12.4 Reaction Kinetics: Rate Laws, Arrhenius & Half-Life

Thermodynamics (Sections 12.1–12.3) tells you whether a reaction is favorable — it says nothing about how fast that reaction happens. Chemical kinetics is the study of reaction rates and the molecular-level factors that control them. A reaction can be enormously favorable (a very negative ΔG°) and still proceed at an imperceptible rate at room temperature — diamond converting to graphite is thermodynamically favorable but kinetically frozen. The MCAT tests kinetics as a distinct axis from thermodynamics, and mixing the two together is one of the most common conceptual errors on Content Category 5E. Enzyme kinetics (Chapter 11) is a specialized application of the same ideas; this section covers the general chemistry foundation: rate laws, Arrhenius temperature dependence, half-life, catalysts, and energy profiles.

Reaction Rate

Reaction rate is the change in concentration of a reactant or product per unit time, typically expressed in M/s. For a general reaction aA + bB → cC + dD, the rate can be written in terms of any species in the equation, but each term must be divided by its own stoichiometric coefficient so that every expression yields the same numerical rate:

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

Reactant concentrations decrease over time, so a negative sign converts that decrease into a positive rate value; product concentrations increase, so their terms are already positive. Initial rate methods measure rate near t = 0 before product accumulates and reverse reaction becomes significant — the usual experimental design in MCAT rate-law problems.

Rate Laws, Rate Constants, and Reaction Order

The rate law relates reaction rate to reactant concentrations: rate = k[A]^m[B]^n, where k is the rate constant (a value that depends on temperature and on the presence of a catalyst, but never on concentration) and the exponents m and n are the reaction order with respect to each reactant. The overall reaction order is the sum m + n.

The single highest-yield trap in this entire topic: reaction orders must be determined experimentally and cannot be read directly off the balanced equation, unless the step in question is a single elementary reaction. A coefficient of 2 in the balanced equation does not automatically mean the reaction is second order in that species. For an elementary step, molecularity equals order (unimolecular → first order; bimolecular → second order overall), but multi-step mechanisms can produce fractional or unexpected orders.

Worked Example: Determining Order from Rate Data

Consider 2 NO(g) + Br2(g) → 2 NOBr(g), studied at constant temperature:

Compare Trials 1 and 2: [NO] doubles while [Br2] stays constant, and the rate quadruples (12 → 48). Doubling a concentration and getting a 4× (2²) increase in rate means the reaction is second order in NO.

Compare Trials 1 and 3: [Br2] doubles while [NO] stays constant, and the rate simply doubles (12 → 24). That is first order in Br2.

So the rate law is rate = k[NO]²[Br2], and the reaction is third order overall (2 + 1). Solving for k using Trial 1: k = 12 ÷ (0.10² × 0.10) = 12 ÷ 0.001 = 12,000 M⁻²s⁻¹. Notice that the units of k shift with the overall reaction order — a pattern the MCAT sometimes exploits by asking you to back out the order purely from the units given for a rate constant.

Zero-order reactions have rates independent of reactant concentration (rate = k): common when a catalyst or enzyme is saturated, so increasing substrate further does not increase rate until saturation is relieved.

Rate-Determining Step

Many reactions proceed through a multi-step mechanism rather than a single collision. The rate-determining step (RDS) is the slowest step in that mechanism, and it acts as a bottleneck: the overall reaction can never proceed faster than its slowest step, no matter how quickly the remaining steps occur. The rate law predicted by the RDS's molecularity must match the rate law determined experimentally — if a proposed mechanism's slow step predicts a rate law that doesn't match the data, that mechanism must be wrong or incomplete. This is a favorite MCAT passage setup: present a multi-step mechanism and ask which step's rate law is consistent with a given experimental rate law.

Intermediates appear in elementary steps but cancel out of the overall balanced equation; they do not appear in the final rate law if expressed only in terms of reactants (after applying steady-state or pre-equilibrium approximations if needed). Catalysts appear on both sides of the overall reaction net or regenerate after use.

Half-Life and Integrated Rate Laws

Half-life (t1/2) is the time required for reactant concentration to fall to half its initial value. How half-life depends on concentration is a high-yield discriminator among reaction orders:

First-order half-life is independent of starting concentration — after one half-life, half remains; after two, one-quarter remains; after n half-lives, (1/2)^n remains. This is the pattern for radioactive decay, many drug elimination processes approximated as first-order, and many unimolecular elementary steps. Zero-order half-life shortens as concentration falls (because t1/2 is proportional to [A]0); second-order half-life lengthens as concentration falls (t1/2 proportional to 1/[A]0).

Worked example (first-order half-life): A radiotracer used in imaging has a first-order half-life of 6 hours. What fraction remains after 18 hours? 18 h / 6 h = 3 half-lives, so fraction remaining = (1/2)³ = 1/8. The rate constant is k = 0.693 / t1/2 ≈ 0.693/6 h⁻¹ ≈ 0.116 h⁻¹ if a passage asks for k rather than remaining fraction.

