J. Phys. Chem. 1996, 100, 17593-17598
17593
Oxygen Influence on Complex Oscillations in a Closed Belousov-Zhabotinsky Reaction Jichang Wang,† F. Hynne,* P. Graae Sørensen,* and K. Nielsen Department of Chemistry and CATS, H. C. Ørsted Institute, UniVersity of Copenhagen, UniVersitetsparken 5, DK-2100 Copenhagen, Denmark ReceiVed: May 23, 1996; In Final Form: August 16, 1996X
The influence of oxygen on the cerium-catalyzed Belousov-Zhabotinsky reaction is investigated in a system that is closed except for transport of oxygen from the atmosphere across a free surface. Experiments show that the introduction of oxygen increases the complexity of oscillations and shortens their duration. We determine the dependence of the oscillatory pattern on the concentration of oxygen in the reaction mixture, as measured with an oxygen electrode and controlled through the content of oxygen above the free surface. We show that the influence of the stirring rate under aerobic conditions is mainly an oxygen effect and report oscillations of the oxygen concentration. Calculations based on a modified Oregonator model show that the influence of oxygen can be qualitatively accounted for through the effect of two rate constants.
Introduction The influence of oxygen on the oscillating BelousovZhabotinsky reaction has been studied for a long time.1-8 Its importance was first pointed out by Barkin et al.1 in their study of reduction of ceric ions by malonic acid. They found that the rate of this reduction was enhanced in the presence of oxygen. A detailed mechanism has been suggested.1,9,10 Later, Noyes and co-workers5,11 found that oxygen greatly accelerates the release of bromide during oxidation of a mixture of BrMA and MA by Ce(IV) (see also ref 12). Such an effect may result from the influence of oxygen on the reaction between malonic acid (MA) and Ce(IV) if the release of Br- from bromomalonic acid (BrMA) is affected by malonic acid radical. Recently, Fo¨rsterling et al.13 have found that oxygen affects the stoichiometry of bromide production from the oxidation of bromomalonic acid by Ce(IV), and it was suggested that the effect was caused by the influence of oxygen on the decomposition of bromomalonic acid radical. Experimental studies of the influence of oxygen on the individual reactions assumed to occur in the BZ reaction have been quite extensive.1,2,5 In contrast, studies of the effect on oscillations of the full BZ reaction have been limited mainly to basic properties such as amplitude or period of simple oscillations. The present study is motivated by the striking effect of air on the complexity observed in stirred systems with a free surface for special initial conditions and by the well-known influence of air on pattern formation in spatially extended systems. An example is shown in Figure 1 where the absorbance of light at 340 nm is recorded for anaerobic and aerobic conditions in a closed system having a free surface in contact with the atmosphere. The purpose of this paper is to demonstrate experimentally that the effect of air is a direct result of the presence of oxygen and that the degree of complexity increases with the oxygen level. We also show that the effect of oxygen on the oscillatory pattern can be qualitatively accounted for in terms of an oxygen dependence of the effective rate constants of two specific reactions of a model. † Present address: Techn. Chem. Lab., Swiss Federal Institute of Technology, Zu¨rich, Switzerland. X Abstract published in AdVance ACS Abstracts, October 1, 1996.
S0022-3654(96)01531-6 CCC: $12.00
Figure 1. Absorbance of light at 340 nm as a function of time for a closed BZ system with initial concentrations shown in Table 1, run in a well-thermostated (25.0 °C) and well-stirred (740 rpm) rectangular glass cuvette with a free surface area of 4 cm2 and a reaction volume of 7.5 cm3. (a) Anaerobic conditions: pure nitrogen in the gas above the free surface. (b) Aerobic conditions: 20.9% O2 in the gas phase.
