Random Decay and Half-Life

Random events, predictable patterns

An unstable atomic nucleus can emit radiation and become more stable. This change is called radioactive decay. Decay is random: we cannot predict exactly when any particular nucleus will decay.

That does not mean a large radioactive sample behaves without a pattern. Although individual decays are unpredictable, the overall rate of decay in a sample containing very many unstable nuclei can be predicted. As nuclei decay, fewer undecayed unstable nuclei remain, so the rate of further decays decreases.

The activity of a sample is the number of nuclear decays occurring per second. It is measured in becquerels, Bq: an activity of 800 Bq means that 800 nuclei decay each second. As the number of undecayed unstable nuclei falls, the activity falls too.

Half-life describes repeated halving

The half-life of a radioactive isotope is the time taken for half the unstable nuclei in a sample to decay. It is also the time taken for the activity of the sample to fall to half its starting value.

A particular isotope has a constant half-life. Each half-life halves what is still undecayed, rather than removing half the original sample every time. After one half-life, half the original unstable nuclei remain. After two half-lives, half of that half remains: one quarter of the original number. After three, one eighth remains.

This is a prediction about the behaviour of a large sample, not a timetable for individual nuclei. A particular nucleus might decay almost immediately or remain undecayed through several half-lives. Half-life does not tell us which nuclei will decay next.

Reading half-life from an activity–time graph

An activity–time graph shows time on the horizontal axis and activity on the vertical axis. The curve falls steeply at first, when many unstable nuclei remain, then becomes less steep as the activity decreases. Equal time intervals produce equal proportional decreases, not equal numerical decreases.

Suppose a sample starts with an activity of 800 Bq and has a half-life of 5 minutes. Its predicted activity follows this pattern:

Data for Finding half-life from an activity–time graph

To find the half-life from the graph, start at 800 Bq on the activity axis and halve it to 400 Bq. Draw or trace a horizontal line from 400 Bq to the curve, then a vertical line down to the time axis. The activity reaches 400 Bq at 5 minutes, so the half-life is 5 minutes.

You can check using a different pair of activities. The activity falls from 400 Bq at 5 minutes to 200 Bq at 10 minutes. The elapsed time is $10-5=5$ minutes: the same half-life. It then takes another 5 minutes to fall from 200 Bq to 100 Bq.

The successive decreases are 400 Bq, 200 Bq and 100 Bq. They become smaller, even though each decrease takes the same time. This is why the graph is a curve rather than a straight line.

Calculating what remains and what has been lost

For a whole number of half-lives, repeatedly divide the initial activity by two. If $n$ is the number of half-lives, then:

$$\frac{\text{remaining activity}}{\text{initial activity}}=\frac{1}{2^n}$$

The net decline is the amount by which the activity has decreased. First find the remaining activity, then subtract it from the initial activity:

$$\text{net decline}=\text{initial activity}-\text{remaining activity}$$

To express that decline relative to the initial activity:

$$\frac{\text{net decline}}{\text{initial activity}}=1-\frac{1}{2^n}$$

In the graph example, 15 minutes is three half-lives. The activity falls through $800\rightarrow400\rightarrow200\rightarrow100$ Bq. The net decline is therefore $800-100=700$ Bq.

The ratio net decline : initial activity is $700:800$, which simplifies to $7:8$. In other words, seven eighths of the initial activity has been lost. By contrast, the ratio remaining activity : initial activity is $100:800=1:8$. These describe the lost and remaining portions of the same starting activity.

Random decay and half-life

• The decay time of an individual nucleus cannot be predicted.
• Large samples show a predictable overall pattern.
• Half-life is the time for half the unstable nuclei to decay, or for activity to halve.
• A particular isotope has a constant half-life; each interval halves what remains.
• Activity is measured in Bq: 1 Bq = one decay per second.

Whole numbers of half-lives

After $n$ half-lives:

$$\text{fraction remaining}=$$