J . Phys. Chem. 1984, 88, 6424-6429
6424
Theoretical Study of Ag+-Induced Osclllatlons and Excitations in the Classical Homogeneous Belousov-Zhabotinsky Reaction Using the Oregonator Model Peter Ruoff* Rogaland Regional College, Ullandhaug, N - 4001 Stavanger, Norway
and Bengt Schwitters Department of Chemistry, University of Oslo, Blindern, N-0315 Oslo 3, Norway (Received: September 7, 1983; In Final Form: June 29, 1984)
Recently the behavior of a stirred Belousov-Zhabotinsky system in a nonoscillating excitable steady state has been studied by perturbing the system with silver ions. In this paper we present a theoretical study of the irreversible Oregonator model which has been extended by a fast Br--removing reaction. It is shown that this model is semiquantitatively able to describe most of the experimental results, indicating that the essential features of Ag+-perturbed Belousov-Zhabotinsky systems and Ag+-induced oscillations may be understood in terms of the Field-Koros-Noyes mechanism. There is presently no need to postulate additional control mechanisms in Ag+-induced oscillations, as for instance the bromine atom control by Noyes or the generalized Lotka-Volterra mechanism suggested by Noszticzius.
Introduction In 1958 Belousov' first observed repeated chemical oscillations during thb bromination of citric acid by bromate in the presence of Ce(S04)2in diluted sulfuric acid. A few years later, ZhabotinskyZ showed that oscillations take place in acidic solutions containing bromate ions, one equivalent redox couple acting as a catalyst, and an organic compound that can be brominated by an enolization mechanism. Such a system where bromide ions are liberated when an organic bromide reacts with the oxidized form of the catalyst is now generally known as a "classical" Belousov-Zhabotinsky (BZ) ~ y s t e m . ~ Field, Koros, and Noyes4 (FKN) presented in 1972 a mechanistic description of the malonic acid driven classical BZ system. On the basis of this mechanism, 2 years later, Field and Noyes5 presented a five-step model, later to be known as the Oregonator, which has been remarkably successful in its ability to reproduce qualitatively and semiquantitatively many aspects of the BZ reaction. On the basis of further Edelson et al.8*9presented results of detailed numeric integrations on a large set of rate equations. This represents to date the most detailed description of the classical BZ reaction. The description by FKN is now generally accepted as correct in major respects but has from time to time been challenged, especially by Noszticzius and co-workers.'*16 Noszticzius' main arguments against the FKN mechanism are based upon the exBelousov, B. P. Sb. Ref. Radiat. Med. 1959, 145. Zhabotinsky, A. M. Dokl. Akad. Nauk SSSR 1964, 157, 392. Noyes, R. M. J. Am. Chem. SOC.1980, 102, 4644. Field, R. J.; Koros, E.; Noyes, R. M. J . Am. Chem. SOC.1972, 94, (5) (6) (7) (8) 417. (9) 155.
Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877. Jwo, J.-J.; Noyes, R. M. J . Am. Chem. SOC.1975, 97, 5422. Noyes, R. M.; Jwo, J.-J. J . Am. Chem. SOC.1975, 97, 5431. Edelson, D.; Field, R. J.; Noyes, R. M. Znt. J . Chem. Kinet. 1975, 7, Edelson, D.; Noyes, R. M.; Field, R. J. Int. J. Chem. Kinet. 1979, 11,
TABLE I: Numerical Values and Intervals of Rate Constants, f, and Initial Concentrations parameter set I set I1 kol = k1*[H+l2 kl* = 2.094 M-' S-' k l * = 2.094 M-3 s-l koz = k2*[H+] k2* = 2.0 X lo9 M-2 s-l k2* = 2.0 X lo9 M-2 s-I ko3 = k 3 * [ H + ] k3* = 1.0 X lo4 M-2 s-I k3* = 200 M-2 s-l 4.0 x 107 M-* s-I 4.0 x 107 M-1 s-1 k04 ~~
~~~~
k05
0-1 s-1
?
