Rate laws from mechanism up

Undergraduates learn to read a rate law off the coefficients of a balanced equation, then unlearn it a semester later when they meet reactions where the rate depends on a species that does not appear in the overall equation, or where the “second-order” reaction is really a preequilibrium hiding a hydride shift. The correct rule is: a rate law comes from the mechanism, and only in the rare case where the elementary step and the balanced equation coincide are the coefficients also the orders. This guide walks from elementary steps to integrated rate laws, names the diagnostic experiments that pin down each order, and points at the failure modes that break the whole framework.

Elementary steps versus overall reactions

An elementary step is a single molecular event: one bond forms, one breaks, one molecule collides with another. For an elementary step, the rate law follows the molecularity: unimolecular (A → products) is rate = k[A]; bimolecular (A + B → products) is rate = k[A][B]; termolecular is rate = k[A][B][C] but so rare that termolecular steps essentially never appear in practical mechanisms except involving M (a third-body collision partner) in gas-phase recombination.

An overall reaction is the sum of one or more elementary steps. Its rate law is derived from the mechanism, not read off. There is no rule that says “if the balanced equation is 2A + B → products, then rate = k[A]²[B].” That equation might have any rate law from k[A][B] to k[A][B]² or something more complicated involving an intermediate.

The way to see which is which: an elementary step is written with a single arrow and a mechanistic annotation. An overall equation is a stoichiometric summary with no mechanistic content.

The rate-determining step

If a mechanism has multiple steps of very different rates, the overall rate is set almost entirely by the slowest step — the rate-determining step (RDS). Every step after the RDS is fast enough to keep up regardless of concentration; every step before it may or may not appear in the rate law depending on whether it is a preequilibrium or an irreversible commit.

Three canonical mechanism patterns and their rate laws:

Pattern A: single-step bimolecular. $\text{A} + \text{B} \xrightarrow{k} \text{P}$ Rate = k[A][B]. Second order overall, first order in each. This is the trivial case where the balanced equation and the rate law coincide.

Pattern B: slow first step, fast second. $\text{A} \xrightarrow{k_1,\ \text{slow}} \text{I};\ \text{I} + \text{B} \xrightarrow{k_2,\ \text{fast}} \text{P}$ Rate = k₁[A]. The second step happens instantaneously once I is produced, so [B] does not appear in the rate law even though it is in the overall equation. This is a classic SN1: the slow step is ionization of the substrate; the fast step is capture by the nucleophile. Nucleophile concentration does not enter.

Pattern C: fast preequilibrium, slow second step. $\text{A} + \text{B} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{I};\ \text{I} \xrightarrow{k_2,\ \text{slow}} \text{P}$ Apply the preequilibrium approximation: [I] = (k₁/k₋₁) [A][B]. Rate = k₂[I] = (k₁k₂/k₋₁) [A][B]. This looks like a simple bimolecular rate law but the “effective rate constant” is a composite of three microscopic constants. Acid-catalyzed reactions often follow this pattern with H⁺ acting as A or B.

The steady-state approximation

When you cannot easily identify which step is rate-determining, invoke the steady-state approximation (SSA): the concentration of a highly reactive intermediate stays approximately constant because it is consumed as fast as it is produced. Setting d[I]/dt = 0 gives an algebraic expression for [I] in terms of stable species, which you then substitute into the rate law for product formation.

The classic worked case is the Lindemann-Hinshelwood unimolecular decomposition:

$\text{A} + \text{M} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{A}^* + \text{M};\ \text{A}^* \xrightarrow{k_2} \text{P}$

SSA on the vibrationally excited A*: d[A*]/dt = k₁[A][M] − k₋₁[A*][M] − k₂[A*] = 0. Solving: [A*] = k₁[A][M] / (k₋₁[M] + k₂). Rate of product = k₂[A*] = k₁k₂[A][M] / (k₋₁[M] + k₂). Two limits: at high [M], the [M] terms dominate and rate → (k₁k₂/k₋₁)[A] — first-order. At low [M], rate → k₁[A][M] — second-order. This gives the fall-off curve seen in gas-phase unimolecular decompositions. McQuarrie §28 has the full derivation.

