Half-Life

Half-life — the average time taken for half the undecayed nuclei in a sample to decay, or equivalently for the activity of the sample to fall to half its value.

Both definitions are acceptable and both appear in mark schemes. The word average matters, because decay is random — half-life is a statistical statement about a large number of nuclei, not a promise about any particular one.

Why a sample never quite disappears. Each half-life removes half of what remains, not a fixed amount:

Halving repeatedly approaches zero without reaching it, which is why decay curves flatten out but never touch the axis.

Reading a half-life from a graph. Find the initial activity, halve it, and read across to the curve and down to the time axis. Doing this from several starting points and averaging is better practice, because a decay curve is noisy — the half-life measured from 800 to 400 should match the one from 400 to 200, and averaging reduces the effect of scatter.

Calculating with half-lives:

1. Work out how many half-lives have passed: n = total time ÷ half-life
2. Halve the starting quantity n times.

Example: a source of activity 800 Bq has a half-life of 6 hours. After 24 hours, n = 24 ÷ 6 = 4 half-lives have passed, so the activity is 800 → 400 → 200 → 100 → 50 Bq.

Working backwards. If a question asks how long until the activity falls from 960 Bq to 60 Bq, count the halvings: 960 → 480 → 240 → 120 → 60 is four halvings, so four half-lives have passed.

Half-lives vary enormously — from fractions of a second to billions of years. Uranium-238 has a half-life of 4.5 billion years, which is why it is still present in the Earth's crust. Carbon-14's 5730 years makes it useful for dating objects thousands of years old, and no use at all for dating dinosaurs.

Choosing a source by half-life. A medical tracer needs a half-life of hours — long enough for the scan, short enough that the patient is not left radioactive. A smoke alarm needs one of hundreds of years, so it does not need replacing.