104-106 M-I s-I 0-5 0.01-0.22 M
0-0.25 s-' 104- 106 M-' s-1 0-5 0.01-0.22 M
[H+l
0.8-1.0 M
0.8-1.0 M
[AI
perimental fact that Br--sensitive electrodes have been shown to be also sensitive to other intermediates, particularly HOBr. lo He also found that (high frequency) oscillations can still be observed, even if the Br- concentration by addition of AgNO, is forced to very low values." On that basis, a generalized Lotka-Volterra type mechanism for halate-driven oscillators was proposed.16 Ganapathisubramanian and Noyes17 confirmed Noszticzius' observations, agreed with the argument of non-Br- control in Ag+-induced oscillations, and postulated an alternative method of control by bromine atoms at low Br- concentrations. Complementary to Noszticzius' results, Ruoff observed for nonoscillating excitable steady states18-21similar behavior, i.e. the appearance of high-frequency oscillations when AgNO, is added,lg but explained the change in frequencyigand amplitude2' by simple model considerations within the framework of the FKN approach. Further evidence of Br- control in a free-running oscillating BZ reaction without use of a Br--detecting device has also been given.22 This work summarizes results of numerical calculations and theoretical analysis performed on the irreversible Oregonator model which has been extended by a fast Br--removing reaction.23 The theoretical results are compared with corresponding experiments. It will be shown that the main features of Ag+-perturbed BZ systems can be understood in terms of the Oregonator model and that no additional postulates about the method of control in
(IO) Noszticzius, Z.; Noszticzius, E.; Schelly, Z. A. J . Am. Chem. SOC. 1982, 104, 6194. (11) Noszticzius, Z. J . Am. Ckem. SOC.1979, 101, 3360. (12) Noszticzius, Z. Acta Chim. Acad. Sei. Hung. 1981, 106, 347. (13) Noszticzius, Z.; Farkas, H. In "Modelling of Chemical Reaction
Systems"; Ebert, K. H., Deuflhard, P., JBger, W., Eds.; Springer-Verlag: West . Berlin, 1981. (14) Noszticzius, Z. KZm.Kozl. 1980, 54, 79. (15) Noszticzius, Z.; Feller, A. Acta Chim. Acad. Sci. Hung. 1982, 110, 261. (16) Noszticzius, Z.; Noszticzius, E.; Schelly, Z. A. J . Phys. Chem. 1983, 87. 510.
0022-36S4/84/2088-6424$01.50/0
(17) Ganapathisubramanian, N.; Noyes, R. M. J . Phys. Chem. 1982,86, 5155. (18) Ruoff, P. Chem. Phys. Lett. 1982, 90, 76. (19) Ruoff, P. Chem. Phys. Lett. 1982, 92, 239. (20) Ruoff, P. Chem. Phys. Lett. 1983, 96, 374. (21) Ruoff, P. Z . Naturforsch., A 1983, 38, 974. (22) Ruoff, P. J . Phys. Chem. 1984, 88, 2851. (23) This paper presents our main results. A full description of our work will be available as a technical report and can be obtained by writing to the
principal author.
0 1984 American Chemical Society
Ag+-Induced Oscillations and Excitations
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6425
Ag+-induced oscillations are needed.
PIpd
The Model The Oregonator is a skeleton of the FKN mechanism but is believed to show the main trends of this mechanism and to give a reasonable description of oscillatory “~lassical”~ Bi-controlled BZ systems and their dynamical behavior. Our version of this model, given by eq 01-06, differs from the ko1
A+Y-X
(01)
ko2
A
X+Y-P
(02)
+ X 5 2X + Z
(03)
kor
2X-A+P
(04)
Z 3 f Y
(05)
kcx
D+Y-Q
(06)
original version of Field and Noyes5 by an additional precipitation reaction of bromide ions with silver ions (06), where A = Br03-, X = HBr02, Y = Br-, Z = 2Ce(IV), D = Ag’, P = HOBr, and Q = AgBr(s). In most calculations, the dynamical variables (Le. the variables which change with time) are A, D, X, Y, and Z. However, in some calculations the concentrations of A and D were kept constant. The variables P and Q are considered as time independent. We consider two sets of parametrizations, denoted as sets I and 11, respectively, which are shown in Table I. Set I consists of the original Oregonator values of the rate constants kl*...ko4 plus intervals chosen for the remaining parameters. Set I1 differs from set I by a substantial reduction of the value of k3*. This somewhat surprising value of k3* is obtained by taking the FKN mechanism as presented by TysonZ4 in his Table I as a starting point in generating the Oregonator. k3* is then equal to k5*in Tyson’s notation. The difference comes about because R5 in the original FKN mechanism4 R5:
Br03- + HBrO,
+ H+ G 2Br02 + H 2 0
Br03-
+ HBrOZ + H+ G Br2O4 + H 2 0
= kodLs - [Dl [Yl)
which is the final form used in the calculations.
