J. Phys. Chem. 1882, 86,3006-3012
3006
Scaling and Reduclng the Field-Koros-no yes Mechanism of the Belousov-Zhabotinskii Reaction John J. Tyson Department of Bioiqy, Vkginia Polytechnic Institute and State Unlverstfy, Blacksburg, Virginia 2406 1 (Received: February 12, 1982)
A system of six first-order ordinary differential equations are derived from the Field-Koros-Noyes (FKN) mechanism of the Belousov-Zhabotinskii (BZ) reaction. These equations are cast into dimensionless form by a scaling covention that identifies the “natural” time scales on which the six chemical species evolve. Then singular perturbation theory is used to reduce the full system to two or three equations which can be investigated analytically for steady-state stability and limit cycle oscillations. Depending on the assumptions made about specific rate constants and time scales, the full system reduces to a family of different models including the “Oregonator”,“model K”, the “IUator”, and several models not previously considered in the literature. This approach uncovers the advantages and limitations of simple models of the Belousov-Zhabotinskii reaction and establishes that the Oregonator is the best of the simple models. The analysis also shows that reaction 7 of the Field-Koros-Noyes mechanism (BrOz+ Ce4++ H20 = Br03- + Ce3++ 2H+), for which rate constants are unknown, never proceeds at a significant rate in either direction during homogeneous chemical oscillations of the BZ reaction.
Introduction The Belousov-Zhabotinskii (BZ) reaction is the oxidation of malonic acid or bromomalonic acid by bromate ions in acidic solution, catalyzed by transition-metal ions (such as cerium, iron, or ruthenium) with standard reduction potentials of 1-1.5 V. Under various conditions this reaction exhibits sustained oscillations, multiple steady states, multiply periodic oscillations, chaos, and spatially propagating waves of oxidation and reduction.14 In 1972 Field, Koros, and Noyes’ (FKN) presented a detailed mechanism of the Belousov-Zhabotinskii reaction, which is now generally accepted as correct and complete in major respects. The Field-Koros-Noyes mechanism identifies 11 principle reactions among 12 chemical species. Using standard principles of chemical kinetics, Field, Koros, and Noyes were able to give a satisfying qualitative (and, in some respects, quantitative) account of the complex behavior observed during different stages of oscillation in the BZ reaction. However, the mechanism was too complicated and the kinetic arguments too imprecise to prove conclusively that these 11 reactions could generate stable limit cycle oscillations in the concentrations of reaction intermediates. This proof was left to modeling efforts. Edelson, Field, and Noyes’ expanded the FKN mechanism to 20 reactions among 18 chemical species. The large system of firsborder ordinary differential equations (ODES)which result from this mechanism by application of the law of mass action was solved numerically by using the Gear algorithm to handle the wide separation of time scales implicit in the equations. Edelson, Field, and Noyes obtained stable limit cycle oscillations that agree remarkably well with observed features of BZ oscillations. At the other extreme of (1) Field, R. J.; Korb, E.; Noyes, R. M. J.Am. Chem. SOC. 1975,94, 8649-64. (2) Tyson, J. J. ’The Belowov-Zhabotinskii Reaction”,Lecture Notes in Biomathematics; Springer-Verlag: West Berlin, 1976; Vol. 10. (3) Field, R. J.; Noyes, R. M. J. Chem. Phys. 1974, 60, 1877-84. (4) Tomita, K.; Ito, A,; Ohta, T. J. Theor. B i d . 1977, 68, 459-81. (5) Schmidt, S.; Ortoleva, P. J. Chem. Phys. 1981, 74, 4488-500. (6)Vidal, C.; Pacault, A. “Non-Linear Phenomena in Chemical Dynamics”; Springer-Verlag: West Berlin, 1981. (7) Edelson, D.; Field, R. J.; Noyes, R. M. Znt. J. Chem. Kinet. 1975, 7, 417-32. 0022-3654182l2086-300880 1.2510
modeling, Field and Noyes3reduced the FKN mechanism to five steps among three reaction intermediates. (This model reaction network is known as the Oregonator.) The resulting system of three first-order ODES yields to standard methods of mathematical analysis: for certain ranges of parameters there exist stable limit cycle oscillations that compare favorably with observed features of the BZ reaction. Wolfes has shown that the full set of ODES in the Edelson-Field-Noyes mechanism are a “singular perturbation” of the three ODES representing the Oregonator. This means that he could identify a small parameter (e) such that, when e = 0, the full set of ODES reduced to the Oregonator. This is an important result, since it demonstrates that the much simplified Oregonator is a valid