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Class XII ⚛️ Physics ~10 MCQs/year Ch 13 of 14

Nuclei

CUET unit: Nuclei

📌 Snapshot

  • Establishes the nucleus as the tiny (radius ~10⁻¹⁵ m), dense (~10¹⁷ kg/m³) core that holds >99.9% of an atom's mass, with composition described by Z (protons) and N (neutrons).
  • Introduces the atomic mass unit (1 u = 1.660539 × 10⁻²⁷ kg ≡ 931.5 MeV/c²) and the nuclear size law R = R₀A^(1/3), R₀ = 1.2 fm, which implies constant nuclear density.
  • Develops Einstein's mass–energy relation E = mc² and uses it to define mass defect ΔM = [Zmₚ + (A−Z)mₙ] − M and binding energy Eb = ΔM·c²; the BE/nucleon curve (peak ~8.75 MeV near A = 56) explains why fission of heavy nuclei and fusion of light nuclei both release energy.
  • Describes nuclear force (short range ~few fm, strong, charge-independent, attractive beyond ~0.8 fm and strongly repulsive within, saturation) and the three radioactive decays (α, β, γ).
  • Covers nuclear energy: fission of ²³⁵U (~200 MeV per fissioning nucleus, basis of reactors and the atom bomb) and thermonuclear fusion (p-p cycle in the sun, 26.7 MeV per 4¹H → ⁴He; basis of stellar energy and controlled fusion research).
  • CUET tests numerical estimation (BE, nuclear density, Q-value, R-ratio), conceptual recall (definitions, force properties, decay types) and the binding-energy-curve reasoning.