Method of Calculation Calculations have been performed by numeric intergration of the rate equations generated from sets I and 11 and by linear stability analysis (LSA) where [A] and [D] were held constant. A HP85 microcomputer operating in BASIC was used for most of the calculations. Numeric Integration. A partially stiffly stable integration algorithm devised by one of us (B.S.) was used for most integrations. The algorithm has been tested extensively by comparison of computed results with those obtained by use of the stiffly stable Gear algorithm. A BASIC versionz6 of the Gear algorithm was first available to us in the latter stages of the calculations referred to here. Comparison of numeric results using the two algorithmic approaches on the same computer was satisfactory. However, the Gear algorithm turned out to be faster in execution by roughly a factor of 4, or even more for very stiff systems, with use of comparative accuracy criteria. In one of the previous reported experiment~’~ use of a semibatch variable-volume reactor is described. In these experiments an aqueous solution of AgNO, was pumped at a constant flow rate r (mL/min) into a closed reactor which at time t = 0 contained Vo(mL) of a BZ reaction in an excitable steady state. In simulating these experiments, we get, besides the well-known reaction terms It+i,a term correcting for the Ag+ input flow = ( ~ / ( V+Ort))(Cin,i- c+i)
#+i
which together with reaction vessel
+ rt+i(t>
+ k06LS)/(kOl[Al + k02[XlSS + k06[D1) (6b)
in Tyson’s version, which is based on recent experimental results referred to in his paper. Although the formation of AgBr by ( 0 6 ) may be considered as an irreversible process, reversibility of this step kcx
k a
AgBr(s)
and inclusion of the solubility product of AgBr(s) seem to be of some importance, as will be shown in the Results and Discussion section. We express this by [DIM =
W1,
= - k o 6 [ ~ 1 [ y 1+
(1)
k-06
where [D], and [Y], are the contribution of the precipitation/dissolution of AgBr to [D] and [Y]. When reversibility of (06) is considered, we assume that always solid AgBr is present, such that the activity of AgBr(s) is equal to one. In a saturated AgBr solution eq 1 becomes kos[D]q[Y]q
+ k-06
= k-06
(5)
where cintAg is the Ag’ concentration in the inlet flow (all other Cin’i = 0 ) . Linear Stability Analysis (LSA). In the LSA the rate equations are linearized about the steady state (SS) (holding [A] and [D] constant) given by
Br204 G 2Br0,
+ Br-
(4)
gives the variation of the species i in the
t+i(t)= $+i(t)
Cfk03lA1
Ag+
(3)
[YISS =
and 5b:
[PI,
2kozko4[Xlss3 + (cf- l)kozko3[Al + 2ko4([Alko, + k06[D1))[xlSS2 - (cf+ ~ ) ~ o I ~ o-~k02k06LS [ A I ~+ k 0 3 [ A l k 0 6 [ ~ ~ ) [ ~-l skoi[A]ko6Ls s =o (6a)
has been split into 5a:
=
+ ko.&
=O
(2)
where & is the solubility product of the precipitate (= 7.7 X lo-’, M2).25 Inserting eq 2 into eq 1, we obtain (24) Tyson, J. J. J . Phys. Chem. 1982, 86, 3006.
[ z l s s = ko3[Al [Xlss/kos
(6c)
We are primarily concerned with the sign of the real part of the eigenvalues Re ( A j ) (j= 1, 2, 3) associated with the matrix of constant terms of the linearized rate equations. If all Re ( A j ) < 0 for a given set of parameters, the SS will be stable to small perturbations. If at least one of the Re ( A j ) > 0, the SS will be unstable. In the parameter space a boundary surface(s) exists between a stable and unstable ss if Re (A,) = 0 and Re (&j) 5 0 for a l l j and k . Setting Ls = ko6 = 0 reduces the LSA to that of the original O r e g ~ n a t o r .The ~ analysis given by Field and Noyes5 applies in this case. Steady-state values of X, Y, and Z will be denoted in this special case by [X]SS,~,[Y]ss,o, and [Z]ss,o.
Results and Discussion Linear Stability Analysis of the Model. With our choices of parameter spaces (Table I), the LSA will in general reveal three regions as illustrated in Figure 1A. With & = kO6 = 0 one finds that in the wedge-shaped region B the SS is unstable to small perturbations, and B therefore corresponds to the oscillatory ( 2 5 ) “Handbook of Chemistry and Physics”, 60th ed.; CRC Press: Boca Raton, FL, 1980; B-220. (26) Field, R. J. J . Chem. Educ. 1981, 58, 408.
