5300
J . Phys. Chem. 1984, 88, 5300-5304
Reaction 11, in which two radicals react with each other, is slower than diffusion controlled. Therefore, again the observed energy of activation for the slow process, which is lower than that expected for a diffusion-controlled reaction, lends support for the correctness of the interpretation. For the discussion of the amplitudes of both reactions a numerical integration of the set of simultaneous kinetic differential equations for reactions 5 and 8-1 1 has been performed. Also two further possible reactions were taken into account, which, however, have only a small influence on the numerical results: H. + CloH70H CloH7O. H2 (12)
-
-+
+
(C10H70)2 (13) (The dinaphthyl peroxide formed in reation 13 might be a candidate for the species which changes the spectrum of the solution after many flashes.) The initial concentrations for the calculations were obtained from the measured amplitudes. Although the ratio of the amplitude of the fast reaction to that of the slow one, as calculated by integrating the rate equations for reactions 8-1 3, increases with concentrations of @-naphthol, the observed very weak dependence of the amplitude of the slow reaction on concentration is only in qualitative accordance with the calculations. A quantitative agreement with the experiment may be obtained with the assumption that, during illumination of the solution, the radical anion -CloH70His subject to photoionization and loses its electron, which then, although not being in a cage with naphthoxyl or hydrogen ions, reacts according to 2C10H70'
reaction 4 forming the diradical anion, or according to reactions 7 and 8 with the production of molecular hydrogen. Thus, during the flash time (4 ps) no -CloH70H will survive to react with naphthoxyl according to reaction 11. With high concentrations of @-naphtholmost of the hydrogen atoms will then have disappeared as molecular hydrogen (reaction 8). Low concentrations of 0-naphthol lead to reaction times for reactions 8-10 that are long compared with the flash time with a relative increase of the amplitude of the slow reaction 11. A numerical integration with these initial conditions leads to good agreement with the experiments. The best set of parameters is ks = 1.6 X lo4 s-l, k l l = 4 X lo9 M-' s-l , k 4 / k 7 = 1.2, k9 = 5 X lo9 M-ls-l, k , , = lolo M-I s-l, and k 1 2= k , , = lo9 M-' sd. The rate constants k6, k7, and k8 were taken from the literature (see Table I). The error for k5 is about *lo%; that for k l l , which depends on the initial concentrations of the reacting species, is estimated to be *50%. In conclusion, this paper proposes a mechanism to understand the relatively slow reactions occurring in a flashed @-naphthol solution connected with large changes in conductivity. Since the proposed intermediats are radicals, they should also be observable by kinetic spectroscopy. Such experiments using an intense laser pulse are planned.
Acknowledgment. We thank Mrs. K. Weirich and Mr. G. Busse for experimental help and Dr. K.-H. Grellmann for valuable discussions. Registry No. CloH70H,135-19-3.
Spatial Waves in the Reaction of Chlorite with Iodide' David M. Weitz and Irving R. Epstein* Department of Chemistry, Brandeis University, Waltharn. Massachusetts 02254 (Receiued: December 12, 1983)
A thin layer of solution containing chlorite, iodide, and a starch indicator exhibits waves of chemical reactivity. The solution first turns rapidly and almost uniformly blue. Then circular colorless areas spontaneously appear and grow at a much slower rate. At higher pH's a slow moving secondary wave of blue develops within the colorless area. The colorless waves were initiated electrically and their velocity was measured as a function of reactant concentrations. A sharp drop in [I-] appears to be the controlling factor in the wave. A simple numerical simulation gives qualitative, though not quantitative agreement with the observed dependence of the wave velocity on [C102-], [I-], and pH.
