What does half-life mean?

The half-life is the time for half the undecayed nuclei in a sample to decay, or for the activity to halve. After two half-lives a quarter is left, after three an eighth, and so on, whatever amount you start with.

If decay is random, how can the half-life be exact?

Each nucleus has a fixed probability of decaying per second. With a huge number of nuclei the random ups and downs cancel out almost completely, so the fraction left follows the exponential curve very closely. The spread is about √N, which is tiny compared with N when N is huge.

How are activity and the number of nuclei related?

Activity is the number of decays per second: A = λN, where λ is the decay constant. Since λ = ln 2 / T½, a sample with a short half-life has a high activity for the same number of nuclei.

What is a decay chain?

The daughter nucleus produced by a decay can itself be unstable. Then the daughters build up as the parent decays and decay away in turn. If the daughter's half-life is much shorter than the parent's, the two activities become equal after a while: secular equilibrium.

How do I work out a half-life from two readings?

Use T½ = t ln 2 / ln(N₀/N). If a count rate falls from 800 to 200 in 12 days, ln(800/200) = 1.386, so T½ = 12 × 0.693 / 1.386 = 6.0 days. T½ from counts in the Solve for panel does it with the working, for counts, activities or masses.

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Radioactive decay

Nobody can say when one nucleus will decay, but a big crowd of them halves at a steady, predictable rate. That steady rate is the half-life.

Just need the number? Half-life

Nuclei left against time

Readouts

What's happening

Every unstable nucleus has the same small chance of decaying in each second, and that chance never changes with age: a nucleus that has survived for an hour is no more likely to decay than a fresh one. The chance per second is the decay constant λ. With N nuclei, about λN decay each second, which is the activity A in becquerels. Because the number decaying is proportional to the number left, N falls exponentially, N = N₀ exp(−λt), and it halves every half-life, T½ = ln 2 / λ. With a few hundred nuclei the count wobbles around that curve: the spread after one half-life is about √(N₀/4), so bigger samples look smoother. Measuring activity over a short window gives a noisy count, like a Geiger counter clicking. In a decay chain the daughter nuclei build up as the parent decays and then decay themselves.

A = λNN = N₀ e^−λtT½ = ln 2 / λA = A₀ e^−λt

GCSE Physics (AQA, Edexcel, OCR): random nature of decay, half-life and activity. A-Level Physics: decay constant, A = λN, N = N₀ exp(−λt) and ln graphs.

Challenge

Predict first: you start with 400 nuclei and a half-life of 5 s. How many are left after 15 s? Type your answer, press Start, and compare it with the real count when the clock passes 15 s. Is your answer exactly what you see?

FAQ

What does half-life mean?: The half-life is the time for half the undecayed nuclei in a sample to decay, or for the activity to halve. After two half-lives a quarter is left, after three an eighth, and so on, whatever amount you start with.
If decay is random, how can the half-life be exact?: Each nucleus has a fixed probability of decaying per second. With a huge number of nuclei the random ups and downs cancel out almost completely, so the fraction left follows the exponential curve very closely. The spread is about √N, which is tiny compared with N when N is huge.
How are activity and the number of nuclei related?: Activity is the number of decays per second: A = λN, where λ is the decay constant. Since λ = ln 2 / T½, a sample with a short half-life has a high activity for the same number of nuclei.
What is a decay chain?: The daughter nucleus produced by a decay can itself be unstable. Then the daughters build up as the parent decays and decay away in turn. If the daughter's half-life is much shorter than the parent's, the two activities become equal after a while: secular equilibrium.
How do I work out a half-life from two readings?: Use T½ = t ln 2 / ln(N₀/N). If a count rate falls from 800 to 200 in 12 days, ln(800/200) = 1.386, so T½ = 12 × 0.693 / 1.386 = 6.0 days. T½ from counts in the Solve for panel does it with the working, for counts, activities or masses.