Dependence of Reaction Rates on Temperature

Rate constants for chemical reactions are known to depend strongly on the reaction temperature. One well known empirical relationship expressing this dependence is the Arrhenius equation (1):

k = A exp(-E_a/RT)      (1)

The important parameters in this relationship are the activation energy E_a (in units of kJ/mol or kcal/mol) and the preexponential factor A. The latter is given in the same units as the rate constant itself ([s^-1] for a first order reaction and [l mol^-1 s^-1] for a second order reaction).

A second expression used to describe the temperature dependence of reaction rate constants is the Eyring equation (2) that results from transition state theory :

k = (k_BT/h) (1/cⁿ) exp(-dG**/RT)      (2)

Here the first part of the equation contains only Boltzmann's constant k_B and Planck's constant h as well as the absolute temperature T (in [K]). The second part accounts for the concentration dependence of the rate constant, c being the reference concentration of the reaction. The exponent n assumes a value of 0 for a first order reaction and of 1 for a second order reaction. The third part of equation (2) contains the activation free enthalpy dG** of the reaction which, according to equ. (3) is directly related to the activation enthalpies and activation entropies:

dG** = dH** - TdS**      (3)

The activation parameters of the Arrhenius and Eyring equations of a first order reaction are related as follows:

E_a = dH** + RT      (4)

A = (c k_BT/h) exp(dS**/R)      (5)

In particular the difference between Arrhenius activation energy E_a and the activation enthalpy dH** are quite small and numerically close to the accuracy attained in most experiments (RT = 2.5 kJ/mol at 298.15K). These two energies are therefore frequently used interchangeably in the literatur to define the activation barrier of a reaction.