The arithmetic method (used in Unit 4 examinations)

(change in concentration)^{order of reaction} = change in rate

If the reaction is first order, then if the concentration doubles the rate will also double. If the concentration triples the rate will also triple, etc.

If the reaction is second order, then if the concentration doubles the rate will quadruple. If the concentration triples the rate will increase ninefold, etc.

If the reaction is zero order, then the change in concentration will have no effect on the rate of reaction.

Example 1

Consider the reaction RX + OH^- à ROH + X^-

The following rate data were obtained at constant temperature:

Initial concentration of RX/ moldm-3	Initial concentration of OH/ moldm-3	Initial rate/ moldm-3 s-1
0.01	0.04	8 x 10-3
0.01	0.02	4 x 10-3
0.005	0.04	4 x 10-3

From expt 2 to expt 1, the concentration of hydroxide ions doubles and the concentration of RX is unchanged. The rate also doubles, so the order of reaction with respect to OH^- is 1.

From expt 3 to expt 1, the concentration of RX doubles and the concentration of hydroxide ions is unchanged. The rate also doubles, so the order of reaction with respect to RX is also 1.

The rate equation can thus be written as follows: rate = k[RX][OH^-]

Having deduced the rate equation, the rate constant can be calculated using the data in one of the experiments.

Eg in expt 1, k = rate/([RX][OH^-]) = 8 x 10^-3/(0.04 x 0.01) = 20 mol^-1dm³s^-1.

Example 2

Consider the reaction PCl₃ + Cl₂ à PCl₅

The following rate data were obtained at constant temperature:

Initial concentration of PCl3/ moldm-3	Initial concentration of Cl2/ moldm-3	Initial rate/ moldm-3 s-1
0.2	0.1	0.0004
0.4	0.1	0.0008
0.8	0.2	0.0064

From expt 1 to expt 2, the concentration of PCl₃ doubles and the concentration of Cl₂ is unchanged. The rate also doubles, so the order of reaction with respect to PCl₃ is 1.

From experiment 2 to experiment 3, the concentration of both reactants doubles. The rate increases eightfold, so the overall order of reaction is three.

The order of reaction with respect to chlorine is therefore 3 – 1 = 2.

The rate equation can thus be written as follows: rate = k[PCl][Cl]²

So k = rate/[PCl₃][Cl₂]² = 0.0004/(0.2 x 0.1²) = 0.2 mol^-2dm⁶s^-1