Kinetics

The reaction rate measures how quickly concentrations change over time. For a general reaction aA + bB → cC + dD, the rate can be expressed from any species:

rate = −(1/a)(Δ[A]/Δt) = −(1/b)(Δ[B]/Δt) = (1/c)(Δ[C]/Δt) = (1/d)(Δ[D]/Δt)

The negative signs for reactants reflect decreasing concentrations; stoichiometric coefficients ensure the rate is the same regardless of which species is monitored. Rates are always positive and have units of M/s (mol·L⁻¹·s⁻¹).

The instantaneous rate at a given time is the slope of the tangent to a concentration-vs.-time curve at that point. The initial rate — measured at t ≈ 0 before products accumulate — is especially useful because it avoids complications from reverse reactions and is the basis of the method of initial rates.

The rate law expresses the rate as a function of reactant concentrations: rate = k[A]^m[B]ⁿ, where k is the rate constant and the exponents m and n are reaction orders. The overall order equals m + n.

Reaction orders are found experimentally, not from the balanced equation. The method of initial rates compares experiments in which one concentration changes while others stay constant:

• If doubling [A] doubles the rate → first order in A (m = 1).
• If doubling [A] quadruples the rate → second order (m = 2).
• If the rate is unchanged → zero order (m = 0).

Once all orders are known, substitute any trial’s data into the rate law to solve for k. Pay close attention to the units of k — they depend on the overall order (e.g., M⁻¹s⁻¹ for second order overall).

Integrated rate laws relate concentration to time, letting you calculate how much reactant remains or how long to reach a target concentration:

For first-order reactions the half-life is constant — independent of concentration. This is characteristic of radioactive decay and many drug-elimination processes. For zero and second order, half-life depends on [A]₀. To determine order graphically, plot data in all three forms; the one yielding a straight line reveals the order.

The rate constant k increases with temperature. The Arrhenius equation quantifies this:

k = A · e^−E_a/RT

• A = frequency (pre-exponential) factor — accounts for collision frequency and molecular orientation
• E_a = activation energy (J/mol)
• R = 8.314 J/(mol·K); T in kelvins

The linear form, ln k = −E_a/R · (1/T) + ln A, gives a straight line when ln k is plotted against 1/T with slope = −E_a/R.

The two-point form ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂) lets you calculate E_a from rate constants at two temperatures or predict k at a new temperature when E_a is known. A useful rule of thumb: for many reactions near room temperature, a 10 °C rise roughly doubles the rate.

The collision model states that for a reaction to occur, molecules must: (1) collide, (2) with sufficient kinetic energy (≥ E_a), and (3) with proper orientation. Only collisions meeting all three criteria — effective collisions — lead to product formation.

An energy diagram (reaction-coordinate diagram) plots potential energy vs. reaction progress:

• The peak is the transition state (activated complex) — a fleeting, high-energy arrangement of atoms that cannot be isolated.
• The height from reactants to the peak is the activation energy (E_a).
• The difference between reactant and product energy levels is ΔH. If products are lower, the reaction is exothermic; if higher, endothermic.

Multi-step mechanisms show multiple peaks on the diagram — one per elementary step — with valleys between them representing reaction intermediates.

A catalyst increases a reaction rate by providing an alternative pathway with a lower activation energy. It participates in intermediate steps but is regenerated and not consumed overall. On an energy diagram, the catalysed path has a lower peak than the uncatalysed one.

Two main categories exist:

• Homogeneous catalysts — present in the same phase as the reactants (e.g., H⁺ catalysing ester hydrolysis in solution).
• Heterogeneous catalysts — in a different phase, typically a solid surface on which gaseous or dissolved reactants adsorb, react, and desorb (e.g., the iron catalyst in the Haber–Bosch process, catalytic converters in vehicles).

Biological catalysts (enzymes) are highly specific proteins that lower E_a dramatically. Crucially, catalysts do not change ΔH or shift the equilibrium position — they only affect how fast equilibrium is reached.

A reaction mechanism proposes a sequence of elementary steps whose sum gives the overall balanced equation. Each elementary step’s rate law can be written directly from its molecularity:

• Unimolecular: A → products, rate = k[A]
• Bimolecular: A + B → products, rate = k[A][B]

The rate-determining step (RDS) is the slowest step — it acts as a bottleneck. The overall rate law matches the RDS’s rate law, after substituting for any intermediates using equilibrium assumptions from faster preceding steps.

A valid mechanism must satisfy two tests: (1) the elementary steps sum to the overall equation, and (2) the predicted rate law matches the experimentally observed one. Intermediates are produced in one step and consumed in a later step — they never appear in the overall equation and should not appear in the final rate law.

Select the right kinetics tool based on what you need to find:

1. Finding the rate law? Use the method of initial rates: compare experiments where only one concentration changes at a time.
2. Finding concentration at a later time? Use the integrated rate law for the correct order (zero, first, or second).
3. Finding half-life? Use the half-life expression for the correct order. Note: only first-order half-life is independent of concentration.
4. Finding activation energy? Use the Arrhenius equation with data at two temperatures.
5. Connecting mechanism to rate law? The rate law is determined by the slow (rate-determining) step. The overall rate law must be consistent with the proposed mechanism.

Common mistakes: assuming reaction order from stoichiometric coefficients (order must be determined experimentally), using the wrong integrated rate law for the reaction order, confusing rate with rate constant (k changes with temperature; rate depends on both k and concentrations), and forgetting that a catalyst lowers activation energy but does not change ΔH or the equilibrium constant.