J. Phys. Chem. 1985. 89. 1019-1022 may play an important role which of course would enter into models for the absorption spectrum and for the excitation profiles in different ways. We conclude that the high-pressure excitation profile data may show some evidence of inhomogeneous broadening that was not present in our earlier studies at 1 bar. The question remains then as to what causes the large red shift in the electronic spectrum of 0-carotene a's pressure is applied. Although the use of configurational coordinate models cannot be ruled out by the present work, our opinion in this case is that probably the solvent theories would be the most appropriate. The solvent models have not been worked out in enough detail to explain pressure-induced solvent changes on resonance Raman excitation profiles. Since both the Tonks and Page transform test and our own multimode modeling give only a fair fit to the experimental excitation profiles at high pressure, we are led to suggest that more theoretical modeling is needed with the inclusion of inhomogeneous broadening in the model. This might be able to explain the fact that our experimental excitation profiles are not in very good agreement with the present modeling. Summary In this work, experiments have been performed on the resonance Raman excitation profiles of the two strongest vibrational modes,
1019
u1 and u2, of
&carotene in a diamond anvil cell at a pressure of 70 kbar. The 0-carotene was dissolved in a 1:l pentaneisopentane mixture. The large red shift in the absorption spectra is reflected in the excitation profiles. The shape and details of the excitation profiles at high pressure are similar to the profiles at low pressure and room temperature except for a slight amount of broadening. We have applied a multimode model to the absorption data and have concluded that only rather small decreases in the linear coupling parameters are necessary to fit the data. These parameters when used to calculate the excitation profiles give a fair agreement with the experiment. We have also used the transform technique of Tonks and Page to compare the absorption spectrum with the excitation profile in a model free way. This transform method fails suggesting that inhomogeneous broadening may be present in the high-pressure experiment. Acknowledgment. This work was supported by NSF, the ASU Faculty Grant-in-Aid program, and by the ASU Provost's research fund. The experiments were performed in the light scattering facility of the Center for Solid State Science at ASU. Particular acknowledgment is made to C. K. Chan for applying the transform technique to our data. Discussions with T. A. Moore, J. B. Page, D. Tonks, and Y. Fujimura were very helpful. Registry No. @Carotene,7235-40-7.
A Thorough Study of Bromide Control In Bromate Oscillators. 2. Simulation by the Oregonator Model of the Behavior of Reacting Beiousov-Zhabotinsky Systems Perturbed by BromeComplex-Forming Metal Ions Margit Varga, Liiszld Gyorgyi, and Endre Koros* Institute of Inorganic and Analytical Chemistry, L. Eotvos University, 1443-Budapest, Hungary (Received: June 28, 1984)
The behavior of BZ oscillators in the presence of thallium(II1)-a bromide-removingion-can be simulated by the Oregonator model if the dynamic bromide concentration conditions prevailing in the reacting systems, the rate of formation and dissociation of thallium(II1) bromo complexes, are considered. The respective rate constants were obtained from correlations known for ligand substitution reactions. Preliminary studies on mercury(I1)-containing BZ systems also indicate that the effects induced by this ion on the BZ oscillators can likely be simulated by the same model.
Introduction In our previous paper1 we have reported how thallium(II1) ions influence the behavior of bromate oscillators. Our investigations revealed that at low thallium(II1) concentrations (1 X 10-5-6 X M) the time of oscillation increased with an increase in thallium(II1) concentration and about 8 X M thallium(II1) completely quenched the oscillation. At high thallium(II1) concentrations (M), however, the reacting systems again showed oscillatory behavior: high-frequency oscillations were recorded which lasted only for a short period of time. The effect induced by thallium(II1) ions was explained in terms of their bromo-complex-forming ability considering the stability constants of thallium(II1) bromo complexes. The time increasing effect of low concentrations of thallium(II1) has been attributed to the buffering of the bromide ion concentration by the bromo complexes of thallium(III), and a qualitative explanation has been put forward. However, the phenomenon appearing at high thallium(111) concentrations could not be accounted for. In our previous paper we regarded the formation of the bromo complexes of thallium(II1) as a much faster process than the formation of bromide ions in the autocatalytic step of the Below-Zhabotinsky (BZ) reaction and, therefore, considered only equilibrium bromide concentrations. These were calculated from (1) Kerb,
known formation constant values. The calculated equilibrium bromide concentrations were always below the critical bromide concentration and therefore we could not explain the occurrence of oscillations at high thallium(II1) concentrations. Based mainly on the experimental results described in our previous paper' we thought that chemical oscillations observed in the presence of bromide-removing ions (T13+,HgZ+,and Ag') can be accounted for in terms of bromide control, and thus the Oregonator model can be used if the dynamic bromide concentration conditions prevailing in the reacting systems, the rate of formation and dissociation of the metal bromo complexes, are considered.
