Rate of Reaction & Rate Law: Orders Explained

The rate of a reaction is how fast concentration changes: rate = −Δ[reactant]/Δt = +Δ[product]/Δt, in mol·L^−1·s^−1. The rate law is the experimental equation rate = k[A]^m[B]^n linking that speed to concentrations, where k is the rate constant and m, n are the orders with respect to each reactant.

Rate laws matter because balanced equations say nothing about speed or mechanism. Orders are found only by experiment — they reveal which molecules actually collide in the slow step, and they let industry size reactors and pharmacists set medicine expiry dates.

Key Concepts

• Average rate over an interval versus instantaneous rate (the tangent slope at one moment); rates usually fall as reactants are consumed.
• Order: the exponent in the rate law (m for A, n for B); the overall order is m + n. Orders are typically 0, 1, or 2 and need not be whole numbers.
• Molecularity (1, 2, 3 molecules in an elementary step) differs from order — order is experimental, molecularity is mechanistic.
• Rate constant k: its units depend on overall order — s^−1 for first order, L·mol^−1·s^−1 for second order. k changes only with temperature (and catalysts).

Formulas

• rate = k[A]^m[B]^n (differential rate law).
• First order integrated: ln[A] = ln[A]0 − kt; half-life t1/2 = 0.693/k (independent of starting concentration).
• Second order (one reactant): 1/[A] = 1/[A]0 + kt.

Worked Example

Initial-rate data for a reaction of A with B:

1. Exp 1: [A] = 0.10, [B] = 0.20 → rate = 4.0 × 10^−3 M/s.
2. Exp 2: [A] = 0.20, [B] = 0.20 → rate = 8.0 × 10^−3 M/s. Doubling A doubles the rate → m = 1.
3. Exp 3: [A] = 0.10, [B] = 0.40 → rate = 1.6 × 10^−2 M/s. Doubling B quadruples the rate → n = 2.
4. Rate law: rate = k[A][B]², overall order 3.
5. k = 4.0 × 10^−3 / (0.10 × 0.20²) = 4.0 × 10^−3 / 0.0040 = 1.0 L²·mol^−2·s^−1.

Common Misconceptions

• "Order equals the stoichiometric coefficients." Only for a single elementary step. For the overall reaction, orders come from experiment alone — the example above has order 3 with no 3 in any balanced equation.
• "k changes with concentration." k is constant at fixed temperature; concentration changes the rate, not k.
• "Reaction rates always slow down." Usually true, but autocatalytic reactions accelerate as a product catalyses the reaction.

Real-World Applications

• Pharmaceuticals: rate laws predict drug degradation and set expiry dates via accelerated stability testing.
• Chemical industry: the Haber and contact processes are engineered around measured rate laws for maximum throughput.
• Atmospheric chemistry: ozone-depletion models are built from elementary-step rate laws.

The rate law is the reaction's fingerprint: measure how the rate responds to each concentration, read off the orders, and you hold the key to the mechanism — and to controlling the reaction's speed in the real world.