First-Order Half-Life Calculator

At a glance:The half-life (symbol t½) is the time required for the reactant concentration (or amount of radioactive substance) to fall to half of its original value; it is an intuitive measure of how fast a reaction proceeds. For a 'first-order reaction' — one whose rate is proportional to the first power of a single reactant concentration, rate = k[A] — the half-life has a very special and important property: it is a constant, completely independent of the initial reactant concentration. The formula is t½ = ln2 / k ≈ 0.693 / k, where k is the first-order rate constant (unit 1/time, most commonly 1/s). It derives from the first-order integrated rate law ln([A]₀/[A]) = kt: when [A] = [A]₀/2 (exactly half remains), ln(2) = k·t½, which rearranges to t½ = ln2/k. The reason 'constant half-life' is the hallmark of first-order reactions is that no matter what concentration you start from, after each t½ the concentration halves again: after 1 half-life 1/2 remains, after 2 half-lives 1/4, after 3 half-lives 1/8 …, and after n half-lives (1/2)ⁿ remains. Using this tool's default: k = 0.0693 /s, so t½ = 0.693 / 0.0693 ≈ 10 s, meaning every 10 s the reactant concentration is halved. This property — 'the half-life does not change with the initial amount' — is exactly the scientific basis of radioactive isotope dating (e.g. carbon-14 dating): radioactive decay is always a first-order process, and every isotope has a fixed half-life (carbon-14 ≈ 5730 years, iodine-131 ≈ 8 days, uranium-238 ≈ 4.5 billion years). By measuring the remaining amount against the known half-life, scientists work back to the sample's age. The half-life is inversely proportional to the rate constant k: a larger k (faster reaction) gives a shorter half-life; a smaller k (slower reaction) gives a longer one. It must be emphasised that 'half-life independent of initial concentration' holds only for first-order reactions. For a zero-order reaction t½ = [A]₀/(2k), proportional to the initial concentration; for a second-order reaction t½ = 1/(k[A]₀), inversely proportional to it. Thus, if an experiment shows a reaction's half-life does not change with starting concentration, that is strong evidence it is first-order. Half-life has very broad applications: (1) radioactive dating and nuclear medicine (clearance of radioactive drugs, radiotherapy dose planning); (2) pharmacokinetics — most drugs are metabolised by an approximately first-order process, and the half-life determines dosing intervals (after about 4–5 half-lives the drug is essentially cleared); (3) assessing degradation rates of chemicals in the environment; (4) estimating shelf life of food and reagents. Using this calculator: first, k must be the first-order rate constant, unit 1/time (e.g. 1/s, 1/min, 1/year), and the half-life unit corresponds to k's time unit (k in 1/s → t½ in seconds). Second, k must be greater than 0, otherwise the reaction does not proceed and the half-life is undefined (this tool returns 0). Third, this formula applies only to first-order reactions; for zero- and second-order reactions use the corresponding formulas. Fourth, ln2 ≈ 0.6931 is the natural logarithm of 2, not to be confused with log₁₀2 ≈ 0.301. In short, this calculator lets you quickly find the constant half-life from the first-order rate constant — a core tool for chemical kinetics and radioactive-decay study.