Chemical Kinetics

Rate Laws and Reaction Rates

The rate of a chemical reaction measures how quickly reactants are consumed or products are formed, typically expressed in units of molarity per second (M/s). For a generic reaction aA + bB -> cC + dD, the rate can be written in terms of any species: Rate = -(1/a)(Delta[A]/Delta t) = -(1/b)(Delta[B]/Delta t) = +(1/c)(Delta[C]/Delta t) = +(1/d)(Delta[D]/Delta t). The negative sign for reactants reflects their decreasing concentration; the stoichiometric coefficient in the denominator normalizes different species to the same numerical rate. The experimentally determined rate law expresses how the rate depends on reactant concentrations: Rate = k[A]^m[B]^n, where k is the rate constant, and m and n are the reaction orders with respect to A and B. Crucially, m and n are NOT the stoichiometric coefficients a and b -- they are determined experimentally and may be integers, fractions, or even zero. The overall reaction order is the sum m + n. The rate constant k has units that depend on the overall order: for zero-order, k has units of M/s; for first-order, 1/s; for second-order, 1/(Ms); and in general, M^(1-n)s^(-1) for nth-order.

Determining Reaction Order Experimentally

Reaction order is determined by the method of initial rates: measure the initial rate of reaction for several trials where one reactant concentration is varied while all others are held constant. If doubling [A] doubles the rate, the reaction is first-order in A (m = 1). If doubling [A] quadruples the rate, it is second-order in A (m = 2). If changing [A] has no effect on the rate, it is zero-order in A (m = 0). The MCAT frequently provides a table of initial-rate data and asks you to deduce m and n by comparing trials. Key pattern: rate2/rate1 = ([A]2/[A]1)^m * ([B]2/[B]1)^n -- isolate one variable at a time by choosing trials where only one concentration changes. Once the order is known, k can be calculated by plugging any trial's data into the rate law. The rate-determining step (slowest step in a mechanism) dictates the overall rate law: the rate law reflects the molecularity of the slow step and the concentrations of species appearing in or before it.

Integrated Rate Laws

Integrated rate laws express concentration as a function of time, allowing you to calculate how much reactant remains after a given interval or how long a reaction takes to reach a target concentration. Zero-order: [A]t = [A]0 - kt. A plot of [A] versus time yields a straight line with slope = -k. Zero-order kinetics occur when a catalyst or enzyme is saturated -- the rate is independent of reactant concentration because the catalyst is the bottleneck. First-order: ln[A]t = ln[A]0 - kt, or equivalently [A]t = [A]0*e^(-kt). A plot of ln[A] versus time gives a straight line with slope = -k. Radioactive decay, drug elimination, and many unimolecular processes follow first-order kinetics. Second-order (single reactant): 1/[A]t = 1/[A]0 + kt. A plot of 1/[A] versus time is linear with slope = +k. For second-order reactions involving two different reactants with equal initial concentrations or A + A -> products, the same form applies. Recognizing which plot yields a straight line is the most direct MCAT method for identifying reaction order from experimental data.

Half-Life

Half-life (t1/2) is the time required for the reactant concentration to decrease to half its initial value. The dependence of half-life on initial concentration is a diagnostic for reaction order. Zero-order: t1/2 = [A]0/(2k) -- half-life decreases as the reaction proceeds because it is proportional to the initial concentration. First-order: t1/2 = ln(2)/k = 0.693/k -- half-life is CONSTANT, independent of initial concentration. This is the hallmark of first-order kinetics: every successive half-life period reduces the concentration by another factor of two. Second-order: t1/2 = 1/(k[A]0) -- half-life INCREASES as the reaction proceeds because it is inversely proportional to initial concentration; each successive half-life is twice as long as the previous one. The MCAT commonly tests: (1) calculating half-life from k, (2) determining how many half-lives are needed to reach a specified fraction of [A]0 (e.g., three half-lives leave 1/8 = 12.5%), and (3) distinguishing reaction orders by comparing how t1/2 changes as [A]0 varies.

Activation Energy and the Arrhenius Equation

Not every molecular collision produces a reaction -- only those with sufficient energy and proper orientation can overcome the activation-energy barrier (Ea). The Arrhenius equation quantifies the temperature dependence of the rate constant: k = Ae^(-Ea/RT), where A is the frequency factor (related to collision frequency and orientation probability), Ea is the activation energy (J/mol), R = 8.314 J/(molK), and T is absolute temperature in Kelvin. 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, allowing experimental determination of Ea. The two-point form is especially MCAT-relevant: ln(k2/k1) = -(Ea/R)(1/T2 - 1/T1), which lets you calculate the new rate constant when temperature changes, or solve for Ea given two rate constants at two temperatures. Key insight: a small increase in temperature produces a disproportionately large increase in rate because Ea appears in the exponent. A reaction with a high Ea is very temperature-sensitive; a reaction with a low Ea is less affected by temperature changes. Transition-state theory extends this picture: reactants must pass through a high-energy transition state (activated complex), and Ea is the energy difference between reactants and this transition state.

