7032
J. Phys. Chem. 1992, 96, 7032-1037
Chem. 1988,60,989. ( b ) Furuuchi, H.; Arai, T.; Kuriyama, Y.; Sakuragi, H.; Tokumaru, K. Chem. Phys. Lert. 1989, 162, 211. (24) (a) see ref. 2. (b) Wismontski-Knittel, T.; Das, P. K. J . Phys. Chem. 1984,88, 2803. ( c ) Bhattacharyya, K.;Chattopadhyay, S. K.; Baral-Tosh,
S.; Das, P. K. J. Phys. Chem. 1986, 90, 2646. (25) Arai, T.; Karatsu, T.; Sakuragi, H.; Tokumaru, K. Chem. Lett. 1981, 1377. (26) Sandros, K.; Becker, H.-D. J . Phorochem. 1987, 39, 301.
Dynamical Study of the Chlorine Dioxide-Iodide Open System Oscillator' Istvdn Lengyel? Jing Li, and Irving R. Epstein* Department of Chemistry and Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02254-9110 (Received: March 6, 1992; In Final Form: April 27, 1992)
The reaction between chlorine dioxide and iodide in a flow reactor has been studied both experimentally and theoretically. Oscillation occurs in a wide range of flow rate (ko) and input iodide concentration ([I-lO). In a narrower range of [I-lO, bistability between steady and oscillatory states can be observed. Simple bistability between two steady states was found only in a very narrow region of the ko-[I-Io plane. An empirical rate law model containing two reactions and three variable concentrations was used successfully to simulate the observed behavior. Sub and supercritical Hopf, saddle loop dimension one and neutral saddle loop codimension two bifurcations have been found.
Introduction The capability, developed during the past decade, for the systematic design of chemical oscillators3has led to the discovery of many new oscillators, most of which operate in a continuous flow stirred tank reactor (CSTR) in order to maintain the necessary far-from-equilibrium condition^.^-^ While detailed mechanisms have been developed for about half of the known oscillators, only a handful have been successfully described by comparatively simple models like the Oregonator model' of the prototypical Belousov-Zhabotinsky oscillator. The reaction of chlorite ion (ClO,), iodide ion (I-), and malonic acid (MA) displays oscillatory behavior not only in a CSTR but also, transiently,in a closed system! This reaction has been widely used to study spatial pattern formation9-" in open systems. In attempting to unravel the mechanism of the C102--I--MA reaction, we recently discoveredI2that chlorine dioxide (C102)plays a key role in the closed system oscillation: after a rapid initial consumption of C102- and I-, reactions involving C102, 12,and MA are responsible for the oscillations observed in the absorbance of 13-. We were able to construct a very simple but comparatively accurate empirical rate law model for the C102-1,-MA reaction consisting of three component processes: a reaction between I2 and MA to produce I-, a reaction between C102 and I- that produces ClO;, and an iodide-inhibited feedback process between C102- and I-. This model suggests that malonic acid can be replaced by other species that react with iodine in a similar fashion to malonic acid (e.g., ethyl acetoacetate) or that MA can be eliminated completely and replaced by an iodide inflow into a CSTR. The first of these predictions has been confirmed experimentally.I2 In this paper, we investigate the latter case,the reaction between C102 and I- in a CSTR. We present both experimental observations and computer simulations in order to check the validity of a simplified model based on two overall reactions. We use both numerical simulation and numerical bifurcation methods to compute one- and two-parameter bifurcation diagrams which are then compared with our experiments. Experiments Materials. Analytical grade NaI.2H20, H2SO4, and MA (all Aldrich) were used without further purification. Chlorine dioxide was prepared as described previously,12and the stock solution was kept slightly acidified (pH = 4, H2S04)in the dark at 5 OC. No measurable change in its absorbance was observed for several weeks. Solutions of C102 were freshly diluted prior to use and
were kept in an ice bath during the experiments. Low boiling point hydrocarbons (n-pentane, petroleum ether) were added to the solution to cover the surface and retard the evaporation of C102. By taking these precautions, we were able to reduce the concentration change in C102to less than 3% per day; without special care the daily concentration change could exceed 10%. At some compositions, the reaction between C102 and I- is very sensitive to the ratio of the reactants, so good control over [ClO,] is essential for obtaining reproducible dpamical behavior. The concentration of CIOz was determined spectrophotometrically from its molar absorptivity of 1250 M-' cm-' at 360 nm.') Methods. A thermostated glass reactor of volume 20.2 cm3 fitted with a Teflon cap was used. The reactor was fed with a Rainin Rabbit-Plus peristaltic pump that gave pulsations of less than 0.5%. An iodide-selective electrode (Orion, 94-53A) was used to monitor the progress of the reaction. The reference electrode was a saturated calomel (Radiometer) connected by a NaNO, salt bridge. The electrode potential was amplified and recorded on a personal computer equipped with a DAS-8 A/D converter, set to sampling rates of 0.5-5 Hz depending on the rate of the reaction using the Labtech Notebook program. In some cases, the potential of the iodide-selective electrode was outside the range where the electrode responds solely to [I-] (50 mV). At low flow rates, the period of oscillation, a few seconds, is shorter than the response time of the iodide-selective electrode. Consequently, part of the oscillatory waveform may be lost, and the amplitude and period may not be accurate. We also monitored the oscillations spectrophotometrically by following the high absorptivity of triiodide at 280 and 350 nm. The temperature in most of the experiments was 25 f 0.2 OC.