6426
Ruoff and Schwitters
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
...
-2
\
t\
C
A 0
*
1
3
f-
Figure 3. Steady-state concentration values of X, Y, and Z when ko6 = 0. Solid lines show set I parametrization (ko5 = 1 s-l), and dashed lines show set I1 parametrization (ko5 = 0.25 s-l). In both cases [A] = constant = 0.06 M and [H'] = constant = 0.8 M. In [ZJ
-81 L
I
8
Figure 1. (A) Schematic representation of LSA in k,-f plane. (B) Expansion of oscillatory domain into the excitable region when ko6[D] > 0. (Note: k5 = ko5.)
0
10
0
20s
20
40
60s
A
I
1 0 g & m 3 * j ~ mV
-4-1
A
t
ioMlnt
2
B Figure 4. (A) Effect of increasing amount of silver ions added to Oregonator in an excitable steady state. Perturbant is added at f = 0. Conditions: set I parametrization with [A] = constant = 0.06 M, [H'] = constant = 0.8 M, kos = 1 s-I, ko6 = lo6 M-' s-I ,f=2, Ls = 0 M2. The excitation threshold was found to be [DIth= (2.05 0.02) X
*
0
1
2
B
3
Figure 2. (A) Boundary surfaces in the ks-f plane obtained by LSA using set I parametrization. [A] = constant = 0.06 M, [H'] = constant = 0.8 M, and Ls = 0 MZ.The numbers on the LSA curves show k6 (= ka[D]) values. (B) Boundry surfaces in the k5-fplane obtained by LSA
using set I1 parametrization. Otherwise, same conditions as in (A). (Note: k, = kos.) domain. The two regions A and C, separated by the dashed line, differ in the following respect: for any k6 = ko6[D] > 0 the SS of region A is stable, while c shows instability for some k6 > k6tcrit. C is the excitable region, but it should be noted that the demarcation line between A and C is dependent upon k6. For some given value of k6 > 0 the LSA will predict an expansion of the oscillating region B into C as illustrated in Figure 1B by the fully drawn curve. The hatched segment D is gained, while the dotted segment of B is lost to A. Upon increasing k6 the oscillatory region expands into C. Parts A and B of Figure 2 show the boundary surfaces obtained from LSA (with & = 0) for sets I and 11, respectively. The LSA for these two sets shows boundary surfaces essentially similar in
M Ag'. Numbering of curves corresponds to initial concentrationsof Agt as follows: (1) 1.0 X M, (2) 1.5 X M, (3) 2.0 X loT7M, (4)2.5 X M. (B) Excitability in a Ce(1V)-catalyzedmethylmalonic acid BZ oscillator. Arrows indicate addition of AgNO,. Numbering of curves corresponds to concentration of unreacted Ag' after addition: (1) 2.5 X M, (2) 3.3 X M. Initial concentrations: emethy~malonic acid = 0.28 M, C?(NH4)2Cc(N03)s = 2.1 X M, eKBr03 = 0.1 M. Reaction volume 30 mL, T = 25 OC (see also ref 29).
shape. The most striking difference, however, is found in the maximum range kos can have within the unstable region. This range is 0 Ikos I446.2 and 0 Ikos I8.25 for sets I and 11, respectively. Figure 3 compares the steady-state values [X]ss,o, [Y]ss,o, and [Z]m,ofor sets I and 11. These values will be of importance when excitation threshold values will be discussed in the next section. Single Excitation Spikes and Their Thresholds. When the Oregonator is in its nonoscillating excitable steady state (region C, Figure lA), perturbation by a finite amount of AgNO, results in the appearance of a single fully developed Ce(1V) spike when the amount of added AgNO, exceeds a certain threshold. Figure 4A illustrates this behavior. Althugh theoretically the excitability property of the irreversible2' and reversible**Oregonator has been known for a while, it was first recently that perturbation exper(27) Field, R. J.; Noyes, R. M. Faraday Symp. Chem. SOC.1974, 9, 21. (28) Field, R. J. J . Chem. Phys. 1975, 63, 2289.
Ag+-Induced Oscillations and Excitations
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6427
H
6
i0
40
6C
0
20
40
60
7 6Min. Figure 5. Comparison between thresholds determined by LSA and the integration method using set I parametrization. Conditions: [A] = constant = 0.06 M, [H+] = constant = 0.8 M, ko5 = 1 s-l, ko6 = lo4 M-' s-', 4 = 0 MZ.
