Half-life calculator

How do I calculate how much remains after decay?

N = N₀ × (½)^(t ÷ T½). 100 grams with a 10-day half-life leaves 50 g after 10 days, 25 g after 20 and 12.5 g after 30.

How do I find the half-life from measurements?

T½ = t × ln 2 ÷ ln(N₀ ÷ N): if 100 g drops to 25 g in 20 days, the half-life is 10 days.

How long until almost nothing is left?

After 10 half-lives about 0.1% remains (0.0977%). Decay never reaches exactly zero in theory.

Does it work for medicines?

Drug elimination often follows the same curve, with the drug's half-life in the body — but follow the dosing your doctor or the label gives.

What is the decay constant?

λ = ln 2 ÷ T½ — the fraction decaying per unit time; the calculator shows it too.

How long does it take to decay to a given fraction?

Solve the decay law for the time: t = T½ × log₂(N₀ ÷ N), which is T½ × ln(N₀ ÷ N) ÷ ln 2. For carbon-14 (T½ = 5,730 years) with 80% of the original amount left, t = 5,730 × ln(100 ÷ 80) ÷ ln 2 = 1,845 years; OpenStax University Physics, Volume 3, Example 10.6, gets 1,844 years with a rounded logarithm. Enter 100, 80 and that time in the half-life mode and you get 5,730 years back.

What is the difference between half-life and mean lifetime?

The mean lifetime is 1 ÷ λ, where λ is the decay constant (OpenStax University Physics, Volume 3, section 10.3, Eq. 10.16). Because λ = ln 2 ÷ T½, the mean lifetime is T½ ÷ ln 2, about 1.44 times the half-life. After one mean lifetime, 1 ÷ e, about 37%, of the amount is still there.

Do temperature or chemistry change the half-life?

For radioactive decay, no. OpenStax notes that the radiation of a nuclide does not vary with its chemical state, temperature or pressure (University Physics, Volume 3, section 10.3), so a half-life is a fixed property of the nuclide.

The formula and what it assumes

The calculator uses N = N₀ × (½)^(t ÷ T½), the same law as N = N₀e^(−λt) with λ = 0.693 ÷ T½ (OpenStax University Physics, Volume 3, section 10.3, Eq. 10.11 and 10.15). After n whole half-lives the amount is N₀ ÷ 2ⁿ (Eq. 10.12). The law assumes the loss rate at any moment is proportional to the amount present, which is why the answer does not depend on how much you start with: grams, atoms, becquerels or a percentage all work. The half-life and the elapsed time must be in the same unit; the time-unit menu applies to both.

Worked example: carbon-14 after 10,000 years

Carbon-14 has a half-life of 5,730 years. Start with 100 (grams, or percent) and wait 10,000 years, which is 10,000 ÷ 5,730 = 1.745 half-lives. The amount left is 100 × ½^1.745 = 29.8. Run the check the other way: with 100 at the start, 29.8 left and 10,000 years elapsed, the half-life mode returns 5,730 years. After exactly two half-lives, 11,460 years, 25 is left, not zero.

Common mistakes

Reading two half-lives as “everything is gone” is the most common slip: each half-life halves what is left, so after two half-lives a quarter remains. Mixing time units is next, for example a half-life in years with an elapsed time in days; the exponent is then off by a factor of 365. For dating, remember the method’s limit: OpenStax puts the usable range of carbon-14 dating at tissues up to about 50,000 years old, since too little carbon-14 remains after that. Finally, the amount that remains is not the amount that has vanished; the decayed part has turned into other nuclides.

Sources: OpenStax University Physics, Volume 3, section 10.3 (radioactive decay, Eq. 10.11, 10.12, 10.15 and 10.16; carbon-14 dating and Example 10.6). Related: carbon dating calculator, exponential growth calculator, integrated rate law calculator and log calculator.