Half-life: what it means, reading the graph, and calculations

Half-life is the time taken for half the radioactive nuclei of an isotope in a sample to decay, which is also the time for the sample's count rate or activity to fall to half. After one half-life, half is left; after two, a quarter; after three, an eighth. Every isotope has its own half-life, from fractions of a second to billions of years.

What half-life means

Radioactive decay is random: you can't predict when any one nucleus will decay. But with a huge number of nuclei, the same fraction decays in every equal time interval, so the number left falls in a predictable pattern. Half-life is the time for half of them to go.

• It doesn't depend on how much you start with. A 1 g sample and a 1 kg sample of the same isotope have the same half-life.
• It can't be changed by heating, pressure or chemical reactions.
• It never reaches zero. Each half-life halves what is left, so in theory some always remains.

AQA's GCSE specification asks you to explain half-life and how it relates to the random nature of decay, and to find a half-life from given information.

The half-life graph

A decay curve plots the count rate (or the number of nuclei) against time. Here the count rate starts at 800 counts per minute, and the half-life is 2 hours, so x is the time in hours and y is the count rate:

How to find the half-life from a graph

1. Read the starting value on the y-axis (here 800).
2. Halve it (400) and draw a horizontal line across to the curve.
3. Drop a vertical line down to the time axis and read the time (2 hours).
4. Check by halving again: 200 should come 2 hours later, at 4 hours.

You can start from any point, not only t = 0: the time for any value to halve is always the same. If the count rate includes background radiation, subtract the background first.

Half-life calculations

How much is left after a number of half-lives

A source has an activity of 960 Bq and a half-life of 5 hours. After 15 hours, n = 15 ÷ 5 = 3 half-lives, so the activity is 960 × (½)³ = 960 ÷ 8 = 120 Bq.

Net decline (higher tier)

AQA's higher tier asks for the net decline, as a ratio, after a number of half-lives. After 3 half-lives, ⅛ of the original remains, so the activity has declined by ⅞, or the ratio of the final to the starting activity is 1 : 8.

Working out a time

How long until only 1/16 of an isotope with an 8-day half-life is left? 1/16 = (½)⁴, so it takes 4 half-lives: 4 × 8 = 32 days.

Practise half-life sums in our Nuclear Energy course, then press Mark to check your answer in the course:

Half-life at A-level: the decay constant

At A-level (AQA 3.8.1.3), each nucleus has a constant probability of decaying per second, the decay constant λ. That leads to exponential decay and a link between λ and the half-life:

For an isotope with a half-life of 6.0 hours, λ = ln 2 ÷ (6.0 × 3600 s) = 3.21 × 10⁻⁵ s⁻¹. A sample of 2.0 × 10¹⁵ nuclei has activity A = λN ≈ 6.42 × 10¹⁰ Bq. After 24 hours, four half-lives, the activity is a sixteenth of that, about 4.01 × 10⁹ Bq.

AQA also asks you to find a half-life from decay curves and from log graphs. Plotting ln A against t gives a straight line with gradient −λ, which is more accurate than reading a curve.

Why half-life matters

• Medical tracers use isotopes with short half-lives, so they are detectable during a scan but decay quickly afterwards.
• Radioactive waste with a long half-life stays hazardous for a long time, so it needs long-term storage.
• Radioactive dating compares how much of an isotope is left with how much there was, then counts the half-lives.

For the wider topic, including activity and the types of radiation, see our radioactive decay guide, and check answers in the half-life calculator.

Common mistakes

• Thinking two half-lives means all gone. After two half-lives a quarter remains.
• Subtracting instead of halving. Each half-life halves what is left, not the original amount.
• Forgetting background radiation when reading a measured count rate.
• Mixing units at A-level. Use seconds for λ in s⁻¹ and activity in becquerels.

Practise with a short timed paper

Half-life is in GCSE Physics and in the second year of A-level Physics. The daily GCSE Physics Higher paper, and the The daily A-level Physics: Year 2 paper are short timed papers of 8 questions, with a new one every day.

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Physics · Atoms and particlesHalf-life calculator: amount left, half-life and timeFree half-life calculator: find the amount left after any time, the half-life from your data, or the time to decay, with every step and a decay curve.