It would be expected that a uniform, homogenous solution such as that of the Belousov-Zhabotinsky Reaction would remain in just that state. However, the concentrations of the myriad chemical species are given to stochastic fluctuations in small regions. That is, there is a probability that in some small pocket of the solution the concentration of reactants will be higher than their average concentration in the medium. When a fluctuation favorable to initiation occurs, a clear center will appear in the sea of red. Carried radially outward by diffusion, the center will promote the reaction with its neighbors. However, as it advances, the interior of the center oscillates back, turn back to red. Allowed to continue uninterrupted, waves of reaction propagate across the medium, forming growth patterns both aesthetically pleasing and striking for their familiar geometry.
The Belousov-Zhabotinsky Reaction remained little
more than a curiosity, suitable primarily for dramatic lecture demonstrations,
until the mechanism was examined in earnest in the early 1970s by Richard
M. Noyes, Richard J. Field, and Endre Koros at the University of Oregon.
They proposed a robust mechanism consisting of eighteen reactions and twenty-one
distinct chemical species, resulting in an impressive and intimidating
system of rate equations (differential equations that describe the rate
of reactions). Both the chemistry and mathematics of this schema
are beyond the reach and grasp of this article. Therefore, the model
considered here is a simplification of the Fields-Koros-Noyes model, where
it has been placed in a crucible and the volatiles have boiled away, leaving
only the essentials: the Oregonator. (Its name is in honor of their
patron institution, the University of Oregon.) However, despite its
relative simplicity, it nonetheless demonstrates the qualitative behavior
of the Belousov-Zhabotinsky Reaction. The Oregonator model gives
the following reaction steps and associated rate equations:
HBrO2 + HOBr |
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2HOBr |
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2HBrO2 + 2Ce4+ |
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BrO3- + HOBr |
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1/2fBr- |
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where B represents all oxidizable organic species present and f is stoichiometric factor that encapsulates the organic chemistry involved. Concerning the notation of the rate equations, recall the enclosing some species in brackets merely indicates that we are measuring the concentration the species, usually in units of moles per liter -- that is, the amount of the species per unit of volume. For example, [HBrO2] merely represents the concentration of HBrO2 (bromous acid). Additionally, recall that the ks are simply rate constants that depend upon temperature and must be experimentally determined.
For the sake of concise notation, which will prove
valuable in developing a system of differential equations, let the various
chemical entities be given by
| A | BrO3- |
| B | All oxidizable organic species |
| P | HOBr |
| X | HBrO2 |
| Y | Br- |
| Z | Ce4+ |
If we treat the concentrations of the reactants A and B
as constant, the rate equations for X, Y, and Z become
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The rate constants, as determined by Fields, Noyes, and Koros, are given
to be
| k1 | 1.28M-1s-1 |
| k2 | 2.4 x 106M-1s-1 |
| k3 | 33.6M-1s-1 |
| k4 | 3 x 103M-1s-1 |
| kc | 1M-1s-1 |
We may further simplify by transforming the concentrations X, Y, and Z into dimensionless variables. Replace X, Y, Z, and t, respectively, with the variables x, y, z, and t, given by
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Moreover, let the dimensionless parameters e,
e', and q be give by
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Therefore, the system of differential equations simplifies to
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Of course, by rendering the system dimensionless, direct physical significance
is lost. However, the dimensionless system may be interpreted to
yield the relative concentrations of each species -- that is, the
concentration of one in relation to another.
This system of differential equations proves intractable
to the analytical approaches, but more amenable to numerical analysis.
Where the initial concentration of A is taken to be 0.06M and that
of B is 0.02M and the stoichiometric factor f is 1, oscillatory
behavior is indeed observed (as seen is the below visual aide). The
clear wave of reaction corresponds to the concentration of Ce4+
as it approaches zero; elsewhere, corresponds to the red sea of the medium.