Catalysts

A catalyst increases the rate of a reaction without being consumed in the process. It achieves this by providing an alternative reaction pathway with a lower activation energy (Ea). Critically, a catalyst does NOT change the thermodynamics of the reaction: Delta G, Delta H, the equilibrium constant (Keq), and the equilibrium position all remain unchanged. The catalyst lowers Ea for BOTH the forward and reverse reactions equally, so it accelerates the approach to equilibrium but does not shift where equilibrium lies. On a reaction-coordinate diagram, a catalyzed pathway shows a lower energy peak (lower Ea) compared to the uncatalyzed pathway, while the energies of reactants and products remain identical. Homogeneous catalysts are in the same phase as the reactants (e.g., an aqueous acid catalyzing an aqueous reaction). Heterogeneous catalysts are in a different phase (e.g., a solid platinum surface catalyzing gas-phase hydrogenation). Enzymes are biological catalysts -- typically proteins -- that operate with extraordinary specificity and efficiency, often achieving rate enhancements of 10^6 to 10^12. The MCAT expects you to recognize that adding a catalyst increases k (the rate constant), which increases the rate for a given set of concentrations, but does not affect the equilibrium constant, the concentrations at equilibrium, or Delta G.

Reaction Mechanisms

A reaction mechanism is the step-by-step sequence of elementary reactions by which an overall chemical change occurs. Each elementary step has a molecularity -- the number of reactant molecules involved: unimolecular (one molecule), bimolecular (two molecules colliding), or termolecular (three, which is rare). The rate law of an elementary step can be written directly from its stoichiometry: for A -> products, rate = k[A]; for A + B -> products, rate = k[A][B]; for 2A -> products, rate = k[A]^2. The overall rate law is determined by the rate-determining step (RDS) -- the slowest step in the mechanism, which acts as a kinetic bottleneck. Species that appear in the mechanism but not in the overall balanced equation are intermediates: they are produced in one step and consumed in a subsequent step. Intermediates do not appear in the overall rate law, but they can appear in the rate law derived from the RDS -- in that case, the steady-state approximation or pre-equilibrium approximation is used to express the intermediate's concentration in terms of reactants. A proposed mechanism must satisfy two criteria: (1) the elementary steps must sum to the overall balanced equation, and (2) the mechanism must predict a rate law consistent with experimental data. The MCAT often presents a proposed mechanism and asks you to identify the rate-determining step, recognize intermediates, verify that the steps sum correctly, or derive the rate law from the slow step.

How it works

Chemical kinetics operates on three layers. First, the rate law -- determined experimentally by the method of initial rates -- tells you the reaction order and the rate constant k. Second, the integrated rate law converts this into a time-dependent picture: zero-order gives linear [A] vs. t, first-order gives linear ln[A] vs. t, and second-order gives linear 1/[A] vs. t. The half-life pattern then confirms the order: constant for first-order, proportional to [A]0 for zero-order, and inversely proportional to [A]0 for second-order. Third, the Arrhenius equation connects k to temperature via Ea: a higher Ea means a steeper temperature dependence. Catalysts add a new mechanistic pathway with lower Ea -- they don't change thermodynamics, only kinetics. On the MCAT, every kinetics problem reduces to (a) identifying order from data or graphs, (b) applying the correct integrated rate law or half-life formula, or (c) reasoning about how Ea, T, and catalysts affect k and rate.

How it works

Chemical kinetics operates on three layers. First, the rate law -- determined experimentally by the method of initial rates -- tells you the reaction order and the rate constant k. Second, the integrated rate law converts this into a time-dependent picture: zero-order gives linear [A] vs. t, first-order gives linear ln[A] vs. t, and second-order gives linear 1/[A] vs. t. The half-life pattern then confirms the order: constant for first-order, proportional to [A]0 for zero-order, and inversely proportional to [A]0 for second-order. Third, the Arrhenius equation connects k to temperature via Ea: a higher Ea means a steeper temperature dependence. Catalysts add a new mechanistic pathway with lower Ea -- they don't change thermodynamics, only kinetics. On the MCAT, every kinetics problem reduces to (a) identifying order from data or graphs, (b) applying the correct integrated rate law or half-life formula, or (c) reasoning about how Ea, T, and catalysts affect k and rate.