The Model The chlorine dioxideiodide reaction is a subset of the previously studied C102-12-MA reaction. We therefore attempted to use the same empirical rate law approach to describe the open dynamics of the C102-I- reaction. For this system, we need only two processes and their experimentally determined rate laws, the reaction between C102and I- and the reaction between C102-and I- ions: ClO2 + Ic102- + '/zIz r ] = k,[CI021[1-1 (1) C102- + 41- + 4H' I2 C1- 2 H 2 0
-
-
0022-3654/92/2096-7032%03.00/0 0 1992 American Chemical Society
+
+
The Journal of Physical Chemistry, Vol. 96, No. 17, 1992 7033
Chlorine Dioxide-Iodide Open System Oscillator The values of the parameters are12 kl = 6 X lo3 M-' s-I k 2a M2 at 25 OC. = 460 M-2 s-l, k2b= 2.65 X s-I, and u = Using overall stoichiometric processes rather than elementary steps for describing a complex kinetic system will yield valid results only if there is no significant interaction between the intermediates of the component processes, i.e., no cross-reactions, and if no intermediates build up to high concentrations. If these conditions hold, we can preserve the accuracy and simplicity of formal kinetic rate equations without having to assume rate constants for immeasurable elementary processes. This method has proven fruitful in describing several complex system^^^'^-'^ including the ancestor of this open system oscillator, the closed system oscillationsin the C1OZ-I2-MA reaction.12 The kinetics of reaction 1 were determined by Lengyel et a1.,12 and those of reaction 2 were determined by Kern and Kim1' and modified by Lengyel et a1.I2 This second reaction shows oscillatory behavior in a flow reactor at high input concentrations of CIOzand IM), but no oscillations occur at the lower concentrations employed in this study, because the key intermediates cannot accumulate to sufficiently high levels. If one describes a reaction like (2) by a single overall stoichiometry, there is only one independent variable, and the model cannot give limit cycle oscillations. Under our experimentalconditions the concentrations M in the reactor. This conof intermediates never exceed clusion is based on the fact that the stoichiometry, as monitored by following [C102], [I2],and [I3-] spectrophotometrically, never deviates by more than 10% from the sum of reactions 1 and 2 during the course of the reaction. Consequently, the simple rate equation 2 gives a comparatively accurate description of the behavior of the system. Under all of our experimental conditions, the fmt term in r2is less than 1% of the second. We have therefore simplified the model by using only the second term to characterize the hetics of reaction 2. From a dynamical point of view, reaction 2 is the crucial feedback process; it is autocatalytic in iodine and inhibited by the reactant iodide. Because under our conditions [I2] is always high and almost constant, the autocatalytic feature plays no significant role in the dynamics. However, [I-] can change by several orders of magnitude, as shown by the potential of the iodide-selective electrode (Figure la), and its inhibitory role is crucial. The differential rate equations 3 corresponding to reactions 1 and 2 contain three variables, the concentrationsof C102, I-, and C102-. The concentration of I2 can be calculated from the stoichiometric relation [Iz] = ( [I-], - [1-])/2, where [I-I0 is the inflow concentration of iodide to the reactor. 9
d[ClOy] -dr
For finding dimension two bifurcation points, we used both the original forms of the differential equations and a dimensionless form also in which a rescaling was employed that preserves the meaning of the original flow parameters and separates the experimentally controllable parameters from the kinetic parameters. For ease of comparison with experiment,we present all bifurcation diagrams in terms of the original variables and parameters. 0 8 6 Numerical integration and continuation with the ~ ~ ~ package1* were used to find steady states and to study periodic orbits of the differential equation system. In some cam, to detect dimension two bifurcations that cannot be recognized by the continuation program, we calculated the trace, determinant, and eigenvalues of the Jacobian matrix separately along the curves of steady-state
