6.23–6.26Half-lives and the random nature of radioactive decay
Radioactive decay is random.
The half-life of a radioactive isotope is the time it takes for the number of nuclei of the isotope in a sample to halve, or the time it takes for the count rate — or activity — from a sample containing the isotope to fall to half its initial level.
You should be able to explain the concept of half-life and how it is related to the random nature of radioactive decay.
The link between the two is the point of this section. It is impossible to say when any individual nucleus will decay — that is what random means. But a sample contains an enormous number of nuclei, and although each one is unpredictable, the proportion decaying in a given time is highly predictable. Half-life is a statement about that reliable average behaviour of a huge number, not a prediction about any one nucleus.
You should be able to determine the half-life of a radioactive isotope from given information — usually a table or a graph of count-rate against time. Find the initial value, halve it, and read off the time at which the curve reaches that level.
Worked example
A source has a count-rate of 800 counts per second. After 12 hours it has fallen to 100 counts per second. Calculate the half-life.
- Halve repeatedly from the starting value: 800 → 400 → 200 → 100.
- That took three halvings.
- Three half-lives occupy 12 hours.
- One half-life = 12 ÷ 3.
Answer: 4 hours
- Half-life
- the time it takes for the number of nuclei of a radioactive isotope in a sample to halve, or for the count rate from the sample to fall to half its initial level
6.27Net decline after several half-livesHigher tier
Higher tier students should be able to calculate the net decline, expressed as a ratio, in a radioactive emission after a given number of half-lives.
After n half-lives the fraction remaining is 1 ÷ 2ⁿ. The net decline expressed as a ratio compares what is left with what there was at the start.
Worked example
Calculate the net decline in the count-rate of a source after 4 half-lives, as a ratio.
- After each half-life the count-rate halves.
- After 4 half-lives the fraction remaining is 1 ÷ 2⁴.
- 2⁴ = 16, so one sixteenth remains.
- As a ratio of remaining to original, that is 1 : 16.
- Check by halving: 1 → ½ → ¼ → ⅛ → 1/16.
Answer: 1 : 16, or one sixteenth of the original count-rate remains.
Common mistake
Four half-lives does not mean the activity falls to zero, and it does not mean it falls by four halves. Each half-life halves what is left at that moment, so the decline is 1 ÷ 2ⁿ, never all the way to nothing.
6.27Different half-lives of radioactive isotopes
Physics only — not on the Combined Science specification
Radioactive isotopes have a very wide range of half-life values — from fractions of a second to billions of years. You should be able to use data presented in standard form, which is how such a range has to be written.
You should be able to explain why the hazards associated with radioactive material differ according to the half-life involved.
The answer to "which is more hazardous" is therefore "it depends what you mean", and a good answer says so. A short half-life means intense radiation now but a source that soon becomes safe. A long half-life means weaker radiation but a hazard that persists for generations — which is why radioactive waste is so difficult to deal with.
This explains the choice of isotopes in medicine: a tracer put into a patient should have a short half-life, so that it stops emitting soon after the scan is finished and the patient's total dose stays low.
Before you practise
- Activity is the rate at which nuclei decay, measured in becquerel (Bq) — one decay per second.
- Half-life is the time for half the undecayed nuclei to decay, or for the activity to halve.
- Decay is random: you cannot say when a particular nucleus will decay, only what a large number will do on average.
- After n half-lives the fraction remaining is 1 ÷ 2ⁿ.
- Read a half-life off a decay curve by finding the time for the activity to fall from any value to half of it — it is the same wherever you start.
- A short half-life means a high activity that falls quickly; a long half-life means a lower activity that persists.
- Subtract the background count before working with a measured count rate.