Half-life: the definition of half-life, the random nature of decay, and using half-life to calculate the activity or amount of radioactive material remaining.

What this dot point is asking

Edexcel wants you to define the half-life of a radioactive isotope, to understand that radioactive decay is a random process, to know that activity is measured in becquerels, and to use half-life to calculate the activity or number of undecayed nuclei remaining after a whole number of half-lives.

Defining half-life

Half-life is a fixed property of each isotope, ranging from fractions of a second to billions of years. Because exactly half decay in each half-life, the amount never reaches zero; it just keeps halving. Either definition (half the nuclei, or half the activity) is accepted, because the activity is proportional to the number of undecayed nuclei.

The random nature of decay

Randomness is why we work with averages and half-lives rather than exact times. It is like rolling a huge number of dice: you cannot say which die will show a six next, but you can predict roughly how many sixes appear overall. This is also why the decay of a single nucleus cannot be predicted, only the statistical behaviour of many.

Activity and the becquerel

Activity falls in step with the number of undecayed nuclei, so the same halving rule applies to both. A detector such as a Geiger-Muller tube measures the count rate, which is related to the activity (after subtracting the background count). The greater the activity, the more decays are detected each second.

Calculating what remains

The reliable method is: first work out how many half-lives have passed (total time divided by the half-life), then halve the starting amount that many times. This works for the number of nuclei, the mass, or the activity, since all fall by the same factor.

How Edexcel examines this

This dot point is examined on both tiers and is heavily tested. The classic calculation gives a half-life, a starting activity (or number of nuclei) and a total time, and asks for the amount remaining; the mark scheme rewards finding the number of half-lives, halving the amount that many times, and the correct final value. The most common error, dividing by the number of half-lives instead of halving repeatedly, loses the marks, so always halve step by step. Definition questions reward stating the half-life as the time for half the undecayed nuclei to decay (or for the activity to halve) and explaining randomness as being unable to predict which nucleus decays or exactly when. Higher-tier questions may give a decay graph or a table and ask you to read off the half-life (the time for the count to fall to half) or to find a fraction such as "what fraction remains after 3 half-lives" ($\frac{1}{8}$). Working in standard form is common when the number of nuclei is large, so keep the powers of ten consistent.

Try this

Q1. Define the half-life of a radioactive isotope. [1 mark]

• Cue. The time taken for half the undecayed nuclei in a sample to decay (or for the activity to halve).

Q2. A sample has an activity of $240\,\text{Bq}$ and a half-life of 3 hours. Calculate its activity after 6 hours. [2 marks]

• Cue. 2 half-lives: $240 \rightarrow 120 \rightarrow 60\,\text{Bq}$.