The half-life of a radioactive isotope is the time it takes for the number…

The idea

Half-life has two equivalent definitions and the board accepts either: the time for the number of undecayed nuclei of that isotope in the sample to halve, or the time for the count rate or activity from the sample to fall to half its starting level. The two match because each nucleus has a fixed chance of decaying each second, so the number of decays per second is proportional to the number of nuclei still left. Halve the nuclei and you halve the activity. That is why the decline is repeated halving rather than a steady fall: one half-life leaves a half, two leave a quarter, three leave an eighth. It also means a long half-life gives a low activity. Polonium-209, half-life 125 years, is far less active than polonium-210, half-life 138 days, for the same number of atoms.

What earns the marks

• The time taken for the number of undecayed nuclei of the isotope in the sample to halve.
• The alternative definition, the time for the count rate or activity to fall to half its initial level.
• Halving the corrected initial count rate and reading across to the time axis on a decay curve.
• Stating that the isotope with the longer half-life has the lower activity for the same number of atoms.

Common mistakes and how to put them right

• Defining half-life as the time for the whole sample to decay, or for half the mass to disappear.

• Halving the time axis instead of halving the count rate when reading a decay curve.

• Forgetting to subtract the background count rate before halving the reading.

• Saying the half-life gets longer as the sample runs down, when it is fixed for that isotope.

• Assuming two half-lives leave nothing, instead of a quarter of the original.

Exam tips

• Write 'undecayed nuclei of the isotope', not just 'atoms', so the definition cannot be read as the mass of the sample.
• On a graph question, mark the initial value and its half on the axis in pencil before reading across, so your working is visible.
• If the question compares two isotopes, state the activity comparison first and the half-life reason second, because the reason mark usually depends on it.