The effects of a linearly increasing ambient temperature on an exothermic reaction in a closed system

Thermal analysis in general, and differential scanning calorimetry in particular, offer rapid and automated techniques for studying exothermic decompositions. When reaction is not too complex, values for the exothermicities and Arrhenius parameters may be extracted. The basis of most mathematical treatments is not deep and quite commonly rests on analyses of ‘static’ situations, in which the responses to constant ambient temperatures are the starting point. In this paper we give an economical analysis of the dynamic process of heating a reactive sample continuously. The reduced variables of thermal explosion theory provide a compact representation of the standard case of a single reaction obeying a first-order rate law with an Arrhenius temperature dependence. An asymptotic treatment is used to refine the equations of mass and energy conservation to a practically important form. The natural variables that arise in this way lead to a model of the system in terms of the rate of the chemical reaction, rather than the more usual temperature of the sample. This new treatment leads to a clear and compact description of the behaviour of the reaction for all useful operating conditions. The well-known Kissinger relation normally derived from static treatments is seen to be relevant in this model, but care must be taken in choosing the precise physical quantities to which it relates.