Explain how the type of radiation emitted and the half-life of an isotope determine which isotope is used for applications including: (a) household fire (smoke) alarms (b) irradiating food to kill bacteria (c) sterilisation of equipment using gamma r
5.2.4 Half‑life
Definition (AO1)
In the Cambridge IGCSE context the half‑life, \(t_{1/2}\), is the time required for the number of radioactive nuclei (or the activity) to fall to one‑half of its original value.
Mathematical form
The decay constant \(\lambda\) and the half‑life are related by
The number of undecayed nuclei after a time \(t\) is
Units used in the syllabus (AO1)
- Activity – measured in becquerels (Bq), where 1 Bq = 1 decay s⁻¹.
- Count rate – the number of decays recorded per second, written as counts s⁻¹ (or cps). This is the experimental quantity; it must be corrected for background radiation.
- Background radiation – the count rate measured with no source present. Subtract it from the raw count rate to obtain the corrected count rate.
Reading a decay curve (AO2)
A typical exponential decay curve is shown below. The point where the curve has fallen to ½ of its initial (background‑corrected) value corresponds to one half‑life.
In an exam you may be given a table of counts versus time. Locate the time at which the corrected count rate is ½ of the initial value.
| Time (s) | Raw count rate (cps) | Background (cps) | Corrected count rate (cps) |
|---|---|---|---|
| 0 | 1200 | 200 | 1000 |
| 30 | 900 | 200 | 700 |
| 60 | 680 | 200 | 480 |
| 90 | 520 | 200 | 320 |
| 120 | 400 | 200 | 200 |
In this example the initial corrected count rate is 1000 cps. Half of this is 500 cps. The corrected count rate falls to ≈500 cps between 60 s and 90 s, so the half‑life is roughly 75 s.
Worked examples (AO2)
- Core example (definition only)
Problem: A sample of an isotope has a half‑life of 12 h. How many half‑lives elapse in 36 h? What fraction of the original nuclei remain?
Solution:
- Number of half‑lives = \(\dfrac{36\ \text{h}}{12\ \text{h}} = 3\).
- After each half‑life the amount is halved, so after three half‑lives the remaining fraction is \((\tfrac12)^3 = \tfrac18\).
- Example using the decay constant
Problem: A sample contains \(8\times10^{6}\) nuclei. After 30 days a background‑corrected measurement shows \(2\times10^{6}\) nuclei remaining. Find the half‑life.
Solution:
- The sample has been reduced to \(\dfrac{2}{8}= \dfrac14\) of its original amount.
- Two half‑lives give a factor of \((\tfrac12)^2 = \tfrac14\); therefore 30 days = 2 half‑lives.
- \(t_{1/2}= \dfrac{30\ \text{days}}{2}=15\ \text{days}\).
Key isotopes used in common applications (AO1)
| Isotope | Radiation emitted | Typical energy (MeV) | Half‑life | Reason for selection |
|---|---|---|---|---|
| Americium‑241 (241Am) | α particles + low‑energy γ (0.06 MeV) | 5.5 α, 0.06 γ | 5.5 years | α particles ionise air but are stopped by a thin metal wall → safe in a sealed chamber. |
| Cobalt‑60 (60Co) | γ rays (two photons 1.17 MeV & 1.33 MeV) | 1.17, 1.33 | 5.27 years | High‑energy γ rays penetrate deeply, giving a uniform dose to packaged food or equipment. |
| Cesium‑137 (137Cs) | β particles + 0.662 MeV γ | 0.512 β, 0.662 γ | 30.1 years | Long half‑life provides a steady source for industrial radiography and some sterilisation plants. |
| Iodine‑131 (131I) | β particles + 0.364 MeV γ | 0.606 β, 0.364 γ | 8.0 days | Short half‑life limits patient exposure after thyroid treatment. |
How radiation type and half‑life determine the choice of isotope
Household fire (smoke) alarms
Isotope used: Americium‑241
- Radiation type: α particles have a range of only a few centimetres in air and are stopped by the thin metal wall of the ionisation chamber. This makes the source safe for a domestic environment.
- Half‑life: 5.5 years gives a stable activity for many years, so the alarm does not need frequent replacement, yet the activity is not so high that the detector becomes overly sensitive.
Irradiating food to kill bacteria (food‑preservation)
Isotope used: Cobalt‑60 (occasionally Cesium‑137 for lower‑energy needs)
- Radiation type: High‑energy γ photons penetrate several centimetres of food, allowing a uniform dose to be delivered through sealed packages.
- Half‑life: ≈5 years provides a high initial dose‑rate, ideal for commercial plants that treat large volumes. The gradual decay means the source remains useful for 3–5 years before replacement, balancing cost and safety.
Sterilisation of medical and laboratory equipment using γ rays
Isotope used: Cobalt‑60 (or Cesium‑137 for lower‑energy applications)
- Radiation type: γ rays easily pass through metal casings, plastic packaging and complex instrument geometries, ensuring every surface receives the required dose.
- Half‑life: A few‑year half‑life gives a high dose‑rate at the start of a sterilisation campaign and a predictable decline, allowing facilities to schedule source replacement without sudden loss of capability.
Why half‑life matters for each application (summary)
- Penetration power of the radiation – Determines whether the radiation can reach the target (α → air ionisation only; γ → deep food or equipment).
- Activity decay rate – A short half‑life gives a high initial activity but requires frequent source changes; a longer half‑life gives a steadier output over years, reducing maintenance.
- Safety and waste considerations – Low‑penetrating radiation (α) can be safely sealed; short‑lived isotopes minimise long‑term radioactive waste.
Additional points required by the Cambridge IGCSE syllabus (AO1/AO2)
- Background radiation must be measured and subtracted from the raw count rate to obtain the true activity of a source.
- The syllabus uses the becquerel (Bq) for activity, but exam questions often give count rates in counts s⁻¹. Be comfortable converting between the two when a detector efficiency is supplied.
- When interpreting a decay curve, the half‑life is the time taken for the curve to fall to ½ of its initial (background‑corrected) value.
- Remember that “half‑life” refers to the nuclei (or activity), not to the energy of the emitted radiation.
Suggested diagram for the notes
