Half-Life & Radioactive Decay
Project how much of a radioactive sample is left after a given time, or work backwards to the elapsed time for radiometric dating. Decay follows first-order kinetics, N = N₀e^(−λt), with λ = ln2/t½. Common isotope half-lives are built in.
How to use this tool
The Decay tab gives the amount left after a time t. The Elapsed time tab gives the time to fall from N₀ to N — the basis of carbon dating. Pick an isotope to load its half-life, or type your own.
What to enter
- Amount: atoms, mass or activity — anything, since only the ratio matters. Use the same unit for N₀ and N.
- Half-life and time: in the same time unit (set by the unit selector).
Reading the result
Decay mode reports the remaining amount, the percentage left, the number of half-lives elapsed, the decay constant λ and the mean lifetime τ = 1/λ. Elapsed-time mode returns t and how many half-lives that is.
Worked example
Carbon-14 (t½ = 5730 yr): a sample at 25% of its original ¹⁴C has decayed for two half-lives, t = 11 460 yr.
Result
Each half-life halves the amount; after n half-lives a fraction (½)ⁿ remains. The decay constant λ and mean lifetime τ describe the same process per unit time.
Methodology
Radioactive decay is first order: N = N₀ e^(−λt) with decay constant λ = ln2 / t½ and mean lifetime τ = 1/λ. The number of half-lives elapsed is t/t½, and the surviving fraction is (½)^(t/t½). Solving for time gives t = (t½/ln2) · ln(N₀/N) — the relation used in radiometric dating. Activity A = λN decays with the same half-life, so an activity ratio may be used in place of an amount ratio.
Sources
- NNDC / NUBASE evaluated half-lives.
- Atkins, P. & de Paula, J. Physical Chemistry — radioactive decay kinetics.
Known limits
- Single-step decay only; for a decay chain (parent → daughter → …) the secular/transient-equilibrium build-up of daughters is not modelled.
- Carbon dating assumes a constant initial ¹⁴C/¹²C ratio; real dates apply a calibration curve.