In this study the influx of oxygen is controlled in two different ways. The basic method is to control the concentration of oxygen in the gas phase with mass flow controllers at a fixed stirring rate. This method has previously been applied in studies of oxygen-consuming reactions, such as the peroxidase-oxidase oscillating reaction14 and the benzaldehyde-cobalt reaction,15 and it seems reliable. The rate of oxygen transport is estimated by measuring the transient change of the oxygen concentration in 1 M sulfuric acid from an initial equilibrium to a new equilibrium in response to a change of the oxygen concentration in the gas phase above the solution; see the Appendix. The other way to control the influx of oxygen is through the stirring rate at a constant oxygen concentration in the gas phase. This study is made in order to demonstrate that the influence of the stirring rate under aerobic conditions is mainly an effect of oxygen. The dependence of the oxygen transfer on stirring rate has been determined quantitatively by measuring the equilibration response for each chosen stirring rate. This question is further discussed in the subsection “influence of stirring rate”. © 1996 American Chemical Society
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TABLE 1: Initial Concentrations of Reactants, Used for the Experiments as Well as the Simulations (in which [H+] ) 1 M) reactant
concn/M
bromate malonic acid cerium(IV) sulfuric acid
0.0467 0.444 0.0022 1
Experimental Section The experiments have been performed in a well thermostated (25.0 ( 0.1 °C) square glass cuvette with a free surface area of 4 cm2. The volume of the reaction mixture was 7.5 mL in these experiments. The absorption of light at 340 nm was measured with an HP8452 diode array spectrophotometer from HewlettPackard, and the oxygen concentration was measured by an MI730 oxygen microelectrode from Microelectrodes Inc., connected to a Keithley 825 picoamperometer. The reaction mixture was stirred by a motor-driven threebladed Teflon impeller. The stirring rate can be varied continuously by servo control of the motor. The useful stirring rates in these experiments were from 420 to 740 rpm. Stirring rates in this interval are sufficiently high to keep the solution homogeneous and sufficiently low to let any small gas bubbles escape. The oxygen concentration in the gas phase was controlled by two model 5850E mass flow controllers from Brooks Instrument B.V., one connected to a nitrogen supply, the other to atmospheric air. The rate of transport of oxygen from the gas phase to the liquid was determined as described in the Appendix. It depends on the stirring rate and is 0.0050 min-1 at 740 rpm and 0.0028 min-1 at 420 rpm. The ratio between the equilibrium concentration of oxygen dissolved in 1 M sulfuric acid and the concentration in the gas phase is 0.025 at the temperature of the experiments, 25 °C. All chemicals were of analytical grade and were used without further purification. The solutions were prepared with doubly ion-exchanged water. The initial concentrations of the reactants in the reaction mixture are listed in Table 1. These were used in all the experiments reported here (and in the simulations reported below). Immediately before starting an experiment, each of the solutions of MA, BrO3-, and Ce(IV) were bubbled with nitrogen for about 10 min in order to reduce the concentration of dissolved oxygen as much as possible. Experimental Results Influence of the Concentration of O2 in the Gas Phase. Figure 2 shows the result of experiments with varying concentrations of oxygen in the atmosphere above the free surface. In all four experiments, the stirring rate is kept constant at 740 rpm. When the gas phase contains only nitrogen (Figure 2a), only simple oscillations are observed, and the amplitude of the oscillations decreases slowly with time. Introduction of 6.0% of oxygen in the gas phase (Figure 2b), makes the transient oscillations slightly complex. The amplitude decreases faster, and the oscillations undergo a “transient period doubling bifurcation” to two-peak oscillations lasting about 1000s (15 min), after which a reverse “bifurcation” returns the system to simple oscillations. As the concentration of oxygen in the gas phase is increased, the onset of complexity becomes more abrupt, the duration of the complex oscillations increases and the difference in magnitude between large and small peaks becomes larger, as in Figure 2c for 9.0% oxygen.
