Radioactive Decay Calculator
Mean lifetime is 1.44 half-lives, not one.
Mean lifetime is 1.44 half-lives, not one. How much of a radioactive sample remains after any elapsed time, with the decay constant and mean lifetime.
Amount remaining
25
25% left after 2 half-lives
Decay is exponential: each half-life removes half of whatever is left, so ten half-lives leave about one part in a thousand and twenty leave one in a million. It never reaches zero, which is why contamination is described in terms of half-lives rather than an end date. The mean lifetime is longer than the half-life, by a factor of 1/ln2 — about 1.44. That catches people out, because intuition says the average should be the halfway point. The distribution has a long tail: a few nuclei survive far longer than the half-life and drag the mean up. The units are yours. Enter the half-life and elapsed time in the same units (years for carbon-14, seconds for a medical tracer) and the answer comes back in them.
How the Radioactive Decay Calculator works
How much of a radioactive sample remains after any elapsed time, with the decay constant and mean lifetime. The mean lifetime is longer than the half-life by a factor of 1.44, which catches almost everyone out.
Also known as: how much is left after n half lives · carbon dating age calculator · decay constant from half life · when is radioactive waste safe
Where to go next
The Radioactive Decay question rarely arrives on its own. These are the ones that usually come with it:
- Nuclear Binding Energy Calculator — Why fusion works below iron and fission above it.
- Length Contraction & Time Dilation Calculator — One factor, γ, does all of it.
- Relativistic Momentum Calculator — γmv, and the ratio to classical is exactly γ.
- Speed Distance Time Calculator — Any one of the three from the other two, in four units.
Frequently asked questions
How does radioactive decay work?
Each half-life removes half of whatever remains. Ten half-lives leave about one part in a thousand and twenty leave one in a million, but it never reaches exactly zero.
Why is the mean lifetime longer than the half-life?
Because the distribution has a long tail. A few nuclei survive far longer than the half-life and pull the average up — the mean is the half-life divided by ln 2, about 1.44 times as long.
What is the decay constant?
The probability per unit time that any given nucleus decays, equal to ln 2 divided by the half-life. It is the reciprocal of the mean lifetime.
How does carbon dating use this?
Carbon-14 has a half-life of 5,730 years and stops being replenished when an organism dies. Measuring what fraction is left gives the elapsed time, which works reliably out to about ten half-lives — roughly 50,000 years.
How long until radioactive waste is safe?
It depends entirely on the isotope. Iodine-131 has an 8-day half-life and is gone in months; plutonium-239's is 24,000 years. "Safe" is usually taken as around ten half-lives, when about a thousandth remains.
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