Half-Life Calculator
Exponential decay, and why it never reaches zero.
Work out Half-Life. Exponential decay, and why it never reaches zero. Compounding frequency is an input, not an assumption.
5,730 years for carbon-14.
Amount remaining
50
50.000% left after 1 half-lives
The defining property is that the fraction lost per unit time is constant, so the halving time is the same whatever you start from — 100 to 50 takes exactly as long as 50 to 25. That is why the quantity never mathematically reaches zero, and why "time until 1% remains" is a more useful practical threshold than "time until gone". The mean lifetime is always longer than the half-life, by exactly 1/ln 2.
How the Half-Life Calculator works
The defining property of a half-life is that the fraction lost per unit time is constant, so the halving time is the same whatever you start from — 100 to 50 takes exactly as long as 50 to 25. That is why the quantity never mathematically reaches zero.
Also known as: radioactive decay calculator · decay constant calculator · exponential decay calculator · half life formula
Why the halving time is constant
Decay is proportional to how much is present, so the fraction lost per unit time never changes. That is what makes the halving time constant regardless of the starting amount.
100 to 50 takes exactly as long as 50 to 25, and as 25 to 12.5. Carbon-14 halves every 5,730 years whether you start with a gram or a tonne.
It never reaches zero
Mathematically the quantity approaches zero asymptotically and never arrives, which is why the question of when it is gone has no well-defined answer.
Practical thresholds are used instead. Seven half-lives leaves under 1%; ten leaves under 0.1%. For radioactive waste the usual working figure is ten half-lives, which is why a 24,000-year isotope is a 240,000-year problem.
Mean lifetime — the average survival time of an individual atom — is 1/λ, or the half-life divided by ln 2. It is always about 44% longer than the half-life, which surprises people who expect the two to coincide.
Frequently asked questions
What is a half-life?
The time for half of something to decay. Carbon-14's is 5,730 years, so a sample loses half its C-14 in that time and half of what remains in the next.
How much is left after n half-lives?
(½)ⁿ of the original. Two half-lives leave a quarter, three leave an eighth, and seven leave under one percent.
What is the decay constant?
λ = ln(2) ÷ half-life. It is the instantaneous rate rather than the halving time, and it appears in the exponential form of the decay equation.
What is mean lifetime?
The average time an individual atom survives — 1/λ, which is the half-life divided by ln 2. It is always longer than the half-life, by about 44%.
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