Biological framing: plasma half-life of a drug (often treated as first-order elimination) and radioactive half-life of isotopes used in PET or nuclear medicine share the same math, even though the microscopic processes differ.

Temperature Dependence and the Arrhenius Equation

Raising temperature increases reaction rate because more molecules collide with enough energy to react — not because reactant concentrations increase. The energy threshold a collision must clear is the activation energy (Ea), the energy barrier separating reactants from products. The Arrhenius equation quantifies how the rate constant, temperature, and activation energy relate:

k = Ae^(−Ea/RT)

where A is the pre-exponential (frequency) factor tied to collision frequency and orientation, R is the gas constant, and T is absolute temperature. Two qualitative relationships matter most for the MCAT: k increases as T increases, because more molecules exceed Ea at higher temperature, and k decreases as Ea increases, because fewer molecules clear a higher barrier at a given temperature. A catalyst works by lowering Ea, which, per the Arrhenius equation, increases k without requiring any change in temperature at all.

A common rule of thumb (not exact) is that many reactions roughly double in rate for a 10°C temperature rise near room temperature — a useful mental check, not a law. In medicine, elevated body temperature (fever) can modestly increase rates of enzymatic reactions, while hypothermia slows metabolism; both are Arrhenius-type effects on biological rate constants.

Taking the natural log of Arrhenius gives a linear form used in experiments:

ln k = ln A − (Ea/R)(1/T)

A plot of ln k versus 1/T is a straight line with slope −Ea/R; passages may ask you to extract Ea from such a graph without solving the exponential form by hand.

Interpreting Energy Profile Diagrams

An energy (reaction coordinate) diagram plots potential energy on the y-axis against reaction progress on the x-axis. Four features recur on every MCAT version of this diagram:

• Reactants and products sit at the energy levels at the start and end of the curve, reflecting their relative stability.
• The activated complex, or transition state, sits at the very peak of the curve — the highest-energy, most unstable arrangement along the pathway, too short-lived to ever be isolated.
• Ea (forward) is the energy gap climbing from reactants up to the peak; Ea (reverse) is the energy gap climbing from products up to that same peak.
• ΔH for the reaction is the energy gap between reactants and products (products minus reactants). An exothermic reaction has products sitting lower than reactants (negative ΔH); an endothermic reaction has products sitting higher than reactants (positive ΔH). Ea itself is always a positive value, regardless of whether the overall reaction is exothermic or endothermic.

For a multi-step mechanism, the energy profile shows multiple peaks (transition states) and valleys (intermediates). The highest peak relative to the reactants typically corresponds to the rate-determining step.

Catalysts

A catalyst speeds up a reaction by providing an alternative pathway with a lower activation energy, without being consumed in the process. Because it lowers Ea for the forward and reverse reactions by an equal amount, a catalyst speeds up how quickly a reaction reaches equilibrium but does not shift the equilibrium position and does not change ΔG°, ΔH, or Keq. This is one of the most heavily tested traps on this topic: students often assume a catalyst increases product yield — it doesn't. It only gets the system to the same eventual yield faster. Biological catalysts, or enzymes, work by the same principle, lowering Ea for reactions that would otherwise be far too slow to sustain life at normal body temperature. Enzymes do not alter the free-energy difference between free substrate and free product; they only lower the barrier.

Kinetic Control versus Thermodynamic Control

When a reaction can form more than one product, the actual outcome depends on the reaction conditions. The kinetic product forms fastest because its pathway has the lower activation energy; the thermodynamic product is the more stable, lower-energy product, even when its pathway carries a higher Ea. At low temperature or over a short reaction time, the kinetic product dominates, because most molecules simply don't have enough energy or time to climb the higher barrier leading to the more stable product. At high temperature, over a long reaction time, or under reversible conditions, the system has enough energy to interconvert products and eventually settle into the lower-energy thermodynamic product. A classic MCAT example is electrophilic addition to a conjugated diene such as 1,3-butadiene: low temperature favors the kinetic (1,2-addition) product, while high temperature favors the thermodynamic (1,4-addition) product.

Common MCAT Traps

• Confusing a large rate constant with a favorable equilibrium. Kinetics (how fast) and thermodynamics (how favorable) are independent; a reaction can be fast and unfavorable, or slow and highly favorable.
• Assuming a catalyst changes reaction yield or Keq. A catalyst only changes the rate at which equilibrium is reached — the equilibrium position, ΔG°, ΔH, and Keq are all untouched.
• Reading reaction order off stoichiometric coefficients. Order must come from experimental rate data, or be given for a known elementary step — never assume it matches the balanced equation's coefficients.
• Forgetting that Ea is always positive. Even a strongly exothermic reaction still has a positive activation energy barrier; only ΔH can be negative.
• Treating all half-lives as constant. Only first-order half-life is independent of concentration; zero- and second-order half-lives depend on [A]0.