Nuclei

Concept overview

A nucleus contains Z protons and N neutrons for a total mass number A = Z + N. Nuclear radius scales as R = R₀A^(1/3) with R₀ ≈ 1.2 fm, and nuclear density is roughly constant across all nuclei.

Mass defect Δm = Zm_p + Nm_n − M_nucleus is the difference between the mass of separated nucleons and the actual nuclear mass. The corresponding binding energy is E_B = Δm × c². Binding energy per nucleon E_B/A is a measure of nuclear stability; it peaks at ~8.8 MeV/nucleon around iron (A ≈ 56) and drops off for lighter and heavier nuclei — the reason both fission (heavy) and fusion (light) release energy.

Radioactive decay is a first-order process: dN/dt = −λN, giving N(t) = N₀ e^(−λt). Half-life t_½ = ln(2)/λ = 0.693/λ is the time for half the sample to decay. Alpha decay reduces A by 4 and Z by 2; beta decay changes n → p + e⁻ + ν̄ (raising Z by 1); gamma decay is a photon emission from an excited nucleus with no change in A or Z.

Fission and fusion are the two energy-releasing nuclear reactions. Fission of U-235 by slow neutrons splits it into two mid-mass fragments, releasing ~200 MeV per fission. Fusion of light nuclei (deuterium-tritium) releases ~17.6 MeV per event and powers stars.

Key formulas & concepts

• Nuclear radius: R = R₀ A^(1/3), R₀ ≈ 1.2 fm
• Binding energy: E_B = Δm × c²; peak at Fe-56 with ~8.8 MeV/A
• Radioactive decay law: N(t) = N₀ e^(−λt)
• Half-life: t_½ = 0.693/λ
• Mean life: τ = 1/λ = t_½/ln 2
• Alpha decay: A → A − 4; Z → Z − 2
• Beta decay: A unchanged; Z → Z + 1 (β⁻) or Z − 1 (β⁺)
• Activity: A = λN; units: becquerel (Bq) or curie (Ci)

Common mistakes to avoid

• Confusing half-life with mean life — τ = t_½ / ln 2 ≈ 1.44 × t_½.
• Using Δm in kg but forgetting to multiply by c² for binding energy in joules — or forgetting to convert to MeV.
• Applying fission energy releases to heavy elements only — fusion applies to light elements; both release energy up to iron.
• Ignoring the antineutrino in beta decay — charge and mass conservation appear to fail without accounting for it.

JEE Main relevance

JEE Main asks 1–2 nuclei problems every paper — usually one half-life or activity calculation and one binding-energy / mass-defect numerical.

JEE Advanced relevance

Advanced problems combine radioactive decay chains with equilibrium (secular / transient) or compute Q-values of specific nuclear reactions. Fission-neutron-economy problems for reactor physics also appear.

Previous-year questions (PYQ)

Half-life numericals are the highest-frequency PYQ archetype in this chapter. Drill 15 of them in a chapter-wise session.