Half-Life Calculator

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

N = N0 times (1/2)^(t/t_half). After 1,2,3 half-lives: 50%, 25%, 12.5% remain.

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

Decay constant lambda = ln2/t_half; N = N0 e^(-lambda t).

Q: Does half-life depend on the amount of sample?

Half-life is independent of sample size in the constant-rate model used here. Some decay modes depend on electronic environment.

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

An initial 100 g of carbon-14, with a half-life of 5,730 years, has 25 g of carbon-14 remaining after 11,460 years (two half-lives): N = 100 × (½)² = 25. The other nuclei have decayed into daughter products.

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.

For the underlying physics, read our explanation of quantum tunneling in radioactive decay.

If you know the decay constant instead of the half-life, use the radioactive decay calculator to find the remaining quantity.

How to use this calculator

Select remaining amount, half-life, elapsed time or initial amount as the unknown.
Use the same amount unit for both quantities: grams of the undecayed isotope, number of nuclei, or activity.
Use the same time unit for half-life and elapsed time, such as seconds or years.
Calculate to see the remaining percentage and a table from zero to five half-lives.

Assumptions and limitations

This model assumes a constant decay rate for a single radioactive isotope in a closed sample, with no additional production or removal. It gives the expected remaining quantity; individual decay events are random.

The remaining amount refers to the original isotope, not the total mass of all decay products. Initial amount and half-life must be positive; elapsed time and remaining amount must be non-negative. A finite sample does not reach exactly zero in this continuous exponential model.

Sources

OpenStax University Physics: Radioactive Decay

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?

For ordinary radioactive-decay calculations, half-life is treated as a property of the isotope and is independent of sample size. Some decay modes can be affected by their electronic environment; this calculator assumes a constant half-life.