Reaction Order and Half-Life: How Chemists Predict the Speed of a Reaction Over Time

Rate versus feasibility: two separate questions

Thermodynamics answers whether a reaction can happen at all — whether the products are energetically favoured over the reactants. Kinetics answers a completely separate question: how fast it happens, if it happens. The two are independent; a thermodynamically favourable reaction can still proceed so slowly under normal conditions as to be practically unobservable (diamond converting to graphite is the standard textbook example — thermodynamically favoured, kinetically frozen at room temperature). Reaction kinetics is the study of that second question: the rate at which reactant concentration falls (or product concentration rises) over time, and what determines that rate.

Rate laws and reaction order

For a reaction, the rate law expresses reaction rate as a function of reactant concentrations, typically in the form rate = k[A]^m[B]^n, where k is the rate constant and the exponents m and n (the "order" with respect to each reactant, and their sum the overall reaction order) must be determined experimentally — they are not generally predictable from the reaction's stoichiometric equation alone, which is a common point of confusion for people encountering kinetics for the first time. A reaction can be first-order in one reactant and second-order in another, or its rate might not depend on a particular reactant's concentration at all despite that reactant appearing in the balanced equation, if that reactant isn't involved in the rate-determining step of the reaction's actual mechanism.

Zero, first, and second order: what each one predicts

Zero-order reactions (rate = k, independent of concentration) occur when something other than reactant concentration limits the rate — commonly, a catalyst or enzyme that's saturated with substrate, so adding more reactant doesn't speed the reaction because the limiting factor is the fixed number of catalytic sites available, not the reactant supply. Concentration falls linearly over time: [A] = [A]₀ − kt.

First-order reactions (rate = k[A]) have concentration decaying exponentially: [A] = [A]₀e^(−kt). This is the same mathematical form as radioactive decay and the exponential probability distribution — not a coincidence, since all three describe a quantity whose rate of decrease is proportional to how much of it currently remains. Many single-molecule decomposition reactions and radioactive decay itself follow this pattern.

Second-order reactions (rate = k[A]² for a single reactant, or rate = k[A][B] for two) arise when the rate-determining step requires two molecules to collide and react together, making the rate proportional to the product of both concentrations (or the square of one, if only one species is involved in that step). Concentration follows 1/[A] = 1/[A]₀ + kt, which decreases the fastest at first and progressively slows more sharply than first-order decay does.

Half-life behaves completely differently across orders

The clearest practical distinction between reaction orders shows up in how half-life — the time for concentration to drop to half its starting value — behaves. For a zero-order reaction, t₁/₂ = [A]₀/(2k): half-life depends on starting concentration, so a more concentrated batch takes proportionally longer to reach half strength. For a first-order reaction, t₁/₂ = ln(2)/k ≈ 0.693/k: half-life is constant, completely independent of starting concentration — this is exactly why radioactive isotopes have a single, fixed half-life regardless of how much material you start with, since radioactive decay is a first-order process. For a second-order reaction, t₁/₂ = 1/(k[A]₀): half-life is inversely proportional to starting concentration, so a more concentrated batch actually reaches half strength faster, not slower — the opposite direction from the zero-order case.

This makes half-life behaviour itself a useful diagnostic: measuring how a reaction's half-life changes (or doesn't) as you vary the starting concentration is one practical way to experimentally determine which order a reaction actually follows, without needing to fit the full concentration-versus-time curve.

Temperature: the other major lever on rate

Reaction rate depends on both concentration (captured in the rate law) and temperature (captured separately, in the rate constant k's own dependence on temperature, described by the Arrhenius equation: k = Ae^(−Ea/RT), where Ea is the activation energy — the minimum energy colliding molecules need for a reaction to proceed — and T is absolute temperature). Because temperature appears inside an exponential, its effect on rate is disproportionately large: a common rule of thumb is that reaction rate roughly doubles for every 10°C increase near room temperature for many reactions, though the exact factor depends on the specific activation energy involved. This exponential temperature sensitivity is the reason refrigeration meaningfully slows spoilage reactions, and why controlling reaction temperature precisely matters far more in industrial chemistry than controlling concentration by a similar relative amount.