Reaction Kinetics: Rate Laws, Activation Energy, and Catalysis

Thermodynamics Predicts Whether; Kinetics Determines When

Diamonds are thermodynamically unstable relative to graphite at room temperature and pressure — ΔG for the diamond-to-graphite conversion is negative. Yet diamonds persist for billions of years. The reason: kinetics. The activation energy barrier for the diamond-to-graphite conversion is so large at room temperature that the rate is effectively zero on any human timescale. Thermodynamics and kinetics are complementary frameworks — the first tells us where a reaction wants to go, the second tells us how long it takes to get there. Both are essential for understanding chemistry.

Rate Laws and the Method of Initial Rates

The rate of a chemical reaction is the change in concentration of a reactant or product per unit time. For a reaction aA + bB → products, the rate law takes the form:

rate = k[A]^m[B]^n

where k is the rate constant, m and n are the partial orders, and m+n is the overall order. Critically, m and n are determined experimentally — they are not necessarily equal to stoichiometric coefficients a and b. The method of initial rates uses experiments varying one concentration at a time to determine each exponent:

• If doubling [A] doubles the rate: first order in A (m = 1).
• If doubling [A] quadruples the rate: second order in A (m = 2).
• If doubling [A] has no effect: zero order in A (m = 0).

Rate constants (k) carry units that depend on overall order: first-order reactions have k in s−1; second-order in M−1s−1. The rate constant is temperature-dependent — determined by the Arrhenius equation.

Integrated Rate Laws and Half-Lives

First-order half-life independence from concentration has profound consequences: radioactive decay (always first-order), drug clearance, and many other natural processes behave identically regardless of initial amount. Doubling the dose of a first-order elimination drug does not halve the clearance time — it extends it by exactly one additional half-life. Pharmacokinetics depends on this fact.

The Arrhenius Equation

The Arrhenius equation quantifies the temperature dependence of rate constants:

k = Ae^(−Ea/RT)

where A is the pre-exponential frequency factor, Ea is the activation energy (J/mol), R is the gas constant (8.314 J/mol·K), and T is absolute temperature. Taking the natural log: ln k = ln A − Ea/RT. A plot of ln k versus 1/T yields a straight line with slope −Ea/R — the standard method for experimental activation energy determination.

The exponential relationship means that small temperature increases produce large rate increases. For many reactions near room temperature, the rate roughly doubles for every 10°C increase — corresponding to an Ea of approximately 50 kJ/mol. Combustion reactions (Ea ~ 200 kJ/mol) are far more temperature-sensitive — the difference between a cold engine that won't start and a hot one that runs smoothly is purely kinetic.

Collision Theory and Transition State Theory

Collision theory holds that reaction occurs when molecules collide with sufficient energy and proper orientation. The rate is therefore proportional to collision frequency (Z), the fraction of collisions with energy exceeding Ea (the Boltzmann factor e^(−Ea/RT)), and a steric factor (p) accounting for geometric requirements:

rate = pZe^(−Ea/RT)

Transition state theory (activated complex theory), developed by Eyring in 1935, provides a more sophisticated framework. Reactants pass through a high-energy transition state (activated complex, denoted [AB]‡) at the top of the energy barrier before forming products. The transition state is a saddle point on the potential energy surface — not a stable species, but a configuration that exists momentarily during the reaction. The Hammond postulate relates transition state structure to reactant and product stability: for exothermic reactions, the transition state resembles reactants (early TS); for endothermic reactions, it resembles products (late TS). This matters for predicting how structural changes affect reaction rates.

Catalysis: Homogeneous and Heterogeneous

A catalyst increases reaction rate by providing an alternative reaction pathway with lower activation energy. The catalyst is not consumed — it is regenerated in the catalytic cycle.

Enzyme Kinetics: Michaelis-Menten Model

Enzymes are biological catalysts — proteins that reduce activation energies of biochemical reactions by factors of 10^6 to 10^17. The Michaelis-Menten model (1913) describes enzyme kinetics through a two-step mechanism:

E + S ⇌ ES → E + P

where E is enzyme, S is substrate, ES is the enzyme-substrate complex, and P is product. The rate equation:

v = Vmax[S] / (Km + [S])

where Vmax is the maximum velocity (when all enzyme is saturated with substrate) and Km (Michaelis constant) is the substrate concentration at half-maximal velocity — a measure of enzyme-substrate affinity (lower Km = tighter binding).

• When [S] << Km: rate ≈ (Vmax/Km)[S] — first-order in substrate.
• When [S] >> Km: rate ≈ Vmax — zero-order (enzyme saturated).

Enzyme inhibitors are characterized by their effect on Km and Vmax: competitive inhibitors increase apparent Km without affecting Vmax; non-competitive inhibitors decrease Vmax without affecting Km; uncompetitive inhibitors decrease both. Many drugs are designed as enzyme inhibitors — statins (HMG-CoA reductase inhibitors), ACE inhibitors, HIV protease inhibitors — and their pharmacology is Michaelis-Menten kinetics applied to medicine.

Chain Reactions and Radical Mechanisms

Chain reactions involve the propagation of reactive intermediates (radicals, ions) through successive steps, each regenerating the intermediate. Combustion of hydrocarbons is the most consequential chain reaction in human civilization. The mechanism involves three stages:

• Initiation: Homolytic cleavage of a bond generates radicals (e.g., Br2 → 2Br• under UV light; HO• from peroxide decomposition in combustion).
• Propagation: Radicals react with stable molecules, consuming one radical and producing another, consuming fuel and producing heat: CH4 + HO• → CH3• + H2O; CH3• + O2 → CH3OO• (and subsequently...). Chain lengths can reach 10^4–10^6 steps before termination.
• Termination: Two radicals combine to form a stable molecule — removing chain carriers and ending the chain: 2HO• → H2O2.

Chain length — the average number of propagation steps per initiation event — determines reaction efficiency. Explosions occur when chain branching (one radical produces more than one new radical per step) causes runaway acceleration. Combustion engines, explosives, and polymer initiation all exploit radical chain kinetics with precise control. Understanding the mechanism enables controlling the rate.