TABLE I: Symbols Used in the Calculated Bifurcation Diagrams
symbol
-
___ 0 0
description stable steady state in 1-parameter bifurcation diagrams or line of limit points in 2-parameter bifurcation diagrams unstable steady state in 1-parameter bifurcation diagrams or Hopf bifurcation line in 2-parameter bifurcation diagrams Hopf bifurcation point amplitude of stable limit cycle amplitude of unstable limit cycle
solutions. Because of the wide range of parameters and the very large variations in the values of the variables, we carried out the continuation calculations using a logarithmic transformation of both the differential equations and the parameters. This rescaling helped to make the calculationsnumerically more stable. In Table I we summarize the symbols used in our calculated bifurcation diagrams. RWlltS Stoichiometry and Closed System Behavior. The overall stoichiometry of the CIOz-I- reaction depends strongly upon the concentration ratio of the reactants. At high iodide excess, the stoichiometry of the reaction is the sum of reactions 1 and 2:
C102
+ 51- + 4H'
= C1-
+ 2.512 + 2H20
(4)
If r = [I-]0/[C102]0is greater than 1, then processes 1 and 2 are well separated in time. Reaction 1 is fast, and because I- is in stoichiometric excess over C102 with respect to reaction 1, the remaining I- inhibits reaction 2, causing this step to be much slower. If the I- remaining from reaction 1 is not in stoichiometric excess with respect to reaction 2 (r < 5), the reaction between C102- and I- will accelerate as the inhibitor [I-] decreases until essentially all the iodide ions are consumed. Stoichiometries (I), (2), and (4) describe these events accurately up to the total consumption of iodide ions. In the presence of iodide, we can ignore any reaction between C10, and 12,because even if iodine were to react rapidly with chlorite, the intermediates HOI, HI02, and HOC1 would react even faster with iodide to regenerate iodine immediately. Until iodide is consumed completely, the concentration of iodine does not decrease. However, once the iodide is gone, C10; ions formed in reaction 1 can react with iodine, and [I2]can drop quite rapidly. In the chlorite-iodine reaction, the concentrations of intermediates can reach relatively high levels, as evidenced by the back-formation of iodine from the disproportionation of I(1) (HOI or ICl) and HIOz.19 Our simple model (3) based only on reactions 1 and 2 neglects these processes, but we believe they do not significantly affect the dynamics under our conditions, because the relevant intermediates are present at very low concentrations. In Figure 1, we show potentiometric and spectrophotometric traces of the closed system reaction of C102 and I- for reactant ratios r between 1 and 5. The first, very rapid decrease in [I-] and increase in [I2] result from reaction 1. We then see a slower decrease in iodide concentration that accelerates as the inhibition by iodide is released by the consumption of I- in reaction 2. If r < 1, the two processes are not separated, and both reactions are very fast, with a time scale in the stopped flow range. If r > 5 , reaction 2 is slow, and since significant amounts of I- always remain, the feedback process 2 cannot be switched on. In the open system, this 5:l ratio based on stoichiometry (4) gives an upper limit for the appearance of such dynamical behavior as bistability or oscillation. No interesting dynamics can be expected above r = 5. Open System Behavior. The behavior of the chlorine dioxideiodide reaction in a CSTR differs from most other systems that have been studied in this configuration. Usually, as one sweeps through a range of parameters, the behavior (other than a single stable steady state) that is most frequently encountered is bistability between two steady states. Oscillatory behavior tends to be somewhat rarer, and bistability between steady and oscillatory states, rarer still. In the present system, at fixed [C102],, oscillations occur over a wide range of flow rate (k,) and [I-],.
7034 The Journal of Physical Chemistry, Vol. 96, No. 17, 1992 250 I
Lengyel et al.
