A half-life is the time for half of a sample to decay, and it drives the most famous pattern in nature — exponential decay. The amount left is N = N₀ × (½)^(t/t½): raise one-half to the number of half-lives that have passed and multiply by what you started with. After one half-life half remains, after two a quarter, after three an eighth. Crucially the half-life is a constant of the substance, independent of how much you have, which is what makes it a dependable clock for everything from carbon dating to drug dosing.
Reviewed: June 19, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the exponential decay law N = N₀e^(−λt).
The half-life equations
The exponent t/t½ counts the half-lives elapsed; it need not be a whole number. The half-life form and the natural-exponential form N = N₀e^(−λt) are identical, linked by λ = ln2/t½. To recover an unknown half-life, take logs of the ratio N₀/N; to find the time for a target amount, t = t½ × log₂(N₀/N). Time and half-life must share a unit so the exponent is dimensionless.
Worked example — three half-lives
Scenario: You start with 100 g of an isotope whose half-life is 10 days. How much remains after 30 days?
Three half-lives leave one-eighth of the sample: 50 g after 10 days, 25 g after 20, and 12.5 g after 30. The decay constant is λ = 0.693/10 = 0.0693 per day, and the mean lifetime 1/λ ≈ 14.4 days. Notice the absolute loss shrinks each step — 50 g, then 25 g, then 12.5 g — yet the time to halve stays a flat 10 days, the signature of exponential decay.
Frequently Asked Questions
N = N₀ × (½)^(t/t½). 100 g, t½ 10 d, after 30 d = 3 half-lives = 100 × 0.125 = 12.5 g.
The time for half a quantity to decay. After 1 → 50%, 2 → 25%, 3 → 12.5% remain.
λ = ln2/t½ ≈ 0.693/t½. It gives the e-form N = N₀e^(−λt). Mean life is 1/λ.
t½ = t·ln2/ln(N₀/N). 100 g → 25 g in 20 d gives t½ = 20×0.693/1.386 = 10 d.
No — half-life is fixed for the substance. A gram or a tonne takes the same time to halve.