Measuring half-life
The time taken for the activity of a radioactive source to reduce by half is called the half-life of the source. The half-life could be measured using the aparatus shown below:
Before the source is used the background count rate is measured using a Geiger Muller tube connected to a counter. The count rate from the source is then measured at regular fixed intervals over a period of time.
The background count rate is subtracted from each measurement of the count rate and so the actual count rate from the source is calculated (known as the 'corrected count rate'). An example of this is shown in the table below.
| Time (hours) | Corrected count rate (counts per minute) |
| 0 | 200 |
| 1 | 110 |
| 2 | 57 |
| 3 | 35 |
| 4 | 20 |
| 5 | 13 |
| Time (hours) | 0 |
|---|---|
| Corrected count rate (counts per minute) | 200 |
| Time (hours) | 1 |
|---|---|
| Corrected count rate (counts per minute) | 110 |
| Time (hours) | 2 |
|---|---|
| Corrected count rate (counts per minute) | 57 |
| Time (hours) | 3 |
|---|---|
| Corrected count rate (counts per minute) | 35 |
| Time (hours) | 4 |
|---|---|
| Corrected count rate (counts per minute) | 20 |
| Time (hours) | 5 |
|---|---|
| Corrected count rate (counts per minute) | 13 |
A graph of the count rate of the source against time is plotted.
From the graph, the time taken for the count rate to fall by half is measured. A number of measurements are made and an average value is calculated. The average value is the half-life of the radioactive source.
For example:
\(200 \rightarrow 100 = 1.2\:hours\)
\(100 \rightarrow 50 \left (2.4 - 1.2 \right) = 1.2\:hours\)
\(50 \rightarrow 25 \left (3.6 - 2.4 \right) = 1.2\:hours\)
\(average = 1.2\:hours\)
\(half\:life\:of\:source = 1.2\:hours\)
Nuclear half-life
The graph above shows the activity of a radioactive source over a period of time.
Question
Calculate the half-life of the source used in the experiment above
The half-life is the time taken for the activity to reduce by half.
From the graph you can see that the activity goes from \(80\) to \(40 kBq\) in 6 days. It also goes from \(40\) to \(20 kBq\) in 6 days and from \(20\) to \(10 kBq\) in around 6 days.
The half-life is therefore 6 days.
Question
A radioactive source has a half-life of 15 minutes.
At a particular time the activity of the source is \(16 kBq\). What is the activity of the source one hour later?
In order to calculate the activity of the source one hour later you need to take the following steps:
1 hour = 60 minutes
60 minutes = 4 x 15 minutes = 4 half-lives
Activity after 1 half-life = 16 x 0.5 = \(8 kBq\)
Activity after 2 half-lives = 8 x 0.5 = \(4 kBq\)
Activity after 3 half-lives = 4 x 0.5 = \(2 kBq\)
Activity after 4 half-lives = 2 x 0.5 = \(1 kBq\)
Activity of the radioactive source 1 hour later = \(1 kBq\)