Introduction One of the most fascinating outgrowths of the burst of interest in chemical oscillation and dynamic instabilities has been the discovery and study of spatial structures and wave propagation in initially homogeneous but excitable media. The most-studied system of this type (and also the most studied chemical oscillator) is the Belousov-Zhabotinskii (BZ) reaction, in which propagating pulses are repeatedly generated at pacemaker centers.2 The BZ system has given rise to more complex phenomena such as scroll waves,3 multiarmed vortices: and three-dimensional structuress as well as to chemical variants: Several calculation^^^ have been (1) Part 20 in the series "Systematic Design of Chemical Oscillators". Part 19: Alamgir, M.; Epstein, I. R. J . Phys. Chem. 1984, 88, 2848. (2) (a) Zaikin, A. N.; Zhabotinskii, A. M. Nature (London) 1970, 225, 535-7. (b) Winfree, A. T. Science 1972, 175, 634-6. (3) Winfree, A. T. Sci. Amer. 1974, 230, 82-95. (4) Agladze, K. I.; Krinsky, V. I. Nature (London) 1982, 296, 424-6. (5) Welsh, B. 9.;Gomatam, J.; Burgess, A. E. Nature (London) 1983, 304, 61 1-49 (6) Jessen, W.; Busse, H. G.; Havsteen, B. Angew. Chem., Int. Ed. Engl. 1976. 15. 689. ( 7 ) Reusser, E. J.; Field, R. J. J. A m . Chem. SOC.1979, 101, 1063-71. (8) Schmidt, S.; Ortoleva, P. J. Chem. Phys. 1980, 72, 2733-6. (9) Rinzel, J.; Ermentrout, G. B. J. Phys. Chem. 1982, 86, 2954-8.
0022-3654/84/2088-5300$01.50/0
carried out to explain at least the simpler types of waves. More recently, Showalter and collaboratorslOJ1have carried out detailed studies of the single propagating front of chemical reactivity in the iodate oxidation of arsenous acid. By combining extensive experimental data on the dependence of the propagation velocity on reactant concentrations with an elegant mathematical treatment of a simplified reaction mechanism in the presence of diffusion, they have gained significant insights into the nature of the waves. The two systems mentioned above differ fundamentally from one another in that the BZ reaction, which is capable of undergoing sustained oscillation in a closed system, supports repeated pulses, while the iodate-arsenous acid reaction, which is bistable in a flow system but not oscillatory, exhibits only a single front. Another less thoroughly studied system of the oscillatory, repeated wave type is the reaction of chlorite, iodate, and malonic acid.12 The bromate-ferroin13 and ferrous-nitric acid14reactions are other (10) Gribshaw, T. A,; Showalter, K.; Banville, D.; Epstein, I. R. J . Phys. Chem. 1981, 85, 2152-5. (11) Hanna, A.; Saul, A.; Showalter, K. J . Am. Chem. SOC.1982, 104, 3838-44. (12) De Kepper, P.; Epstein, I. R.; Kustin, K.; Orban, M. J . Phys. Chem. 1982, 86, 170-1.
0 1984 American Chemical Society
Reaction of Chlorite with Iodide
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5301
examples of single front propagation. Here we report on the investigation of traveling waves in a system intermediate in character between the BZ and iodatearsenous acid reactions. The reaction between chlorite and iodide ions, while not oscillatory in a closed system, shows both bistability and oscillation in a stirred tank reactor.I5 It is of interest to determine what sorts of spatial phenomena may occur in such a system. We find wave behavior more nearly analogous to single front propagation, though with some of the character of the repeated pulses of the BZ reaction. We discuss the nature of the waves and the dependence of their velocity on the reactant concentrations. We also present a preliminary effort at mathematically modeling this wave behavior. Experimental Section
