How it works
Radioactive nuclei decay at random, but a large sample loses a fixed fraction per unit time. The half-life is the time for half of what is left to decay. After two half-lives a quarter remains, after three an eighth, and so on.
The same exponential law describes drug elimination from the body, discharge of a capacitor and many other first-order processes. Use any amount unit you like; the result comes back in the same unit.
Carbon-14, used for radiocarbon dating, has a half-life of 5,730 ± 40 years. After 11,460 years, 25% of the original carbon-14 remains.
Formula
N = N₀ × (½)^(t ÷ T½) λ = ln 2 ÷ T½ τ = 1 ÷ λ
Questions people ask
How many half-lives until a sample is gone?
Mathematically it never reaches zero. After 7 half-lives less than 1% remains, and after 10 about 0.1%, which is often treated as negligible.
How do I find age from the remaining fraction?
Rearrange: t = T½ × log₂(N₀ ÷ N). If 12.5% remains, that is 3 half-lives, or 17,190 years for carbon-14.
What is the decay constant?
λ = ln 2 ÷ T½ is the probability per unit time that a nucleus decays. Activity (decays per second) equals λ times the number of atoms.
What is mean lifetime?
The average time a nucleus survives, τ = 1/λ = T½ ÷ ln 2 ≈ 1.443 × T½.