Consider the reaction:

A + B → C + D

The rate equation can be expressed in several ways:

To find how the rate varies with [A] or [B] it is necessary to perform a series of experiments in which one of the concentrations is kept constant and the other varied.

Usually it is found that:

This is called the rate equation for the reaction.

k is the rate constant at a given temperature. x and y are orders of reaction with respect to A and B. They are usually whole numbers - 0, 1, 2, 3, but sometimes they can be fractions.

The order of a reaction with respect to a given reactant is defined as: 'power of its concentration in the rate equation'.

For example:

(CH₃)₃CCl_(aq) + OH^-_(aq) → (CH₃)₃COH_(aq) + Cl^-_(aq)

It has been found experimentally that the rate equation for the reaction is:

Therefore, the reaction is first order with respect to (CH₃)₃CCl_(aq) (x = 1) and zero order with respect to OH^- (y = 0).

For example, changing [OH^-_(aq)] has no effect on the reaction rate and so it does not appear in the rate equation.

The overall order of a reaction is defined as 'the sum of the powers of the concentrations of the reactants which appear in the rate equation'.

For the above reaction overall order = x + y = 1 + 0 = 1. This is a first order reaction.

Note: Rate equation and orders must be found experimentally. They cannot simply be deduced from the equation.

To find the order of a reaction with respect to one of the reactants A,

1. Plot [A] against time.
2. Calculate the rate at 5 or 6 different times by drawing tangents to the curve at these times and finding the gradients.
3. Plot the rate against [A]. If this is a straight line then the reaction is first order in A. If not a straight line then, plot rate against [A]². A straight line shows the reaction is second order in A.

Three examples of this type of graph (showing zero order, first order, then second order) are shown below:

This method is the same as the above. The rate is found for different concentrations of A. Then rate is plotted against [A], [A]²... until a straight line is obtained.

Since rate = k[A]

The value of k is found by taking the gradient of the graph.

The half-life (t1/2) of a reactant is the time taken for the initial concentration of a reactant to fall by half.

First order reactions have a constant half-life which is independent of the initial concentration.

Note: All radioactive decay processes are first order and have a constant half-life.

For second order reactions the half-life does depend upon the initial concentration. It can be shown that it is inversely proportional to it.