Half-life calculator
Half-life, mean lifetime and decay constant are three ways of saying the same thing. Give any one and this gives the others — along with the mistake that comes from thinking two of them are the same. Nothing you type here leaves your browser — it is worked out on your own device, so there is nothing to send and nothing to log.
τ is not t½, and the difference is 44%
Radioactive decay gets written two ways, and both are correct:
N = N₀ × ½t/t½ N = N₀ × e−t/τ
The first uses the half-life, the time for half of it to go. The second uses the mean lifetime, which is the average time an individual atom survives. They are not the same number:
τ = t½ / ln2 = 1.442695 × t½
So the mean lifetime is 44% longer than the half-life. Put a half-life into the exponential form and you get an answer that is wrong by 26% and looks entirely reasonable: after one half-life you would be told 36.79% remains instead of 50%.
36.79% is 1/e. It is the right answer to a different question — how much is left after one mean lifetime — which is what makes it so hard to spot. It is not a nonsense number. It is a correct number in the wrong place.
| Half-lives | Actually left | If τ confused |
|---|---|---|
| 1 | 50.00% | 36.79% |
| 2 | 25.00% | 13.53% |
| 3 | 12.50% | 4.98% |
| 10 | 0.098% | 0.0045% |
The error compounds. By ten half-lives the two answers differ by a factor of twenty-two.
Why there are three names at all
The decay constant λ is the fundamental one: the probability per second that any given atom decays. Everything else is derived from it. τ = 1/λ is its reciprocal, and t½ = ln2/λ is the more human way of saying it, because "half of it is gone" is something you can picture and "the probability per second is 3.8×10−12" is not.