For the reaction

, the rate is

,

with and ,

where and are the pre-exponential Arrhenius constants for the forward and reverse reactions, and are the energies of activation, is the universal gas constant, is the absolute temperature, , , , and are the initial concentration of the reactants, and is the conversion of species . The temperature function that gives the maximum reaction rate satisfies the condition at each point in time; this function has an analytical solution for this reaction [1]

,

with

,

, and

;

the initial concentrations of and are taken equal to 0.5, and 1.0.

One complication can occur: for low conversions, may have a value sufficiently small enough to make ; then the equation for gives a value (or negative); in practice the temperature is limited by the reactor materials or the catalyst's physical properties. The optimum temperature profile and the concentration of the reactants as a function of time are shown for user-set values of reaction time, maximum allowable temperature, and parameters and .