Half-lives
Revise Half-lives for GCSE Physics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA, Edexcel and OCR. The half-life is the time taken for the number of unstable nuclei, or the activity of a source, to halve.
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Topic overview
The half-life of a radioactive isotope is the time taken for the number of nuclei of the isotope in a sample to halve. It is also the time taken for the count-rate to fall to half its initial level.
Half-life is constant for a given isotope, however much of the sample remains. Starting with 800 counts per second and a half-life of 2 hours, after 2 hours there are 400, after 4 hours 200, and after 6 hours 100.
The net decline is often expressed as a ratio. After three half-lives the activity has fallen to one eighth of its original value, so the ratio of final to initial activity is 1:8. Counting the number of half-lives elapsed is the key step in any calculation.
Revision notes
What half-life means
The half-life is the time taken for the number of nuclei of the isotope in a sample to halve.
Equivalently it is the time for the count-rate to fall to half its initial level. It is constant for a given isotope regardless of how much remains.
Doing the calculation
Work out how many half-lives have elapsed, then halve the activity that many times.
With a half-life of 2 hours and an initial count-rate of 800 per second: after 2 hours 400, after 4 hours 200, after 6 hours 100. Three half-lives have elapsed after 6 hours.
Expressing the decline as a ratio
After n half-lives the activity is (½)ⁿ of the original.
After 1 half-life it is ½, after 2 it is ¼, after 3 it is ⅛. The ratio of final to initial activity after 3 half-lives is therefore 1:8.
Key points
- Half-life is the time for the number of nuclei to halve.
- It is also the time for count-rate to halve.
- Half-life is constant for a given isotope.
- After n half-lives the activity is (½)ⁿ of the original.
- After 3 half-lives the activity is one eighth.
- Count the number of half-lives first.
Worked examples
Example 1
A sample has a count-rate of 800 counts per second and a half-life of 2 hours. Calculate the count-rate after 6 hours. [3 marks]
Model answer
6 hours is 3 half-lives (1 mark). After each half-life the count-rate halves: 800 to 400 to 200 (1 mark), then to 100 counts per second (1 mark).
Example 2
Define half-life. [2 marks]
Model answer
The half-life is the time taken for the number of nuclei of an isotope in a sample to halve (1 mark), or equivalently the time for the count-rate to fall to half its initial level (1 mark).
Example 3
The activity of a source falls to one eighth of its original value. Calculate how many half-lives have passed. [2 marks]
Model answer
One eighth is (½)³ (1 mark), so 3 half-lives have passed (1 mark).
Common mistakes
Halving the time instead of the activity.
The half-life is a fixed time; it is the activity that halves each time.
Thinking half-life changes as the sample decays.
It is constant for a given isotope regardless of how much remains.
Miscounting the number of half-lives.
Divide the total time by the half-life before halving anything.
Saying the activity reaches zero.
It halves repeatedly, approaching zero but never reaching it exactly.
Exam tips
- Work out the number of half-lives before doing any halving.
- Show each halving step for method marks.
- Use (½)ⁿ when the number of half-lives is large.
- Express ratios as final:initial, such as 1:8.
Key terms
- Half-life
- The time for the number of nuclei or the count-rate to halve.
- Count-rate
- The decays per second recorded by a detector.
- Isotope
- An atom with the same protons but different neutrons.
- Activity
- The rate of decay of a source, in becquerels.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.