Radioactivity and Nuclear Energy

The nucleus

Rutherford's alpha-scattering experiment showed the atom is mostly empty space with a tiny, dense, positive nucleus (most alpha particles passed through; a few deflected sharply). The nucleus contains protons and neutrons (nucleons); nuclear radius is of order 10⁻¹⁵ m (femtometres), with R = r₀A^(1/3), so nuclear density is roughly constant.

Radioactive decay

Unstable nuclei decay, emitting radiation:

• Alpha (α): a helium nucleus; highly ionising, stopped by paper.
• Beta-minus (β⁻): a fast electron (a neutron → proton + electron + antineutrino); stopped by aluminium.
• Gamma (γ): high-energy EM wave; very penetrating, reduced by lead.

Decay is random and spontaneous — you can't predict which nucleus decays or when, but you can treat large numbers statistically.

Half-life and the decay equation

• Activity A = λN (λ = decay constant, N = number of undecayed nuclei), in becquerels.
• N = N₀e^(−λt); activity A = A₀e^(−λt).
• Half-life t½ = ln2 ÷ λ — the time for half the nuclei (or activity) to decay. Used in radioactive dating (e.g. carbon-14).

Mass–energy and nuclear reactions

Einstein's E = mc² links mass and energy. In nuclear reactions, a small mass difference (mass defect) is converted to energy.

• Binding energy = energy needed to separate a nucleus into its nucleons (= the energy released when it formed).
• Binding energy per nucleon is greatest around iron-56 — the most stable nuclei.

Fission and fusion

• Fission: a large nucleus splits into smaller nuclei (+ neutrons), releasing energy — used in nuclear reactors, controlled with moderators and control rods.
• Fusion: small nuclei join to form a larger one, releasing even more energy per nucleon — powers stars; requires extreme temperature/pressure to overcome repulsion.

Both release energy because the products have higher binding energy per nucleon than the reactants.

Worked example

A radioactive source has a half-life of 8 days and an initial activity of 400 Bq. What is its activity after 24 days?

• 24 days = 3 half-lives → 400 → 200 → 100 → 50 Bq. ✓

Common mistakes

• Confusing the penetrating powers of α, β, γ.
• Forgetting decay is random — half-life is a statistical average.
• Not linking energy release to binding energy per nucleon changes.

Exam tips

• Learn the decay equations (A = λN, N = N₀e^(−λt), t½ = ln2/λ).
• Explain fission/fusion using the binding-energy-per-nucleon curve (peak at iron).
• Use E = mc² with the mass defect for energy released.

Key facts to remember

• Nucleus = protons + neutrons (Rutherford scattering); α/β/γ radiation differ in ionising/penetrating power.
• Decay is random: A = λN, N = N₀e^(−λt), t½ = ln2/λ.
• E = mc²: mass defect → binding energy; binding energy per nucleon peaks at iron-56; fission and fusion both move products toward this peak, releasing energy.