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Half-Life Calculator

Work out how much of a radioactive sample remains after a given time using N = N₀ (½)^(t/t½) — or solve for the half-life or the elapsed time.

 

Formula

$$ N = N_0\left(\tfrac{1}{2}\right)^{t/t_{1/2}} \qquad t_{1/2}=\frac{t\,\ln 2}{\ln(N_0/N)} \qquad t=t_{1/2}\,\log_2\!\frac{N_0}{N} $$

Worked example

Carbon-14 has a half-life of 5,730 years. After 11,460 years (two half-lives) a 100 g sample decays to \( 100\times(½)^{2}=25\ \text{g} \) — a quarter remains.

How it works

Radioactive decay is exponential: every half-life, half of the remaining unstable nuclei decay. Starting from N₀, the amount left after time t is N = N₀ (½)^(t/t½), where t½ is the half-life.

The calculator can solve for any variable. Keep N and N₀ in the same unit, and t and t½ in the same time unit (the calculator only uses their ratio). It also reports the number of half-lives elapsed and the fraction remaining.

Frequently asked questions

How do you calculate the amount remaining after n half-lives?

Multiply the initial amount by (½) raised to the number of half-lives: N = N₀ (½)^(t/t½). After 1, 2, 3 half-lives you have 50%, 25%, 12.5% left.

How do you find the half-life from decay data?

Use t½ = t·ln2 / ln(N₀/N). Set 'Solve for' to Half-life and enter the initial amount, remaining amount and elapsed time.

What is the relationship between half-life and the decay constant?

The decay constant λ = ln2 / t½ ≈ 0.693 / t½. The amount can equivalently be written N = N₀ e^(−λt).

Does half-life depend on the amount of sample?

No. Half-life is a fixed property of the isotope and does not change with the sample size, temperature or chemical form.

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