Half Life Calculator
Find how much of a radioactive substance is left, how long decay took, the half-life or the starting amount. Pick an isotope or enter your own half-life.
Amount left after each half-life
| Half-lives | Time | Fraction left | Amount left |
|---|
How to use the Half Life Calculator
- Choose what to solve for: remaining amount, initial amount, half-life or time.
- Pick an isotope to fill in its half-life, or type your own and choose the unit.
- Enter the other values – amounts in any unit and the elapsed time.
- Press Calculate decay to see the answer, the percentage left and the decay constant.
- Check the decay curve and the half-life table, and download the chart if you need it.
Half-life formulas
Radioactive decay is exponential: in each half-life, half of the remaining nuclei decay. With initial amount N₀, half-life t½ and elapsed time t:
- N(t) = N₀ × (½)t ÷ t½ – amount remaining
- N₀ = N × 2t ÷ t½ – initial amount
- t = t½ × log₂(N₀ ÷ N) – time elapsed
- t½ = t × ln 2 ÷ ln(N₀ ÷ N) – half-life
The same law is often written N = N₀e−λt with the decay constant λ = ln 2 ÷ t½ ≈ 0.693 ÷ t½. The mean lifetime τ = 1 ÷ λ = t½ ÷ ln 2 ≈ 1.443 × t½.
Worked examples
- Carbon dating: a sample has 25% of its original carbon-14. log₂(100 ÷ 25) = 2 half-lives, so t = 2 × 5,730 = 11,460 years.
- Medical isotope: 80 mg of iodine-131 (t½ = 8.02 days) after 30 days: 80 × (½)30/8.02 = 5.97 mg remain.
- Finding a half-life: a sample drops from 80 to 10 units in 24 hours. N₀/N = 8 = 2³, so three half-lives passed and t½ = 8 hours.
Half-lives of common isotopes
| Isotope | Half-life | Typical use |
|---|---|---|
| Fluorine-18 | 109.8 minutes | PET scans |
| Technetium-99m | 6.01 hours | Medical imaging |
| Radon-222 | 3.82 days | Indoor radon gas |
| Iodine-131 | 8.02 days | Thyroid treatment |
| Cobalt-60 | 5.27 years | Radiotherapy, sterilization |
| Tritium (H-3) | 12.32 years | Glow signs, fusion research |
| Cesium-137 | 30.08 years | Industrial gauges, fallout |
| Carbon-14 | 5,730 years | Radiocarbon dating |
| Plutonium-239 | 24,110 years | Nuclear reactors |
| Uranium-238 | 4.47 billion years | Dating rocks and the Earth |
After 10 half-lives less than 0.1% (1/1,024) of the original material remains, a common rule of thumb for when a radioactive source has effectively decayed away. The same formulas apply to drug elimination in pharmacology and to any first-order process.
Frequently asked questions
What is a half-life?
The half-life is the time it takes for half of the radioactive nuclei in a sample to decay. It is constant for each isotope: after one half-life 50% remains, after two 25%, after three 12.5%, and so on.
How do you calculate the remaining amount?
Multiply the starting amount by one half raised to the number of half-lives elapsed: N = N₀ × 0.5^(t ÷ t½). For 100 g of carbon-14 after 10,000 years that is about 29.8 g.
How does carbon dating work?
Living things keep a constant ratio of carbon-14 to carbon-12. After death the carbon-14 decays with a half-life of 5,730 years, so measuring the remaining fraction gives the age: t = 5,730 × log₂(N₀ ÷ N).
What is the decay constant?
The decay constant λ is the probability per unit time that a nucleus decays. It is related to the half-life by λ = ln 2 ÷ t½ and appears in the form N = N₀e^(−λt).
Does a substance ever fully decay?
Mathematically the amount approaches zero but never reaches it. In practice, after 10 half-lives less than 0.1% remains and after 20 half-lives less than one millionth.
Can I use this for drug half-life?
Yes. Drug elimination usually follows first-order kinetics, so the same formulas estimate how much remains after a given time. It is for information only, not dosing advice.