Define half-life and use it to calculate remaining activity or the number of undecayed nuclei

What this dot point is asking

SEAB wants you to define half-life, to understand the random nature of decay, to read a decay curve, and to calculate the remaining activity or number of undecayed nuclei after a whole number of half-lives. The big idea is that although individual decays are unpredictable, a large sample halves in a fixed, characteristic time.

The answer

The random nature of decay

Radioactive decay is random: you cannot say when any one nucleus will decay. But in a large sample, a predictable fraction decays each second, so the behaviour of the whole sample is regular even though each nucleus is unpredictable.

Activity

The activity of a source is the number of nuclei that decay per second, measured in becquerels ($\text{Bq}$), where $1\ \text{Bq}$ is one decay per second. As the undecayed nuclei run out, the activity falls.

Half-life

The half-life is the time taken for half of the undecayed nuclei in a sample to decay. Equivalently, it is the time for the activity to fall to half its value. Each isotope has its own characteristic half-life, ranging from fractions of a second to billions of years.

Using half-life in calculations

After each half-life, the number of undecayed nuclei (and the activity) halves:

• After 1 half-life: $\tfrac{1}{2}$ remains.
• After 2 half-lives: $\tfrac{1}{4}$ remains.
• After 3 half-lives: $\tfrac{1}{8}$ remains.

To solve a problem, find the number of half-lives (total time divided by the half-life), then halve the starting value that many times.

The decay curve

A graph of activity (or undecayed nuclei) against time is a curve that falls steeply at first and then more gently, halving over each half-life and approaching, but never quite reaching, zero.

Examples in context

Example 1. Carbon dating. Living things contain a fixed proportion of radioactive carbon-14, which has a half-life of about $5700$ years. When an organism dies it stops taking in carbon, and the carbon-14 decays. By measuring how much remains, scientists count the half-lives that have passed and estimate the age of ancient wood, bone, or cloth.

Example 2. Choosing a medical tracer. A medical tracer should have a short half-life, long enough to do its scan but short enough that the radioactivity soon falls to a safe level inside the patient. An isotope with a half-life of a few hours has mostly decayed within a day, limiting the dose the patient receives.

Try this

Q1. Define the half-life of a radioactive isotope. [2 marks]

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

Q2. A source of activity $640\ \text{Bq}$ has a half-life of $2$ hours. Find its activity after $6$ hours. [2 marks]

• Cue. $6$ hours is $3$ half-lives: $640 \to 320 \to 160 \to 80\ \text{Bq}$.

Q3. A sample has $4.8 \times 10^{6}$ undecayed nuclei. After $3$ half-lives, how many remain? [2 marks]

• Cue. Halve three times: $4.8 \times 10^{6} \to 2.4 \times 10^{6} \to 1.2 \times 10^{6} \to 6.0 \times 10^{5}$ remaining.