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)

The half‑life (t_½) of a radioactive isotope is the time required for half of the nuclei in a sample to decay.

Decay law, decay constant and activity (AO1)

The number of undecayed nuclei N after a time t is given by

where N₀ is the initial number of nuclei.

The same law can be written with the decay constant λ:

The activity A (decays s⁻¹, measured in becquerels, Bq) is related to the number of nuclei by

Thus a short half‑life → large λ → high activity; a long half‑life → small λ → low, steady activity.

Quick‑check calculations (AO2)

Example 1 – integer number of half‑lives

• Sample: 8 × 10⁶ atoms, t_½ = 2 h, elapsed time = 6 h.
• Number of half‑lives: \( \frac{6}{2}=3 \).
• Remaining atoms: \( N = 8\times10^{6}\left(\frac12\right)^{3}=1\times10^{6} \) atoms.

Example 2 – non‑integer half‑life

• Sample: 5 × 10⁶ atoms, t_½ = 3 h, elapsed time = 5 h.
• Use the exponential form:
  

  \( N = N_{0}e^{-\lambda t}=5\times10^{6}\,e^{-\ln2\,(5/3)} \)

• Calculate \( \lambda =\frac{\ln2}{3}=0.231\ \text{h}^{-1} \) and then
  

  \( N \approx 5\times10^{6}\,e^{-0.231\times5}\approx5\times10^{6}\,e^{-1.155}\approx5\times10^{6}\times0.315\approx1.6\times10^{6} \) atoms.

Reading decay curves and tables (AO2)

• Decay curve: a graph of activity (or count rate) versus time. The curve is exponential; the time taken for the curve to fall to half its initial value is the half‑life.
• Decay table: a table giving the activity (or number of nuclei) at regular time intervals. The half‑life can be found by locating the interval where the activity has fallen to ½ of the starting value.

Practice question – A decay table shows that a source has an activity of 800 Bq at 0 h and 200 Bq at 6 h. Estimate the half‑life.

Solution outline – The activity has reduced to ¼ of its original value in 6 h, i.e. two half‑lives have elapsed. Therefore the half‑life ≈ 3 h.

Why half‑life matters for practical uses

• Activity level: Determines how much radiation is available for the intended purpose.
• Service life: A long half‑life means the source can be used for years without replacement; a short half‑life gives a high dose but must be replaced frequently.
• Safety and handling: Short‑lived isotopes become harmless quickly, whereas long‑lived isotopes remain hazardous and need robust shielding.
• Radiation type: The emitted particle or photon (α, β, γ, X‑ray) decides the penetration ability and the kind of shielding required.

Radiation types and their properties

Isotopes commonly used in the applications below

Application (a): Household fire (smoke) alarms

• Isotope used: Americium‑241 (α emitter).
• How it works: α particles ionise the air in a sealed chamber, producing a small electric current. Smoke particles attach to the ions, reducing the current and triggering the alarm.
• Why the half‑life matters: 432 years means the source remains effective for many decades, so the alarm never needs a radioactive‑source replacement.
• Why α radiation is chosen: Its very low penetration ensures the radiation cannot escape the detector, making the device safe for domestic use.

Application (b): Irradiating food to kill bacteria

• Isotope used: Cobalt‑60 (γ emitter).
• Mechanism: High‑energy γ photons pass through the packed food, breaking DNA strands in bacteria and other microorganisms, thereby sterilising the product without heating it.
• Half‑life relevance: 5.27 years provides a stable dose rate for several years; after this the source is replaced in a planned maintenance schedule.
• Radiation choice: γ rays have the required penetration to give a uniform dose throughout bulk items (spices, meat, dried herbs).
• Safety: Thick lead or concrete shielding protects operators; the food receives only the intended dose.

Application (c): Sterilisation of medical and laboratory equipment (gamma sterilisation)

• Isotope used: Cobalt‑60 (γ emitter).
• Process: Items are placed in a sealed “gamma chamber”. The γ photons penetrate packaging and destroy bacterial spores, allowing sterilisation of heat‑sensitive instruments.
• Half‑life benefit: The 5.27‑year half‑life gives a predictable, long‑term source; dose rates can be accurately calculated for each batch.
• Typical dose: About 25 kGy is required to achieve a 10⁶ reduction in spore count.
• Shielding & safety: The chamber is surrounded by lead and concrete; remote handling eliminates occupational exposure.

Key take‑away points

1. The half‑life determines how long a radioactive source remains useful and how its activity changes over time.
2. The type of radiation (α, β, γ) dictates penetration ability, shielding requirements, and suitability for a particular application.
3. For each practical use we select an isotope whose half‑life and radiation type give the optimum balance of safety, effectiveness, and economic lifespan.
4. Understanding the decay law, the activity‑half‑life relationship, and the attenuation equation \(I = I_{0}e^{-\mu x}\) enables accurate dose calculations and appropriate shielding design.

Summary table – Matching isotope, half‑life and radiation to the three applications