Radioactive Decay and Half-Life: Complete A-Level Guide
Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. The rate of decay is described by the half-life — a constant characteristic of each isotope — and is entirely independent of temperature, pressure, chemical state, or any other external factor.
The Half-Life Equation
N = N₀ × (½)^(t/t½)
Where N is the number of undecayed nuclei remaining, N₀ is the initial number, t is the elapsed time, and t½ is the half-life. This applies equally to number of atoms, activity (Bq), or mass of radioactive material.
Exponential Decay Form
N = N₀ × e^(−λt) where λ = ln(2)/t½ ≈ 0.693/t½
λ is the decay constant (s⁻¹), representing the probability of decay per unit time for a single nucleus. Both equations are mathematically identical — use whichever is more convenient. The exponential form is preferred for A-Level calculations involving non-integer multiples of the half-life.
Activity: A = λN
A = λN (Becquerels, Bq)
Activity A is the number of decays per second. 1 Becquerel = 1 decay per second. Activity falls exponentially at the same rate as N: A = A₀ × e^(−λt). Because both A and N decrease at the same rate, the half-life of activity is the same as the half-life of the number of nuclei.
Half-Lives of Important Isotopes
| Isotope | Half-Life | Decay Type | Application |
|---|
| Carbon-14 (¹⁴C) | 5,730 years | β⁻ | Archaeological dating |
| Uranium-238 (²³⁸U) | 4.5 billion years | α | Geological dating |
| Potassium-40 (⁴⁰K) | 1.25 billion years | β⁻ | Rock dating, natural radioactivity in food |
| Cobalt-60 (⁶⁰Co) | 5.27 years | γ | Radiotherapy, food irradiation |
| Iodine-131 (¹³¹I) | 8.02 days | β⁻ γ | Thyroid cancer treatment |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | γ | Medical imaging (PET/SPECT) |
| Radon-222 (²²²Rn) | 3.82 days | α | Background radiation (buildings) |
| Uranium-235 (²³⁵U) | 703 million years | α | Nuclear fission fuel |
Types of Radioactive Decay
Alpha (α) decay: Nucleus emits a helium-4 nucleus (²⁴He: 2 protons + 2 neutrons). Atomic number decreases by 2, mass number decreases by 4. Alpha particles are the most ionising but least penetrating (stopped by paper or a few cm of air). Example: ²³⁸U → ²³⁴Th + ⁴He.
Beta-minus (β⁻) decay: A neutron converts to a proton + electron (emitted) + antineutrino. Atomic number increases by 1, mass number unchanged. Moderately ionising, stopped by a few mm of aluminium. Example: ¹⁴C → ¹⁴N + e⁻ + ν̄.
Gamma (γ) decay: Nucleus emits a high-energy photon to transition from an excited to ground state. No change in atomic number or mass. Least ionising, most penetrating (requires lead or thick concrete). Usually accompanies alpha or beta decay.
Carbon Dating
Living organisms continuously exchange carbon with the atmosphere, maintaining a constant ratio of C-14 to C-12. At death, absorption stops and C-14 decays at the known rate (t½ = 5,730 yr). Measuring the remaining C-14/C-12 ratio and applying N = N₀ × (½)^(t/t½) gives the time since death. Effective for dating up to ~50,000 years. Calibration using tree rings (dendrochronology) corrects for historical variations in atmospheric C-14.
Medical Applications
Medical isotopes need short half-lives so activity drops quickly after diagnosis, minimising patient dose. Technetium-99m (t½ = 6 hours) is the most widely used diagnostic isotope: injected into the patient, it emits gamma rays detected by a gamma camera to image bone, organs, and blood flow. Iodine-131 (t½ = 8 days) is used to treat thyroid cancer — the thyroid selectively absorbs iodine, and the beta radiation destroys cancer cells locally.
Nuclear Fission and Power
Uranium-235 undergoes fission when struck by a slow (thermal) neutron, splitting into two smaller nuclei plus 2–3 neutrons and releasing ~200 MeV per fission. The neutrons trigger further fissions — a chain reaction. In a nuclear reactor, control rods (boron) absorb excess neutrons to maintain a controlled chain reaction. One kilogram of U-235 contains the energy equivalent of ~3,000 tonnes of coal.
Worked GCSE Example — Half-Life
Q: A sample of Iodine-131 (t½ = 8 days) has an initial activity of 800 Bq. What is the activity after 32 days?
