Science / Radioactive decay
Half-Life Calculator
Work out the amount left, the starting amount, the time elapsed, or the half-life from the other three, with the decay constant and a decay curve.
Half-Life Calculator: Radioactive material decays so that half of what is left goes in each half-life: N = N₀ × ½^(t ÷ T). The calculator rearranges that formula for whichever value is missing, using logarithms to find a time or a half-life, and shows the decay constant λ = ln 2 ÷ T and the mean lifetime 1 ÷ λ. Runs 100% locally in your browser with zero server file uploads.
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Fill in any three and leave the fourth empty. Use the same unit of time for t and T, and the same unit of amount for N₀ and N.
N = N₀ × ½^(t ÷ T): every half-life halves what is left, so after two half-lives a quarter remains and after ten about a thousandth. Carbon-14 has a half-life of about 5,730 years.
A worked example
Iodine-131, used in thyroid treatment, has a half-life of about 8.02 days. After 24 days, three half-lives, ½ × ½ × ½ = 12.5% would be left with a half-life of exactly 8 days; with 8.02 days it is 12.56%.
For powers and logarithms by themselves, use the log calculator; for very large or small numbers, the scientific notation converter.
Half-life and the decay constant
The same decay can be written N = N₀ × e^(−λt), where λ = ln 2 ÷ T is the decay constant: the fraction decaying per unit of time, while it is small. The mean lifetime, 1 ÷ λ, is about 1.44 half-lives.
How to use it
- Fill in three of the starting amount, the amount left, the time elapsed, and the half-life.
- Leave the one you want empty.
- Read the answer, the fraction left, and the decay curve.
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Frequently asked questions
Does the unit of time matter?
Only that t and T use the same one: years and years, or days and days. The amounts can be in any unit, or a percentage, as long as both use it.
How is carbon dating done with it?
The fraction of carbon-14 left in a sample gives its age: with 25% left and a half-life of 5,730 years, two half-lives, or 11,460 years, have passed. Laboratories still report radiocarbon ages with Libby's older half-life of 5,568 years, by convention, and then calibrate them.
Does anything ever decay completely?
In theory the amount only halves, so it never reaches zero; in practice, after ten half-lives less than a thousandth is left, and after a few dozen there may be no atoms left at all.
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