5.2.4 Half-life
- Syllabus
- 0625–2026–2027
- Topic
- 5.2.4
- Level
- —
Learning objectives
5.2.4.1Half-life as the time for half the• Define half-life as the time for half the nuclei in any sample to decay; use it in simple calculations with tables or decay curves, excluding background radiation5.2.4.2Half-life from data or decay curves• Calculate half-life from data or decay curves from which background radiation has not been subtracted5.2.4.3Isotope choice using radiation type• Explain isotope choice using radiation type and half-life for: (a) smoke alarms (b) food irradiation (c) gamma sterilisation of equipment (d) material thickness control linked to penetration/absorption (e) cancer diagnosis/treatment using gamma rays
Define and use half-life
The half-life of a particular isotope is the time taken for half the radioactive nuclei in any sample of that isotope to decay.
Because activity and corrected count rate are proportional to the number of undecayed nuclei, each also halves in one half-life when the same measurement arrangement is used.
| Number of half-lives | Fraction remaining | Fraction decayed |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 1/2 | 1/2 |
| 2 | 1/4 | 3/4 |
| 3 | 1/8 | 7/8 |
For an elapsed time t and half-life T, first find the number of half-lives n = t ÷ T, then halve the starting number, mass, activity or corrected count rate n times. The amount decayed equals initial amount minus amount remaining.
A source has a half-life of 5 days and an initial mass of 400 mg. After 10 days, two half-lives have passed: 400 → 200 → 100 mg, so 100 mg remains and 300 mg has decayed.
The sample does not become zero after one or two half-lives. For the simple calculations in this objective, use source-only or corrected values: background radiation is excluded.
Find half-life when background is included
An uncorrected detector reading M contains source count rate S and background count rate B: M = S + B. Only S halves; B is treated as constant.
| Step | From raw data or a curve |
|---|---|
| 1 | identify B from the stated background or the late-time plateau |
| 2 | at time t₁, calculate source rate S₁ = M₁ − B |
| 3 | calculate the half-source target S₂ = S₁/2 |
| 4 | convert back to a detector reading M₂ = S₂ + B |
| 5 | read the time t₂ at M₂; half-life = t₂ − t₁ |
Equivalently, the detector reading after one half-life is B + (M₁ − B)/2 = (M₁ + B)/2. It is not usually M₁/2.
If the measured rate starts at 100 counts/min and background is 20 counts/min, the source rate is 80. After one half-life it is 40, so the detector reads 40 + 20 = 60 counts/min. The time for the curve to fall from 100 to 60 is one half-life.
Use more than one halving interval when the data allow it. Similar intervals support the estimate; small differences are expected because radioactive counts fluctuate randomly and graph readings have uncertainty.
Do not halve the background and do not subtract it twice. An uncorrected decay curve approaches the background level, not zero.
Choose isotopes for practical applications
Choose both the radiation type and the half-life. The radiation must penetrate or be absorbed by the intended amount of material, and the half-life must keep the source useful for long enough without causing unnecessary prolonged exposure or replacement.
| Application | Suitable radiation and why | Suitable half-life and why |
|---|---|---|
| household smoke alarm | alpha: strongly ionises air, short range, and smoke absorption changes the current | long: reliable for many years with little replacement |
| irradiating food to kill bacteria | gamma: penetrates food and packaging and kills microorganisms | long enough for repeated practical use and stable output |
| sterilising equipment | gamma: penetrates equipment and packaging and destroys microorganisms | long enough for an industrial source to remain useful |
| measuring/controlling thin material | beta for paper or thin aluminium: partly absorbed, so detector rate changes with thickness | long: output stays usable and the source is not replaced often |
| cancer diagnosis | gamma: escapes the body for external detection and is weakly ionising | short enough to reduce dose, but long enough to complete the procedure |
| cancer treatment | gamma: penetrates tissue and can destroy cancer cells | chosen to provide a controllable useful treatment source without unnecessary persistence |
For thickness control, alpha would be absorbed by even a very thin sheet, while gamma may pass through with too little change. Beta gives a measurable change in detector count when thickness changes.
A diagnostic tracer must not decay before it reaches the target and measurements are completed, but should decay soon afterwards. Treatment and sterilisation require sufficient penetration and dose, so the source arrangement and service life also matter.
There is no universal rule that every application needs the shortest or longest half-life. State why the chosen duration fits the actual operating time and exposure risk.