📖 Detailed Notes

2.1 Core concepts

  • The nucleus is ~10⁴ times smaller in radius than the atom; its volume is ~10⁻¹² times the atomic volume, yet it carries more than 99.9% of the atom's mass (NCERT §13.1, p. 306).
  • Atomic mass unit (u) is defined as 1/12 the mass of a ¹²C atom; 1 u = 1.660539 × 10⁻²⁷ kg (NCERT §13.2, p. 306–307, Eq. 13.1).
  • Atomic masses are measured by a mass spectrometer, which also reveals isotopes — atoms of the same element with the same chemical properties but different masses (NCERT §13.2, p. 307).
  • Chlorine has two isotopes of masses 34.98 u and 36.98 u with abundances 75.4% and 24.6%; weighted average ≈ 35.47 u (NCERT §13.2, p. 307).
  • Hydrogen has three isotopes (1.0078 u, 2.0141 u, 3.0160 u): protium, deuterium, tritium; tritium is unstable and made artificially (NCERT §13.2, p. 307).
  • Proton mass mₚ = 1.00727 u = 1.67262 × 10⁻²⁷ kg; carries +e; number of protons = Z (atomic number) (NCERT §13.2, p. 307–308, Eq. 13.2).
  • Neutron was discovered by James Chadwick (1932) when α-bombardment of beryllium produced neutral radiation that could knock protons out of light nuclei; neutron mass mₙ = 1.00866 u = 1.6749 × 10⁻²⁷ kg (NCERT §13.2, p. 308, Eq. 13.3).
  • A free neutron is unstable (mean life ~1000 s) and decays into a proton, an electron and an antineutrino; it is stable inside the nucleus (NCERT §13.2, p. 308).
  • Composition: Z = number of protons, N = number of neutrons, A = Z + N = mass number = total number of nucleons; nuclide notation ᴬ_ZX (NCERT §13.2, p. 308, Eq. 13.4).
  • Isotopes (same Z, different N), isobars (same A), isotones (same N); gold has 32 isotopes from A = 173 to 204 (NCERT §13.2, p. 309).
  • Size of nucleus: from electron and α-scattering, R = R₀A^(1/3) with R₀ = 1.2 fm; therefore volume ∝ A and nuclear density is constant, ≈ 2.3 × 10¹⁷ kg/m³, independent of A (NCERT §13.3, p. 309, Eq. 13.5).
  • Nuclear matter density (~10¹⁷ kg/m³) is comparable to that of neutron stars and ~10¹⁴ times that of water (10³ kg/m³) (NCERT §13.3, p. 309 and Example 13.1, p. 310).
  • Einstein's mass–energy relation E = mc² (c ≈ 3 × 10⁸ m/s) shows mass is a form of energy; 1 g of matter is equivalent to 9 × 10¹³ J (NCERT §13.4.1, p. 310, Eq. 13.6 and Example 13.2).
  • Mass defect ΔM = [Zmₚ + (A − Z)mₙ] − M; for ¹⁶O the measured nuclear mass (15.99053 u) is 0.13691 u less than constituent total (16.12744 u) (NCERT §13.4.2, p. 310–311, Eq. 13.7).
  • Binding energy Eb = ΔM·c²; energy needed to break a nucleus into its constituent nucleons. 1 u = 931.5 MeV/c²; ΔM for ¹⁶O ≡ 127.5 MeV (NCERT §13.4.2, p. 311, Eq. 13.8 and Example 13.3).
  • Binding energy per nucleon Ebn = Eb/A; for the curve Ebn vs A: maximum ~8.75 MeV at A = 56, and 7.6 MeV at A = 238; Ebn is practically constant for 30 < A < 170 and lower for light (A < 30) and heavy (A > 170) nuclei (NCERT §13.4.2, p. 312, Eq. 13.9, Fig. 13.1).
  • Constancy of Ebn in the middle range is a consequence of saturation: a given nucleon interacts only with its near neighbours within the short range of the nuclear force (NCERT §13.4.2, p. 312–313).
  • A heavy A = 240 nucleus splitting into two A = 120 fragments releases energy (fission); two light nuclei (A ≤ 10) fusing into a heavier one also release energy (fusion — source of the sun's energy) (NCERT §13.4.2, p. 313).
  • Nuclear force: (i) much stronger than Coulomb and gravitational forces between the same particles; (ii) very short range — falls rapidly to zero beyond a few fm, leading to saturation; potential energy minimum near r₀ ≈ 0.8 fm — attractive for r > 0.8 fm, strongly repulsive for r < 0.8 fm; (iii) charge-independent (n-n, p-n and p-p nuclear forces are approximately equal); has no simple mathematical form (NCERT §13.5, p. 313–314, Fig. 13.2).
  • Radioactivity was discovered by A. H. Becquerel in 1896 by accident — uranium-potassium sulphate blackened a wrapped photographic plate (NCERT §13.6, p. 314).
  • Three natural radioactive decays: (i) α-decay (emission of a helium nucleus ⁴₂He); (ii) β-decay (emission of electrons or positrons — particles with the same mass as electrons but opposite charge); (iii) γ-decay (emission of high-energy photons of hundreds of keV or more) (NCERT §13.6, p. 314).
  • For the same mass of fuel, nuclear sources produce ~10⁶ times more energy than chemical sources: 1 kg of ²³⁵U fission yields ~10¹⁴ J versus ~10⁷ J from 1 kg of coal burning (NCERT §13.7, p. 314).
  • Fission of ²³⁵U by a thermal neutron produces intermediate-mass fragments and 2–4 more neutrons; e.g. ¹₀n + ²³⁵₉₂U → ²³⁶₉₂U → ¹⁴⁴₅₆Ba + ⁸⁹₃₆Kr + 3 ¹₀n; fragments are radioactive and emit β to reach stable end products. Q ≈ 200 MeV per fission (estimated as 240 × 0.9 ≈ 216 MeV from the Ebn curve) (NCERT §13.7.1, p. 315, Eqs. 13.10–13.12).
  • Disintegration energy in fission first appears as kinetic energy of fragments and neutrons, then transfers to surroundings as heat; nuclear reactors use controlled fission, atom bombs use uncontrolled fission (NCERT §13.7.1, p. 315).
  • Fusion releases energy when light nuclei combine: e.g. ¹₁H + ¹₁H → ²₁H + e⁺ + ν + 0.42 MeV; ²₁H + ²₁H → ³₂He + n + 3.27 MeV; ²₁H + ²₁H → ³₁H + ¹₁H + 4.03 MeV (NCERT §13.7.2, p. 315, Eq. 13.13).
  • For fusion the two positively charged nuclei must overcome the Coulomb barrier (~400 keV for two protons), requiring temperatures ~3 × 10⁹ K from (3/2)kT ≈ 400 keV; this is called thermonuclear fusion (NCERT §13.7.2, p. 316).
  • The sun's interior is at 1.5 × 10⁷ K — lower than the average value above, so fusion in the sun involves high-energy protons of the tail of the distribution; the p–p cycle effectively converts 4 ¹H + 2e⁻ → ⁴₂He + 2ν + 6γ + 26.7 MeV (NCERT §13.7.2, p. 316, Eqs. 13.14–13.15).
  • Once the sun's hydrogen depletes (~5 × 10⁹ y from now), the core will collapse, heat further, and ignite helium → carbon fusion (~10⁸ K); successively higher elements can form in stellar cores, but no element heavier than the Ebn-peak region can be produced by fusion (NCERT §13.7.2, p. 316–317).
  • Controlled thermonuclear fusion aims at steady power at ~10⁸ K; the fuel becomes a plasma (ions + electrons) that must be confined without a material container — a challenge being pursued internationally including India (NCERT §13.7.3, p. 317).
  • A summary table lists the physical quantities atomic mass unit (u), decay constant λ (s⁻¹), half-life T₁/₂ (s), mean life τ (s) — the time at which the number of nuclei is reduced to e⁻¹ of the initial value — and activity R (Bq) (NCERT Summary table, p. 319).
  • Stability requires the n:p ratio to be near 1:1 for light nuclei, rising to about 3:2 for heavy nuclei (extra neutrons offset proton repulsion); only ~10% of known isotopes are stable (NCERT Points to Ponder #8, p. 320).
  • The chain-reaction picture for ²³⁵U: each fission releases 2–3 neutrons that can in turn fission other ²³⁵U nuclei; if at least one neutron per fission causes the next, the reaction sustains. A nuclear reactor uses moderators (e.g. heavy water) to slow neutrons to thermal energies where the ²³⁵U fission cross-section is largest, and control rods (e.g. cadmium) to absorb excess neutrons (NCERT §13.7.1, p. 315; Summary p. 319).
  • The thermonuclear-fusion path in the sun's core (the proton–proton chain) is slow because it requires the weak-interaction conversion of two protons into a deuteron with positron and neutrino emission; that slowness gives the sun a stable ~10¹⁰-y main-sequence lifetime (NCERT §13.7.2, p. 316).
  • Q-value of a nuclear reaction is the c² times the difference between initial and final rest masses; Q > 0 indicates an exothermic (energy-releasing) reaction, Q < 0 an endothermic one that requires energy input. The same definition applies to α, β, γ decays and to fission/fusion (NCERT Summary point 9, p. 319).
  • Activity of a radioactive sample R = λN decays per second; SI unit becquerel (Bq). The older unit curie (Ci) ≡ 3.7 × 10¹⁰ Bq, originally the activity of 1 g of ²²⁶Ra. Half-life T₁/₂ and mean life τ are related by T₁/₂ = τ ln 2 ≈ 0.693 τ (NCERT Summary table p. 319).

