A half-life calculator for radioactive decay that solves the decay equation for any one unknown: the remaining amount, the elapsed time, the half-life, or the initial amount. Alongside the answer it reports the decay constant, the mean lifetime, how many half-lives have elapsed and the fraction remaining, then plots the full exponential decay curve. It is built for students, lab work, radiometric dating and nuclear-medicine dosing, and it runs entirely in your browser — no figures leave your device.
How it works
Radioactive decay is a first-order process: the rate at which atoms decay is proportional to how many are left. That gives the exponential law
N(t) = N₀ · (½)^(t ⁄ T) = N₀ · e^(−λt)
where N₀ is the starting quantity, N is what remains after time t, T is the half-life (the
time for half the sample to decay), and λ is the decay constant. The half-life and decay
constant are two views of the same number, linked by λ = ln(2) / T, using ln(2) ≈ 0.6931. The
mean lifetime is τ = T / ln(2) = 1/λ, roughly 1.44 half-lives.
Because those four quantities sit in one equation, knowing any three fixes the fourth. The
calculator rearranges algebraically rather than guessing: to find time it uses
t = T · ln(N₀/N) / ln(2); to find a half-life from data it uses T = t · ln(2) / ln(N₀/N); and
to recover an initial amount it uses N₀ = N · 2^(t/T). Quantity can be anything proportional to
the number of atoms — mass, becquerels of activity, counts per minute or moles — as long as N₀ and
N share the same units. A quick-pick menu loads accepted half-lives for common isotopes such as
Carbon-14 (5,730 years), Iodine-131 (8.02 days) and Technetium-99m (6.01 hours).
Worked example
A medical sample starts with 100 mg of an isotope whose half-life is 8 days. How much remains after 16 days? Sixteen days is exactly two half-lives, so the amount halves twice: 100 → 50 → 25. The formula agrees: N = 100 · (½)^(16/8) = 100 · (½)² = 25 mg, a fraction remaining of 25%. The decay constant is λ = ln(2)/8 days ≈ 0.0866 per day, and the mean lifetime is τ = 8/ln(2) ≈ 11.5 days. Running the calculator in reverse — solving for time with N₀ = 100, N = 25 and T = 8 days — returns t = 8 · ln(4)/ln(2) = 16 days, confirming the round trip.
| Half-lives elapsed | Fraction remaining | Percent |
|---|---|---|
| 0 | 1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 5 | 1/32 | 3.125% |
| 10 | 1/1024 | ~0.098% |
Formula note
Use a consistent quantity for N₀ and N (mass, activity, or atom count) and consistent ideas of time. The model assumes a single decay channel with a constant decay constant, which holds for the overwhelming majority of practical problems; it does not model decay chains where a daughter isotope is itself radioactive, nor branching ratios. After about 10 half-lives less than 0.1% of the original sample remains, which is the usual rule of thumb for treating a source as effectively gone.
Every calculation happens locally in your browser — nothing is uploaded or stored on a server.