Figure 2. Absorbance of light at 340 nm as a function of time for a closed BZ system with varying concentrations of oxygen in the gas phase above the free surface: (a) 0%, (b) 6.0%, (c) 9.0%, and (d) 12.6%. Otherwise, the experimental conditions are the same in all four experiments and are described with Figure 1. In particular, the initial concentrations (Table 1) and the stirring rate (740 rpm) are the same.
When the concentration of oxygen is increased still further as in Figure 2d for 12.6% oxygen, the oscillations become more complex (four-peak oscillations can be seen between 4500 and 5000 s) and also become more irregular. In summary, the presence of oxygen in the gas phase above an otherwise closed, stirred BZ system can shorten the duration of the oscillations and induce complexity. If the system of Figure 2 is exposed to ordinary air, the oscillations last only about 1 h, as Figure 1b shows. Complex oscillations similar to those induced by oxygen can also be seen under anaerobic conditions if the initial concentrations are changed.12,16,17 Influence of Stirring Rate. Under anaerobic conditions, even quite complex oscillations are independent of the stirring rate provided it is so fast that the system stays homogeneous.12,16,17 The three time series in Figure 3 show the result of running the same experiment with three different stirring rates but with the same concentration of oxygen in the atmosphere above the free surface. In Figure 3a the stirring rate is 420 rpm. The pattern looks much like in Figure 2a but the amplitude decreases much faster. With a stirring rate of 570 rpm (Figure 3b), the decrease of amplitude is faster than in Figure 3a and the system develops complex oscillations through a “transient period doubling” occurring about 2800 s (45 min) after the start of the experiment. At a stirring rate of 740 rpm (Figure 3c) the oscillations become more complex and irregular, starting already after 2200 s (35 min) through a transient period doubling. Comparison of Figures 2 and 3 shows that qualitively the same changes of oscillation pattern and bifurcation structure
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Figure 5. Effect of changing the stirring rate during an experiment with a closed BZ system: simultaneous recordings of (a) the absorbance of light at 340 nm and (b) the current of the oxygen electrode as functions of time. The concentration of oxygen in the gas above the free surface is 20.9%. The stirring rate is decreased from 740 to 420 rpm about 2 h after the start of the experiment, resulting in sharp drop in the oxygen concentration and a reappearence of oscillations. Other experimental conditions are as given in the text of Figure 1. Figure 3. Absorbance of light at 340 nm as a function of time for a closed BZ system with varying stirring rates: (a) 420, (b) 570, and (c) 740 rpm. Otherwise, the experimental conditions are the same in all three experiments: the concentration of oxygen in the gas phase above the free surface is 12.6%, and the initial concentrations are given in Table 1.
Figure 4. Simultaneous recordings of (a) the absorbance of light at 340 nm, and (b) the current of the oxygen electrode as functions of time for an experiment on a closed BZ system. The concentration of oxygen in the gas above the free surface is 10.5%. The initial concentrations are shown in Table 1. Other experimental conditions are as given in the text to Figure 1. The first 300 s, the signal from the oxygen electrode is not significant.