In
TABLE II: Corrected Threshold Values, [D]guA, as a Function o f f , koL and kos"
(a) Set I (ko5 = 1 ko6, M-' s-I [Dl$.;SA, M
lo6 3.06-16.3'
s-l,
1.8 < f < 2.8)
105 4.27-88.9'
104 22.3-816'
20-500'~'
-21
(b) Set I (kos = 0.05 s-I, 2.4 < f < 3.4) M-' s-I [D]$,rsA, M
kO6,
kO6,
M-'
s-I
lo6 7.93-20.9'
105 11.6-97.4'
104 44.7-849'
60-400',d
(c) Set I1 (kol = 0.25 s-', 2.2 < f < 3.2) 105 104 lo6
[D]$rLsA, M
3.28-5.03'
3.17-6.10'
B
3.26-13.4e O.2-5.Oc*'
(d) Set I1 (ko5 = 0.02 s-I, 2.4 < f < 3.4) 105 104 ko6, M-' s-' lo6 [D]$:LsA, M 3.52-5.07" 3.66-6.48e 4.15-14.9e 0.6-4.0dse
" Thresholds are given for the end points of the f range f,,. < f < f,,, for different kO6 values. Parameters in common to all entries: [A] = 0.10 M, [H+] = 1.0 M, 4 = 7.7 X lo-'' MZ. bAll entries to be multiplied by CExperimentalvalues (see ref 18). dExperimental values (see ref 29). eAll entries to be multiplied by iments with Br--removing reagents have been realized to exhibit this sort of excitability in homogeneous classical BZ systems (Figure 4B).18,29 A natural quantity to be measured in an excitable BZ system is its excitation threshold, i.e. the smallest amount of added perturbant (here Age ions) which is able to induce a single fully developed Ce(1V) pulse. Figure 5 shows numerically determined thresholds (Le. determined by numerical integration of the rate equations), denoted as [D]*, compared with thresholds determined by LSA as a function off. When using LSA for determining thresholds, we simply note that ko6[DIcnt= k6,critrepresents the perturbation at the boundary surface between the stable and unstable regimes, given that all Re (A,) < 0 for k6 = ko6[D] = 0. To accentuate the approximative nature of a threshold determined by LSA, we denote it by [DIth,~sA.From Figure 5 we see that the difference between [Illthand [D]th,LSA increases with highly increasingf, making the correspondence [DIth [D]th,LSA approximative for f > 3. But nevertheless, the simplicity of evaluation combined with the generalized insight the LSA affords makes up for its lack of accuracy, and our main concern will therefore be with [D]th,LSA. When comparing experimentally determined thresholds with theoretical values, we have also to consider the effect of the solubility product of AgBr in our model. In this case, experimental values have to be compared with the following corrected thresholds
-
and
LD1l'
UA
S
[Dlth - LS/[ylSS,O
(7b)
where the second term on the right-hand side is the correction (29) Ruoff, P.; Schwitters,B. Z.Phys. Chem. (Wiesbaden) 1982,132,125; 1983,135, 171.
Figure 6. (A) Oscillations induced by addition of an excess of Ag+ ions into an excitable Ce(1V)-catalyzedBZ reaction. Same initial concentrations as in Figure 4B, but malonic acid used as organic substrate. Reaction volume 150 mL and T = 25 "C. (B) Oscillations induced by addition of Ag' ions to an excitable Oregonator. Conditions: set I parametrizations, [A] = constant = 0.06 M, [H'] = constant = 0.8 M, [D]initlal = 5.0 X lo-' M, ko5 = 1 S-I, ko6 = lo6 M-I s-' ,f=2, Ls = 0 MZ.