Calculations The irreversible Oregonator2 consists of the following five steps: A + Y Z X + P
E.;Varga, M.J . Phys. Chem. 1984,88,4116-21. 0022-3654/85/2089-1019$01.50/0
0 1985 American Chemical Society
(01)
1020 The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
Varga et al.
TABLE I Composition of the Reaction Mixture (0.05 M KBrOh 0.20 M Malonic Acid, 1.0 M HfiO,, and 0.002 M Catalyst)
relative time' Oregonator exptl measuredb modelC
10S[T13+],M 1 .o 2.0 3.0 4.0 5.0
1.8 2.0 2.8 3.4 4.2
1.1 1.6 2.8 3.6 4.6
'Relative period time = (time of the two oscillation in the presence of thallium(III))/(time of the second oscillation). Ce3+catalyzed system. CTheperiod of the Oregonator is 48.75 s. where A = BrO), X HBr02, Y Br-, Z 2Ce4+, and P and Q are products. A = 0.06 M . kol = 1.34 M-' s-I; ko2 = 1.6 X lo9 M-ls-l; ko3 = 8 X lo3 M-' s-l; ko4 = 4 X lo7 M-l s-l* k 0 5 = 1.0 s-l; a n d f = 1. In the presence of thallium(II1) ions Y (Br-) formed in reaction 0 5 reacts with thallium(II1) and therefore the following equilibria should be considered: 9
k
T13++ Br- -I,TIBrZ+
(1)
k-1
k
TlBr2 + Br-
TIBr2+
(2)
& TlBr3 k-3
(3)
+ Br- & TlBr4k+
(4)
k-2
TlBr2+ + BrTlBr3
k
In order to calculate the temporal change of bromide concentration and that of the various thallium(II1) bromo complexes eight rate constants, kl-k4 and k-1-k-4, were needed. The estimation of thallium(II1) bromo complex rate constants is described in the Appendix. From the following set of rate constants, all phenomena induced by thallium(II1) of different concentrations could be simulated by the Oregonator model: k , = lo5 M-ls-'; k-, = 3.1
X
s-l
k 2 -- lo4 M-l s-l; k-2 = 3.6
X
s-l
k3 = 2.5
X
lo2 M-'
k4 = lo2 M-' s-I;
t
s-l;
k-3 = 7.4
k-4 = 7.3
X
50
150
io0
ttme
/si
Figure 1. Simulation of the behavior of a BZ system by the Oregonator model: (a) log [Br-] vs. time plot; (b) redox potential vs. time plot. Simulation of the behavior of a thallium(II1)-perturbed BZ system, M; (c) log [Br-]frocvs. time plot; (d) log [Br-I,,, [T13+],od= 3 X vs. time plot; (e) redox potential vs. time plot. The following concentration values were taken from the limit cycle: [HBr02] = 2.415 X M; [Ce4+]= 1.138 X lo-' M; [Br-] = 3.787 X lo-' M; [ T P ] = 4.579 X lo* M; [T1Br2+]= 8.166 X lo-' M; [TIBr2+]= 2.460 X lod M; M; [TIBri] = 5.234 X l o d M. The composition [TlBrJ = 2.148 X corresponds the minimal potential.
s-I
X s-l
-5
Differential equations were described for the intermediates of the Oregonator and the complex formation reactions applying the law of mass action. The system of differential equations was solved by a fourth-order semiimplicit Rung-Kutta method which proved to be very effective for the integration of these and similar stiff s y ~ t e m s . The ~ initial concentrations are given in the respective figures.
