Sustained oscillations in isothermal branched-chain reactions

Abstract

The kinetics behaviour of a wide range of autocatalytic systems can be well approximated by the prototype step A + B → 2B, rate =k₁ab.

A similar rate law is also used extensively when dealing with the important class of branched-chain reactions. The above step is considered in the present paper for a system in which the catalyst B is not infinitely stable but undergoes a removal or decay whose rate increases less than linearly with the catalyst concentration: B → C, rate =k₂b/(1 +rb). Such an expression may reflect deactivation of B at a surface where the number of free sites available for adsorption may become significantly reduced.

There exists a unique, physically acceptable stationary state at all residence times: multiplicity is not a feature of this system. However, the local stability of the stationary state changes as the residence time varies. At shortest residence times the solution is a stable node and small perturbations decay monotonically. At longer residence times, damped oscillatory responses are found characteristic of a stable focus. For surfaces such that r > k₁/k₂, the stable focus becomes unstable at some residence time t^*_res. At this point of Hopf bifurcation the system begins to display sustained oscillations in the concentrations of A and B. The amplitude of the oscillations increases from zero as t_res is increased beyond t^*_res, growing as the square root of this difference.

The conditions for Hopf bifurcation and hence for oscillatory reaction are evaluated numerically in terms of the parameters k₁, k₂, r and a₀(the concentration of reactant A in the inflow to the reactor). Finally, some qualitative connections are drawn between the above model scheme and the varied patterns of behaviour observed during the oxidation of carbon monoxide.