Half-Life From Decay Rate Calculator
Turn a continuous decay constant λ into half-life and mean lifetime, and read off how much remains after a chosen time.
How it works
Half-life t½ = ln(2) / λ · mean lifetime τ = 1 / λ · fraction remaining = e^(−λt)
In exponential decay the quantity falls by a fixed fraction per unit time, governed by the decay constant λ. The half-life — the time for it to drop to 50% — is ln(2)/λ, and it is constant no matter how much you start with. The mean lifetime τ = 1/λ is the average time a single particle survives and is always a bit longer than the half-life, since a few long-lived stragglers pull the average up.
Worked example
With λ = 0.1 per unit time, the half-life is ln(2)/0.1 ≈ 6.93 units and the mean lifetime is 1/0.1 = 10 units. After 10 units of time, e^(−0.1 × 10) = e^(−1) ≈ 36.8% of the original amount remains.
Sources: Exponential decay — Wikipedia
Frequently asked questions
What is the difference between half-life and mean lifetime?
Half-life is when half the amount is gone; mean lifetime τ = 1/λ is the average survival time of an individual unit. They are related by t½ = τ·ln(2) ≈ 0.693·τ, so the mean lifetime is always the longer of the two.
How do I get λ from a percentage decline?
If a fraction p decays per unit time continuously, λ = −ln(1 − p). For a small percentage the two are close, but for larger declines the logarithm matters — a 50% continuous decline per unit time is λ = ln(2) ≈ 0.693, not 0.5.
Does the starting amount change the half-life?
No. Exponential decay is scale-free: the time to fall by half is the same whether you begin with a gram or a tonne, which is what makes half-life such a convenient descriptor.
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This calculator is for educational purposes. Double-check important results before acting on them.