Reaction Rate & Half-Life Calculator

Reaction Half-Life Explained

Half-life is the time it takes for a reactant's concentration to drop to half its starting value — how it's calculated depends on the reaction's order, which describes how rate depends on concentration.

Zero order: $$t_{1/2} = \frac{[A]_0}{2k}$$

First order: $$t_{1/2} = \frac{\ln 2}{k}$$

Second order: $$t_{1/2} = \frac{1}{k[A]_0}$$

Notice that only first-order half-life is independent of the starting concentration — a distinctive signature that helps identify reaction order from experimental data.

Worked Example: First-Order Decay

Using the calculator's default first-order values — a rate constant of 0.05 s⁻¹ — half-life is $t_{1/2} = \ln(2)/0.05 \approx 13.86$ s, completely independent of the starting concentration of 1.0 mol/L. After one half-life, concentration drops to 0.5 mol/L; after two half-lives (about 27.7 s), it drops to 0.25 mol/L, and so on — the same pattern of repeated halving seen in radioactive decay.

Why Reaction Order Matters

The order of a reaction isn't something you can guess from the balanced chemical equation — it must be determined experimentally, and it fundamentally changes how concentration evolves over time. A zero-order reaction proceeds at a constant rate regardless of concentration (common in reactions limited by a catalyst's surface area), a first-order reaction's rate is directly proportional to one reactant's concentration, and a second-order reaction's rate depends on the square of a concentration (or the product of two different concentrations).

A Brief History of Chemical Kinetics

Ludwig Wilhelmy conducted some of the earliest quantitative rate studies in 1850, measuring how sugar inversion proceeded over time — an early example of what would become first-order kinetics analysis. Svante Arrhenius's later work in the 1880s connecting rate constants to temperature (the Arrhenius equation) helped establish chemical kinetics as a rigorous quantitative field, building directly on this same idea of a well-defined rate constant governing how fast a reaction proceeds.

Common Reaction Order Mistakes

Assuming every reaction is first order (since it's the most commonly taught case) is a common error — using the wrong order's half-life formula gives a systematically wrong answer, since the formulas aren't interchangeable. Forgetting that zero- and second-order half-life change as the reaction proceeds (since they depend on the current concentration, not just the starting one) is another frequent mistake — those two half-life formulas only give the half-life for the very first half-life period, not every subsequent one. Confusing rate constant units across orders (which differ by order, unlike first order's simple s⁻¹) is a third common slip.

Chemical Kinetics Terms You Should Know

Reaction Order — how a reaction's rate depends on reactant concentration, determined experimentally.

Rate Constant (k) — the proportionality constant in a rate law, with units that depend on reaction order.

Half-Life (t½) — the time for concentration to drop to half its value.

Integrated Rate Law — the equation describing concentration as a function of time, derived from the differential rate law.

These formulas describe simple single-reactant kinetics; reactions with multiple reactants or more complex mechanisms may require more advanced treatment.