Integrated rate equations

The essentials

Integrated rate equations relate concentration directly to time, which is easier to measure than instantaneous rates. They also let us find k and the half-life (t½), the time for the concentration of a reactant to fall to half its initial value.

Key points

• Zero order: [R] = [R]₀ − kt. A plot of [R] against t is a straight line with slope −k. t½ = [R]₀ ÷ 2k, so it is proportional to the initial concentration. Example: decomposition of ammonia on a hot platinum surface at high pressure.
• First order: k = (2.303/t) log([R]₀/[R]), or ln[R] = ln[R]₀ − kt. A plot of log[R] against t is a straight line with slope −k/2.303. t½ = 0.693 ÷ k, which is independent of initial concentration. Examples: decomposition of N₂O₅, natural radioactive decay, hydrogenation of ethene.
• For a first-order reaction, 75% completion takes 2 half-lives and 87.5% takes 3; 99.9% takes about 10 half-lives.
• Pseudo first-order reactions are really of higher order, but one reactant is present in large excess, so its concentration hardly changes. Examples: acid hydrolysis of ethyl acetate in excess water, and inversion of cane sugar.

Exam tips

• Example: t½ = 10 min gives k = 0.693 ÷ 10 = 0.0693 min⁻¹.
• For a zero-order reaction with [R]₀ = 0.2 M and k = 0.01 mol L⁻¹ s⁻¹, t½ = 0.2 ÷ 0.02 = 10 s.

Common mistakes

• Using the first-order half-life formula for a zero-order reaction.
• Saying that pseudo first-order reactions are truly first order in every reactant.