Half-Life Calculator (radioactive decay)
Work out how much of a substance is left after a given time, how long a decay takes, or the half-life itself — with a table of common isotopes.
What's left after n half-lives
| Half-lives | Fraction left | Percentage left |
|---|---|---|
| 0 | 1 / 1 | 100% |
| 1 | 1 / 2 | 50% |
| 2 | 1 / 4 | 25% |
| 3 | 1 / 8 | 12.5% |
| 4 | 1 / 16 | 6.25% |
| 5 | 1 / 32 | 3.125% |
| 6 | 1 / 64 | 1.5625% |
| 7 | 1 / 128 | 0.78125% |
| 8 | 1 / 256 | 0.39063% |
| 9 | 1 / 512 | 0.19531% |
| 10 | 1 / 1024 | 0.097656% |
Half-lives of common isotopes
| Isotope | Half-life | Left after 1 year | Usually met in |
|---|---|---|---|
| Technetium-99m | 6.007 hours | <0.001% | Medical imaging |
| Iodine-131 | 8.025 days | <0.001% | Thyroid treatment |
| Cobalt-60 | 5.271 years | 87.68% | Sterilisation, radiotherapy |
| Tritium (H-3) | 12.32 years | 94.53% | Self-luminous signs |
| Strontium-90 | 28.9 years | 97.63% | Fallout |
| Caesium-137 | 30.08 years | 97.72% | Fallout, gauges |
| Radium-226 | 1 600 years | 99.96% | Historic luminous paint |
| Carbon-14 | 5 730 years | 99.99% | Radiocarbon dating |
| Plutonium-239 | 24 110 years | 100% | Reactor fuel, waste |
| Uranium-235 | 704 000 000 years | 100% | Fissile fuel |
| Potassium-40 | 1 248 000 000 years | 100% | Natural background |
| Uranium-238 | 4 468 000 000 years | 100% | Natural uranium, dating |
Published half-lives differ slightly between sources in the last digit or two; these are the commonly cited values.
Half-life and radioactive decay
Radioactive decay is random for any individual atom but utterly predictable for a large number of them. In a fixed interval, a fixed proportiondecays — never a fixed amount. The half-life is the interval over which that proportion is exactly one half.
N = N₀ · (½)^(t / T) · λ = ln2 / T · τ = 1 / λ
Because the fraction is constant, decay never reaches zero. One half-life leaves 50%, two leave 25%, ten leave about 0.1%. The rule of thumb in radiation safety is that ten half-lives is "gone" — not because nothing is left, but because a thousandth of the original is usually below anything that matters.
Worked example
Carbon-14 has a half-life of 5 730 years. A sample with 25% of its original C-14 has been through two half-lives, so it is about 11 460 years old. That is radiocarbon dating in one line — enter 100 and 25 above with a half-life of 5 730 years and switch to "time elapsed" to see it.
Half-life, decay constant, and mean lifetime
Three numbers describe the same thing. The decay constant λ is the probability per unit time that any given atom decays, and it relates to the half-life by λ = ln2 / T. The mean lifetime τ is the average time an atom survives, τ = 1/λ, and it is always longer than the half-life — by a factor of about 1.44 — because the few atoms that last a very long time drag the average up.
Not just radioactivity
The same exponential describes anything that decays by a constant proportion: a drug clearing from the bloodstream, a capacitor discharging through a resistor, the intensity of light through a filter. Pharmacology uses the identical half-life language, and the reason five half-lives is the usual clinical rule for a drug being cleared is the same arithmetic that puts ten half-lives in the radiation guidance — 5 half-lives leaves about 3%.
For the electrical version of this curve, see theRC time constant calculator, where the time constant τ plays exactly the role of the mean lifetime here.
Where this model stops working
This assumes a single isotope decaying to a stable product. Many real chains go through several radioactive steps, and the intermediate products build up and decay on their own schedules — so the total activity of a sample does not follow one clean exponential. Dating methods also assume the initial amount is known, which for radiocarbon means correcting for changes in atmospheric C-14 over time.