for Ag' ions initially in the solution due to the presence of excess AgBr(s). Table I1 compiles our calculated [D]&A values. The values of kos have been chosen so as to approximately reproduce the different period lengths for malonic acid BZ systems (entries a and c) and methylmalonic acid BZ systems (entries b and d), where the latter has considerably longer period lengths for the Ce(1V)-catalyzed ~ystem.2~ Corresponding experimental threshold values are given in the respective entries. For set I1 parametrizations, the inclusion of L, exhibits [D]*,L~A and [DIELsAvalues which are considerably higher compared with experimental thresholds, while set I parametrizations compare best. The main reason for these high values in set I1 parametrizations is found by comparing [Y]ss,o of the two sets in Figure 3 and taking into account that the product [Y]ss,o[D]th,LsA2 & must be satisfied in the presence of solid AgBr. Set I parametrizations compare best with experimentally determined thresholds when ko6 = lo4 M-' sd. We note that this range of ko6 values has already been predicted from much simpler model consideration^.'^ Furthermore, using set I, the thresholds are predicted to be virtually independent of kos. Also, this seems to stand in good agreement with experiments, because although the methylmalonic acid BZ period lengths are considerably longer than corresponding malonic acid period lengths (which in the Oregonator is realized by lower kos values), the experimentally determined thresholds between these two systems are found to be approximately e q ~ a l . ' ~ , ~ ~ Multiple Excitation Spikes. We have seen that the Oregonator is not only able to simulate single Ag+-induced excitation spikes but also to give reasonable values of the excitation thresholds. When increasing the amount of initially added silver ions in excess compared to the excitation threshold, we observe the appearance of high-frequency oscillations. Also, this is found by experiment. Figure 6B shows a series of numerically obtained oscillations, while Figure 6A gives the corresponding experimental situation. Using set I parametrization, both frequency and amplitude of the Ag+-induced oscillations are highly dependent upon the value of kos The frequency decreases and the amplitude increases when
6428
Ruoff and Schwitters
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
TABLE 111: Flow Simulations and Oscillatory Periods at Different Flow Conditions" (a) Set I (kos = 1 s?, [ A ] = 0.1 M, [H'] = constant = 1.0 M, Ls = 0 M 2 , f = 2) r F -1.6/1/ -1.81 -1.8-
,/
0.5 21.0 2.0 17.2 8 .O 12.7 (b) Set I (kos = 1 s-l, [ A ] = 0.06 M, [H+]= constant = 0.8 M, Ls = 0 M 2 , f = 2)
*Calculated values oExperimental OExperimental values
I
-2.0-
-2.2-
-2.4r 0
P
r
0.5 33.5 2.0 22.6 8.0 12.3 (c) Set I1 (kos = 0.25 s-I, [A] = 0.06 M, [H+]= constant = 0.8 M, Lr = 0 MZ. f = 2)
2
4
(mL/Min) 6
8
Figure 7. Logarithm of frequency (s-l) as a function of flow rate r (mL/min) with a 0.05 M AgNO, solution. Experimental points from ref 19. Conditions for calculated values: set I, [A]initisl= 0.06 M, [H+] = constant = 0.8 M , f = 2, & = 0 M2, ko5 = 1 sd, ko6 = lo4 M-I s-', V, = 150 mL, [D] = 0.05 M in input flow.
ko6 decreases. Set I1 parametrization shows essentially the same type of behavior; however, since both [Y]ss,o and [Z]ss,o are considerably reduced in the f region of interest as compared with set I, Ag+ (and also bromate) consumption is strongly reduced. This results in a sort of "perpetuum mobile" situation where up to several hundred high-frequency oscillations occur before the strongly excited system settles back to the original (excitable) steady state. However, such behavior, as far as we can see, has not been observed experimentally, indicating the superiority of set I for semiquantitative comparison. If one takes the restrictions of this relatively simple model into account, these results compare favorably with experirnent~.'~,'~ Oscillations Generated by Silver Zon Flow. In experiment, Ag+-induced oscillations are the natural consequence of the excitability property in BZ systerm18~'9With this we mean that Ag+-induced oscillations have their explanation in that they are the result of a steadily activated excitable steady state; Le., the bromide concentration is always lower than the critical4 concentration of this species. The same is also true for the Oregonator model. In terms of LSA the activation of Ag+ ions was seen to correspond to an extension of the oscillating region B into the excitable region C (Figure 1B). Experimentally, Ag+-induced oscillations may be obtained by pumping an aqueous solution of AgN03 into the closed excitable BZ ~ y s t e m . ' ~Simulation ~ ~ ~ ~ * ~calculations using set I parametrizations were performed as to mimic the experimental conditions as closely as possible (eq 4 and 5 ) . In Figure 7 experimental and theoretical results are given. Experimental datal9 and one series of calculated values are shown as log (PI) as a function of the flow rate r. The data given in Table 111are an extension of these shown in Figure 7 chosen to span a reasonable range in [A] and [H+]. Obviously, the effect of calculated periods is small. The period is only slightly dependent upon f. The calculated curve in Figure 7 is strongly reminiscent of the theoretical curve in Figure 2 of ref 19, indicating that the assumptions this simplified calculation was based upon are essentially correct. However, also in the calculations using the Oregonator there is a significant discrepancy with experiment when using low flow rates, because experimental periods at low Ag+ flow rates are markedly longer than in the normal unperturbed oscillating state, while, on the other hand, calculations always show shorter period lengths for Ag+-induced oscillations than for the unperturbed oscillating state. Using set I1 parametrizations, we found that silver ion concentration increases steadily, resulting in an increasing frequency and decreasing amplitude of the oscillations. This is due to the very ineffective silver ion consumption already mentioned in the previous multiple excitation spikes section. Comparing period lengths as a function of flow rate (Table 111) shows that set I1 has little variation in period lengths compared with set I.