reduction (in the limit t 0) of the complete mechanism. Therefore, we can have some confidence that results derived from the Oregonator are applicable to the BZ reaction. Like Wolfe’s work, this paper applies singular perturbation theory to a detailed mechanism of the BZ reaction in order to simplify the mathematical description of this complex reaction. However, the scaling conventions that I use differ from Wolfe’s, and this difference has several important consequences. First of all, my equations are more flexible, so that taking the singular limit can be done in various ways. This generates a family of different models, including not only the Oregonator but also model K of Tomita, Ito, and Ohta,4 the IUator of Schmidt and Ortolevat and others which have not previously appeared in the literature. Secondly, the scaling that I adopt singles out the metal-ion catalyst as the “slow” variable, which not
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(8) Wolfe, R. J. Arch. Ration. Mech. Anal. 1978, 67, 225-50. (9) Tyson, J. J. Ann. N . Y. Acad. Sci. 1979, 316, 279-95. (10) w o n , J. J. J. Math. B i d . 1978,5, 351-62. (11) Showalter, K.; Noyes, R. M.; Bar-Eli, K. J. Chem. Phys. 1978,69, 2514-24. (12) Noyes, R. M.; Jwo, J. J. J. Am. Chem. SOC. 1977, 97, 5431-3. (13) In system E l , if b is small ( b < 1/4g or thereabouts), it can happen that z(t) exceeds 1, which is chemically impossible. As long as b is large, z(t) stays small, and the factor 1 - 22 is not too important. (14) Tyson, J. J. In “on-Linear Phenomena in Chemical Dynamics”; Vidal, C., Pacault, A,, Eds.; Springer-Verlag: Weat Berlin, 1981; pp 222-7. (15) Fhterling, H. D.; Lamberz, H.; Schreiber, H. 2.Naturjorsch. A 1980, 35, 329-31, 1354-60.
0 1982 American Chemical Society
Field-Koros-Noyes Mechanism of the BZ Reaction
The Journal of Physical Chemisrty, Vol. 86, No. 15, 1982 3007
only seems to be correct physically but also leads to allebraic simplifications in the reduced systems. Finally, by comparison of the models on common ground, their relative merits are more easily assesed. We shall find that the Oregonator is not only the simplest model but also the most successful in accounting for the period and the amplitude of observed oscillations. The reason for the Oregonator's success can be attributed to the fact that it singles out exactly those reactions of importance in generating oscillations: reactions that proceed very much faster are incorporated into pseudo-steady-state approximations, and reactions that proceed very much slower are neglected entirely. This paper is restricted to a discussion of models of the "classical" BZ reaction. These models will be compared in light of well-known properties of the bromate-malonic acid-cerium system. I shall not attempt to model recent modifications of the BZ reaction, which use different organic substrates, different catalysts, or no catalyst at all. An excellent mechanistic discussion of these bromate/ bromide-dependent oscillators has been given by Noyes.le Furthermore, I shall not attempt in this paper to discuss chemical waves in unstirred, thin layers of BZ reagent since these waves are adequately described in an earlier publication.l7
Scaling the FKN Mechanism The 11 reactions of the FKN mechanism are given in Table I. In order to write the rate equations determined by this mechanism, we first introduce the following abbreviations: A = [BrOc], C = [Gel,,, H = [H'], M = [CH,(COOH),], S = [Br204],U = [BrO,], V = [Br,], W = [HOBr], X = [HBr02],Y = [Br-],Z = [Ce4+]. The total cerium concentration, C, is a constant, and the hydrogen ion concentration, H , is nearly constant under the highly acidic conditions of the BZ reaction. Furthermore, we assume that A and M , concentrations of major reactants, are also constants. With these assumptions the rate equations read
Rs = k5,HAX - (k-5, + k5b)S + k-5bV
RU = 2k5bS - 2k-5bV - k&U(C - 2) + k,XZ
-
k7UZ + k-,@A(C - 2)
Rv = klHWY - k - l V - k&M Rw = -klHWY
+ k-1V + 2kzHXY + k 3 P A Y -
k-3 W X + k4X2
Rx = -k2HXY + k 3 P A Y - k-3WX - 2k4X2k5,HAX
R y = -k1HWY RZ = k&U(C
+ k-5,S + k&U(C
+ k-1V-
- 2) - k+XZ
+
k2HXY - k3lPAY k-3WX + k&M
- Z)- k+XZ -
k7UZ
+ klJ
+ k_7PA(C - Z)-
(6k9 + 4k10)Z
where Rs = dS/dT, etc., and T = time. To scale these equations, we introduce the dimensionless variables s = S/So, u = U / U o ,u = V / V o ,w = W / Wo,x = X / X o , y = Y/Yo, z = Z/Zo, t = TITo,where So, ...,2, are arbitrary constants of dimensions mol/L and Tois an (16) Noyes, R. M. J. Am. Chem. SOC.1980,102,4644-9. (17) Tyson, J. J.; Fife, P. C. J. Chem. Phys. 1980, 73, 2224-37. (18)Field, R. J.; Raghavan, N. V.; Brummer, J. G. J. Phys. Chem. 1982,86, 2443.