I
100-
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50-
E
v
-
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'
I1
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I
> E
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Y
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--i ..I
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b
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00
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system at several initial reactant ratios. Voltage of iodide-selective electrode (A) and absorbance, mainly due to 13-, at 280 nm (B). r = [I-]o/[CIOz]o. r = 1.05 (a), 2.03 (b), 3.0 (c), 3.95 (d), and 4.85 (e). [CIOz]o= 1 X lo4 M,[HZSO4]= 5 X lo-' M. T = 18 OC.
-a -
-50-150
-2 5
-50
150
-50
j v v, ,w i, ./ , v u, y
-150,
0
I
I
60
,
I
I
120
180
-2 0
Figure 3. Experimental one-parameter bifurcation diagram obtained by varying k,,. Arrows indicate transitions between steady state. Solid lime indicate stable steady states, and dashed lines the maximum and the minimum of oscillationin potential of iodideselectiveelectrode. [CIOz]o = 1 X lo4 M; [I-lO= 1 X lo4 M (a) and 3.2 X lo4 M (b), and [HzSO,] = 5 X M.
In Figure 2 we show experimental oscillatory curves at several flow rates in the CSTR. As ko increases, the amplitude of oscillation grows, while the frequency decreases. At low input M) and low flow rates near the boundary concentrations (of the oscillatory range, the amplitude is only a few millivolts and the period about 3-5 s. Because of the slow response of our iodide-selective electrode, these experiments give only qualitative determinations of the waveform, but they do indicate the approximate position of the Hopf bifurcation. Figure 3 shows two bifurcation diagrams obtained by changing the flow rate at different levels of [I-],. In Figure 3a, a supercritical Hopf bifurcation is seen at the left. When the flow rate is increased in small steps near the bifurcation point, oscillations appear with increasing amplitude, and there is no hysteresis. On the right side of Figure 3a, oscillations disappear abruptly with increasing flow rate and reappear at the same ko with the same waveform when the flow rate is decreased. In the neighborhood there is a rather sharp rise in the amplitude of log (k,) = of the oscillation as ko increases. A similar, but less dramatic, phenomenon is observed in Figure 3b at somewhat lower koa We attribute these effects to a change in the species that determine the potential of the iodideselective electrode rather than to a true dynamical effect. Below about 0 mV, [I-] is so low that the iodide-selective electrode cannot be calibrated using an iodide solution, and the electrode responds to other species such as HOI. However, some type of dynamical cause (e.g., a canard bifurcation) cannot be. excluded. In Figure 3b the Hopf bifurcation at the left now appears to be supercritical;oscillation starts with a finite amplitude, though our limited experimental resolution did not permit us to detect any hysteresis. There is a clear region of bistability between a steady and an oscillatory state at high flow rates. On the right side of Figure 3b, before the ascillations disappear at high ko,the amplitude grows and the period becomes long and uncertain, as the waveform appears quasi-periodic, perhaps indicating the proximity of a homoclinic orbit emanating from an unstable steady
i
50
w
I
-3 0
log(ko)
Figure 1. Behavior of the chlorine dioxide-iodide reaction in a closed
50
'
240
300
360
Time (s)
Figure 2. Oscillatory curves in the C102-I- reaction in a CSTR at different flow rates. Potential of iodide-selective electrode against standard calomel reference. [ClOzl0=-I X lo4 M, [I-lO = 3 X lo4 M, (a), 2.5 X M, and ko = 1.5 X (b), 5 X [H2S0,I0 = 5 X s-I (d). (c), and 7.5 X
Bistability between two steady states exists only in a very narrow range of parameters. that is almost impossible to find without preliminary model calculations. The range of bistability between steady state and limit cycle is somewhat wider. In the remainder of this section we present a more detailed overview of the dynamics of the system.
The Journal of Physical Chemistry, Vol. 96, No. 17, I992 7035
Chlorine Dioxide-Iodide Open System Oscillator
-
-4-
n 0
1-
Y
W
0,
0 -5 _ I ,
-2 0
-2 5
- 1' 0
-1'5
lodko) -j 0
ib
c- _- _- _----_
-1
5
-1 0
Figure 5. Experimental two-parameter bifurcation diagram in the [I-],-k, phase plane. Solid line bounds region of bistability. Dashed line encloses region of oscillation. [C102]o= 1 X IO4 M, [H2S04]= 5 X 10-3
-71
-2 5
I
I/
M.
*.
