Oscillations of an exothermic reaction in a closed system - I. Approximate (exponential) representation of Arrhenius temperature-dependence

In a closed system in which matter is conserved the only true steady state is the final chemical equilibrium state. This means that potential oscillatory behaviour in models appropriate to such systems is not as easily open to analysis as in flow systems. Oscillatory trains cannot last indefinitely and they often begin and end well before the equilibrium point. Here we study such transitory behaviour via the `pool chemical approximation'. We study the simple prototype $P \rightarrow A \text{rate} = k_0p; E_0 = Q_0 = 0,$ $A \rightarrow B + \text{heat rate} = k_1 a, k_1 = A_1 \exp (-E_1/RT),$ in which only the step $A \rightarrow B$ evolves heat and is in any way responsive to temperature. Heat losses obey newtonian cooling: heat release provides thermal feedback. If the initial concentration of precursor exceeds a certain value, oscillations may be seen. Analytic expressions can be found for the major features (such as duration of pre-oscillatory incubation periods, post-oscillatory decays, etc.). Primitive treatments of these problems ignore the consumption of the precursor. The significance and validity of this `pool chemical' assumption are analysed qualitatively and quantitatively. Properly applied, it is an approximation of the greatest utility.