1
I Is/ 50
Results
BZ System of Low Thallium(III) Concentrations. With low thallium(II1) concentrations (1 X 10-5-5 X M ) the relative times measured by and calculated by the Oregonator model including the bromo-complex-forming reactions (Eq 1-4) are given in Table I. The data in Table I show very good agreement between the experimentally measured and calculated values. In Figure 1 a and b, the log [Br-] and the redox potential, respectively, vs. time plots of a thallium(II1)-free BZ system are shown, simulated by the Oregonator model. In Figure 1, c and d, the free (2) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877-84. ( 3 ) Gottwald, B. A.; Wanner, G. Computing 1981, 26, 355-60. (4) KBros, E.; Varga, M. Read. Kinet. Catal. Leu. 1982, 21, 521-6.
100
150
ttma
Figure 2. The logarithm of the sum of the rates of formation (curve b) and the rates of dissociation (curve a) of the thallium(II1) bromo complexes vs. time plots. [TI'+],,, = 3 X 10" M.
and total bromide concentrations and in Figure 2 the logarithm of the sum of the rates of formation and dissociation, respectively, of thallium(II1) bromo complexes vs. time plots are shown. Vl+ = kl [TI3+][Br-] Vl- = k-I[TIBr2+] etc. Figure l e shows the temporal change in the redox potential calculated by the thallium(II1)-modified Oregonator model. The computed curve is practically identical with that recorded ex-
The Journal of Physical Chemistry, Vol. 89, No. 6,1985 1021
Bromide Control in Bromate Oscillators
is far from equilibrium since log (ul+
+ uz- + u3- + u4-).
+ u2+ + u3+ + u4+) + log
(01-
f
I 0
[&] '3
-
0.005 [C'"]
-4 -2
' \ I
30
60
time
Irl
c
Figure 3. Simulation of the behavior of a thallium(II1)-perturbed BZ system by the Oregonator model. [BrO,] = 0.10 M, [Tl3'+ltaal = 1.5 X M. The logarithm of the sum of the rate M; =1X of formation (curve a) and the rate of dissociation (curve b) of the thallium(II1) bromo complexes vs. time plots: (c) log [Br-I,,, vs. time plot; (d) log [Bill, vs. time plot; (e) redox potential vs. time plot. Concentration values at the start of the reaction calculated from the complex equilibria are as follows: [HBr02] = 8.00 X 10" M;[Ce"] = 1.606 X lW3M;[Br-] = 1.784 X lo-" M;[T13+] = 1.400 X M; [TlBF] = 9.900 X 10-4 M, [TlBr2+]= 4.916 X 1W' M;[TlBrJ = 2.923 X 1WI2 M; [TlBr4-] = 7.196 X M.
[Bi],,,
perimentally. (See Figure l a in our previous paper.') BZ Systems of High Thallium(I1I) Concentrations. The experimental and computed curves agree rather well also in the case of high thallium(II1) concentrations. In Figure 3, d and c, the free and total bromide concentrations and in Figure 3, a and b, the logarithm of the sum of the rates of formation and dissociation, respectively, of thallium(II1) bromo complexes vs. time plots are shown. Figure 3e shows the temporal change of the redox potential calculated by the thallium(II1)-modified Oregonator model.
