Activity and half-life

However, a source usually contains a very large number of nuclei. Although the individual events are random, their combined behaviour follows a predictable pattern. We can predict how the activity of a large sample will decrease without knowing which nuclei will decay next. Small samples and short measurements show greater random fluctuations.

Half-life is a constant halving time

The half-life of a radioactive isotope is the time taken for half its undecayed nuclei to decay. Equivalently, it is the time taken for the activity of a source to fall to half its value.

Each isotope has its own half-life. For a particular isotope, the half-life stays constant: halving the activity takes the same time whether the activity is initially high or already much lower.

For example, consider a source with an initial activity of 800 Bq and a half-life of 2 hours:

Each 2-hour interval halves the activity remaining. The decreases become smaller—400 Bq, then 200 Bq, then 100 Bq—but the time needed for each halving is unchanged.

Reading half-life from a graph

The graph below shows this source’s predicted activity.

Data for Equal half-lives on an activity–time graph

To read the half-life, choose an activity on the vertical axis, move horizontally to the curve, then vertically down to read the time. Repeat for half that activity and subtract the two times.

Here, 800 Bq occurs at 0 hours and 400 Bq at 2 hours, so the half-life is 2 hours. You can check this using another pair: 200 Bq occurs at 4 hours and 100 Bq at 6 hours. The interval is again 2 hours.

A graph of the number of undecayed nuclei against time can be read in exactly the same way: find how long the number takes to halve.

Calculating how much remains

First work out how many half-lives have passed:

$$\text{number of half-lives}=\frac{\text{elapsed time}}{\text{half-life}}$$

Then halve the starting activity or number of undecayed nuclei that many times. After one, two and three half-lives, the fractions remaining are $\frac12$, $\frac14$ and $\frac18$ respectively.

For a sample containing initially 80 000 undecayed nuclei, with a half-life of 15 minutes, 45 minutes represents $45\div15=3$ half-lives. Repeated halving gives:

$$80\,000\rightarrow40\,000\rightarrow20\,000\rightarrow10\,000$$

So 10 000 nuclei of the original isotope remain undecayed. The number that has decayed is $80\,000-10\,000=70\,000$, or $\frac78$ of the initial number. These are predicted values for the sample, not a timetable for individual nuclei.

You can also work backwards to find a half-life. If a source’s activity falls from 1200 Bq to 300 Bq in 10 hours, the sequence $1200\rightarrow600\rightarrow300$ shows two halvings. Its half-life is therefore $10\div2=5$ hours.

• Half-life: time for half the undecayed nuclei to decay, or for activity to halve.
• Constant for a particular isotope.
• Individual decay times are unpredictable; the activity of a very large population can be predicted.

Calculations and graphs

$$\text{number of half-lives}=\frac{\text{elapsed time}}{\text{half-life}}$$

After 1, 2, 3 and 4 half-lives, the fractions remaining are $\frac12$, $\frac14$, $\frac18$ and $\frac1{16}$.

• Repeatedly halve the starting activity or number of undecayed nuclei.
• Number decayed = initial number − number remaining.
• On a graph, half-life is the time interval between any value and half that value.