r
P
0.5 2.0
21.8 15.9 12.8
8.0 (d) Set I (kos = 1 s-I, [A] = 0.06 M, [H+]= constant = 0.8 M, Ls = 4.1 X lo-" M2, r = 1 mL/min)
f
P
1.52 1.60 1.70 1.80 1.90 2.00
27.57 27.70 27.72 27.79 27.81 27.90
'Common to all entries: Periods IPl in seconds. flow rate (rl in mL/min, initial volume V, = 150 mL, [ D ] in input flow is 0.05 M, ko6 = 104 M-I s-l. \
,
\
,
Summing up, oscillations induced by Ag+ flow are semiquantitatively modeled with set I parametrization and flow rates greater than 1 mL/min with use of a 0.05 M AgNO, solution. The Question of Control in Silver Ion Induced Oscillations. It was first Noszticzius'l who reported about AgN0,-enforced oscillations when perturbing a classical malonic acid BZ oscillator with AgN0, and concluded that Ag+-induced oscillations are not controlled by Br- ions. In his subsequent papers he even considers Br- ions to play no control function at a11.12-16 Ganapathisubramanian and NoyesI7 agreed in Noszticzius' interpretation of non-Br- control at low Br- concentrations, but they consider Brions to be still the important control intermediate in the normally oscillating BZ reaction. When Ag+ ions decrease the Br- concentration, then they postulate that the control is gradually taken over by bromine atoms. Adding a Br--removing reaction to the five-step Oregonator model, we have shown that single excitation spikes, their thresholds, as well as Ag+-induced oscillations can be semiquantitatively modeled. When a Br--removing step is included in the Oregonator model, the oscillatory region B is shown to expand into the excitable region C (Figure lB), indicating that there is principally no difference between an oscillating state enforced by Ag+ ions or driven by Br03- ions. This immediately raises the question about the Br- control in Ag+-induced oscillations. The Oregonator still favorably describes AgNO,-perturbed BZ systems, which indicates that the same type of control mechanisms as proposed in the normal oscillating state can also account for Ag+-induced oscillations. Certainly, the question of whether there are additional intermediates which may react with HBr02, like bromine atoms as proposed by Ganapathisubramanian and Noyes,17 cannot be excluded by such a model consideration, but there seems presently no need to consider the much more complicated bromine atom control. On the other hand, as might be expected from such a simple model, there are some discrepancies between theoretical prediction and observed experimental behavior. An important discrepancy concerns the behavior of the Y variable in the Oregonator and the dynamical behavior of a Br--detecting electrode when per-
J . Phys. Chem. 1984, 88, 6429-6435
I
P=..
r'--..
log IEr-I
(C)
(d)
Figure 8. (a) Idealized behavior of a Br--removing pulse. (b) Behavior when pulse in (a) is recorded by a Br--detecting device with proposed delay. (c) Concentration-time profile of Br- concentration in Oregonator model when Agt ions are added (see also Figure 6B). (d) Record of Brconcentration shown in (c) by using a Br--detecting device with proposed delay (see also Figure 6A).
turbing the BZ reaction by AgN03. While the platinum electrode, in agreement with presented calculations, shows high-frequency oscillations, the Br--detecting electrode shows sometimes,"J but not always,19a monotonic stepwise increase of the potential when the BZ reaction is treated with AgN03. Figure 6 illustrates this discrepancy. It is the absence of clear oscillations at the Br-detecting electrode which makes Noszticzius and Ganapathisubrarnanian and Noyes postulate alternative mechanisms of control.1~,~6,17 However, there seems to exist possible explanations other than changing the entire control mechanism. An alternative explanation of the behavior of the Br--detecting electrode is qualitatively
6429
indicated in Figure 8. The proposal is that when the Br- concentration in BZ systems is driven to very low values, the Br-detecting electrode may show a delay in its response. Figure 8a shows the idealized situation when Br- ions are suddenly removed and then suddenly added again, while Figure 8b shows the (also idealized) corresponding situation when the Br- concentration is now recorded by a Br--detecting device with proposed delay. Figure 8c,d illustrates how the experimental potential of the Br--detecting electrode in Ag+-perturbed BZ systems (see Figure 6A) can now be explained by such a delay mechanism. We intend to test such assumptions critically by further experimental and theoretical work.
