📌 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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Q4. The binding energy per nucleon curve (Fig. 13.1) has its maximum value of approximately 8.75 MeV near
▸ Show answer & explanation
Answer: B
NCERT states "the curve has a maximum of about 8.75 MeV for A = 56" (iron region); at A = 238 it falls to 7.6 MeV.
Q5. Given mₚ = 1.00727 u, mₙ = 1.00866 u and the mass of a ⁴₂He nucleus = 4.001506 u, the binding energy of a ⁴₂He nucleus is approximately (use 1 u ≡ 931.5 MeV/c²)
▸ Show answer & explanation
Answer: C
ΔM = 2(1.00727) + 2(1.00866) − 4.001506 = 4.03186 − 4.001506 = 0.030354 u. Eb = 0.030354 × 931.5 ≈ 28.28 MeV. Option (A) is BE per nucleon, (B) and (D) are scaling traps.
Q6. Which of the following pairs is a pair of isotones?
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Answer: B
Isotones have the same N but different Z. ¹⁹⁸₈₀Hg has N = 118 and ¹⁹⁷₇₉Au has N = 118 — NCERT's own example. (A) is isobars; (C) and (D) are isotopes.
Q7. The density of nuclear matter is
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Answer: C
Since R = R₀A^(1/3), volume ∝ A and so mass/volume is constant. NCERT quotes ≈ 2.3 × 10¹⁷ kg/m³ for iron and notes it equals the order of magnitude for neutron-star matter.
Q8. Assertion (A): Energy is released when a very heavy nucleus undergoes fission into two intermediate-mass fragments. Reason (R): The binding energy per nucleon is larger for intermediate-mass nuclei (A ≈ 120) than for very heavy nuclei (A ≈ 240).
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Answer: A
NCERT explicitly states that for A = 240 → 2(A = 120), nucleons become more tightly bound (Ebn rises from ~7.6 to ~8.5 MeV) and the total gain of ~216 MeV is released. R is the exact reason for A.
Q9. Match the column: | Column I | Column II | |---|---| | (P) α-decay | (1) emission of a high-energy photon | | (Q) β-decay | (2) emission of a ⁴₂He nucleus | | (R) γ-decay | (3) emission of an electron or positron | | (S) Fission of ²³⁵U by a thermal neutron | (4) intermediate-mass fragments and 2–3 free neutrons |
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Answer: A
α-decay → ⁴₂He; β-decay → electron or positron; γ-decay → high-energy photons; thermal-neutron-induced ²³⁵U fission → e.g. ¹⁴⁴Ba + ⁸⁹Kr + 3 neutrons.
Q10. The Q-value of the reaction ¹₁H + ³₁H → ²₁H + ²₁H is best computed as
▸ Show answer & explanation
Answer: A
By definition, Q = (sum of initial masses − sum of final masses)·c². Initial: m(¹H) + m(³H); final: 2 m(²H). Hence option (A). Option (B) is the sign-flipped version.
Q11. The Q-value released in the complete proton–proton fusion cycle in the sun is approximately
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Answer: C
The net p–p chain converts 4 ¹H + 2 e⁻ into ⁴He + 2ν + 6γ with a total energy release of 26.7 MeV per ⁴He produced. 200 MeV is the ²³⁵U fission Q-value.
Q12. The half-life of a radioactive nuclide is 10 minutes. Its mean life is approximately
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Answer: C
τ = T₁/₂ / ln 2 = 10 / 0.693 ≈ 14.4 min. Option (A) is the inverted form τ ln 2, which equals T₁/₂.
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