Consider the first-order reversible reaction involving chemical species

and

:

.

The reaction rate for

is given by:

, where

is the fraction of

converted,

and

are the reaction rate constants, and

and

are the initial concentrations of species

and

* *with

.

When this reaction takes place in a perfectly mixed batch reactor, the mass balance is given by the following equation:

,

where

,

and

are the final reaction time and the time variable,

,

for

and

,

and

are the activation energy and pre-exponential factor,

is the ratio of the activation energies, and

is the dimensionless temperature.

This mass balance can be integrated to give:

with

.

The value of the optimum temperature

or

can be found for given values of

,

, and

from:

at

. After the optimum value of

is found, the actual optimal temperature is given by:

; the reaction is assumed to be exothermic (i.e.,

). This assumption is important for the Demonstration to perform correctly because only for exothermic reactions can one find an optimum temperature. Indeed, high

* *values increase the rate constant but lower the equilibrium conversion. Thus, one must reconcile between competing thermodynamic and kinetic considerations by choosing an appropriate optimal temperature. High

* *is better in the beginning where equilibrium limitations are not important and low

* ** *is better later when one approaches equilibrium

*.*The Demonstration plots the optimal dimensionless temperature,

, and the corresponding final conversion,

, versus parameter

for user-set values of the ratio of the activation energies,

.