Conclusion Although the Oregonator is only a skeleton of the FKN mechqnism, the model reproduces qualitatively or semiquantitatively experimentally observed excitation thresholds, single and multiple excitation spikes, as well as period lengths in systems where A g N 0 3 is pumped into the reaction vessel. Due to the simplicity of the model, there are second-order effects the model cannot account for, such as the frequencies at low Ag+ flow rates and the nonoscillatory behavior of Br--detecting electrodes when the Br- concentration by addition of AgN03 is forced to very low values. However, besides these discrepancies, it seems fair to say that our results show that Ag+-induced oscillations and excitation phenomena can be explained and modeled within the framework of the FKN theory. Thus, there is presently no need to postulate other control mechanisms as done by N o s z t i c ~ i u s l ~ -and ' ~ by Ganapathisubramanian and Noyes." Registry No. Ag, 7440-22-4;BrO,-, 15541-45-4;Ce, 7440-45-1.
Negative Activation Energies and Curved Arrhenius Plots. 1. Theory of Reactions over Potential Wells Michael Mozurkewich and Sidney W. Benson* Donald P . and Katherine B. Loker Hydrocarbon Research Institute, University of Southern California, Los Angeles, California 90089 (Received: March 22, 1984)
The Occurrence of negative activation energies and strongly curved Arrhenius plots in apparently elementary, gas-phase reactions is explained as being due to the formation of an intermediate complex. Activation energies as negative as -1.2 to -1.8 kcal mol-', for reactions near 300 K, may be explained by this mechanism. At higher temperatures, the activation energies should become increasingly positive. We present here a general procedure, based on RRKM theory, for calculating the rate constants of reactions of this kind.
I. Introduction In recent years a number of gas-phase reactions which were believed to be elementary have been shown to have either strongly curved Arrhenius plots or strongly negative activation energies. For example, the reaction of hydroxyl radical with carbon monoxide has a strongly curved Arrhenius plot; the activation energy at 300 K is near zero, but at 2000 K it is about 7 kcal mol-'.'S2 The reactions of hydroxyl radical with nitric acid3 and peroxynitric acid4 and the disproportionation of hydroperoxy radicals3 have negative activation energies of about -1.5 kcal mol-'. In this series of papers we seek to explain these effects.
(1) Baulch, D. L.; Drysdale, D. D. Combust. Flame 1974, 23, 215. (2) Smith, I. W. M. Chem. Phys. Len. 1977, 49, 112. (3) DeMore, W. B.; Molina, M. J.; Watson, R. T.; Golden, D. M.; Hampson, R. F.; Kurylo, M. J.; Howard, C. J.; Ravishankara, A. R. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, 1983, JPL Publ. 83-62. (4) Smith, C. A,; Molina, L. T.; Lamb, J. J.; Molina, M. J. Int. J . Chem. Kiner. 1984, 16, 41.
0022-3654/84/2088-6429$01 .50/0
Dryer et aL5have explained the curvature of the Arrhenius plot for the reaction OH C O as being due to a tight transition state with a low threshold energy. Smith and Zellner6 interpreted this in terms of a stable intermediate complex. Margitan and Watson' have proposed that a similar model may explain the negative temperature dependence for the reaction O H "0,. Patrick, Barker, and Golden8 and Kircher and Sander9 have carried out calculations of the negative temperature dependence of the self-reaction of HOP In section I1 of this paper we present physical arguments to explain how the formation of a stable intermediate can produce negative action energies and curved Arrhenius plots. In section I11 we develop quantitative expressions, based on RRKM theory, that may be used to calculate the rates of these
+
+
( 5 ) Dryer, F.; Nageli, D.; Glassman, I. Combust. Flame 1970, 71, 270. (6) Smith, I. W. M.; Zellner, R. J. Chem. SOC.,Faraday Trans. 2 1973,
69, 1617. (7) Margitan, J. J.; Watson R. T. J . Phys Chem. 1982, 86, 3819. (8) Patrick, R.; Barker, J. R.; Golden, D. M. J. Phys. Chem. 1984.88, 128. (9) Kircher, C. C.; Sander, S. P. J . Phys. Chem. 1984,88, 2082.
0 1984 American Chemical Society
