Rate equation and order

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Rate of reaction is the change in concentration of a reactant or product per unit time, measured in mol dm⁻³ s⁻¹. For a reaction A + B → products, the rate is expressed by the rate equation: rate = k[A]ᵐ[B]ⁿ, where the exponents m and n are found by experiment, never from the stoichiometric equation. Order of reaction with respect to each reactant is the power in the rate law; the overall order is the sum of m + n. A first-order reaction has a constant half-life t₁/₂ = 0.693/k, independent of starting concentration. The Arrhenius equation, k = Ae^(-Ea/RT), shows that the rate constant rises with temperature because the fraction of molecules exceeding activation energy (Ea) grows. Catalysts speed up reactions by lowering Ea through an alternative pathway but are not consumed and do not shift equilibrium. Enzymes are biological catalysts that denature above ~40 °C or outside their optimum pH. UACE must-knows: the initial-rates method for finding orders, the half-life distinction between first and second order, sketching and labelling Boltzmann distribution curves with and without a catalyst, and the Arrhenius two-temperature equation for calculating Ea from rate constants at two temperatures.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definition and measurement of rate

Rate is defined as the decrease in concentration of a reactant or increase in concentration of a product per unit time: rate = −Δ[Reactant]/Δt = Δ[Product]/Δt Units are mol dm⁻³ s⁻¹ (or min⁻¹, h⁻¹). Two experimental approaches are tested at UACE: the initial rates method (varying initial concentrations and measuring the slope of [product] vs time, or timing a fixed change such as a given volume of gas), and the continuous monitoring method (following concentration by titration at intervals, colorimetry, conductivity, or gas volume).

Rate equation and order

For A + B → products, rate = k[A]ᵐ[B]ⁿ. The exponents are determined experimentally:

Order with respect to A	Effect of doubling [A] on rate
Zero	No change
First	Rate doubles
Second	Rate quadruples

Common patterns UACE sets: a reaction is first order overall if halving [A] doubles t₁/₂; it is second order in A if t₁/₂ is proportional to 1/[A]₀. Zero order rate depends only on k (e.g., surface-catalysed reactions at saturation).

Half-life

For a first-order reaction: t₁/₂ = 0.693/k (constant, independent of [A]₀). For a second-order reaction in a single reactant: t₁/₂ = 1/(k[A]₀). Half-life is the time taken for the concentration of a reactant to fall to half its original value.

Temperature, activation energy and the Arrhenius equation

The Arrhenius equation is k = Ae^(−Ea/RT), where A is the pre-exponential factor, Ea the activation energy in J mol⁻¹, R = 8.314 J K⁻¹ mol⁻¹, and T the absolute temperature. A useful linear form is ln k = ln A − Ea/RT, giving a straight line of slope −Ea/R when ln k is plotted against 1/T. A two-point version used at UACE is:

ln(k₂/k₁) = (Ea/R)(1/T₁ − 1/T₂)

A 10 K rise near room temperature typically doubles the rate of many reactions because a larger fraction of molecules then possesses energy ≥ Ea (visible as an increased area under the high-energy tail of the Maxwell–Boltzmann distribution).

Catalysis

A catalyst provides an alternative reaction pathway with a lower Ea, increasing the rate constant. It is regenerated in the reaction, is needed only in small amounts, and is specific. Catalysts do not alter the position of equilibrium or the value of ΔH — they only let it be reached faster. Homogeneous catalysis has reactants and catalyst in the same phase (e.g., NO(g) in the lead-chamber process for H₂SO₄). Heterogeneous catalysis involves different phases; reaction proceeds by adsorption of reactants onto active sites on the solid surface, reaction at the surface, then desorption of products. Enzyme catalysis is highly specific (lock-and-key / induced-fit), works under mild conditions of temperature and pH, and is destroyed by denaturation at high temperatures or extreme pH.