Discussion BZ System of Low Thallium(III) Concentrations. A comparison of Figure l a with Figure IC shows that in the thallium(111)-containingBZ system the bromide ion concentration reaches the critical value ( 5 X 104[Br0;] = 3 X lo-' M) after a long period of time, since below a certain bromide concentration the rate of consumption of bromide, in the bromate-bromide reaction, becomes very low. This can be interpreted in terms of a bromide buffering effect of the system which contains different bromo complexes of thallium(II1). As we have reported earlier' predominantly TlBr3 and TIBr4- are present. The higher is the thallium(II1) concentration the longer is the time period. BZ systems of low thallium(II1) concentrations show limit cycle behavior. As seen in Figure 2 both the rate of formation and the rate of dissociation of thallium(II1) bromo complexes change periodically. The change is naturally more pronounced in the formation rate due to the autocatalytic generation of bromide ions. Also it can be clearly seen in Figure 2 that the reacting system
BZ System of High Thallium(1II) Concentrations. The computed behavior of a BZ system of high thallium(II1) concentration is in qualitative agreement with its experimental behavior. (Compare Figure 3 with Figure 3 in ref 1.) This means that at high thallium(II1) concentrations (M) high-frequency oscillations appear according to both experiments and computations. The difference is that in the computations the times are shorter than those recorded experimentally. As seen in Figure 3 the formation and dissociation rates approach and finally (after a rather short period of time) become equal. Also it should be pointed out that the rates of complex formation (see Figure 3a) are periodic. This can be attributed to the fact that the they depend on the concentration of bromide, which changes periodically with time. On the other hand, the rates of complex dissociation show a stepwise change with time. Bromide-selective electrodes cannot sense the change in bromide concentration, since the changes are between lod and M, and this is below the sensitivity limit of the electrodes. Simulation of the Mercury(Il)-Containing BZ Systems (Preliminary Studies). We have reported' that mercury(I1) ions a t low concentrations exert similar effect on the reacting BZ system as thallium(III), i.e., they increase the time of oscillations. However, mercury(I1) ions in high concentration do not induce oscillations at all. The rates of formation of the mercury(I1) bromo complexes are of many orders of magnitude higher than those of the corresponding thallium(II1) complexes. The water exchange rate constant for mercury(I1) is 3 X lo9 s-I, and K, is probably the same as for thallium(II1) since the double charge on mercury(I1) is partly shielded by the presence of hydrogen sulfate ions in the inner coordination sphere. HertzS reported the rate constants for the reactions HgBrz BrHgBrC and HgBr3- BrHgBr4". His values are on the order of IO8 and lo9. Simulations using these k values, respective k values two orders of magnitude lower, and the formation constants of the bromo complexes of mercury(II)6 show that at low mercury(I1) concentrations there is an increase in the time of oscillations. Above lo4 M, mercury(I1) inhibits the oscillations, which do not reappear at high (M) mercury(I1) concentrations, contrasted with the behavior of the thallium(II1)-containing systems. These are in qualitative agreement with our experimental results. Details on the mercury(I1)-containing BZ systems will be reported in a forthcoming paper.
+
+
Conclusion In recent years some authors questioned the importance of bromide ions in the function of bromate oscillators. Modified Oregonator models and chemical mechanisms were published in the literature with the aim in mind to give an acceptable interpretation of the experiments. This issue has been discussed in the Introduction of our previous paper.' Our results reported in this paper provide substantial evidences that it is not necessary to modify the Oregonator model and the skeleton of the chemical mechanism. Bromide ion should not be replaced by an other control intermediate (e.g., HOBr or Br), and more than one autocatalytic reaction should not be considered. Namely, the original Oregonator can simulate very well the experimentally recorded redox potential vs. time curves (with low thallium(II1) concentrations also the bromide concentration vs. time curves) if the proper dynamics of the bromide removing reactions are considered. Our opinion is strongly supported by a recent paper of Ruoff,' who could reproduce by the Oregonator model the main trends of the oscillation parameters in a silver(1)-perturbed BZ system. Silver(1) is also a bromide-removing ion. (5) Hertz, H. G. Z . Elektrochem. 1961, 65, 36-50. (6) "Stability Constants of Metal Ion Complexes"; Sillen, L. G., Martell, A. E., JUS.;The Chemical Society: London, 1964. (7) Ruoff, P. Z . Naturforsch., A 1983, 38A, 974-9.
