Radioactive Decay
Stochastic decay of atoms vs theoretical N(t) = N₀e^(−λt) with λ = ln2/T½.
Radioactive decay: unstable nuclei spontaneously emit α, β, or γ radiation and transform into another nucleus.
α-decay: emits ⁴₂He nucleus. (A, Z) → (A−4, Z−2) + α. Common in heavy nuclei (Z ≥ 83).
β⁻-decay: a neutron converts to proton + electron + antineutrino. (A, Z) → (A, Z+1) + e⁻ + ν̄. Occurs in neutron-rich nuclei.
β⁺-decay (positron emission): proton → neutron + e⁺ + ν. Occurs in proton-rich nuclei.
γ-decay: nucleus de-excites by emitting a γ-ray photon (no change in A or Z) — analogous to atomic transitions.
Decay is RANDOM: any single nucleus may decay at any time. Only the STATISTICAL behaviour is predictable.
Decay law: dN/dt = −λN ⇒ N(t) = N₀ e^(−λt). λ = decay constant. τ = 1/λ = mean life.
Half-life T_½ = (ln 2)/λ — time for half the nuclei to decay.
Decay law (exponential)
N₀ = initial number; λ = decay constant (s⁻¹).
Decay rate (activity)
Units: becquerel (Bq) = 1 disintegration/s. Curie (Ci) = 3.7 × 10¹⁰ Bq.
Half-life
Time for half the nuclei to decay.
Mean life
Average lifetime of a nucleus before decaying.
Radioactive decay is a NUCLEAR property — not affected by chemistry, temperature, or pressure.
α-particles are highly ionising but short range (~few cm in air; stopped by paper). β-particles penetrate further (~mm of aluminum). γ-rays are most penetrating.
Health hazard: α inside the body (inhaled/ingested) is dangerous; outside skin, paper stops it. γ from outside is the main external hazard.
Decay constants vary over many orders of magnitude: U-238 T₁/₂ ≈ 4.5 × 10⁹ years; carbon-14 ≈ 5730 years; iodine-131 ≈ 8 days.
Activity (Bq) does NOT equal mass — a tiny amount of short-lived isotope can have HUGE activity.
Common pitfall: confusing 'decay constant' (s⁻¹) with 'half-life' (s). They're related by ln 2.