2.2 Definitions to memorise

Term Definition Page
Atomic mass unit (u) 1/12 of the mass of one ¹²C atom; 1 u = 1.660539 × 10⁻²⁷ kg = 931.5 MeV/c² 306–307, 311
Isotope Atomic species of the same element (same Z) differing in mass (different N) 307, 309
Isobar Nuclides with the same mass number A (e.g. ³₁H and ³₂He) 309
Isotone Nuclides with the same neutron number N but different Z (e.g. ¹⁹⁸₈₀Hg and ¹⁹⁷₇₉Au) 309
Nucleon A proton or a neutron; total nucleons = mass number A 308
Mass number A A = Z + N, total number of protons and neutrons 308
Nuclear radius R = R₀A^(1/3); R₀ = 1.2 fm 309
Mass defect ΔM ΔM = [Zmₚ + (A − Z)mₙ] − M (always positive) 311
Binding energy Eb Eb = ΔM·c² — energy needed to separate a nucleus into its nucleons 311
Binding energy per nucleon Ebn Eb / A — average energy per nucleon to separate nucleus 312
Nuclear force Strong, short-range, charge-independent attractive force binding nucleons; saturates 313–314
α-decay Decay emitting ⁴₂He 314
β-decay Decay emitting an electron or a positron 314
γ-decay Decay emitting a high-energy photon (hundreds of keV or more) 314
Fission Heavy nucleus splits into two intermediate-mass fragments + neutrons 315
Thermonuclear fusion Fusion at very high temperature so nuclei overcome Coulomb barrier 316
Q-value Q = (sum of initial masses − sum of final masses)c² 319
Half-life T₁/₂ Time taken for one-half of the initial nuclei to decay 319
Mean life τ Time at which N is reduced to e⁻¹ of initial value 319
Activity R Measure of activity of a radioactive source (Bq = s⁻¹) 319
Decay constant λ Disintegration constant (s⁻¹) 319
Proton (mₚ) Hydrogen nucleus; charge +e, mass 1.00727 u 307–308
Neutron (mₙ) Neutral nucleon discovered by Chadwick (1932); mass 1.00866 u 308
Z (atomic number) Number of protons in the nucleus 308
N (neutron number) Number of neutrons in the nucleus 308
Chain reaction Self-sustaining series of fissions in which neutrons from one event trigger the next 315
Moderator Material (e.g. heavy water) used in a reactor to slow fast neutrons to thermal energies 319
Saturation of nuclear force Each nucleon interacts only with its nearest neighbours within the short range of the strong force 312–314

