This Demonstration shows the temperature trajectory that maximizes the reaction rate of a reversible reaction.

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]

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

.