Quick answer: After two half-lives a quarter is left: 100 g of carbon-14 (half-life 5,730 years) leaves 100 × 0.5^(11,460 ÷ 5,730) = 25 g after 11,460 years.
How it's calculated
Exponential decay halves the amount every half-life T½: N = N₀ × (1/2)^(t/T½). Rearranged, the time is t = T½ × log₂(N₀/N) and the half-life is T½ = t × ln 2 ÷ ln(N₀/N).
The decay constant λ = ln 2 ÷ T½ is the fraction that decays per unit time, and the mean life τ = 1/λ = T½ ÷ ln 2 is the average lifetime of one atom.
The same law applies to any first-order process, such as how a medicine's level falls in the blood. Keep time and half-life in the same unit.
Worked examples
- 100 g, T½ 5,730 years, 11,460 years: N = 100 × 0.5^2 = 25 g
- Iodine-131 (T½ 8.02 days), 80 → 10 units: t = 8.02 × log₂(8) = 24.06 days
- 1,000 → 125 in 30 minutes: T½ = 30 × ln 2 ÷ ln 8 = 10 minutes
Half-lives of common isotopes
| Isotope | Half-life |
|---|---|
| Carbon-14 | 5,730 years |
| Iodine-131 | 8.02 days |
| Cobalt-60 | 5.27 years |
| Caesium-137 | 30.1 years |
| Radon-222 | 3.82 days |
| Uranium-238 | 4.47 billion years |
Frequently asked questions
How much is left after one half-life?
Half. After two half-lives a quarter is left, after three an eighth, and after ten about 0.1%.
What is the difference between half-life and mean life?
The mean life τ is the average lifetime of a single atom; it equals the half-life divided by ln 2, so it is about 1.44 times longer.
Does the starting amount change the half-life?
No. The half-life is a property of the substance, so the same fraction decays in each half-life whatever the amount.
How long until less than 1% is left?
A little under 7 half-lives (6.64, to be exact). After 7 you're down to 0.78%, and after 10 to about 0.1%.
What is the half-life of carbon-14?
About 5,730 years. That's what makes radiocarbon dating work for things up to around 50,000 years old.
How does half-life apply to medicines?
It tells you how quickly a drug's level in the blood falls. After four or five half-lives, almost all of a single dose (94 to 97%) has gone. Always stick to the dosing your doctor or pharmacist gives you.