can be observed in the closed BZ system either by changing the concentration of oxygen in the gas phase above the free surface or by changing the stirring rate. This fact indicates that the dependence on the stirring rate is due to oxygen: the rate of transport of oxygen into the reaction mixture depends on the stirring rate. We shall verify this interpretation below by measuring the rate of transport of oxygen from the gas phase to the liquid as a function of the stirring rate (see the Appendix). Oscillations of the Oxygen Concentration. The time dependence of the concentration of oxygen has been investigated by using an oxygen electrode. Figure 4 shows the result of an experiment where the absorbance at 340 nm (proportional to [Ce(IV)]) and the output from an oxygen electrode (proportional
to [O2]) were measured simultaneously at conditions where the system exhibits complex oscillations. Oxygen oscillations appear at the onset of complexity (as seen in the light absorption). Before that, oxygen oscillations cannot be distinguished from noise. In the first 300 s of a run, the signal is unreproducible, depending on the precise experimental conditions (bubbles on the electrode membrane, etc.) and so is not significant. One may wonder if the response of the oxygen electrode could be caused by other species involved in the BZ reaction. However, an experiment (not shown) under the same experimental conditions as in Figure 4 except that the concentration of oxygen in the gas phase is zero, showed no oscillations of the oxygen electrode signal although the absorbance at 340 nm was oscillatory as in Figure 1a. This fact strongly suggests that the oscillations, (Figure 4b), detected by the oxygen electrode, do represent oscillations of [O2]. Most of the oxygen transferred to the reaction solution from the gas phase by the stirring is consumed by the chemical reactions. The average oxygen concentration is largest under the large-amplitude oxygen oscillations where it is about 0.6 µM (1 pA corresponds to about 0.1 µM of O2.) At 25 °C, the concentration of oxygen in 1 M sulfuric acid is 0.11 mM in equilibrium with an oxygen gas of concentration 4.3 mM (10.5%). So the average concentration is less than 1% saturated. Even though there is a net average consumption of oxygen in the reaction, an interesting question is whether oxygen is also produced: can the growth of [O2] in the rising part of an oscillation be accounted for by the inflow of oxygen across the surface? The transfer rate of oxygen is about 0.4 µM s-1 so during the approximately 3 s the concentration is rising, there is transferred about 1.2 µM. That amount is more than enough to account for the change of concentration of about 0.7 µM during the rise. So there is a net oxygen consumption during the rising part as well, and there need not be a production of oxygen in the reaction. Figure 5 shows the result of an experiment with a controlled gas phase corresponding to atmospheric air and the effect of a change of the stirring rate after the oscillations have stopped. The initial conditions are the same as in Figure 1b, and the output from an oxygen electrode is measured simultaneously with the absorbance at 340 nm as in Figure 4.
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TABLE 2: Modified Oregonator Used for Simulating the Effect of Oxygen in the BZ Reaction no.
reaction
rate constant kn
ref
1 2 3 4 5 6 7 8 9 10 11
BrO3- + Br- + 2H+ f HBrO2 + BrMA HBrO2 + Br- + H+ f 2BrMA 2HBrO2 f BrO3- + BrMA + H+ BrO3- + HBrO2 + H+ f 2HBrO2 + 2Ce4+ Ce4+ + BrMA f BrMA• Ce4+ + MA f MA• 2MA• f MA + P MA• + BrMA• f Br2BrMA• f Br- + BrMA BrMA• f BrBrMA f P
2 M-3 s-1 3× 106 M-2 s-1 3× 103 M-1 s-1 42 M-2 s-1 30 M-1 s-1 0.3 M-1 s-1 3.2 × 109 M-1 s-1 1 × 109 M-1 s-1 1 × 108 M-1 s-1 7 s-1 4 × 10-4 s-1
18 18 18 18 19 19 20 20 20 a a
a
Adjustable in this work.