Figure 4. Calculated bifurcation diagrams at the same conditions as in Figure 3.
state. This view is supported by the fact that the uncertain period appears only if bistability exists between oscillatory and steady states. If the limit cycle is close to the separatrix, a small perturbation can cause a significant effect on the period. In Figure 4, we show the results of modeling these experiments usipg eq 3. Figure 4a contains two Hopf bifurcations, as in the experiments, with the lower one supercritical and the upper subcritical. The picture in Figure 4b is more complicated; there are two limit or saddle-node bifurcation points on the solution curve. Oscillations appear at low ko via a subcritical Hopf bifurcation as observed in Figure 3b. They disappear in a collision of a saddle point with the limit cycle to form a homoclinic orbit. This phenomenon is referred to as a saddle-loop codimension one bifurcation in the dictionary introduced by Guckenheimer.zo The qualitative picture in terms of the patterns of states, bistability, and type of bifurcations agrees with our experiments, but there are some quantitative differences. In the model, the first Hopf bifurcation appears and the oscillations disappear at lower flow rates than in the experiments. The calculated range of bistability is somewhat wider than the experimental one. The period of the oscillations in the model is 1.5 times shorter than the experimental period. We shall return to these systematic differences between experiments and model in the Discussion section. In F w 5, we show an experimental tweparameter bifurcation diagram in the k0-[I-I0 phase plane. The region inside the solid lines corresponds to bistability, while the dashed curve marks the boundary of oscillation. A calculated bifurcation diagram at the same parameters is shown in Figure 6. There are two special points, shown in more detail in the two smaller panels, where the curve of Hopf bifurcations and the curve of limit points point meet and the curve of Hopf bifurcations terminates. At these points the trace of the Jacobian is zero, but the determinant is not. Such points are codimension two bifurcations known as trace zero saddle loop or neutral saddle loop bihucations.lg Between these two points the line of codimension one saddle loop bifurcations runs. Both the experimental and calculated behaviors in the neighborhood of these bifurcation points are consistent with bifurcation theory with respect to the appearance of a saddle point of the periodic solution (subcritical Hopf bifurcation) and a d i m e n s i o n one saddle loop bifurcation between the two neutral saddle loop points.
-
-5.5
1 ~
-3.0
I
-2.5
I
1
-1.5
- .o
Figure 6. Calculated two-parameter bifurcation diagrams at the same parameters as in Figure 5 . indicates trace zero or neutral saddle loop codimension two bifurcation points. Labels are for paths shown in Figure I.
In Figure 7, we give a full list of the various one-parameter bifurcation diagrams that can occur in this system as ko is varied with [TI0 fixed at the values labeled a-f in Figure 6. At low [TI0 only oscillation occurs between the two Hopf bifurcation points (Figure 7a). At higher [I-lO,in Figure 7b, a narrow region of bistability between the two steady states appears beyond the high flow rate Hopf bifurcation point. At somewhat higher [I-lO (Figure 7c), the limit point at low flow rate occurs before the higher Hopf bifurcation point, leading to two regions of bistability, one between oscillation and steady state and one between two steady states. After the neutral saddle loop codimension two bifurcation point bistability occurs only between oscillation and steady state (Figure 7d), and the oscillation ceases at a d i mension one saddle loop bifurcation, corresponding to the behavior shown on the left of Figure 4b. At still higher [I-I0, above the second neutral saddle loop point (Figure 7e), the second Hopf bifurcation appears again at high flow rates and gives a similar type of bifurcation diagram as in Figure 7c. Finally, in a very
Lengyel et al.
7036 The Journal of Physical Chemistry, Vol. 96, No. 17, I992 -4,
1
-51
-25
-20
-1 5
-1 0 I
-5.5 -I 18
-3
le
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- 1 41
-3,
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-I 42
- 1 44
I
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-3,
7 1
If
I
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-20
-1 6
1
/------
I
1
1-5-
-2 7
-2 5
-2 3
-63
-2 8
-2 6
-2 4
Figure 7. One-parameter bifurcation diagrams in the model of CI02-Ireaction at [I-]0values labeled in Figure 6. Horizontal and vertical axes are assigned to log (k,) and log ([I-]), respectively.
narrow range of (Figure 70, we find only simple bistability between two steady states. The range and the shape of the calculated curves of Hopf bifurcation and limit points are very similar to the experiments. Experimentally, scenarios a, d, and f have been detected unambiguously. In the other cases, the range of flow rate over which the intermediate behavior occurs appears to be too narrow to detect them experimentally.