UACE question patterns

Sketch the Boltzmann distribution, label the activation energy with and without a catalyst, and shade the area representing molecules that react at a higher temperature.
Use data of initial rate vs concentration to deduce the order with respect to each reactant.
Calculate Ea from k at two temperatures using the Arrhenius two-temperature equation.
Distinguish between heterogeneous and homogeneous catalysis with named examples.
Explain why enzymes lose activity at temperatures above about 40 °C.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Collision theory and the activated complex

Reactant particles must (1) collide, (2) possess combined kinetic energy ≥ Ea, and (3) have the correct orientation. On collision, a transient, high-energy activated complex (transition state) forms at the peak of the energy profile; its decomposition to products releases energy. Catalysts stabilise this complex and lower the energy barrier but do not change ΔH of reaction or the equilibrium constant Kc.

Rate-determining step and mechanisms

In a multi-step mechanism, the slow (rate-determining) step governs the overall rate. The rate equation contains the concentrations of species appearing in this step (and those in rapid pre-equilibria that feed it). This is why experimental orders need not match stoichiometric coefficients — UACE tests the ability to spot this and to deduce which proposed elementary steps are consistent with a measured rate law.

Half-life in detail

First order: t₁/₂ = 0.693/k, constant at all concentrations. Useful for radioisotope dating (¹⁴C, K–Ar).
Second order (single reactant): t₁/₂ = 1/(k[A]₀), increases as reaction proceeds.
Zero order in a single reactant: t₁/₂ = [A]₀/(2k), also concentration-dependent.

A common UACE trap: assuming all reactions have t₁/₂ independent of concentration. Only first order behaves that way.

Heterogeneous catalysis — surface mechanism

Steps are: (i) diffusion of reactants to the surface, (ii) adsorption onto active sites (often physisorption then chemisorption), (iii) surface reaction between adsorbed species, (iv) desorption of products. The Langmuir–Hinshelwood picture has two adsorbed species reacting on the surface; in the Eley–Rideal mechanism a gas-phase species collides with an adsorbed one. Industrial examples tested: Fe in the Haber process, Pt/Rh in Ostwald’s process, V₂O₅ in the Contact process, Pt/Pd in catalytic converters. Catalyst poisoning by species such as sulfur or lead blocks active sites and reduces activity.

Common mistakes and exam traps

Writing orders from the balanced equation instead of from experimental data.
Forgetting that doubling the concentration of a zero-order reactant changes nothing.
Using °C in the Arrhenius equation instead of K.
Confusing the effect of a catalyst on Ea with its (non-)effect on ΔH and Kc.
Stating that catalysts shift the position of equilibrium — they do not; both forward and reverse rates are increased equally.
Treating enzyme denaturation as a reversible change — at high T or extreme pH, denaturation is irreversible.

Worked micro-example

The decomposition of N₂O₅ in CCl₄ is first order with k = 6.0 × 10⁻⁴ s⁻¹ at 320 K. Find the time for [N₂O₅] to fall to one-eighth of its initial value. t₁/₂ = 0.693/k = 0.693 / (6.0 × 10⁻⁴) = 1155 s. One-eighth = (½)³, so t = 3 × t₁/₂ = 3465 s ≈ 57.8 min.

Practice prompts

For the reaction 2A + B → products, the following initial rates were obtained: [A] = 0.10, [B] = 0.10 → rate = r; [A] = 0.20, [B] = 0.10 → rate = 2r; [A] = 0.10, [B] = 0.30 → rate = 3r. Determine the order in A, in B, the overall order, and the rate constant (with units) when [A] = [B] = 0.10 mol dm⁻³.
A reaction has k = 1.5 × 10⁻³ s⁻¹ at 300 K and k = 6.0 × 10⁻³ s⁻¹ at 320 K. Calculate Ea in kJ mol⁻¹ and comment on whether the value is consistent with a typical catalysed or uncatalysed reaction.

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Rate equation and order