Comparisons

• C/P (Rate laws): Determining reaction order from initial-rate data tables; calculating k from rate data; deducing units of k from overall order.
• C/P (Integrated rate laws): Identifying reaction order from linear plots ([A] vs. t, ln[A] vs. t, 1/[A] vs. t); calculating concentration at a given time using integrated equations.
• C/P (Half-life): Calculating t1/2 for zero-, first-, and second-order reactions; using half-life patterns to identify reaction order; relating k to half-life for first-order processes.
• C/P (Arrhenius): Two-point Arrhenius calculations; predicting the effect of temperature on rate; interpreting ln(k) vs. 1/T plots to extract Ea.
• C/P (Catalysts): Distinguishing kinetic from thermodynamic effects; interpreting reaction-coordinate diagrams with and without catalysts; recognizing that catalysts do not change Delta G, Delta H, or Keq.
• B/B (Enzyme kinetics): Michaelis-Menten kinetics is chemical kinetics applied to enzymes -- zero-order at saturating [S] (Vmax), first-order at low [S]; competitive inhibition mirrors increased apparent Km.
• B/B (Pharmacology): Drug elimination kinetics: first-order elimination (constant fraction per time) vs. zero-order elimination (constant amount per time, e.g., ethanol, phenytoin, aspirin at high doses).

Common confusions

• Assuming reaction order equals stoichiometric coefficients: m and n in rate = k[A]^m[B]^n are determined EXPERIMENTALLY, not from the balanced equation. The MCAT will give you a stoichiometric equation and a separate data table -- use the data, not the coefficients.
• Confusing zero-order half-life direction: t1/2 = [A]0/(2k) -- half-life DECREASES as [A]0 decreases. Each successive half-life is shorter, not longer. Second-order is the one where each half-life DOUBLES.
• Forgetting that catalyst does NOT change thermodynamics: a catalyst lowers Ea but does not affect Delta G, Delta H, Keq, or the equilibrium position. It only affects how fast equilibrium is reached.
• Using the wrong integrated rate law plot: zero-order = [A] vs. t (linear); first-order = ln[A] vs. t (linear); second-order = 1/[A] vs. t (linear). Mix these up and you'll misidentify the reaction order every time.
• Arrhenius unit errors: Ea must be in J/mol when using R = 8.314 J/(mol*K). The MCAT may provide Ea in kJ/mol -- convert to J/mol before plugging in. Also, T must be in Kelvin.
• Misidentifying intermediates vs. catalysts: an intermediate is produced then consumed (appears in mechanism, not in overall equation); a catalyst is consumed then regenerated (appears in mechanism, canceled in overall equation). Both are absent from the overall balanced equation.
• Forgetting that the rate-determining step controls the rate law: if the slow step involves an intermediate, you must use the steady-state or pre-equilibrium approximation to express [intermediate] in terms of reactants.
• Temperature and rate: a 10-degree-C increase roughly doubles the rate for many reactions (Q10 ~ 2), but this is an empirical rule of thumb -- the actual factor depends on Ea. Know how to calculate with Arrhenius, don't just memorize the rule.

Quick review

• Rate law: Rate = k[A]^m[B]^n; m, n determined experimentally, NOT from stoichiometry.
• Order from units of k: zero-order -> M/s; first-order -> 1/s; second-order -> 1/(M*s); nth-order -> M^(1-n)/s.
• Zero-order: [A] = [A]0 - kt; linear [A] vs. t; t1/2 = [A]0/(2k); half-life decreases with time.
• First-order: ln[A] = ln[A]0 - kt; linear ln[A] vs. t; t1/2 = ln(2)/k = 0.693/k; constant half-life.
• Second-order: 1/[A] = 1/[A]0 + kt; linear 1/[A] vs. t; t1/2 = 1/(k[A]0); half-life increases with time.
• Arrhenius: k = A*e^(-Ea/RT); two-point: ln(k2/k1) = -(Ea/R)(1/T2 - 1/T1); Ea in J/mol, R = 8.314, T in K.
• Catalyst: lowers Ea, increases k, does NOT change Delta G, Delta H, Keq, or equilibrium position.
• Reaction mechanism: elementary steps sum to overall equation; rate-determining step (slowest) controls rate law.
• Intermediates: produced then consumed, appear in mechanism but NOT in overall equation.
• Method of initial rates: compare trials where only one [reactant] changes; rate2/rate1 = ([conc]2/[conc]1)^order.
• Arrhenius plot: ln(k) vs. 1/T -> slope = -Ea/R, y-intercept = ln(A).
• Transition state: highest-energy species along reaction coordinate; Ea = E(transition state) - E(reactants).