How radioactive decay works
A radioactive isotope has unstable nuclei that change into other elements by emitting radiation. Each nucleus decays at random, but in a large sample the pattern is very regular: in a fixed time, the half-life, half of the remaining nuclei decay. That makes radioactivity a reliable clock, used to date fossils and rocks and to plan medical treatments.
The formulas
- Remaining amount N = N0 × (1/2)^(t ÷ t½)
- Equivalently N = N0 × e^(−λt), with the decay constant λ = ln 2 ÷ t½
- Age from the share remaining t = t½ × log2(N0 ÷ N)
- Activity, in becquerels or curies, follows the same curve as the number of nuclei
Worked example: carbon-14
A sample starts with 100 units of carbon-14 (half-life 5,730 years). How much is left after 1,000 years?
- Half-lives elapsed: 1,000 ÷ 5,730 = 0.175.
- Remaining: 100 × 0.5^0.175 = 88.6.
- 11.4% has decayed.
- It takes about 38,000 years to fall below 1%.
Radiocarbon dating
While alive, plants and animals take in carbon with the same carbon-14 share as the atmosphere. After death, the carbon-14 decays and is not replaced. Measuring the share left gives the age:
| Carbon-14 left | Age (years) |
|---|---|
| 90% | about 870 |
| 75% | about 2,380 |
| 50% | 5,730 |
| 25% | 11,460 |
| 10% | about 19,000 |
| 1% | about 38,000 |
Beyond roughly 50,000 years, too little is left to measure. Labs also calibrate raw ages against tree rings and other records, because atmospheric carbon-14 has varied over time.
Half-lives of common isotopes
| Isotope | Half-life | Use or source |
|---|---|---|
| Technetium-99m | 6.0 hours | Medical imaging |
| Iodine-131 | 8.0 days | Thyroid treatment, nuclear accidents |
| Cobalt-60 | 5.3 years | Radiotherapy, sterilization |
| Tritium | 12.3 years | Glow-in-the-dark signs, fusion research |
| Cesium-137 | 30.1 years | Fallout from Chernobyl and Fukushima |
| Carbon-14 | 5,730 years | Archaeological dating |
| Uranium-238 | 4.47 billion years | Dating the age of the Earth |
Practical rules of thumb
- After 7 half-lives, less than 1% remains; after 10, less than 0.1%.
- Short half-lives mean intense but brief radioactivity, which is why medical isotopes like technetium-99m are chosen.
- Long half-lives mean weak but lasting radioactivity: uranium-238 has decayed only by about half since the Earth formed.