 3. (c) A r k , F.; UngvBrai-Nagy, Zs. J . fhys. Chem. 1986, 90, 1496. (d) Field, R. J.; Farsterling, H. D.J . fhys. Chem. 1986, 90, 5400. (e) Fersterling, H. D.; Murinyi, S.; Schreiber, H. 2.Nofurforsch. 1989, 4 4 4 555.
6580
J. Phys. Chem. 1991, 95, 6580-6585
conclusion is that a t least for some compositions the mechanism of the Belousov-Zhabotinsky reaction apparently includes some steps that are virtually diffusion controlled and probably of radical-radical type whether the Radicalator model is valid in detail or not. (ii) Diffusion-controlled reactions can cause hitherto unsuspected stirring effects even in batch reactors. Conversely, a stirring effect observed in a batch or a continuously fed stirred tank reactor can indicate the presence of a diffusion-controlled reaction. The reactants of such a reaction are not necessarily radicals, but both reactants should be present in low concentrations (lod M or less) and the diffusion-controlled step should play an important role in the whole mechanism. (iii) A uniform ultrasonic stirring can substitute for the conventional mechanical stirring in experiments with the BZ reaction. This is im rtant in the case of delicate nonlinear dynamic experimentsp(observation of different bifurcation^,^^.^^^^ strange (49) (a) Swinney, H. L.; Roux, J. C. In Nonequilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1984; p 124. (b) Noszticzius, Z.; McConnick, W. D.; Swinney, H. L. J. Phys. Chem. 1989, 93,2796. (c) Argoul, F.; A m d o , A,; Richetti, P.; Roux, J. C.;Swinney, H. L. Acc. Chem. Res. 1987, 20, 436.
attract or^?^ etc.) where the nonuniform mechanical stirring can create different reaction within the otherwise uniform reactor. (The rate of a diffusion-controlled reaction'"wou1d be higher at the stirring propeller than at the wall of the reactor.53) Acknowledgment. We thank Prof. H. D. FBrsterling, Prof. H. L. Swinney, and I. Noszticzius for their help in computer hardware and software and M. Noszticzius and E. Kovfics for their help in preparing the manuscript. This work was partially supported by the Department of Energy Office of Basic Energy Sciences, the Robert A. Welch foundation, and OTKA grants. Registry No. BrOC, 15541-45-4; Ce. 7440-45-1; malonic acid, 14182-2.
(50) Bar-Eli, K.; Noyes, R. M. J . Chem. Phys. 1987,86, 1927. (51) GispBr, V.;Showalter, K. J . Chem. Phys. 1988,88, 778. (52) Gyargyi, L.; Field, R. J. J . Phys. Chem. 1989, 93, 2865. (53) Nagata, S.Mixing; Wiley: New York, 1975; Figures 3.32 and 3.35. (54) Some new features of stirring sensitivities of the BZ reaction were reported just recently: Lopez-Tomas, L.; Sagues, F. J. Phys. Chem. 1991, 95, 701.
Entropy-Driven ProtowTransfer Reactions Michael Meot-Ner (Mautner) Chemical Kinetics Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (Received: February 14, 1991)
The relation between kinetics and thermochemistry in fast reactions is examined, including reactions with substantial entropy changes. Rate constants for such reactions, in the range of (0.02-3.0) X l e cmz s-l, were measured by pulsed high-pressure mass spectrometry. The following relations were observed: (1) The reaction eficiency in either direction is controlled uniquely and completely by the overall reaction free energy change. Specifically, the efficiency r is determined by the equilibrium constant according to r = K/(1 + K ) . (2) The sum of reaction efficienciesin the forward (exergonic) and reverse (endergonic) directions is near unity (rl rr zz 1). These relations are observed in anionic and cationic systems, in reactions with AHo up to 12 kcal/mol and with ASoup to 15 cal/(mol K). Consistent with ( l ) , reactions that are endothermic up to 7 kcal/mol can nevertheless proceed near the collision rate, when positive entropy changes make the reactions exergonic. The entropy changes are effective regardless of their structural origin. Relations analogous to (1) and (2) are also derived for reactions with multiple channels that proceed without significant barriers through a common intermediate.
+
Introduction Most fast ion-molecule reactions are exothermic processes with negligible entropy changes. However, some bimolecular exchange reactions do involve substantial entropy changes, and even small entropy changes can have significant effects at high temperatures. It is necessary therefore to generalize the relation between kinetics and thermochemistry in fast ion-molecule reactions, to include reactions with significant entropy changes. This is the objective of the present work. Usually, fast ion-molecule reactions proceed in the exothermic (and exergonic) direction near the collision rate, i.e., k = kmll. In other words, the reaction efficiency is near unity, eq 1. r = k/k,ll z 1 (1) In the reverse direction these processes are slowed by the thermochemical factor, eq 2. k = kWllexp(-AGO/RT) = kWllexp(-AHO/RT) (2) Bohme demonstrated these relations for a set of reactions,' and they are accurate for reactions with substantial overall free energy ( I ) (a) Bohme. D.; Mackay, G. 1.; Schiff, H. I. J . Chem. Phys. 1980, 73. 4976. (b) Bohme, D. In Ionic Processes in the Gas Phase; Almoster-Ferreira, M. A., Ed.; Reidel: Dordecht, The Netherlands, 1984; p 11 1.
changes, i.e. where [AGO1 > 4 kcal/mol. However, for a general treatment, fast reactions with small overall free energy changes should be also examined. This paper will focus on a set of such reactions. The reactions in Bohme's study had small entropy changes and AGO and AHo were indistinguishable within the accuracy of the thermochemical data.* We observed, however, some protontransfer and charge-transfer reactions with large entropy changes, up to 15 cal/(mol K).394 In these cases the kinetics were clearly determined by AGO rather than AHo. This was evident in particular when the magnitude of T U o was large and AH" and AGO had opposite signs. In particular, we found endothermic reactions that proceeded near unit efficiency even when the enthalpy factor of exp(-AHO/RT) would reduce the efficiency by 5 orders of magnitude. This occurred where T A S O was positive and large enough to render AGO n e g a t i ~ e . ~ ? ~ (2) Henchman, M. In Structure, Reactivity and Thermochemistry of Ions; Ausloos, P., Lias, S.G., as.; Reidel: Dordecht, The Netherlands, 1987; p 381, (3) Meot-Ner (Mautner), M.; Hamlet, P.; Hunter, E. P.; Field, F. H. J . Am. Chem. Soc. 1980,102,6393. (4) Sicck, L. W.; Meot-Ner (Mautner), M. J . Phys. Chem. 1982.86,3646.
Meot-Ner (Mautner), M. In Structure, Reactivity and Thermochemistry of
low; Ausloos. P., Lias, S.G., Eds.;Reidel: Dordrecht, The Netherlands, 1987; p 383.
This article not subject to U.S. Copyright. Published 1991 by the American Chemical Society
