Chemical Kinetics

Rate of reaction and rate laws

For aA + bB → products, the average rate is −Δ[A]/(aΔt) = Δ[product]/Δt. The rate law is rate = k[A]^m[B]ⁿ; m and n are orders found experimentally, not from stoichiometry. Overall order is m + n. For 2 N₂O₅(g) → 4 NO₂(g) + O₂(g), the rate law is first order in N₂O₅ despite the coefficient of 2. Atkins Chapter 22 and FSc XI Chapter 11 list common units of k: zero order has k in M/s, first order in s⁻¹, second order in M⁻¹s⁻¹.

Integrated rate laws and half-life

Zero order: [A] = [A]₀ − kt; t_½ = [A]₀/(2k) (depends on starting concentration). First order: ln[A] = ln[A]₀ − kt; t_½ = 0.693/k (independent of starting concentration — diagnostic of first order). Second order: 1/[A] = 1/[A]₀ + kt; t_½ = 1/(k[A]₀). Radioactive decay obeys first-order kinetics — ¹⁴C has t_½ = 5730 years, used in archaeological dating. After three half-lives only (1/2)³ = 1/8 of the original remains.

The Arrhenius equation and activation energy

Rate constants rise sharply with temperature: k = A e^−E_a/RT, where A is the pre-exponential (collision frequency factor) and E_a is activation energy. Taking logarithms: ln k = ln A − E_a/RT. A plot of ln k vs 1/T has slope −E_a/R. A typical reaction with E_a = 50 kJ/mol roughly doubles in rate for a 10 K rise near room temperature — the rule of thumb students often quote without proof. The transition-state theory (Eyring) refines this picture: reactants pass through an activated complex of higher energy.

Catalysts and mechanism

Catalysts provide an alternative pathway with lower E_a; they are regenerated and do not appear in the overall stoichiometry. Homogeneous catalysts (acid catalysis of esterification) share the phase of reactants; heterogeneous (Pt for hydrogenation, V₂O₅ in the Contact process) operate on a surface. Enzymes are biological catalysts with rate enhancements of 10⁶–10¹². A multistep mechanism has elementary steps; the slowest is the rate-determining step (RDS), and its molecularity equals its order. This is why an experimentally observed rate law often differs from the overall stoichiometric coefficients.

Factors influencing rate — and worked numerical

Concentration: more reactant → more collisions. Surface area: powders react faster than lumps. Light: photochemical reactions (H₂ + Cl₂ chain). Temperature: dominant factor through k. Suppose a first-order reaction has k = 2.0×10⁻³ s⁻¹ at 25 °C. After how long is 75% of the reactant consumed? [A]/[A]₀ = 0.25, ln(0.25) = −kt, t = ln 4 / k = 1.386 / 2.0×10⁻³ = 693 s. Equivalently this is two half-lives, each 0.693/k ≈ 347 s. Always check by computing both ways — a habit Clayden recommends for organic mechanisms.