# DEPENDENCE ON TEMPERATURE: THE ARRHENIUS EQUATION

The Arrhenius model for rates of reactions assumes that molecules must undergo collisions before they can react. The number of collisions per unit time is A, the frequency factor. This model also assumes that not all collisions will result in or lead to a reaction. Rather, only those with sufficient energy, called the activation energy, Ea (to allow molecules to reach an activated complex), will achieve this result.

The rate constant k is a function of temperature and can be expressed theoretically as the Arrhenius equation:

where

A = the frequency factor, a measure of the number of collisions per second,

*Ea* = the activation energy in joules or calories per mole,

R = the universal gas constant in units consistent with those for Ea,

*T* = the absolute temperature in kelvins.

A more useful form of the equation can be written as:

The rate constants k1 and k2 at two different temperatures, T and *T _{2}, *respectively, can be written as two separate equations, using equation 2.24. The equations can be divided by each other and the result rearranged such that the rate constants are related as follows:

*Example 2.20*

The following reaction between nitric oxide and ozone is important in the chemistry of air pollution as discussed in Chapter 2 in Applied Chemistry for Environmental Engineering:

The frequency factor *A =* 8.70 x 10** ^{12}** sec

**and the rate constant k = 300 sec**

^{1}**at 75**

^{-1}^{o}C.

A. Find the activation energy, Ea, in joules per mole for this reaction.

B. Find the rate constant k of this reaction at 0^{o}C, assuming Ea to be constant.

*Solution:*

A. Use the Arrhenius equation and solve for Ea.

Use *R* = 8.31 J/K x mol and *T* = 273 + 75^{o}C = 348 K. Then,

*Ea* = 69,700 J/mol or 69.7 kJ/mol

B. Denote the rate constant given at 75^{o}C as k**1** and the rate constant at 0^{o}C as k**2**, and use equation 2.25:

Substitute the above information, and solve for k**2**.

# CATALYSIS

Substances that increase the rate of reaction without themselves being consumed in the reaction are called catalysts. They work by lowering the activation energy, *E*a, or the barrier required for the reaction to proceed. A catalyst may be in the same physical state as the other species in the reaction (homogeneous catalysis) or in a different physical state (heterogeneous catalysis).