Materials and Apparatus. Sodium chlorite and potassium iodide solutions were prepared and stored as described earlier.15 In addition to the chlorite and iodide, solutions contained an acetic acid-acetate buffer to maintain the desired pH, 0.05% starch as an indicator, and 0.05% Liqui-nox, a detergent that served as a surfactant. The electrochemically generated waves were studied in a plastic petri dish placed over a sheet of graph paper ruled with a 1-mm grid. The petri dish was covered with a plexiglass sheet which held the platinum initiation electrodes. Procedure. Reaction mixtures were prepared by pipetting stock solutions into a beaker and bringing the volume up to 20 mL. NaC102 was added last from a rapid delivery pipette and complete delivery was defined as time zero. The solution was mixed for 1 min and then 10 mL was poured into the petri dish to give a layer of 0.8-1.0-mm depth. A pH meter was used to monitor the remaining 10 mL of solution. The cover with the central electrode (biased at 7.5 V with respect to the negative edge electrode) was put in place immediately. At t = 2.5 min, the edge electrode was inserted, setting up a potential difference. The edge electrode was removed when a clear region surrounded by a blue film was observed at the central electrode. After the wave had moved some distance from the central electrode ( t = 5-6 min) the position of the wave was recorded visually. The wave is not always perfectly circular, and positions at opposite ends of a diameter were averaged. The accuracy of the position determination is estimated to be dz0.25 mm. In some cases the electrodes were omitted and the spontaneous development of the waves was observed starting at randomly distributed loci. Some measurements were also made of the iodide concentration at a fixed location in the solution using an iodide-sensitive electrode (Orion). All experiments were carried out at a temperature of 25.0 f 1 O C . Results
If solutions of chlorite and iodide are thoroughly mixed, starch indicator added, and the pH adjusted to about 4 then a thin, initially colorless layer of solution turns blue rapidly and almost uniformly after an induction period of about 5 min. After a somewhat longer time of the order of tens of minutes, one or more colorless spots develop in the blue solution and slowly begin to expand, forming roughly circular regions. Later, a second blue spot may form and grow within the colorless region. The spontaneous initiation points and times of the colorless waves are unpredictable, but waves may be induced at a desired location by adding a drop of chlorite or silver nitrate. In order to study the wave velocities systematically, we employed a technique similar to that of Showalter," initiating the waves electrically at the center of a petri dish. As in the case of (13) Showalter, K. J. Phys. Chem. 1981, 85, 440-7. (14) Bazsa, G . ; Epstein, I. R., unpublished work. (15) Dateo, C.; Orban, M.; De Kepper, P.; Epstein, I. R. J . Am. Chem. SOC.1982, 104, 504-9. (16) Bray, W. C. Z . Phys. Chem. 1906,54, 731-49. (17) De Meeus, J.; Sigalla, J. J. Chim. Phys. Phys.-Chim. Biol. 1966, 63, 453-9.
Figure 1. Propagating waves in the chlorite-iodide reaction. (a) Colorless wave 12 min after initiation. (b) Secondary blue wave (dark) within colorless region 30 min after initiation. [ClOf] = 3.75 X M, [I-] = 7.5 X M, [H'] = 1.0 X M. B indicates blue and C indicates colorless areas.
spontaneous initiation, the initial colorless to blue transition occurs too rapidly to follow quantitatively. However, the colorless wave then spreads at a rate convenient for accurate monitoring as described in the Experimental Section. Figure l a shows a typical nearly circular colorless wave after it has spread somewhat from the initiating electrode. In Figure 1b we see the growth of the secondary blue wave. Quantitative measurements were made only on the propagation of the colorless wave. As Figure 2 shows, the wave travels at a constant velocity for about the first 3-4 cm or 15 min. After this, there is a noticeable acceleration which may result from the depletion of the initial reactants and/or from boundary effects as discussed below. In our discussion of the dependence of wave velocity on reactant concentration, we consider only the constant velocity exhibited during the early part of the reaction. The wave velocity v, measured as a function of the initial iodide, chlorite, and hydrogen ion concentrations, is plotted in Figure 3.
-I
-
6
1
5-
P
a.
4-
Y
I
3-
>
n
P
0
2-
P I
I-
O
I
i
4
5
4
lo4 M (A),and 2.25 X lo4 (0).[I-] = 5.0 X IO4 M, pH 3.9. Potential differenceinitiated at t = 2.5 mm. In (b), the time scale is expanded to
n
show constant velocity during early part of reaction.
