Decay Constant Calculator
Enter a half-life to get the radioactive decay constant λ in inverse time units — plus the mean lifetime τ — straight from λ = ln(2) / t½.
What is the decay constant?
The probability of decay per unit time
This decay constant calculator turns a single measurement — an isotope's half-life — into the decay constant λ (lambda), the probability per unit time that any given atom decays. A bigger λ means a faster-decaying, less stable isotope; a smaller λ means a slower, more stable one. The constant is the heart of the exponential decay law N(t) = N₀ · e^(−λt), and it links directly to the half-life through λ = ln(2) / t½. Alongside λ, the calculator reports the mean lifetime τ — the average time an atom survives before it decays — which is simply the reciprocal of the decay constant.
Enter a half-life in any time unit to get the decay constant λ per that unit and the mean lifetime τ instantly.
The decay constant is the natural logarithm of two divided by the half-life, and the mean lifetime is simply the reciprocal of that constant.
λ = ln(2) / t½Because ln(2) ≈ 0.693, the decay constant is always a little under 0.7 divided by the half-life. The mean lifetime then follows from τ = 1 / λ, which is the same as t½ / ln(2) — always longer than the half-life by the factor 1 / ln(2) ≈ 1.4427.
Take carbon-14, the isotope behind radiocarbon dating, with a half-life of 5730 years:
Divide ln(2) by the half-life
0.693147 ÷ 5730 = 0.00012097 per year — the decay constant λ.
Take the reciprocal for the mean lifetime
1 ÷ 0.00012097 = 8266.64 years — the mean lifetime τ, longer than the 5730-year half-life.
So roughly 0.0121 % of any carbon-14 atoms decay each year, and an average atom lasts about 8267 years before it does.
The formula is exact for any single isotope, but a couple of practical points are worth keeping in mind.
One isotope, one consistent time unit
This calculator describes the simple exponential decay of a single radioactive isotope; it does not model decay chains where a daughter product is itself radioactive. Keep the half-life and any later time values in the same unit — if you enter the half-life in years, λ is per year and any time you later use in e^(−λt) must also be in years, or the result will be wrong.