Half-life, explained with examples
Half-life is the time it takes for half of the radioactive nuclei in a sample to decay. It lets physicists predict how a radioactive source changes over time, even though each individual decay is random.
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Radioactive decay is random
Unstable nuclei give out radiation to become more stable. This is radioactive decay. For any single nucleus, decay is random: there is no way to predict when it will happen, and nothing we do, such as heating or pressing it, changes the chance. But in a sample of billions of nuclei, the overall pattern is very predictable. That predictable pattern is described by the half-life.
Alpha, beta and gamma
Half-life questions often mention the type of radiation, so it helps to know the three main types. Alpha particles are two protons and two neutrons; they are highly ionising but stopped by paper or a few centimetres of air. Beta particles are fast electrons from the nucleus; they are moderately ionising and stopped by a few millimetres of aluminium. Gamma rays are electromagnetic waves; they are weakly ionising and need thick lead or concrete to reduce them.
Two definitions of half-life
The half-life of a radioactive isotope is:
- the time it takes for the number of unstable nuclei in a sample to halve, or
- the time it takes for the count rate (or activity) from a sample to fall to half its initial level.
Both definitions are accepted in the exam. Activity is the rate at which a source decays, measured in becquerels (Bq), where 1 Bq is one decay per second. Count rate is the number of decays recorded each second by a detector such as a Geiger-Müller tube.
Half-lives vary enormously: polonium-214 has a half-life of under a millisecond, iodine-131 about 8 days, carbon-14 about 5,700 years, and uranium-238 about 4.5 billion years.
Calculating with half-lives
After each half-life, the amount remaining halves. After 1 half-life, 1/2 remains; after 2, 1/4; after 3, 1/8; after n half-lives, (1/2)n remains.
Example 1: activity after several half-lives
A sample has an activity of 800 Bq and a half-life of 3 hours. What is its activity after 9 hours?
- 9 hours is 3 half-lives.
- 800 → 400 → 200 → 100 Bq.
Example 2: finding the half-life
The count rate from a source falls from 1200 counts per minute to 150 counts per minute in 24 days. Find the half-life.
- 1200 → 600 → 300 → 150 is 3 halvings.
- 24 days ÷ 3 = 8 days.
Example 3: mass remaining
A sample contains 64 g of a radioactive isotope with a half-life of 5 years. How much of the isotope remains after 20 years?
- 20 years is 4 half-lives: 64 → 32 → 16 → 8 → 4 g.
- Note: the decayed material has not vanished; it has become a different element.
Example 4: net decline as a ratio (Higher)
What fraction of the original activity has been lost after 3 half-lives? Give the ratio of final to initial activity.
- After 3 half-lives, 1/8 remains, so 7/8 has been lost.
- The net decline, expressed as a ratio of final to initial, is 1 : 8.
Example 5: background radiation
A detector reads 520 counts per minute near a source. The background count is 20 counts per minute. After 2 half-lives, what will the detector read?
- Subtract background first: 520 − 20 = 500 from the source.
- After 2 half-lives: 500 → 250 → 125. Add background back: 145 counts per minute.
Reading half-life from a graph
A graph of activity (or count rate, or number of nuclei) against time is a smooth curve that falls quickly at first and then more slowly, never quite reaching zero. To find the half-life:
- Read the starting value on the vertical axis.
- Halve it, and draw a horizontal line from that value to the curve.
- Draw a vertical line down to the time axis and read the time.
- For a more accurate answer, repeat from another starting point and take the average.
The time to fall from 100% to 50% is the same as from 50% to 25%, and from 25% to 12.5%. That equal spacing is the signature of radioactive decay.
Uses, and choosing the right half-life
Choosing an isotope for a job means matching its half-life and type of radiation to the task.
- Medical tracers, such as technetium-99m, need a short half-life (about 6 hours) so the patient is not exposed for long, and gamma radiation so the signal can be detected outside the body.
- Smoke alarms use americium-241, an alpha source with a half-life of over 400 years, so the alarm keeps working for its whole life.
- Thickness gauges in paper factories use beta sources with long half-lives, so the readings stay steady.
- Carbon dating uses carbon-14, with a half-life of about 5,700 years, to estimate the age of once-living material up to about 50,000 years old.
Safety: contamination and irradiation
Irradiation means being exposed to radiation from a source outside the body; it stops when you move away. Contamination means radioactive atoms get onto or into something, so exposure continues. Sources with long half-lives can stay hazardous for a very long time, which is why nuclear waste must be stored safely for thousands of years. Shielding, distance and limiting exposure time are the three main ways to reduce the dose.
Common mistakes
- Thinking the whole sample decays after two half-lives; after two, a quarter remains.
- Forgetting to subtract background radiation before halving.
- Saying the decayed nuclei disappear; they become nuclei of a different element.
- Reading the half-life from the time axis without starting from the initial value.
Practice questions
Try each one before opening the answer.
Q1. A source has activity 2400 Bq and half-life 2 days. Find its activity after 8 days.
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Q2. The activity of a source falls from 400 Bq to 50 Bq in 12 minutes. Find its half-life.
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Q3. What fraction of a sample remains after 5 half-lives?
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Q4. Why is a short half-life chosen for a medical tracer?
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Frequently asked questions
Is half-life in Combined Science?
Yes. Half-life and simple calculations are in atomic structure for both Combined Science and separate Physics. Net decline as a ratio is a Higher tier skill.
Can you predict when one nucleus will decay?
No. Radioactive decay is random. Half-life describes the behaviour of a large number of nuclei, not a single one.
Key takeaways
- Half-life is the time for the number of unstable nuclei, or the activity, to halve.
- After n half-lives, (1/2)n remains.
- Subtract background count before calculating.
- Read half-life from a graph by halving the starting value.
- Match the half-life of an isotope to its use.