- Number of half-lives: n = 32/8 = 4
- A = A₀ × (½)^n = 800 × (½)⁴
- A = 800 × 1/16 = 800/16 = 50 Bq
Worked A-Level Example — Using Decay Equation
Q: Carbon-14 (t½ = 5,730 yr) in a wood sample is found to be 25% of its original level. How old is the sample?
- N/N₀ = 0.25 = (½)^(t/t½)
- ln(0.25) = (t/t½) × ln(0.5)
- −1.3863 = (t/5730) × (−0.6931)
- t/5730 = 1.3863/0.6931 = 2.000
- t = 2 × 5,730 = 11,460 years
Frequently Asked Questions
What is the half-life formula for radioactive decay?
The half-life equation is N = N₀ × (½)^(t/t½), where N is the remaining quantity, N₀ is the initial quantity, t is time elapsed, and t½ is the half-life. The alternative exponential form is N = N₀ × e^(−λt), where λ = ln(2)/t½ ≈ 0.693/t½ is the decay constant. Both give identical results. The second form is more convenient when t is not a simple multiple of t½.
What is carbon dating and how does it use half-life?
Carbon-14 is continually produced in the upper atmosphere by cosmic ray neutrons reacting with nitrogen-14. Living organisms absorb C-14 at the same rate it decays, maintaining a constant C-14/C-12 ratio. When an organism dies, absorption stops and C-14 decays with t½ = 5,730 years. By measuring the remaining C-14/C-12 ratio and applying the decay equation, archaeologists can date organic materials up to about 50,000 years old. The technique was developed by Willard Libby in 1949, for which he received the Nobel Prize in Chemistry in 1960.
What is the difference between alpha, beta and gamma decay?
Alpha (α): emits ²₄He nucleus. Atomic number −2, mass −4. Most ionising, stopped by paper or skin. Dangerous if inhaled/ingested. Example: ²³⁸U → ²³⁴Th + α. Beta-minus (β⁻): neutron → proton + electron + antineutrino. Atomic number +1. Stopped by aluminium (few mm). Example: ¹⁴C → ¹⁴N + β⁻. Gamma (γ): high-energy photon from excited nucleus. No change in composition. Penetrates lead/concrete. Used in medical imaging. Often accompanies α or β decay.
What is the decay constant λ?
λ = ln(2)/t½ ≈ 0.693/t½. It is the probability per unit time that any given nucleus will decay. Units: s⁻¹, hr⁻¹, yr⁻¹ (matching the unit of t½). Activity A = λN (Bq), where N is the number of undecayed nuclei. A higher λ means faster decay and a shorter half-life. For Carbon-14: λ = 0.693/5730 yr = 1.21×10⁻⁴ yr⁻¹ = 3.83×10⁻¹² s⁻¹.
Why is the half-life of Carbon-14 useful for dating?
5,730 years is ideal for archaeological timescales: it is short enough to cause measurable changes within the range of human history (up to ~50,000 years = ~8–9 half-lives), yet long enough that a significant fraction remains to measure. For geological ages (millions or billions of years), scientists use longer-lived isotopes: Potassium-40 (1.25 Gyr) for rocks, Uranium-238 (4.5 Gyr) for the oldest Earth materials, and Rubidium-87 (49 Gyr) for stellar chronometry.
What is background radiation?
Background radiation is low-level ionising radiation present everywhere in the natural environment. In the UK, average annual dose is ~2.7 mSv. Sources: Radon gas (50%, from uranium in granite rock), medical procedures (15%, mainly X-rays), gamma from soil/buildings (15%), food and drink (12%, mainly Potassium-40), cosmic rays (12%). Cornwall and Aberdeen have higher-than-average doses due to granite geology. Background must be subtracted from all experimental measurements of source activity.
How is radioactive decay used in nuclear power?
Nuclear power stations use controlled fission of Uranium-235. A slow neutron triggers fission: ²³⁵U + n → fission products + 2–3 neutrons + ~200 MeV energy. Control rods (boron or cadmium) absorb neutrons to control the reaction rate. The heat produced is used to generate steam, which drives turbines. A 1 GW nuclear plant uses about 25 tonnes of enriched uranium per year. By contrast, generating the same energy from coal would require ~3 million tonnes and release ~9 million tonnes of CO₂.