At such high oxygen concentration, the oscillations last only for 1 h, during which the amplitude of oxygen increases while that of Ce(IV) decreases and the period (time between peaks) increases. So the fraction of time the system spends in the reduced state (low [Ce(IV)] to [Ce(III)] ratio) increases with time. We note that the concentration of oxygen increases very fast immediately after a peak of [Ce(IV)]. Subsequently, [O2] decreases roughly linearly until it reaches almost zero, at which point a new peak of [Ce(IV)] appears. At the moment when the oscillations stop, the concentration of oxygen starts a steady increase while the concentration of Ce(IV) stays at a low level. For about 1 h the BZ system shows no oscillations. At that time, the stirring rate is decreased from 740 to 420 rpm in the experiment. This results in a rather steep drop in the oxygen concentration due to a lower rate of transfer of oxygen from the gas phase to balance the consumption of oxygen in the chemical reactions. At the point when [O2] has reached almost zero, the system starts oscillating again. Model Calculations Studies1-8 have indicated that the influence of oxygen in the BZ reaction mainly occurs in the organic reaction part where Br- is produced. It is known1 that oxygen accelerates the reaction between Ce4+ and MA, and as shown by Fo¨rsterling et al.,13 the stoichiometry of the bromide production is also influenced: two Ce4+ ions produce one Br- ion under anaerobic conditions but two Br- ions under aerobic conditions. The authors attributed this stoichiometric effect to an increase of the rate of decomposition of bromomalonic acid radical. To relate the results mentioned above to our observations of the influence of oxygen, we have made simulations with the model defined in Table 2. The model is derived from the Oregonator21 by extending it with reactions describing the regeneration of Br- in more detail. Equations 1-4 form the inorganic part of the Oregonator model except that [BrMA] appears and is treated as a dynamical variable here. Equations 5-11 describe the organic part of the mechanism including the regeneration of bromide. Reaction 6 is known to be accelerated by oxygen. Reactions 8-11 represent possible reactions through which oxygen may change the rate of release of bromide. The characterization of the influence of oxygen is carried out by adjusting the rate constants of appropriate reactions from this set. Because the initial concentration of malonic acid is much higher than that of bromate, [MA] varies little in the course of the reaction and is kept constant in the calculation. The same conclusion applies to [H+], which is fixed at 1 M. Thus, there are seven dynamical variables in the model, namely, the concentrations of cerium (Ce4+), bromate (BrO3-), bromous acid (HBrO2), bromide (Br-), bromomalonic acid (BrMA), bromo-
Figure 6. Simulation of the BZ reaction in a closed system under anaerobic (a) and aerobic (b) conditions. The time series of the cerium(IV) concentration are obtained by numerical integration of the model defined in Table 2 with the initial conditions of bromate, malonic acid, cerium(IV), and sulfuric acid given in Table 1 (which means that [H+] ) 1 M). In (a) the rate constants are as listed in Table 2, but in (b) k6 and k10 are changed to simulate the effect of oxygen, and in (c) and (d), the separate effects of the changes of k6 and k10 are shown. The rate constants k6 and k10 are as follows: (a) k6 ) 0.30 and k10 ) 7, (b) k6 ) 0.40 and k10 ) 20, (c) k6 ) 0.40 and k10 ) 7, and (d) k6 ) 0.30 and k10 ) 20. All other rate constants are as in Table 2.
malonic acid radical (BrMA•), and malonic acid radical (MA•); P represents inactive products such as CO2. The time evolution is obtained by numerical integration of the mass action rate equations obtained from the model (Table 2) with the seven dynamical variables mentioned above. Figure 6 shows the result of simulations of the BZ reaction carried out in a closed system under anaerobic (Figure 6a) and aerobic (Figure 6b) conditions. The anaerobic run Figure 6a is calculated with the rate constants given in Table 2 and the initial conditions of Table 1. The influence of oxygen is simulated in Figure 6b by increasing the rate constants of reactions 6 and 10 from the values shown in Table 2: k6 is increased to 0.4 M-1 s-1, and k10 is increased to 20 s-1. All other rate constants and initial conditions are as in Figure 6a. The result is simple oscillations lasting shorter than in Figure 6a interrupted by a section of complex oscillations. Figure 6c,d shows the separate effects of increasing k6 and k10 individually. In Figure 6c k6 is increased to 0.4 M-1 s-1, with the result that the oscillations last shorter than in Figure 6a. In Figure 6d, k10 is increased to 20 s-1, with all other rate constants as in Figure 6a. The result here is that the simple oscillations are interrupted by an interval of complex oscillations and that the oscillations last longer than in Figure 6a. The effect of the two changes combined is a shorter duration of the oscillations (the effect of k6 on the duration dominates)