2.3 Diagrams / processes to remember

  • Fig. 13.1 — Binding energy per nucleon vs mass number A: peak ~8.75 MeV at A = 56 (Fe-56 region), 7.6 MeV at A = 238, low for A < 30; flat for 30 < A < 170. Use to argue why fission of heavy and fusion of light nuclei release energy (NCERT p. 312).
  • Fig. 13.2 — Nucleon-nucleon potential energy vs separation: minimum near r₀ ≈ 0.8 fm; attractive for r > r₀, strongly repulsive for r < r₀, force vanishes beyond a few fm (NCERT p. 313).
  • Mass spectrometer: experimental basis of accurate atomic-mass measurement and isotope detection (NCERT §13.2, p. 307).
  • Geiger–Marsden α-scattering: distance of closest approach of a 5.5 MeV α to gold ≈ 4.0 × 10⁻¹⁴ m, giving an upper bound on nuclear size (NCERT §13.3, p. 309).
  • Chadwick's experiment: α + ⁹Be → neutron emission; energy-momentum conservation proves neutral particle (not photon) of mass ≈ proton mass (NCERT §13.2, p. 308).
  • ²³⁵U fission scheme: n + ²³⁵U → ²³⁶U* → ¹⁴⁴Ba + ⁸⁹Kr + 3n (and other channels), Q ≈ 200 MeV/fission (NCERT §13.7.1, p. 315).
  • Solar p-p cycle: net reaction 4¹H + 2e⁻ → ⁴He + 2ν + 6γ + 26.7 MeV (NCERT §13.7.2, p. 316).

2.4 Common confusions / NTA trap points

  • Mass defect formula uses the proton and neutron masses (mₚ, mₙ) when M is the nuclear mass; if atomic masses (including electrons) are used, the electron masses must cancel — students often mix the two conventions. NCERT computes ¹⁶O nuclear mass = 15.99053 u after subtracting 8 electron masses from 15.99493 u (NCERT §13.4.2, p. 311).
  • R₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm — distractors often quote 1.2 × 10⁻¹⁴ m or 1.2 × 10⁻¹³ m. R ∝ A^(1/3), so volume ∝ A and density is independent of A (NCERT §13.3, p. 309).
  • Nuclear force is charge-independent (F_nn ≈ F_np ≈ F_pp as nuclear forces). NTA traps say "nuclear force is electromagnetic" or "depends on charge" — both wrong (NCERT §13.5, p. 314).
  • The Ebn peak is near A = 56 at ~8.75 MeV, NOT at uranium or at A = 238 (where Ebn = 7.6 MeV). Confusing the peak with the high-A end is a classic trap (NCERT §13.4.2, p. 312).
  • β-decay emits electrons OR positrons — not "protons" or "alpha"; positrons have the same mass as electrons and opposite charge (NCERT §13.6, p. 314).
  • A free neutron decays with mean life ~1000 s; bound neutrons inside a nucleus are stable. NTA may swap this with "neutron is always stable" or "proton decays" (NCERT §13.2, p. 308).
  • The Coulomb-barrier estimate for two protons (~400 keV) gives an "average" temperature of ~3 × 10⁹ K; the sun's interior is only 1.5 × 10⁷ K — fusion proceeds via the tail of the distribution, not because the average proton energy is sufficient (NCERT §13.7.2, p. 316).
  • Half-life T₁/₂ and mean life τ are different: mean life is the time at which N falls to e⁻¹ of the initial value, not to half (NCERT Summary table, p. 319).
  • Confusing atomic mass (includes electron masses) with nuclear mass (no electrons) when computing ΔM and Eb. Atomic-mass tables include Z electrons; if both sides of the reaction have the same number of electrons, they cancel and atomic masses can be used directly.
  • The Ebn curve is sometimes drawn upside down by students. The peak is at the top (~8.75 MeV at A ≈ 56); deeper binding → larger Ebn → greater stability.
  • For ²³⁵U the fission is induced by a thermal (slow) neutron — not a fast one. Fast neutrons mostly cause inelastic scattering off ²³⁸U.
  • Saturation of the nuclear force is what makes Eb roughly linear in A (so Ebn is constant) — many students wrongly think saturation refers to the n:p ratio.
  • The decay ⁴₂He emitted in α-decay is the doubly charged helium-4 nucleus; the daughter has Z − 2 and A − 4 compared with the parent. Be careful with charge/mass bookkeeping.

