Radioactivity » Half-Life Calculations and Graphs
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 7.13
- How to work out the activity after 1, 2, 3 or more half-lives
- How to find a half-life, or a time, from activities
- How to subtract background radiation before you start
- How to read the half-life from an activity-time graph
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Unlock This CourseHalving again and again
In Activity and Half-Life you learnt that the half-life is the time taken for the activity to halve. Every half-life, the activity halves again. The best way to show this is a halving table.
A source has an activity of 640 Bq. Its half-life is 3 days.
| Time (days) | Half-lives passed | Activity (Bq) |
|---|---|---|
| 0 | 0 | 640 |
| 3 | 1 | 320 |
| 6 | 2 | 160 |
| 9 | 3 | 80 |
| 12 | 4 | 40 |
Notice the pattern. Each row, the time goes up by one half-life and the activity is divided by 2. After 12 days (4 half-lives) the activity is 40 Bq.
Key terms:
- Halving table: a table that halves the activity once for every half-life that passes.
The fraction remaining
The same pattern gives a quick shortcut. After each half-life, the activity is half of what it was.
| Half-lives passed | Fraction of the activity left |
|---|---|
| 0 | 1 (all of it) |
| 1 | 1/2 |
| 2 | 1/4 |
| 3 | 1/8 |
| 4 | 1/16 |
Worked example
A source has an activity of 3200 Bq. What is its activity after 5 half-lives?
3200 → 1600 → 800 → 400 → 200 → 100. That is five halvings.
Check with the fraction: 5 half-lives leaves 1/32, and 3200 ÷ 32 = 100 Bq.
Answer: 100 Bq.
Three types of question
🔢 Find the activity
You are given the half-life and the time. Work out how many half-lives have passed (time ÷ half-life), then halve that many times.
⏳ Find the time
You are given two activities and the half-life. Count how many halvings get you from one activity to the other, then multiply by the half-life.
🔍 Find the half-life
You are given two activities and the time between them. Count the halvings, then divide the time by that number.
Worked example: finding the time
A source has an activity of 2400 Bq. Its half-life is 20 minutes. How long does it take for the activity to fall to 150 Bq?
2400 → 1200 → 600 → 300 → 150. That is 4 halvings, so 4 half-lives.
Time = 4 × 20 = 80 minutes.
Shortcut: 150 ÷ 2400 = 1/16, which is 4 half-lives.
Worked example: finding the half-life
The activity of a sample falls from 1600 Bq to 200 Bq in 36 hours. What is its half-life?
1600 → 800 → 400 → 200. That is 3 half-lives.
Half-life = 36 ÷ 3 = 12 hours.
Worked example: an old sample
A sample of old wood has an activity one quarter of the activity of a fresh piece of the same wood. The half-life of the carbon-14 in it is 5700 years. One quarter is 2 half-lives, so the wood is 2 × 5700 = 11 400 years old.
Subtract the background first
In Detecting Radiation and Background Radiation you learnt that a detector also picks up background radiation. The half-life applies only to the source. So subtract the background count rate from every reading first, then halve.
Worked example
A detector reads 220 counts per minute near a source. The background is 20 counts per minute. Six hours later the detector reads 70 counts per minute. What is the half-life?
Start: 220 − 20 = 200 counts per minute.
Later: 70 − 20 = 50 counts per minute.
200 → 100 → 50. That is 2 half-lives.
Half-life = 6 ÷ 2 = 3 hours.
Common mistakes
- Dividing the time by 2 instead of by the half-life. First work out the number of half-lives.
- Counting the wrong number of halvings. Write the numbers out in a row and count the arrows.
- Halving only once. 3 half-lives means halving 3 times, not dividing by 3.
- Forgetting to subtract the background. Do it to every reading, before you halve.
Reading the half-life from a graph
An activity-time graph has time on the horizontal axis and activity on the vertical axis. It is a smooth curve that starts high and falls steeply, then flattens out. It never quite reaches zero.
Here are some readings from one such graph:
| Time (min) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| Activity (Bq) | 1000 | 794 | 630 | 500 | 397 | 315 | 250 |
To find the half-life from the graph, follow these steps:
- Pick an activity on the curve. Use a point you can read easily, such as 1000 Bq at 0 minutes.
- Halve it. Half of 1000 Bq is 500 Bq.
- Draw a line across from 500 Bq to the curve, then down to the time axis. Read off 6 minutes.
- Subtract the two times: 6 − 0 = 6 minutes. That is the half-life.
- Repeat from a different starting point to check. At 4 minutes the activity is about 630 Bq. Half is about 315 Bq, which is at 10 minutes. 10 − 4 = 6 minutes again.
If the two answers agree, you can be confident. In the exam, draw your lines on the graph so the examiner can see your method.
Common mistakes with graphs
- Reading off the time for half the activity and stopping. You must subtract the two times, because the half-life is the gap between them.
- Subtracting activities instead of times. The half-life is a time.
- Forgetting the background. If the graph shows count rate, subtract the background first.
Exam-style question
A sample of a radioactive isotope has an activity of 2000 Bq. After 15 hours its activity is 250 Bq.
(a) Calculate the half-life of the isotope. [3 marks]
(b) Calculate the activity of the sample after a further 10 hours. [2 marks]
Model answer
(a) 2000 → 1000 → 500 → 250 (1), so 3 half-lives have passed (1). Half-life = 15 ÷ 3 = 5 hours (1).
(b) 10 ÷ 5 = 2 half-lives (1). 250 → 125 → 62.5 Bq (1).
Exam tip
Always show the halving, with the numbers in a row, even if you can do it in your head. The marks are for the method.