We see that u increases essentially linearly with [C102-], but considerably less rapidly with [H']. The velocity is inversely depdendent on [I-], though the relationship is not precisely hyperbolic. If one follows the chlorite-iodide reaction in a stirred s~lution,'~ one observes a nearly constant level of [I-] and an autocatalytically increasing [I2] during the first part of the reaction. In the presence of starch indicator, this period corresponds to the development of the blue color. There is then an extremely rapid decrease in [I-] of some five to seven orders of magnitude accompanied by a drop of 3040% in [I2]. These abrupt changes, which coincide
same magnitude. It coincides with the passage of the colorless wave. The longer duration of the drop results in large measure from the fact the electrode surface over which [I-] is averaged is of macroscopic dimensions while the wave moves relatively slowly. The figure also shows the increase in [I-] which accompanies the passage of the secondary blue wave. This latter observation was made at higher pH where the blue wave is darker and of longer duration.
7
6
C.
-
E
3a
L
E
U
>
a
5 m
l
5
IO-',--e---
,o-6-
~
I
.'
0
CI
10-6-
10-'O
I r 0
b
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5303
Reaction of Chlorite with Iodide This process is autocatalytic in Iz and inhibited by I-, with empirical rate law’7
-1 -d[Iz1 2 dt - (kZa[H+l [I-] + kZb[121 + kZc[IZ] / [I-]) [CloZ-]
1
I
I
I
-
I
(2)
If any chlorite remains, the iodine is further oxidized to iodate in process 3. 5C10;
+ 212 + 2 H z 0 = 5C1- + 4103- + 4H+
(3)
A recent stopped flow studyIs has established the rate law for process 3 in the absence of iodide as
Finally, if iodide and iodate are present simultaneously, they may react via the Dushman reactionI9 IO3- 51- 6H+ = 31z 3 H z 0 (5) with the rate law2o
+
+
+
__-- .
,-
I I
I
c*
Hanna et al.“ were able to combine part of a similar overall rate law for the iodate-arsenous acid reaction with a one-dimensional diffusion term to derive an analytical expression for the wave velocity as a function of concentration. In the chlorite-iodide case, however, the form of the key autocatalytic term, k,[Iz] [CIOz]/ [I-] renders analytic integration of the corresponding equation intractable. An alternative approach is to attempt to integrate numerically the partial differential equations resulting from the addition of diffusion terms to the rate eq 2, 4, and 6. Reusser and Field successfully applied this technique7 to the Oregonator model for the BZ reaction. In their approach, the “method of lines”, one divides the spatial region into n compartments and then converts the three independent partial differential equations, derived form eq 2, 4, and 6 into 3n ordinary differential equations by replacing the spatial derivatives by difference quotients. Field and Reusser7 found that about 250-1 150 compartments (750-3450 coupled differential equations) were needed to eliminate boundary effects. They also noted that numerical “stiffnessnZl of the equations may result in very long computing times in such problems. We have applied the methods of lines to the chlorite-iodide system using the rate 2, 4, and 6 with two modifications. In eq 2, the [I-] denominator in the kzcterm has been replaced by [I-] k i . This change eliminates the physically unrealistic (and numerically unstable) consequence that the rate approaches infinity as [I-] approaches zero. Equation 4 is valid only for very M).18 To properly account for the iodide low [I-] ([I-] Q inhibition of reaction 3 and the fact that the kb term must go to zero as [C102-] 0, we have replaced the rate law (4) by
+
1 d[Iz1 - -2 dt =
[
-
k.la + % / ( l
+ k;[I-]) + 1
The iodide dependence introduced in eq 7 is suggested by preliminary studiesZZon the mechanism of the chloriteiodide reaction. The resulting set of reaction-diffusion equations is displayed in the Appendix. On integrating the equations in the absence of the diffusion terms, one obtains the batch reaction profiles shown (18) Grant, J. L.; De Kepper, P.; Epstein, I. R.; Kustin, K.; Orban, M. Inorg. Chem. 1982, 21, 2192-6. (19) Dushman, S. J. J . Phys. Chem. 1904, 8, 453-82. (20) Liebhafsky, M. A.; Roe, G. M. In?. J . Chem. Kiner. 1979, 1 1 , 693-703. (21) Hirschfelder, .I. 0. J . Chem. Phys. 1952, 26, 271-3. (22) Epstein, I. R.; Kustin, K., submitted to J . A m . Chem. SOC.