2.5 Key formulas table

Quantity Symbol / Formula NCERT reference
Atomic mass unit 1 u = 1.660539 × 10⁻²⁷ kg = 931.5 MeV/c² §13.2, Eq. 13.1, p. 306–307
Mass number A = Z + N §13.2, Eq. 13.4, p. 308
Nuclide notation ᴬ_ZX §13.2, p. 308
Nuclear radius R = R₀ A^(1/3); R₀ = 1.2 fm §13.3, Eq. 13.5, p. 309
Nuclear density ρ = 3 mₙ /(4π R₀³) ≈ 2.3 × 10¹⁷ kg/m³ §13.3, Ex. 13.1, p. 310
Einstein mass–energy E = mc² §13.4.1, Eq. 13.6, p. 310
Mass defect ΔM = [Z mₚ + (A − Z) mₙ] − M §13.4.2, Eq. 13.7, p. 311
Binding energy Eb = ΔM · c² §13.4.2, Eq. 13.8, p. 311
Binding energy per nucleon Ebn = Eb / A §13.4.2, Eq. 13.9, p. 312
Peak Ebn ~8.75 MeV at A ≈ 56 §13.4.2, Fig. 13.1, p. 312
Equilibrium separation in nuclear force r₀ ≈ 0.8 fm §13.5, Fig. 13.2, p. 313
Q-value of nuclear reaction Q = (Σ m_initial − Σ m_final) c² Summary 9, p. 319
²³⁵U fission energy Q ≈ 200 MeV per fission §13.7.1, p. 315
Solar p-p net reaction 4¹H + 2e⁻ → ⁴He + 2ν + 6γ + 26.7 MeV §13.7.2, Eq. 13.14–13.15, p. 316
Coulomb barrier for two protons ≈ 400 keV §13.7.2, p. 316
Required fusion temperature T ≈ 3 × 10⁹ K (avg) §13.7.2, p. 316
Half-life ↔ mean life T₁/₂ = τ ln 2 ≈ 0.693 τ Summary, p. 319
Activity R = λ N Summary, p. 319
SI unit of activity becquerel Bq = decay/s Summary, p. 319
Curie 1 Ci = 3.7 × 10¹⁰ Bq Summary, p. 319
Mass equivalent 1 g ≡ 9 × 10¹³ J §13.4.1, Ex. 13.2, p. 310

🎯 Practice MCQs

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Q1. The radius of a nucleus of mass number A is given by R = R₀A^(1/3) with R₀ = 1.2 fm. The ratio of the nuclear radii of ¹⁹⁷₇₉Au to ¹⁰⁷₄₇Ag is approximately

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Answer: B

R_Au / R_Ag = (197/107)^(1/3) = (1.841)^(1/3) ≈ 1.226. Option (C) is the wrong (linear) ratio 197/107 ≈ 1.84.

Q2. The atomic mass unit is defined as

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Answer: A

By definition, 1 u = (mass of one ¹²C atom)/12 = 1.660539 × 10⁻²⁷ kg. The hydrogen-atom and proton masses (≈ 1.00783 u and 1.00727 u) are close to but not equal to 1 u.

Q3. Which of the following statements about the nuclear force is INCORRECT?

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Answer: C

The nuclear force falls rapidly to zero beyond a few femtometres — it is short-ranged, which is why it saturates. The other three statements are exactly the features listed in §13.5.

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