Dynamic Equilibrium - Relationship Between Equilibrium and Rate Constants

Relationship Between Equilibrium and Rate Constants

In a simple reaction such as the isomerization:

there are two reactions to consider, the forward reaction in which the species A is converted into B and the backward reaction in which B is converted into A. If both reactions are elementary reactions, then the rate of reaction is given by

where k_f is the rate constant for the forward reaction and k_b is the rate constant for the backward reaction and the square brackets, denote concentration. If only A is present at the beginning, time t=0, with a concentration ₀, the sum of the two concentrations, _t and _t, at time t, will be equal to ₀.

The solution to this differential equation is

and is illustrated at the right. As time tends towards infinity, the concentrations _t and _t tend towards constant values. Let t approach infinity, that is, t→∞, in the expression above:

In practice, concentration changes will not be measurable after . Since the concentrations do not change thereafter, they are, by definition, equilibrium concentrations. Now, the equilibrium constant for the reaction is defined as

It follows that the equilibrium constant is numerically equal to the quotient of the rate constants.

In general they may be more than one forward reaction and more than one backward reaction. Atkins states that, for a general reaction, the overall equilibrium constant is related to the rate constants of the elementary reactions by