1022 The Journal of Physical Chemistry, Vol. 89, No. 6, 1985
Acknowledgment. The authors express their thanks to Dr. T. Deutsch and Dr. S. Vajda (Laboratory for Chemical Cybernetics) for the stimulating discussions. Appendix The ligand substitution reactions of the thallium(II1) ion have been vaguely studied. Only the rate constant for the reaction between Tl(Hz0)50HZ+and semixylenolorange, H3SX0, has been reported in the literature.8 In spite of this and on the basis of theoretical relationships pertinent to ligand substitution reactions it is possible to estimate the rate constants for the complex equilibria 1-4. A substitution reaction where the incoming species is a mondentate ligand can be regarded as a two-step process. In the first step, in an association reaction between the ligand and the hydrated metal ion, an outer-sphere complex is formed, and in the second step a water molecule in the inner coordination sphere of the metal ion is replaced by the monodentate ligand, L: M(H20)"'
+ L-&
-
M(H20)xn+L-
k,
M(H20),"+L-
M(H20)x-lL("1)+ + H 2 0
(5)
(6)
The reaction in eq 5 may be considered as a rapid preequilibrium, and K, is known as the outer-sphere association constant. The forward rate constant, kf, is given by d[ML]/dt = kf[M,,"+][L-];
k f = Ko,k,
(7)
The value of k, corresponds to that of the rate constant for exchange of a water molecule from the inner coordination sphere of the'metal ion: M(H,O),"+
+ H2O* * M(H20),,(H20*)"+ + H20
(8)
For the thallium(II1) ion this rate constant is not known, but a good estimate for its value can be obtained. For most of the labile 2+ and 3+ metal ions if ligand field and other effects are absent, a plot of log k, vs. Z / & gives a straight line. Z is the charge of the metal ion and d is related to rl and r2, the separation of the metal ion and the oxygen in the hydrated ion (taken to be the sum (8) 'Coordination Chemistry"; Martell, A. E., Ed.; American Chemical Society: Washington, DC, 1978; ACS Monograph 174.
Varga et al. of the ionic radii of the metal and oxygen) and in the activated complex, respectively, by the following equation: d = (rl + r 2 ) / 2 . For r2 - r l a physically reasonable value 0.05 nm is used. Since the k, values for the ions of the group 13 elements (A13+, Ga3+, In3+) are known,* k, for T13+can be obtained by extrapolation. This value is 2 X lo6 S-I. The K , value for a labile metal ion is rather difficult to determine. In order to estimate KO,it is often sufficient to use the following expression:8 KO,= (Y3Nu3e-*)X
M-'
where N is the Avogadro constant, a is center-to-center distance (in cm) between M and L, and b = ZMZLe:/U€kr ZMand ZL are the formal charges on the reacting species, e, is the electronic charge, t is the dielectric constant of the solvent, k is the Boltzmann constant, and T is the absolute temperature. In our reacting system the incoming ligand is the bromide ion, thus ZL= -1. As regards the value of ZM,however, we are reduced to supposition. In the oscillatory chemical system owing to the high sulfuric acid concentration (1 M) thallium(II1) is present as a hydrogen sulfate complex; it is most likely that T1(HS04)2(H20)4+is the predominant species (log p2 value of this complex is 2.1 14). Thus the formal charge of the cation, ZM,can be taken as 1. Using these values for KO,we obtain 0.06 M-' and kf = k , = lo5 M-' s-I. k-, can be calculated from the relation p1= k,/k-, by taking into account that thallium(II1) is complexed by hydrogen sulfate; PI (109.62)was divided by lo2.". Since the entry of additional negatively charged ions (Br-) into the inner coordination sphere of thallium(II1) is more and more hindered (as found also with the bromo complexes of mercury(I1) and cadmium5) the value of k decreased by one order of magnitude on going from kl to k,; a somewhat greater decrease was supposed on going from kz to k3 because at the entry of the third bromide ion a stereochemical change from octahedral to tetrahedral is likely to occur
+
-
TIBr2(H,0),+ + BrT1Br3(H20) octahedral tetrahedral The k value was only slightly changed on going from k3 to k4. As the simulations indicated the above considerations proved to be correct. Registry No. TI, 7440-28-0; Hg, 7439-97-6; Br03-, 15541-45-4; Ce, 7440-45-1.
