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Class 12 · Physics · Part 2

Chapter 13: Nuclei

Revision notes, practice numericals, a question bank, assertion and reason questions and answers to the NCERT questions. Tap a group to open it; opening one closes the others.

13.1–13.3Composition and size of the nucleusIsotopes, isobars and nuclear radius

Composition

  • The nucleus has protons (charge +e) and neutrons (no charge); together they are nucleons. Atomic number Z = number of protons, neutron number N, mass number A = Z + N. A nuclide is written ᴬ_Z X.
  • Isotopes: same Z, different N. Isobars: same A. Isotones: same N. Atomic masses are measured in u (1/12 of the mass of a ¹²C atom): 1 u = 1.66 × 10⁻²⁷ kg = 931.5 MeV/c².
  • A free neutron is unstable and decays into a proton, an electron and an antineutrino.

Size and density

  • Nuclear radius R = R₀ A^(1/3) with R₀ = 1.2 fm (1 fm = 10⁻¹⁵ m). The volume is proportional to A, so the density is the same for all nuclei: about 2.3 × 10¹⁷ kg/m³, roughly 10¹⁴ times that of water.
  • The nucleus is about 10⁻⁴ of the atom in radius and holds more than 99.9% of its mass.
13.4–13.5Mass–energy and binding energyMass defect, binding energy curve and nuclear force

Mass defect and binding energy

  • The nuclear mass is less than the sum of its nucleons' masses. The mass defect ΔM = [Zmₚ + (A − Z)mₙ] − M. With atomic masses use ΔM = Z m_H + (A − Z) mₙ − M_atom.
  • The binding energy E_b = ΔM c² (= ΔM × 931.5 MeV with ΔM in u) is the energy needed to separate the nucleons.
  • Energies in nuclear processes (MeV per nucleus) are about a million times larger than in chemical processes (eV per atom).

Binding energy per nucleon

  • For 30 < A < 170 the binding energy per nucleon is nearly constant, about 8 MeV, with a maximum of 8.8 MeV near A = 56. It is lower for light nuclei (A < 30) and for heavy nuclei (A > 170).
mass number AB/A (MeV)Fe-56 (8.8)U-238fusion: light → mediumfission: heavy → medium48
Binding energy per nucleon against mass number.

Nuclear force

  • The force that holds nucleons together is strong, short-ranged (a few fm), charge-independent (p–p, n–n and p–n are nearly alike) and does not follow an inverse-square law. It is much stronger than the electrostatic force at short distances.
13.6RadioactivityTypes of decay

Radioactive decay

  • Becquerel discovered radioactivity (1896). An unstable nucleus emits: α-particles (helium nuclei ⁴₂He: A falls by 4, Z by 2), β-particles (electrons or positrons: Z changes by ±1, A is unchanged) or γ-rays (high-energy photons: A and Z unchanged).
  • Example: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He.
13.7Nuclear energyQ-value, fission and fusion

Q-value

  • For a reaction A + b → C + d: Q = [m_A + m_b − m_C − m_d] c² = final kinetic energy − initial kinetic energy. Q > 0 is exothermic (energy released); Q < 0 is endothermic (energy must be supplied).

Fission and fusion

  • Fission: a heavy nucleus such as ²³⁵U, struck by a neutron, splits into two medium nuclei and releases neutrons, with about 200 MeV per fission. 1 kg of ²³⁵U gives about 10¹⁴ J. Neutrons are effective because they carry no charge. A reactor uses a moderator to slow neutrons and control rods to absorb them.
  • Fusion: light nuclei combine into a heavier one (e.g. ²₁H + ²₁H → ³₂He + n + 3.27 MeV). It needs very high temperatures (about 10⁷ to 10⁹ K) to overcome the Coulomb barrier, which is about 0.36 MeV for two deuterons. This is the source of the energy of the Sun and stars.
  • Both processes move the nuclei to higher binding energy per nucleon, so the mass decreases and energy is released.
Quick referenceFormulas at a glanceRevise before the exam

Key results

  • R = 1.2 A^(1/3) fm; ρ ≈ 2.3 × 10¹⁷ kg/m³
  • ΔM = Zm_H + (A − Z)mₙ − M; E_b = ΔM × 931.5 MeV; 1 MeV = 1.6 × 10⁻¹³ J
  • Q = (Σm_initial − Σm_final) × 931.5 MeV
  • α: A − 4, Z − 2; β⁻: Z + 1; γ: no change

Common mistakes

  • Using the mass of the nucleus with mH, or atomic masses and nuclear masses mixed.
  • Forgetting the neutron number (A − Z) in the mass defect.
  • Saying that nuclear density depends on A (it does not).
  • Taking Q > 0 to be endothermic.