Radioactive Half-Life Calculator

What is a half-life?

The half-life is the time it takes for half of the nuclei of a radioactive isotope to decay. With every half-life the remaining amount halves: 50 percent is left after one half-life, 25 percent after two and 12.5 percent after three.

The decay law

N = N₀ × (1/2)^(t / T)

N₀ is the initial amount, T the half-life and t the time elapsed. The same law can be written with the decay constant: N = N₀ × e^(−λ × t), where λ = ln 2 / T. The amount can be a number of nuclei, a mass or an activity, and all of them fall off in the same way.

Example

Carbon-14 has a half-life of 5730 years. After 11 460 years, 11 460 / 5730 = 2 half-lives have passed and (1/2)² = one quarter of the original amount is left.

The other way round, if an old piece of wood has 80 percent of the carbon-14 found in a living tree, the time elapsed is 5730 × log₂(1 / 0.8) = 1845 years. Radiocarbon dating rests on this calculation.

Activity

Activity is the number of decays per second and is measured in becquerels (Bq): A = λ × N. For the older unit, the curie, 1 Ci = 3.7 × 10¹⁰ Bq. One gram of strontium-90, with a half-life of 28.8 years, contains 6.7 × 10²¹ nuclei and has an activity of 5.1 × 10¹² Bq, about 138 Ci. Activity falls with the same half-life.

Half-life from measurements

If the amount or the count rate is known at two different times, T = t × ln 2 / ln(N₀ / N). A sample whose count drops from 1000 to 700 in five hours has a half-life of 9.7 hours.

Limits

The law describes the average behaviour of a large number of nuclei. If the decay products are radioactive too, the total activity is different. The tool has no isotope table, so you need to enter the half-life from your own source. It must not be used for radiation safety or dose calculations.