Half-Life & Exponential Decay
Radioactive decay removes a constant fraction, not a constant number, in equal time intervals. Half-life is the time for one repeated halving.
Each half-life halves what remains
Definition
1 markWhat is meant by the half-life of a radioactive nuclide?
Model answer: The half-life is the time taken for the activity, or number of undecayed nuclei, to decrease to half its value.
(9702/42/F/M/25 Q9(b))
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| Elapsed time | Fraction remaining | If the initial value is 800 |
|---|---|---|
| 0 | 1 | 800 |
| one half-life | 1/2 | 400 |
| two half-lives | 1/4 | 200 |
| three half-lives | 1/8 | 100 |
The amount lost becomes smaller in each interval: 400, then 200, then 100. The fraction lost remains one half. This is why the graph becomes less steep but does not become a straight line.
A corrected count rate is 320 s−1. What is it after two half-lives?
Show worked answer
Two halvings give 320 → 160 → 80 s−1.
A larger decay constant gives a shorter half-life
A large decay constant means each nucleus has a greater probability of decaying per second. The sample therefore reaches half its initial value sooner.
Symbols
- = decay constant (s⁻¹)
- = half-life (s)
- For one nuclide, the half-life is constant.
- It does not depend on how many nuclei are currently present.
- The time unit must match the reciprocal unit of the decay constant.
A nuclide has decay constant . Calculate its half-life.
Show worked answer
The half-life is 23 s.
Common mistake