Integrated rate laws

For a zero-order reaction, rate = k (independent of concentration). Integrating: [A]_t = [A]_0 − kt. Half-life t_1/2 = [A]_0 / (2k). Half-life depends on initial concentration — a diagnostic. Zero-order reactions are common when a catalyst or surface is saturated (a fixed-throughput enzyme in its Vmax plateau; heterogeneous catalysis with substrate coverage 1).

For first order, rate = k[A]. Integrating: ln[A]_t = ln[A]_0 − kt, or [A]_t = [A]_0 e^(−kt). Half-life t_1/2 = ln(2)/k — independent of initial concentration. That constancy is the tell for first-order kinetics. Radioactive decay is the canonical example (see the half-life calculator).

For second order in a single species (2A → products), rate = k[A]². Integrating: 1/[A]_t = 1/[A]_0 + kt. Half-life t_1/2 = 1/(k[A]_0) — inversely proportional to initial concentration. Doubling [A]_0 halves the half-life. Bimolecular dimerization reactions follow this shape.

For a mixed second order (A + B → products with [A]_0 ≠ [B]_0), integrating is uglier: the closed form is (1/([B]_0 − [A]_0)) · ln([B]_0[A]_t / ([A]_0[B]_t)) = kt. In practice, run under pseudo-first-order conditions: put one reagent in massive excess so its concentration is effectively constant, and observe first-order kinetics in the limiting reagent. The observed rate constant is k_obs = k[B]_0. Repeat at several [B]_0 to extract k. This trick is worth memorizing.

Diagnostic plots

Every order has a signature linear plot:

• Zero order: [A] vs t is linear, slope = −k.
• First order: ln[A] vs t is linear, slope = −k.
• Second order: 1/[A] vs t is linear, slope = +k.

Run all three transformations on the same data set and whichever plot is linear tells you the order. Atkins §20 emphasizes this as the diagnostic move for kinetics data.

Worked example — hydrolysis of methyl acetate

Arrhenius: how rate depends on temperature

For most elementary steps, the rate constant depends on temperature as:

$k = A \exp(-E_a / RT)$

with A the pre-exponential factor (units of the rate constant) and E_a the activation energy. Rearrange:

$\ln k = \ln A - E_a / (RT)$

A plot of ln k vs 1/T is linear with slope −E_a/R and intercept ln A. From two measurements:

$\ln(k_2/k_1) = -\frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

Rule of thumb (Laidler Ch. 3): for E_a ≈ 50 kJ/mol near room temperature, a 10 K temperature rise roughly doubles the rate constant. This is not a universal law, only a consequence of the exponential — for very small or very large E_a the factor per 10 K is different.

CODATA 2018 gives R = 8.314 462 618 J K⁻¹ mol⁻¹, so E_a computed from an Arrhenius plot is reliable to four sig figs if the temperatures and rate constants are precise.

Where the framework breaks

• Complex mechanisms with several intermediates at similar rates. Neither RDS nor SSA gives a clean rate law; you need the full system of differential equations. Modern kinetic modeling software (Chemkin, Cantera) handles this.
• Chain reactions. Radical polymerization has initiation, propagation, and termination running simultaneously. The overall rate law is a compound expression involving all three step rates, and dimensional analysis rules out any single-elementary-step interpretation.
• Diffusion-controlled reactions. When the intrinsic bimolecular rate constant is comparable to the diffusion-limited rate (~10¹⁰ M⁻¹s⁻¹ in water at 25 C), the observed rate is limited by encounter rate, not by chemistry. Rate constants become independent of activation energy in this regime.
• Autocatalysis. The product is also a catalyst. The rate law contains [product], and the reaction accelerates over time before slowing as reagents deplete — an S-shaped concentration-vs-time curve. Belousov-Zhabotinsky is the classic case.

Practice problems

Where to go next

• The half-life calculator handles first-order kinetics and radioactive decay directly; it is a good sanity check when working problems.
• The next layer up is transition-state theory, which replaces Arrhenius’s empirical A and E_a with the Eyring equation involving ΔH‡ and ΔS‡. Atkins §21 has the transition to that framework.
• The sig figs cheatsheet matters more for kinetics than for most areas — activation energies computed from two rate measurements are extremely sensitive to the precision of both, and small errors magnify through the exponential.

The overall discipline: never write a rate law without a mechanism, never trust a mechanism without a diagnostic plot, and always specify the temperature. Everything else is bookkeeping.