Discussion We have presented here an experimental and theoretical study of the dynamics of the ClO,-I- reaction. This system and related C102--and C102-containing reactions have been the experimental prototypes for studying Turing patterns in open spatial reactors."'O The chlorine dioxide-iodide system has recently been used to investigate bursting, the characteristic alternation of quiescent and oscillatory behavior found in many neurons?' To understand these important spatial and temporal pattern formation phenomena, it will be necessary to characterize and analyze the dynamics of the relevant reactions in some detail. Having a simple model that accurately describes a system is a considerable advantage in predicting the Occurrence and nature of new dynamical structures and in finding the important parameters that regulate the development and characteristics of these patterns. The forerunner of the C102-I- open system oscillator, the C10z-12-MA closed system oscillator, has recently been shown to be the second experimental example of Turing patterns in a gel reactor?* Batch oscillations in that system can be modeled successfully using an empirical rate law approach, and we have shown here that the relevant portion of that model accurately describes the open system behavior of the C102-I- reaction. In our calculations, we have relied heavily on the tools of numerical bifurcation theory. To use only numerical integration is an inefficient approach for evaluating bifurcation diagrams, because one must then follow the time development of the system a t many different initial concentrations. It is necessary to map the parameter space densely, and finding unstable solutions requires the use of backward integration in time. Continuation is a much more powerful tool for finding steady-state solutions, both
stable and unstable, as well as bifurcation points. Bifurcations of periodic solutions can also be studied. Such methods have been profitably employed, primarily by engineers, for over a decade?3 and their advantages over integration for finding steady states and periodic orbits of complex chemical systems have recently been described.24 Elementary bifurcation analysis is also a useful guide to identifying codimension one and two bifurcation points. The main feedback process in the model is the self-inhibitoryZ5 or substrate-inhibitory reaction 2 between ClOF and I-. In inorganic chemical reactions, inhibition by reactants is not a common feature, but inhibition is found in many enzyme systems, where it plays a regulatory role. Several oscillatory models with substrate inhibition may be found in the literature of mathematical biology, and they show a rich variety of dynamical behavior in both homogeneous and inhomogeneous systems. Among the most interesting are the Degn-Harrison model studied by Velarde et a1.26band the Thomas model.27 The model employed here has three variables. It might appear at first glance that C102 plays only a minor role and that the essence of the dynamics takes place in the ClO,--I- plane, because the feedback process involves only these two species. One might attempt to reduce the differential equation system (3) by replacing [C102] by its steady-state concentration in terms of the other two variables. Such a procedure does not modify the steady solution, but the time-dependent behavior may be changed. Thus, the curve of limit points remains the same as without this simplification. The line of Hopf bifurcation points, however, changes significantly near the curve of limit points in Figure 6. The amplitude of oscillation becomes so large that the maximum in [I-] reaches the first limit point of Figure 3b, causing the codimension two bifurcations to become cusp points where the line of Hopf bifurcations meets the two tips of the curve of limit points. Bistability between the oscillatory and steady states as found in the experiments and in the three-variable model can no longer occur. Thus, CIOz has an important role in governing the oscillatory behavior. Its concentration does not follow instantaneously the concentration of iodide (as is implied in the steady-state replacement) but instead introduces a delay, which is essential in determining the dynamics and the bistability. As we mentioned earlier, there is a systematic quantitative difference between the experimental and calculated bifurcation diagrams. Oscillation appears and disappears at lower flow rates, and the period of oscillation is longer in the experiments than in the calculations. We believe that this discrepancy results from our neglect of any reaction between iodine and chlorite. This omission is valid if iodide is present at concentrations above lo-' M. In the experiments, however, when the amplitude of oscillation is high, this assumption is not fulfilled near the minimum in [I-], and this will cause some change in the dynamics. As we have seen, the chlorite-iodine reaction does not play a major role, but for a more exact description it should be taken into consideration. Unfortunately, it is impossible to include the reaction of C102and I2 in a simple empirical rate law model like ours, because at a more detailed level, this process involves the same elementary steps as the C102--I- reaction, and thus there must necessarily be "cross-talk" between the two processes. To see this, consider Kern and Kim's proposalL6for the mechanism of the ClO