0
----r--100
I
I
I
I
I
200
300
400
500
600
time (s), Figure 5. [I-] (a) and I2 (b) as functions of time calculated by integrating rate eq 2 , 4 , and 7 for a “batch” stirred reaction with initial conditions [H+] = 10”’ M, [ClO,] = 3.75 X 10”’ M, [I-] = 7.5 X M, [I2] = 0.
TABLE I: Parameters Used in Simulation of Chlorite-Iodide Waves ~
parameter kza, M-Zs-l kzb, M-’ kze, s-I
S-’
ki, M k4a,M-’ s-’
value 6.5 X lo2 3.2 X lo-’ 6.5 X 1.0 X 1.1 X 10’ 1.1 X
parameter
~
~
~~~
~
value
x
10-1
kdo s-l k4/, M-’
5.4
k [ , M-l k6a, M-) s-l
7.0 X lo2 8.0 X lo3 6.0 X lo8
1.0 x 10”
k6b, M-4 s-’ 2.0 X D,”cm2 s-’ ‘D is the diffusion coefficient, taken to be the same for all species. k4b, s-’
TABLE II: Calculated Wave Velocities as a Function of Reactant Concentrationso 1041~10,-1,M 10411-1, M 1041H+l,M 103u.cm s-] 3.75 3.75 3.75 3.75 3.75 3.75 3.75 5.0 6.0
3.75 6 .O 7.5 9.0 7.5 7.5 7.5 7.5 7.5
1
1 1 1 0.5 2 4
1 1
3.18 2.67 2.16 1.90 1.72 2.95 4.52 2.77 3.50
“Initial conditions as shown in Figure 6, 50-point grid, 0.1-mm spacing between grid points. Velocity calculated by taking wave front position as first grid point where [I-] falls below 10” M and dividing 0.7 mm by time required for wave front to travel from point 21 to point 28. in Figure 5. Note the autocatalytic rise in [I2] and the extremely rapid falloff of both [I,] and [I-]. The set of differential eq A2-A4 is extremely stiff, requiring large amounts of computer time. For this reason, we were unable to use more than 100 points (300 coupled equations) in the spatial grid. With this many points, end effects are significant. However, calculations using different numbers of and spacings between grid points suggest the wave velocities between the same pair of points in the sample should provide a valid basis for comparing solutions of different compositions. Both the experimental and the (considderably greater) calculated accelerations of the wave result from the fact that the solution farthest from the initiation point must undergo complete reaction in a time less than or equal to that for the stirred solution regardless
5304
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 ===r===
r;
0.
--4-====i== I = 100 5
t=Os
b.
Weitz and Epstein detailed mechanism involving only elementary steps and hence more species but simpler rate laws will yield numerically more tractable equations and better agreement with experiment remains to be seen. Development of such a mechanism is well under way, and preliminary results indicate that this mechanism will differ fundamentally from that of the BZ reaction. Both the calculations and the experimental results ([I-] trace of Figure 4, ability of Ag+ to initiate the wave) point to the sharp drop in iodide concentration, which activates the autocatalytic pathway corresponding to the kzc term of eq 2, as the means of propagation of the wave. The secondary blue wave probably results from the reduction of I2 and/or oxidation states of iodine like HOI and H I 0 2 by Cl-. Since these reactions are not included in our model, the calculated [I-] does not exhibit a secondary rise. Like the arsenous acid-iodate system, the wave front in the chlorite-iodide reaction constitutes a boundary between regions resembling the two bistable steady states of the system which appear under flow conditions. Here, however, it is relatively easy to identify I- as the species responsible for wave propagation, whereas in the former system considerable ambiguity remains” as to whether iodide or iodine is the primary propagating species. Acknowledgment. We thank Gyorgy Bazsa, Mohamed Alamgir, Donald Boyd, and Kenneth Kustin for their assistance and encouragement. This work was supported by the National Science Foundation (CHE 8204085) and by a grant from E.I. Dupont and Co. to the Brandeis Undergraduate Research Participation Program.
