Every radioactive isotope decays at its own characteristic rate. After one half-life, exactly half the original nuclei have decayed; after two, a quarter remain. This predictable pattern underpins carbon dating of ancient artefacts, medical imaging, and cancer treatment — and calculating it is a core GCSE physics skill.
What is radioactive decay and why is it random?
Radioactive nuclei are unstable. They spontaneously emit radiation — alpha particles, beta particles, or gamma rays — as they rearrange into a more stable state. This process is random: you cannot predict when any particular nucleus will decay, nor can you make it decay faster or slower by changing the temperature, pressure, or chemical environment.
However, with a very large number of atoms, statistical behaviour becomes predictable. Even though each individual decay is random, the fraction of nuclei that decay per unit time is constant for a given isotope. This gives radioactive decay its characteristic mathematical pattern.
What is half-life?
The half-life (symbol t½) of a radioactive isotope is:
The time taken for half the radioactive nuclei in a sample to decay.
Or equivalently:
The time taken for the activity (count rate) of the sample to halve.
Both definitions are correct and you may be asked to state either.
The key property of half-life: After every half-life, the remaining number of undecayed nuclei is halved. The fraction remaining after n half-lives is (1/2)ⁿ.
| Number of half-lives elapsed | Fraction of nuclei remaining | Percentage remaining |
|---|---|---|
| 0 | 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% |
| 10 | 1/1,024 | ~0.1% |
Notice that the sample never quite reaches zero — it halves repeatedly, getting smaller and smaller but theoretically never disappearing entirely (though in practice, once only a handful of atoms remain, random fluctuations dominate).
How do you calculate the number of half-lives from count rate data?
Worked example 1 — finding the number of half-lives:
A radioactive sample has an initial count rate of 800 counts per minute (cpm). After 60 minutes the count rate is 100 cpm. How many half-lives have passed?
Track the halving: 800 → 400 → 200 → 100
That is 3 halvings, so 3 half-lives have passed in 60 minutes.
Half-life = 60 ÷ 3 = 20 minutes
Worked example 2 — finding remaining activity:
A radioactive source has an initial activity of 640 Bq (becquerels). Its half-life is 8 days. What is the activity after 40 days?
Number of half-lives = 40 ÷ 8 = 5 half-lives
Activity after 5 half-lives = 640 × (1/2)⁵ = 640 × (1/32) = 20 Bq
Worked example 3 — finding age from fraction remaining:
Carbon-14 has a half-life of 5,730 years. A wooden beam has only 25% of its original carbon-14 remaining. How old is the beam?
25% = 1/4 = (1/2)² — so 2 half-lives have elapsed.
Age = 2 × 5,730 = 11,460 years
How do you read a decay graph?
A decay graph plots activity (or count rate, or number of undecayed nuclei) on the y-axis against time on the x-axis. The curve is an exponential decay — it falls steeply at first and then levels off gradually.
To find the half-life from a graph:
- Pick any starting point on the curve and note the activity value (e.g. 200 Bq).
- Find where the curve reaches half that value (100 Bq).
- Read off the time difference on the x-axis — this is the half-life.
- Check: starting from a different point, the time to halve again should be the same.
The half-life is constant regardless of where on the curve you start — this is a key feature that distinguishes exponential decay from linear decay.
Background radiation: In real experiments, the count rate never falls to zero — background radiation (from cosmic rays, naturally occurring radioactive rocks, etc.) is always present. The background count must be subtracted from all readings before plotting or calculating. Forgetting this is a common exam mistake.
What are the half-lives of important isotopes and what are they used for?
| Isotope | Half-life | Radiation type | Use |
|---|---|---|---|
| Carbon-14 | 5,730 years | Beta | Archaeological carbon dating (organic material up to ~50,000 years old) |
| Technetium-99m | 6 hours | Gamma | Medical imaging (bone scans, kidney function) |
| Iodine-131 | 8 days | Beta and gamma | Thyroid cancer treatment and diagnosis |
| Cobalt-60 | 5.3 years | Gamma | Radiotherapy for cancer; sterilising medical equipment |
| Americium-241 | 432 years | Alpha | Smoke detectors |
| Uranium-238 | 4.5 billion years | Alpha | Geological dating of rocks (alongside lead-206) |
| Radon-222 | 3.82 days | Alpha | Naturally occurring; household health hazard |
Why does the choice of half-life matter for medical uses?
In medicine, the choice of isotope depends critically on matching the half-life to the clinical need:
Technetium-99m (t½ = 6 hours) is ideal for medical imaging. The patient receives only a brief radiation dose — by the time they leave hospital, most of the radioactivity has decayed away. The 6-hour half-life is long enough for the isotope to be delivered and accumulate in target tissue (e.g. bone or kidney), but short enough to minimise long-term exposure. It also emits gamma radiation, which can penetrate the body and be detected externally by a gamma camera.
Iodine-131 (t½ = 8 days) is taken up preferentially by thyroid tissue. For treating thyroid cancer, a longer half-life provides sustained radiation dose to destroy cancerous cells. For diagnosis only, a very short half-life tracer is preferred.
Americium-241 (t½ = 432 years) in smoke detectors needs to last the lifetime of the detector (tens of years) without significant decay — hence the long half-life. It emits alpha particles, which ionise air between two electrodes to create a small electric current. When smoke enters, it disrupts this current, triggering the alarm.
How is carbon dating used and what are its limits?
Carbon dating (radiocarbon dating) was developed by Willard Libby in 1949. Living organisms continuously take in carbon — including radioactive carbon-14 produced by cosmic ray bombardment of nitrogen in the upper atmosphere. When an organism dies it stops taking in carbon; the C-14 begins to decay. By measuring the ratio of C-14 to stable C-12 in a sample, scientists calculate how long ago the organism died.
Limitations:
- Only works for organic material — not rocks or metal artefacts.
- Reliable only for material up to ~50,000 years old; for geological time, uranium–lead dating is used instead.
- Assumes the atmospheric C-14/C-12 ratio has been constant (corrected using tree-ring records — dendrochronology).
Frequently asked questions
Why is radioactive decay random but still predictable?
Radioactive decay is quantum-mechanical: there is a fixed probability per unit time that any given nucleus will decay, but no way to predict exactly when. With millions or billions of atoms in even a tiny sample, the law of large numbers guarantees that the statistical behaviour is extremely predictable — the fraction decaying per second is constant. It is similar to tossing thousands of coins simultaneously: you cannot predict any single coin, but you can reliably predict that roughly half will be heads. The half-life is a precise, measurable property of the isotope.
Does the half-life of an isotope change with temperature or chemical form?
No — this is one of the key features of radioactive decay. Unlike ordinary chemical reactions, which speed up with temperature or depend on chemical bonds, radioactive decay originates in the nucleus and is unaffected by temperature, pressure, or whether the atom is in a compound or an element. This is what makes radioactive dating reliable: the half-life of carbon-14 has been the same since the Earth formed, regardless of the chemical environment of the carbon atoms.
How many half-lives does it take for a radioactive source to become safe?
There is no universal rule — it depends on the isotope, its radiation type, and the original activity. A commonly used guideline is that after 10 half-lives, the activity has fallen to approximately 0.1% of its original level, which is usually considered negligible for most practical purposes (less than background). For Iodine-131 (t½ = 8 days), this means about 80 days. For a nuclear power plant's spent fuel (containing long-lived isotopes with half-lives of thousands to millions of years), safe storage is required for tens of thousands of years — which is why deep geological disposal is being developed.
Professor Newton at aitutors.me can run you through half-life calculation practice, help you read decay graphs step by step, and make sure you are fully confident before your GCSE physics paper.