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Here is a strange fact about radioactivity. Take a single atom of iodine-131 and ask when it will decay. Nobody can tell you. It might be in the next second or a year from now. But take a trillion of them, and you can predict with great precision that in 8 days, half will be gone.
That predictable “half gone” interval is the half-life, and it’s one of the most useful numbers in science — used to date fossils, time medical treatments, and plan how long nuclear waste must be stored.
The definition
The half-life (t½) of a radioactive isotope is the time it takes for half of the nuclei in a sample to decay.
After one half-life, 50% remains. After two, half of that half: 25%. After three, 12.5%. Each half-life halves whatever is left:
| Half-lives elapsed | Fraction remaining |
|---|---|
| 0 | 100% |
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
| 10 | about 0.1% |
The crucial point is that the half-life doesn’t depend on how much you start with. A gram of iodine-131 and a tonne of it both lose half their iodine-131 atoms in the same 8 days.
Why it works that way
Every unstable nucleus has a fixed probability of decaying in any given moment. It doesn’t “age” and it has no memory — a nucleus that has existed for a million years is exactly as likely to decay in the next second as one created a moment ago. With a huge number of nuclei each playing the same dice game, the fraction that decays per unit time is constant, which produces exponential decay. That’s also why half-life can’t be changed by heating, cooling or chemically combining the atom: decay happens in the nucleus, far from the electrons that chemistry acts on.
The formula
N = N₀ × (½)^(t ÷ t½)
- N₀ = starting amount
- N = amount remaining after time t
- t½ = half-life
The amounts can be in any unit — grams, number of atoms, radioactivity in becquerels, percent — as long as N and N₀ use the same one. Likewise t and t½ just need matching time units.
Rearranged to find elapsed time:
t = t½ × log₂(N₀ ÷ N)
Worked example 1: amount remaining
A hospital receives 20 mg of iodine-131 (t½ = 8.02 days). How much is left after 24 days?
24 ÷ 8.02 = 2.99 half-lives, so almost exactly three.
N = 20 × (½)^2.99 ≈ 2.5 mg
Worked example 2: elapsed time
A wooden artifact has 25% of the carbon-14 found in living wood. How old is it? (t½ of C-14 ≈ 5,700 years)
25% remaining is two half-lives: 100 → 50 → 25.
t = 2 × 5,700 = about 11,400 years
For amounts that aren’t neat fractions, use the log formula. If 60% remained: t = 5,700 × log₂(100 ÷ 60) = 5,700 × 0.737 = about 4,200 years. We cover the method, and its limitations, in how carbon-14 dating works.
Worked example 3: finding the half-life
A sample’s activity falls from 800 to 100 counts per minute in 30 hours. What is the half-life?
800 → 400 → 200 → 100 is three halvings. 30 hours ÷ 3 = 10 hours.
Real half-lives
Half-lives range from far less than a second to far longer than the age of the universe. These values come from the IAEA’s nuclear data, which is also what the isotope tables on our element pages use.
| Isotope | Half-life | Where you meet it |
|---|---|---|
| Fluorine-18 | ~110 minutes | PET scans |
| Radon-222 | ~3.8 days | Natural radon gas in homes |
| Iodine-131 | ~8.0 days | Thyroid treatment; fallout |
| Phosphorus-32 | ~14.3 days | Biochemistry research |
| Cobalt-60 | ~5.3 years | Radiotherapy; sterilizing food and equipment |
| Hydrogen-3 (tritium) | ~12.3 years | Glow-in-the-dark exit signs and watch dials |
| Strontium-90 | ~28.9 years | Fission product |
| Cesium-137 | ~30.1 years | Fission product; Chernobyl and Fukushima contamination |
| Americium-241 | ~433 years | Household smoke detectors |
| Radium-226 | ~1,600 years | Historical luminous paint |
| Carbon-14 | ~5,700 years | Radiocarbon dating |
| Plutonium-239 | ~24,100 years | Nuclear fuel and weapons |
| Uranium-235 | ~704 million years | Nuclear reactor fuel |
| Potassium-40 | ~1.25 billion years | Present in every human body; potassium-argon dating |
| Uranium-238 | ~4.47 billion years | About the age of the Earth |
The choice of isotope in medicine is all about half-life. Fluorine-18 lasts long enough to be made, shipped and scanned, but decays within a day, so the patient’s exposure is short.
Short half-life, high activity
A shorter half-life means more decays per second from the same number of atoms. That’s why short-lived isotopes are often the most intensely radioactive, while uranium-238, with its enormous half-life, is only weakly radioactive per gram. The decay rate is linked to half-life through the decay constant, λ = ln 2 ÷ t½.
Quick answers
Does anything ever reach zero? Mathematically, no. In practice, after about ten half-lives less than 0.1% remains, and eventually the last atoms decay.
Can temperature change the half-life? No — nuclear decay is unaffected by ordinary temperature, pressure and chemistry (with tiny exceptions for a few electron-capture isotopes).
Is half-life the same as “how long it’s dangerous”? Not exactly. Danger depends on the type of radiation, how much there is, and how it gets into the body. A common rule of thumb for waste is 10 half-lives.
Calculate it
The half-life calculator has real half-lives for common isotopes built in, solves for any of the four quantities and draws your position on the decay curve. For more on the particles emitted during decay, see alpha, beta and gamma radiation.
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