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262 5. Some worked examples arising from physical problems

5.7 CHEMICAL AND BIOCHEMICAL REACTIONS
The examples that are presented here are intended to show that it is possible to model,
and describe using perturbation theory, some very complex processes that, perhaps fifty
years ago, were thought to be mathematically unresolvable. Certainly, some extensive
simplification is necessary in the development of the model—and this requires consid-
erable skill and knowledge—but the resulting differential equations remain, generally,
quite daunting. We will describe: E5.26 Kinetics of a catalysed reaction; E5.27 Enzyme
kinetics; E5.28 The Belousov-Zhabotinskii reaction.

E5.26 Kinetics of a catalysed reaction
In a model (the Langmuir-Hinshelwood model; see Kapila, 1983) for the kinetics of a
particular type of catalysed reaction, the concentration of the reactant varies according
to the equation

with

(where is a given constant. The parameter is a rate constant and,
for we observe that the equation reduces to a purely algebraic problem which
is readily solved:

We have selected the positive sign so that On x = 1, this gives the value

and so we will require a boundary layer near x = 1; see §§2.6, 2.7. (Note that, from
this solution, c = 1 on x = 0, so we can expect that as in the solution
of the full equation.) Let us introduce and write
the equation for C is then

We seek a solution