Find how much is left after radioactive decay (or a drug leaving the body), how long it took, the starting amount, or the half-life itself.
The half-life formula
N(t) = N₀ × (½)t/T
N₀ is the starting amount, N(t) is what's left after time t, and T is the half-life: the time it takes for half to decay. Rearranged:
t = T × log₂(N₀ ÷ N(t)) T = t × ln 2 ÷ ln(N₀ ÷ N(t)) N₀ = N(t) × 2t/T
The same decay is often written N(t) = N₀e−λt, with the decay constant λ = ln 2 ÷ T. The mean lifetime τ = 1 ÷ λ = T ÷ ln 2 ≈ 1.4427 × T is the average time a single atom survives.
Worked example: carbon-14
Carbon-14 has a half-life of about 5,730 years. Of 100 g, after 10,000 years:
N = 100 × 0.510000/5730 = 100 × 0.51.7452 ≈ 29.829 g
Going the other way, a sample with 25% of its carbon-14 left has been through exactly 2 half-lives, so it's about 2 × 5,730 = 11,460 years old.
Limitations
- Decay is random. The formula gives the expected amount, which is very accurate for the huge numbers of atoms in any real sample but not for a handful of atoms.
- Published half-lives have uncertainty. Carbon-14 is often quoted as 5,730 years (the value used in many textbooks), while current nuclear data lists 5,700 ± 30 years.
- Radiocarbon dating also needs calibration for past changes in atmospheric carbon-14, so a raw half-life age isn't a calendar date.
- For medicines, the half-life varies from person to person. Don't use this to plan doses; ask a pharmacist or doctor.
Sources
How we checked this
Before every update, this calculator is run in a real browser with these known answers, and it must match all of them:
- Half-life of 1 second reads 'Every second' ✓
- 100 g of carbon-14 after 10,000 years (T = 5,730): ≈ 29.829 g ✓
- 25 of 100 left with T = 5,730 years: 11,460 years ✓
- 80 → 10 in 24 hours: half-life 8 hours ✓
- 12.5 left after 3 half-lives: started with 100 ✓
- Decay constant for T = 5,730 years ≈ 0.000120968 ✓
- More left than at the start is rejected ✓
Last checked October 26, 2026. Spotted something wrong? The math is done in your browser with the formula shown on this page, so you can check any result by hand.
Questions people ask
How do I calculate half-life?
Use T = t × ln 2 ÷ ln(N₀ ÷ N(t)). If 80 g decays to 10 g in 24 hours, N₀ ÷ N = 8, ln 8 = 3 × ln 2, so T = 24 ÷ 3 = 8 hours.
How much is left after 3 half-lives?
One-eighth, or 12.5%. Each half-life halves the amount: ½ × ½ × ½ = ⅛.
What is the decay constant?
λ = ln 2 ÷ T, the fraction that decays per unit of time when the time step is very small. For carbon-14 (T = 5,730 years), λ ≈ 0.000121 per year.
Does anything ever fully decay?
In the formula, the amount never reaches exactly zero. In reality, after about 10 half-lives less than 0.1% is left, and eventually the last atoms decay.
What's the difference between half-life and mean lifetime?
Half-life is when half is gone. Mean lifetime is the average time an atom lasts, which is longer: τ = T ÷ ln 2 ≈ 1.44 × T.