25
5
L 25
Appendix. The Reaction-Diffusion Equations The equations of the model are obtained by augmenting the kinetic eq 2,4, and 7 with one-dimensional diffusion terms of the form
5
GRID POINT
(2)
NUMBER
Figure 6. Calculated [I-] (a) and [Iz] (b) as functions of distance from initiating electrode at different times after initiation in an unstirred one-dimensional “dish” of length 0.5 cm. Initial conditions: [H’]= lo4 M M at all grid points, [I-] = 7.5 X M and [CIOz-] = 3.75 X and [Iz] = 0 at points 3-50, [I-] = 0 and [I2] = 3.75 X M at points 1 and 2. Initial conditions at points 3-50 are equivalent to initial conditions of Figure 5, and horizontal arrows at right of each panel indicate concentration in batch reaction at the corresponding time.
diff
=Di-a2Xi ar2
where Di is the diffusion coefficient, xi the concentration of the ith species, and r is the spatial variable. The one-dimensional “dish” of length I is divided into n compartments of length h = I/n and the spatial derivative in eq A1 is replaced by a difference quotient. We thus obtain 3n coupled ordinary differential equations for the concentrations of I-, 12, and C102- in each of then compartments. Letting x1 = [I-], x2 = [I2],and x3 = [ClO;] and adding superscripts j to specify the compartment number, we may combine eq 2, 4, 7 , and A1 to yield
of the progress of the wave. Thus as the wave approaches the outer boundary, it appears to speed up because the reaction which occurs at the wave front has begun to occur anyway even without assistance from species diffusing from behind the wave. dxJ/dt = -4$ - 5 d D~(xJ-’- 2xIj xlj+l)/h2 (A2) In Table I we give the rate parameters used in the calculations whose results are summarized in Table 11. The rate constants dx,”/dt = 2 d 3d - 2Wi + D2.xd-l - 2x2 + X2+l)/h2 (A3) for eq 2,4, and 7 were taken from the experimental l i t e r a t ~ r e , ” ~ ~ ~ ~ ~ ~ dxJ/dt = -d - 5Wr D3(~3j-’- 2x4 x2”)/h2 (A4) while the primed quantities were estimated from the observed lack of iodide inhibition under the conditions employed by Grant et where a1.I8 In Figure 6, we follow the development of the wave in terms of the calculated iodide and iodine concentrations. The clear region u j = k2,Hx4 k2bx4 + behind the wave occurs where [I-] (and hence [I3-]) is very low. XJ + k2‘ The sharp wavefront which is observed experimentally corresponds to the decrease in [I-] of about 4 orders of magnitude which occurs over a very short distance. Note that after the initial period, which is heavily influenced by the starting conditions, [I2] never varies kb k4 b by more than about one order of magnitude over the sample. It X4XJ w, = k4a H(l + k4/x$) i(x3j + k4”xlj) is not clear whether the small secondary maximum calculated in [I-] at t = 500 s is related to the beginning of the secondary blue with H = [H+] assumed constant and cd = [I03-]J = [I-]0 + 2[I& wave . - XIJ - 2x2. While the results in Table I1 are only in qualitative agreement The boundary conditions of no flux across the ends of the “dish” with the experimental dependence of wave velocity on concenare obtained by defining dummy variables xi0 and xrfl such that tration, they do suggest that a scheme which accounts for the three x p = xi1 and xr+l = X: for i = 1, 2, or 3. overall stoichiometric processes 1, 3, and 5 should be adequate Registry No. CIOz-, 14998-27-7; I-, 20461-54-5. to describe wave propagation in this system. Whether a more
+
+
+
+
(
(
+
+
+