TABLE I: Field-Koros-Noyes Mechanism" ~
~
~~~
~
reaction
rate constants
+ Br- + H+ = Br, + H,O (2) HBrO, + Br' + H' 2HOBr (3) Br0,- + Br- + 2H+ = HBrO, + HOBr (4) 2HBr0, Br0,- + HOBr + H+ (5a) Br0,- + HBrO, + H+ =
k , = 8 X l o 9 M-z s-l k-, = l o ' s - ' k , = 2 X lo9 M-, s-l M-l s-, k-, = 5 X k , = 2 M-, s', k-, = l o 4 M-ls-' k , = 4 X lo7M-I s" k-, = 2 X lo-'' M-'S-' k , , = 2 X 10' M-'s-l k-sa = 5 s-' k s b = 7 X 10's" k-sbl= 1.4 X l o 9 M-'
(6) BrO, + Ce3++ H+ = HBrO, + Ce4+ (7) BrO, t Ce4' + H,O = Br0,- + Ce3++ 2H' (8) Br, + CH,(COOH), BrCH(COOH), + Br- + H' (9) 6Ce4++ CH,(COOH), + 2H,O -+ 6Ce3++ HCOOH + 2C0, t 6H+ (10) 4Ce4++ BrCH(COOH), + 2H,O + 4Ce3++ HCOOH + Br- + 2C0, + 5H'
k, = 5 X l o 5M', s-l k-, = 8 X 10, M-ls-' k , unknown k - , = 2 k , X l o - ' M-' r , = k,[H+] [MA] k, = 10-2 M-1 8-1 r s = k9[Ce4+] k, = O.O9s-'[MA] / (0.5M+ [MA]) r,, = kIo[Ce4+] k,, = 0 . 0 2 ~ - ~ [ B r M A ] / (0.2M + [BrMA])
(1)HOBr
-f
-f
Br,O, + H,O (5b) Br,O, = 2Br0,
-f
S-
" Those reactions that operate essentially irreversibly under normal reaction conditions are indicated by a single arrow and those that operate reversibly by an equals sign. Reactions 9 and 1 0 are composite steps, not elementary reactions. Most o f the rate constants were estimated by Field, Koros, and Noyes except (i) those characterizing reactions 5a and 6 were determined by Forsterling, Lamberz, and Schreiber,15 (ii) k 5 b was determined from k- sb and the equilibrium constant measured by Field, Raghavan, and Brummer,l8 and (iii) the equilibrium constant for reaction 7 was determined from the thermodynamic data in Field, Koros, and Noyes and the rate constants tabulated here for reactions 5a and 5b.
3008
Tyson
The Journal of Physical Chemistry, Vol. 86,No. 15, 1982
The form of these equations suuggests that we choose
To = 2k4C/(kbaHA)'
800 s
TABLE 11: Dimensionless Parameters Appearing in the Scaled Rate EquationsQ parameter
order of magnitude
definition
2 x 10-8 2.5 X 1 . 2 x 10-5 3 x 10-5 1.2 x 10-4 5 x lo+ 4 x 10-4 80
Ps Pu Pu Pw Px
PY 4
m
b
Xo
= k5,HA/(2k4) N 1.2 x
Yo = k5,A/k,
N
5x
Psrs = X -
+ K-5bU2 - K - ~ , s
PJu = 2s - 2K-5bu2- u(1 - z )
K- 3 K-sa
mol/L
The approximate numerical values of these constants are determined for H = 1 M, A = 5 X lo-' M, and C = M. (Notice that Yois just the critical bromide ion concentration introduced by Field, Koros, and Noyes in their original paper.) A natural choice of dimensionless parameters is given in Table 11. In these terms the rate equations become
10 0.07 3 x 10-5 I x 10-5 I x 10-3
g
mol/L
K- 5b
800
K-6
2h7 x Ms h , x 10-3 M s
K7
K- 7
a Orders of magnitude are estimated from the rate constants in Table I and the following concentrations: A = 5 x lo-' M, C = M, H = 1 M, [malonic acid] = 10" M, [bromomalonic acid] = M.
-
-
In the limit p s 0, p u 0, and p u 0, Irsl, lrul, and Irul+ except when s = x , u ( 1 - z ) - K+xz = 2s, and wy = u m. Therefore, s, u , and u will change very rapidly (in this limit) in order to maintain the three previous equalities. This allows us to reduce the rate equations to -+
+
+ K+xz
- K ~ U+Z K-7(l - z )
p*r'x = 4 1 - x ) - Y(X - q ) pyrY = bgz - Y ( X + 4 )
rz = 2x pyry = -wy
+ u - xy - qy + K-,wx + m + bgz
r, = ~ (- Zl ) - K+XZ- K ~ U+Z K-7(1 - Z ) - bz (E) We shall refer to this set of ODES as system E. The pi's that appear on the left-hand sides of system E are time-scale constants. If the right-hand side of the equation for piri is O ( l ) ,then the ith component changes at a rate ri = O(p;l). Since the pi's are all
