A rate law (or rate equation) expresses how the reaction rate depends on the concentrations of reactants (and sometimes products or catalysts). A reaction mechanism is the full sequence of molecular-level steps (elementary steps) by which reactants are converted into products. Experimentally determined rate laws often constrain which mechanisms are plausible.

4.1 Experimental Determination of Rate Laws. Consider the general overall reaction: a A + b B → products. The observed (overall) rate law usually takes the form: Rate = k [A]^m [B]^n, where k is the rate constant at the given temperature, m and n are the reaction orders with respect to A and B, respectively, and the overall order is m + n. Importantly, m and n are determined by experiment and are not necessarily equal to the stoichiometric coefficients a and b.

4.2 Reaction Order, Rate Constant, and Units. The order of reaction with respect to each reactant is simply the exponent in the rate law. The overall order is the sum of those exponents. Typical orders are zero, first, and second: - Zero-order (overall order = 0): Rate = k. Units of k: concentration/time (e.g., M·s⁻¹). - First-order (overall order = 1): Rate = k [A]. Units of k: s⁻¹. - Second-order (overall order = 2): Rate = k [A]^2 or Rate = k [A][B]. Units of k: M⁻¹·s⁻¹.

4.3 Common Rate Laws: Zero, First, and Second Order. 4.3.1 Zero-Order Reactions - Rate law: Rate = k. - Differential form: d[A]/dt = –k. - Integrated form: [A]_t = [A]_0 – k·t. - Half-life t₁₋₂ (time to reduce [A] to half of [A]_0): t₁₋₂ = [A]_0 / (2 k). Notice that t₁₋₂ depends on [A]_0. 4.3.2 First-Order Reactions - Rate law: Rate = k [A]. - Differential form: d[A]/dt = –k [A]. - Integrated form: ln([A]_t) = ln([A]_0) – k·t. - Half-life t₁₋₂ is independent of [A]_0: t₁₋₂ = (ln 2) / k ≈ 0.693 / k. 4.3.3 Second-Order Reactions - Two common scenarios yield second-order kinetics: 1. Two molecules of the same reactant: 2A → products, Rate = k [A]^2. 2. One molecule each of two different reactants: A + B → products, Rate = k [A][B].

4.4 Integrated Rate Equations and Half-Life. For convenience, here is a summary of integrated forms and half-life expressions: Order Rate Law Integrated Form Half-Life Graph to Test Zero Rate = k [A]_t = [A]_0 – k·t t₁₋₂ = [A]_0 / (2·k) Plot [A] vs. t (straight line) First Rate = k [A] ln([A]_t) = ln([A]_0) – k·t t₁₋₂ = 0.693 / k Plot ln([A]) vs. t (straight line) Second Rate = k [A]^2 1/[A]_t = 1/[A]_0 + k·t t₁₋₂ = 1 / (k [A]_0) Plot 1/[A] vs. t (straight line)

4.5 Molecularity and Elementary Steps. A reaction mechanism is a sequence of elementary steps, each representing a single-collision event (or a unimolecular rearrangement) that occurs without intermediates. Each elementary step has a well-defined molecularity: - Unimolecular: molecularity = 1. Example: A → products, Rate = k [A]. - Bimolecular: molecularity = 2. Examples: 2A → products, Rate = k [A]^2 or A + B → products, Rate = k [A][B]. - Termolecular: molecularity = 3. Example: A + B + C → products, Rate = k [A][B][C]. Termolecular steps are rare because the chance of three-molecule collisions is very low.

4.6 Rate-Determining Step and the Steady-State Approximation. In many multi-step mechanisms, one step—the rate-determining step (RDS)—is significantly slower than all the others. This 'bottleneck' step controls the overall reaction rate. If the RDS involves only reactants (no unstable intermediates), its rate law often matches the observed overall rate law directly. If the RDS involves an intermediate species I, the steady-state approximation is invoked: Steady-State Approximation Assume the concentration of any intermediate I remains very small and nearly constant during most of the reaction, so d[I]/dt ≈ 0.

4.7 Pre-Equilibrium Approximation. The pre-equilibrium approximation is a special case of the steady-state approach when an early step in the mechanism is so fast that it effectively reaches equilibrium before the slow (rate-determining) step occurs. One treats that early step as if it is at equilibrium, expresses the intermediate’s concentration in terms of reactant concentrations via the equilibrium constant, and substitutes into the rate law for the slow step.

4.8 Complex Mechanisms: Chain Reactions and Catalytic Cycles. 4.8.1 Chain Reactions (Radical Chains). In certain gas-phase or solution-phase reactions, highly reactive radicals serve as intermediates. A classic example is the free-radical chlorination of methane: 1. Initiation (radical formation): Cl₂ → 2 Cl· (by heat or light) 2. Propagation: Cl· + CH₄ → HCl + CH₃· CH₃· + Cl₂ → CH₃Cl + Cl· 3. Termination (radical recombination): Cl· + Cl· → Cl₂ CH₃· + Cl· → CH₃Cl CH₃· + CH₃· → C₂H₆. Because radical concentrations (e.g., [Cl·], [CH₃·]) are very low but nearly constant during the steady portion of the reaction, one applies the steady-state approximation to them. Doing so leads to an overall rate law of the form Rate ≈ k · [Cl₂] · [CH₄], in agreement with experimental observations over a certain range of conditions.