The formula
- General rate law
rate = k[A]^m[B]^n- Overall reaction order
overall order = m + n- Method of initial rates, comparing two runs
rate₂ / rate₁ = ([A]₂ / [A]₁)^m when [B] is held fixed- Units of the rate constant
k has units M^(1 - overall order)·s⁻¹
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
rate | Rate of reaction, as change in concentration per unit time; a numeric subscript names the experimental run | M/s |
k | Rate constant, fixed at a given temperature | depends on the overall order |
[A], [B] | Molar concentrations of the reactants; a numeric subscript names the experimental run, as in [A]₂ | M |
m | Reaction order with respect to A, found experimentally | dimensionless |
n | Reaction order with respect to B, found experimentally | dimensionless |
When it applies
- Orders come from rate data, not from the balanced equation. A coefficient of 2 does not make a reactant second order.
- The method of initial rates needs two runs that differ in one concentration only, so that the ratio isolates a single order.
- The rate law describes the reaction at one temperature. Changing the temperature changes k, which is where the Arrhenius equation takes over.
- Orders are usually small whole numbers but can be zero, fractional or negative, and a zero-order reactant drops out of the rate law entirely.
Worked example
Problem. For A + B → products, three runs give: [A] 0.10 M, [B] 0.10 M, rate 2.0 × 10⁻³ M/s; [A] 0.20 M, [B] 0.10 M, rate 4.0 × 10⁻³ M/s; [A] 0.10 M, [B] 0.20 M, rate 8.0 × 10⁻³ M/s. Find the rate law and k.
- Compare runs 1 and 2, where only [A] changes. Doubling [A] doubles the rate, 4.0 × 10⁻³ / 2.0 × 10⁻³ = 2, and 2 = 2^m gives m = 1. The reaction is first order in A.
- Compare runs 1 and 3, where only [B] changes. Doubling [B] quadruples the rate, 8.0 × 10⁻³ / 2.0 × 10⁻³ = 4, and 4 = 2^n gives n = 2. The reaction is second order in B.
- Write the rate law: rate = k[A][B]². The overall order is 1 + 2 = 3.
- Solve for k from run 1: k = rate / ([A][B]²) = 2.0 × 10⁻³ / (0.10 × 0.10²) = 2.0 × 10⁻³ / 1.0 × 10⁻³ = 2.0.
- Fix the units from the overall order: for a third-order reaction k carries M⁻²·s⁻¹, so k = 2.0 M⁻²·s⁻¹.
Answer. rate = k[A][B]², third order overall, with k = 2.0 M⁻²·s⁻¹.
Common mistakes
- Reading the orders off the balanced equation. The orders are experimental, and for many reactions they do not match the coefficients at all.
- Comparing two runs in which more than one concentration changed, which leaves two unknowns in a single equation.
- Reporting k without units, or carrying over the s⁻¹ units of a first-order constant to a reaction of a different overall order.
- Treating k as a fixed property of the reaction. It is fixed only at one temperature, and it rises steeply as temperature goes up.
Related formulas
- Half-life formula:
t½ = ln 2 / k ≈ 0.693 / k - Arrhenius equation:
k = A·e^(-Ea / RT)