Nuclei Class 12 Notes (2026-27) — CBSE
Class 12 Physics Chapter 13 notes: nuclear size and density, mass defect and binding energy, radioactivity and decay law, half-life, fission and fusion.
Nuclei — Class 12 Physics Notes
Chapter Snapshot
Inside the atom's tiny core: this chapter covers the composition, size and density of the nucleus, why nuclei hold together (mass defect and binding energy), the binding-energy-per-nucleon curve that explains both fission and fusion, radioactive decay with its exponential law and half-life, and finally fission and fusion.
Board relevance: part of the ~12-mark Atoms + Nuclei unit. Expect a binding-energy calculation and a half-life/decay numerical. Remember 1 u = 931.5 MeV — it turns mass defects into energies in one step.
Key Concepts & Definitions
Composition: the nucleus contains protons (Z of them) and neutrons, together called nucleons. Mass number A = Z + N. Notation: ᴬ₂X.
- Isotopes: same Z, different A (e.g. ¹H, ²H, ³H).
- Isobars: same A, different Z.
- Isotones: same number of neutrons.
Atomic mass unit (u): 1 u = 1/12 the mass of a ¹²C atom = 1.66 × 10⁻²⁷ kg, and
1 u = 931.5 MeV/c²
Nuclear size:
R = R₀A^{1/3}, R₀ = 1.2 fm (1.2 × 10⁻¹⁵ m)
Since volume ∝ R³ ∝ A, the nuclear density is independent of A — roughly 2.3 × 10¹⁷ kg/m³, the same for all nuclei (astonishingly dense).
Nuclear force — the strong attractive force binding nucleons. It is charge-independent (acts equally between n-n, p-p, n-p), short-range (~few fm, becoming repulsive below ~0.8 fm), and far stronger than the electrostatic force but only over that tiny range. It saturates — each nucleon interacts only with its nearest neighbours.
Formulas — Mass Defect and Binding Energy
Mass defect (Δm) — the "missing" mass:
Δm = [Zmp + (A − Z)mn] − Mnucleus
Binding energy:
BE = Δm·c², or in practical units BE (MeV) = Δm (u) × 931.5
Binding energy per nucleon = BE/A — the true measure of stability.
The BE-per-nucleon curve
- Rises steeply for light nuclei, peaks around A ≈ 56 (iron) at ~8.8 MeV, then falls slowly for heavy nuclei.
- The middle of the curve is the most stable region.
- Light nuclei fusing → move up the curve → energy released (fusion).
- Heavy nuclei splitting → move up the curve → energy released (fission).
Radioactivity
Spontaneous emission of radiation from unstable nuclei. Three types:
Decay Emission Change
Alpha (α) ⁴₂He nucleus Z → Z − 2, A → A − 4
Beta minus (β⁻) Electron Z → Z + 1, A unchanged (a neutron becomes a proton)
Gamma (γ) High-energy photon No change in Z or A (nucleus de-excites)
Penetrating power: γ β α. Ionising power: α β γ.
Law of radioactive decay
The decay rate is proportional to the number of undecayed nuclei:
dN/dt = −λN ⟹ N = N₀e^{−λt}
Activity R = λN, measured in becquerel (Bq) or curie (1 Ci = 3.7 × 10¹⁰ Bq).
Quantity Formula
Half-life T½ = 0.693/λ
Mean life τ = 1/λ
Relation T½ = 0.693τ
After n half-lives N = N₀/2ⁿ
Fission and Fusion
Nuclear fission — a heavy nucleus splits into lighter ones, releasing ~200 MeV per event:
²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n + energy
The extra neutrons can trigger a chain reaction, controlled in a reactor by moderators (slow the neutrons, e.g. heavy water/graphite) and control rods (absorb neutrons, e.g. cadmium/boron).
Nuclear fusion — light nuclei join to form a heavier one, releasing even more energy per nucleon. It needs extremely high temperature and pressure to overcome the Coulomb repulsion (hence thermonuclear). Fusion of hydrogen into helium powers the Sun and stars.
Worked Examples
Example 1 — Nuclear radius: Find the radius of a nucleus with A = 125.
R = R₀A^{1/3} = 1.2 × (125)^{1/3} = 1.2 × 5 = 6 fm.
Example 2 — Binding energy: For ⁴He, Δm = 0.0303 u. Find the BE and BE per nucleon.
BE = 0.0303 × 931.5 ≈ 28.2 MeV; BE/A = 28.2/4 ≈ 7.06 MeV per nucleon.
Example 3 — Half-life: A sample has a half-life of 10 days. What fraction remains after 40 days?
n = 40/10 = 4 half-lives → N = N₀/2⁴ = N₀/16 (6.25%).
Example 4 — Decay constant: Find λ for T½ = 5 years.
λ = 0.693/T½ = 0.693/5 = 0.1386 per year.
Important Question Patterns
1. Binding energy (3 marks): compute Δm and BE using 1 u = 931.5 MeV; find BE per nucleon; compare stability of two nuclei.
2. BE curve (2–3 marks): sketch/describe it; explain why both fission and fusion release energy.
3. Decay law (3 marks): N = N₀e^{−λt}; fraction remaining after n half-lives; relate λ, T½ and τ.
4. Decay types (2 marks): write nuclear equations for α and β decay; changes in Z and A; compare penetrating/ionising power.
5. Fission/fusion (2–3 marks): the ²³⁵U fission equation; role of moderator and control rods; why fusion needs high temperature.
⚡ Quick Revision
- A = Z + N; isotopes (same Z), isobars (same A), isotones (same N). 1 u = 931.5 MeV.
- R = R₀A^{1/3} (R₀ = 1.2 fm) → nuclear density is the same for all nuclei.
- Nuclear force: short-range, charge-independent, saturating, much stronger than electrostatic.
- Δm = [Zmp + (A−Z)mn] − M; BE = Δm × 931.5 MeV; stability measured by BE/A.
- BE/A curve peaks at A ≈ 56 (~8.8 MeV) → light nuclei fuse, heavy nuclei fission, both releasing energy.
- Decay: α (Z−2, A−4), β⁻ (Z+1, A same), γ (no change). Penetration γβα; ionisation αβγ.
- N = N₀e^{−λt}; T½ = 0.693/λ, τ = 1/λ; after n half-lives N = N₀/2ⁿ.
- Fission: ²³⁵U + n → Ba + Kr + 3n + ~200 MeV; chain reaction controlled by moderator and control rods. Fusion powers the Sun.
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