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with an interval of complex oscillations (from the increased rate of reaction 10) as shown in Figure 6b. On the other hand, reactions 8 and 9 have little influence on either the pattern or duration of the oscillations: if the rate constants of any of these two reactions is doubled, the result does not differ much from that of Figure 6a. Conclusion Effects of oxygen on the Belousov-Zhabotinsky reaction are important because oxygen will enter a system unless special measures are taken to prevent it, whether in stirred systems or in spatial systems with a large open surface bounding a thin layer of BZ reagent. We show that oxygen effects are particularly important in experimental investigations of chaos and chemical turbulence. In closed, well-stirred BZ systems, oxygen has two effects to oscillations. It shortens their duration, and it may make them complex. Both effects increase with the rate of inflow of oxygen. In the experiments, the inflow is carefully controlled through the concentration of O2 in a gas phase above the reaction mixture and through the stirring rate. We demonstrate that an increase of the concentration of O2 in the gas phase at constant stirring rate has the same effect as an increase of the stirring rate at constant gas composition; compare Figures 2 and 3. By controlling the concentration of oxygen in the gas phase, we control the inflow of O2, but not its concentration in the liquid phase. Indeed, the oxygen level in the reaction mixture appears to be very important to the oscillatory behavior, and [O2] oscillates during complex oscillations and when the oscillations of [Ce(IV)] are small, as shown in Figure 4. A sufficiently high level of oxygen seems to keep [Ce(IV)] low and prevent oscillations. Thus, oscillations may reappear, even after a long period in a nonoscillatory state, if we reduce the level of oxygen by decreasing the inflow (see Figure 5). The control of the inflow of oxygen by the concentration of oxygen in the gas phase above the free surface (or by the stirring rate) seems entirely reliable. Nevertheless, we must discuss whether alternative interpretations are possible. The two obvious ones are a removal of volatile species by the gas flow22 and inhomogeneities due to insufficient stirring.23,24 Loss of volatile species at high stirring rates cannot explain the findings since we can induce the effects at a constant stirring rate by increasing only the concentration of oxygen in the gas phase. Note that the rate of the gas flow above the surface is constant in all the experiments. In addition we show that we can get the same series of patterns of complex oscillations and transient bifurcations by changing the stirring rates when there is oxygen in the gas phase, but there is no effect of the stirring rate when the gas is pure nitrogen. Inhomogeneities (or fluctuation/nucleation processes) are not likely to play a major role in the formation of the complex oscillations in the present experiments. For different initial conditions the system can show complex oscillations under anaerobic conditions,12,16,17 and these are similar to those seen here (e.g., transient period doubling bifurcations) and are independent of the stirring rate. The simplest interpretation of the experiments is that the presence of oxygen changes the chemical kinetics and takes the system into a regime of complex oscillations. This view is supported by the model simulations. Possible mechanistic explanations of the effect of oxygen are suggested by simulations that qualitatively reproduce the phenomena observed experimentally. Two reactions known to be affected by oxygen are important in the model: increasing the rate of the reaction between Ce4+ and MA has the effect of
Figure 7. Relaxation curves used for the determination of the rates of transport of oxygen between the gas phase and the reaction solution and for calibration of the oxygen electrode. The experiment was made at 25.0 °C by changing the gas phase above 1 M sulfuric acid between 12.6% oxygen and pure nitrogen. The stirring rate was 740 rpm until 4500 s (marked with a small vertical line on the curve) after which it was 420 rpm.
