Rate of reaction, order and half-life

Chemistry · Reaction kinetics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The rate of reaction is the change in concentration of a reactant or product per unit time. The rate equation shows how the rate depends on concentration, for example rate = k[A]^a[B]^b. The constant k is the rate constant.

The order of reaction is the sum of the powers in the rate equation. It is found by experiment, not from the balanced equation, and it can be zero or a fraction.

The half-life is the time taken for half of the reactant to be used up. In a reaction with several steps, the slowest step is the rate-determining step, and only the species in it appear in the rate equation.

Rules to remember

Rate = change in concentration ÷ time taken; unit mol dm⁻³ s⁻¹.
Specific rate constant: the rate of the reaction when the concentration of each reactant is 1 mol dm⁻³.
Units of k: zero order mol dm⁻³ s⁻¹; first order s⁻¹; second order dm³ mol⁻¹ s⁻¹; third order dm⁶ mol⁻² s⁻¹.
Examples: zero order, photochemical H₂ + Cl₂; first order, decomposition of N₂O₅; second order, NO + O₃ → NO₂ + O₂; third order, 2NO + O₂ → 2NO₂.
Half-life for initial concentration a: first order t_½ = 0.693 ÷ k (independent of a); second order t_½ = 1 ÷ (k × a); in general t_½ is proportional to 1 ÷ aⁿ⁻¹.
Molecularity is the number of particles taking part in one step; it is always a whole number.
Measuring rate: physical methods (spectrometry, electrical conductivity, dilatometry, refractometry, optical rotation) or chemical methods (titration). Order is found by the half-life method or the method of large excess.

Common mistake

Students read the order from the coefficients of the balanced equation. Order comes only from experiment or from the slow step; the coefficients of the overall equation do not give it.

Worked example

A first-order reaction has a half-life of 10 minutes. Find its rate constant and the time for the concentration to fall from 0.80 to 0.10 mol dm⁻³.

k = 0.693 ÷ t_½ = 0.693 ÷ 10 = 0.0693 min⁻¹
Each half-life halves the concentration: 0.80 → 0.40 → 0.20 → 0.10, which is 3 half-lives.
Time = 3 × 10 = 30 minutes (the half-life stays the same because the reaction is first order).

Answer: k = 0.0693 min⁻¹; time taken = 30 minutes.

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