decreasing the duration of the oscillations, whereas increasing the rate of the decomposition of bromomalonic acid radical with the formation of bromide ions results mainly in the appearence of complex oscillations. Appendix: Rates of Transport of Oxygen between Gas Phase and Solution The rate of transport of oxygen from the solution to the gas phase can be determined directly from the current of the oxygen electrode without knowing the proportionality constant between oxygen concentration and current. It is obtained as the timecoefficient, k-, in the exponential relaxation of the current from one equilibrium to another which occurs when the gas phase above 1 M sulfuric acid is changed from nitrogen with 12.6% oxygen to pure nitrogen. (The opposite change has the same characteristic time and may be used as well.) The experimental relaxation curves are shown in Figure 7. The smooth curve shows that the transport is well described by a macroscopic law. In practice, the transfer rate is obtained from the initial slope of the relaxation curve. At the conditions of the experiments, the coefficient of the transfer rate of oxygen from the solution to the gas phase is found to be k- ) 0.20 min-1 at a stirring rate of 740 rpm, and k- ) 0.11 min-1 at 420 rpm. The ratio between the equilibrium concentrations of oxygen in 1 M sulfuric acid and in the gas phase is 0.025 at 25 °C.25 Multiplication of k- by this ratio of equilibrium concentrations immediately yields the coefficient of the transfer rate of oxygen from the gas phase to the solution as k+ ) 0.0050 min-1 at a stirring rate of 740 rpm and k+ ) 0.0028 min-1 at 420 rpm. Acknowledgment. We thank Merete Torpe for assisting in the experiments. We are grateful to the DALOON Foundation and to the Center for Chaos and Turbulence Studies (CATS) for financial support and to the Danish Natural Science Research Council for a research grant. References and Notes (1) Barkin, S.; Bixon, M.; Noyes, R. M.; Bar-Eli, K. Int. J. Chem. Kinet. 1978, 10, 619. (2) Bar-Eli, K.; Haddad, S. J. Phys. Chem. 1979, 83, 2952. (3) Treindl, L.; Fabian, P. Coll. Czech. Chem. Commun. 1980, 45, 1168. (4) Ruoff, P. Chem. Phys. Lett. 1982, 92, 239. (5) Ganapathisubramanian, N.; Noyes, R. M. J. Phys. Chem. 1982, 86, 5158. (6) Menzinger, M.; Jankowski, P. J. Phys. Chem. 1986, 90, 1217 (7) Li, R.; Li, J. Chem. Phys. Lett. 1988, 144, 96. (8) Sevcik, P.; Adamacikova, I. Chem. Phys. Lett. 1988, 146, 419. (9) Field, R. J. In Oscillations and TraVelling WaVes in Chemical systems; Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985; p 65. (10) Ruoff, P.; Noyes, R. M. J. Phys. Chem. 1989, 93, 7394. (11) Jwo, J.-J.; Noyes, R. M. J. Am. Chem. Soc. 1975, 97, 5422. (12) Wang, J. Ph.D. Thesis, Copenhagen University, Danmark, 1994.
17598 J. Phys. Chem., Vol. 100, No. 44, 1996 (13) Fo¨rsterling, H. D.; Stuk, L.; Barr, A.; McCormick, W. D. J. Phys. Chem. 1993, 97, 2623. (14) Olsen, L. F.; Degn, H. Biochim. Biophys. Acta 1978, 523, 321. (15) Jensen J. H. J. Am. Chem. Soc. 1983, 105, 2639. (16) Wang, J.; Sørensen, P. G.; Hynne, F. J. Phys. Chem. 1994, 98, 725. (17) Wang, J.; Sørensen, P. G.; Hynne, F. Z. Phys. Chem. 1995, 192, 63. (18) Field, R. J.; Fo¨rsterling, H.-D. J. Phys. Chem. 1986, 90, 5400. (19) Gyorgyi, L.; Rempe, S. L.; Field, R. J. J. Phys. Chem. 1991, 95, 3159.
Wang et al. (20) Gyorgyi, L.; Turanyi, T.; Field, R. J. J. Phys. Chem. 1990, 94, 7162. (21) Field, R. J.; Noyes, R. M. J. Chem. Phys. 1974, 60, 1877. (22) Noyes, R. M. J. Chem. Soc., Faraday Trans. 1 1986, 82, 2999. (23) Ruoff, P. J. Phys. Chem. 1993, 97, 6405. (24) Vanag, V. K.; Melikhov, D. J. Phys. Chem. 1995, 99, 17372. (25) Geffcken, G. Z. Phys. Chem. 1904, 49, 257.
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