Questions & explanations
1. Which of the following best explains why energy is released when two light nuclei fuse to form a heavier nucleus under suitable conditions?
- The total binding energy of the product nucleus is greater than the sum of binding energies of the reactants, leading to a decrease in mass.
- The total mass of the product nucleus is greater than the sum of masses of the reactants, and the extra mass is converted into energy.
- Fusion is the opposite of fission, and like fission, it always releases energy because nuclear forces are repulsive.
- Energy is released because the strong nuclear force overcomes the Coulomb repulsion, and the excess kinetic energy is radiated away.
Answer: The total binding energy of the product nucleus is greater than the sum of binding energies of the reactants, leading to a decrease in mass.
In nuclear fusion, light nuclei combine to form a heavier nucleus. The binding energy per nucleon increases for light nuclei up to iron, so the product nucleus has a higher total binding energy. This increase in binding energy corresponds to a mass defect, where the mass of the product is less than the sum of the reactant masses. The lost mass is converted into energy according to Einstein's equation E = mc², releasing a large amount of energy. Option B is incorrect because the product mass is less, not greater. Option C is incorrect: fusion of very light nuclei releases energy, but fusion of heavy nuclei would require energy; also nuclear forces are attractive, not repulsive. Option D, while mentioning overcoming Coulomb repulsion, does not correctly describe the fundamental cause of energy release, which is the mass defect and increase in binding energy.
2. In the nuclear reaction \(^{238}_{92}\text{U} \to \,^{234}_{90}\text{Th} + X\), what is the particle X?
- An alpha particle (\(^4_2\text{He}\))
- A beta particle (\(^0_{-1}\text{e}\))
- A gamma photon
- A neutron
Answer: A beta particle (\(^0_{-1}\text{e}\))
The mass number decreases by 4 (238 → 234) and the atomic number decreases by 2 (92 → 90). This is characteristic of alpha decay, where an alpha particle (helium nucleus) is emitted, conserving both mass and charge.
3. The mass of the nucleus of ⁵⁶Fe is 55.9201 u. The masses of a proton and neutron are 1.0073 u and 1.0087 u respectively. The binding energy per nucleon of ⁵⁶Fe is approximately:
- A) 8.8 MeV
- B) 7.8 MeV
- C) 9.8 MeV
- D) 6.8 MeV
Answer: A) 8.8 MeV
Mass of constituents = 26×1.0073 + 30×1.0087 = 56.4508 u. Mass defect = 56.4508 – 55.9201 = 0.5307 u. Binding energy = 0.5307 × 931.5 ≈ 494.4 MeV. Binding energy per nucleon = 494.4 / 56 ≈ 8.83 MeV ≈ 8.8 MeV.
4. Which of the following radioactive emissions has the highest ionising power?
- Alpha (α) rays
- Beta (β) rays
- Gamma (γ) rays
- X-rays
Answer: Alpha (α) rays
Alpha particles are heavy and doubly charged, so they interact strongly with matter and cause the greatest ionisation per unit path length. Beta and gamma radiations have progressively lower ionising power.
5. Which of the following nuclear reactions is an example of nuclear fusion?
- \(^{235}_{92}\text{U} + n \to \,^{141}_{56}\text{Ba} + \,^{92}_{36}\text{Kr} + 3n\)
- \(^{2}_{1}\text{H} + \,^{3}_{1}\text{H} \to \,^{4}_{2}\text{He} + n\)
- \(^{226}_{88}\text{Ra} \to \,^{222}_{86}\text{Rn} + \alpha\)
- \(^{60}_{27}\text{Co} \to \,^{60}_{28}\text{Ni} + \beta^- + \bar{\nu}_e\)
Answer: \(^{2}_{1}\text{H} + \,^{3}_{1}\text{H} \to \,^{4}_{2}\text{He} + n\)
Nuclear fusion involves the combining of two light nuclei to form a heavier nucleus, as in deuterium–tritium fusion. The other options are fission (A), alpha decay (C), and beta decay (D).
6. According to the binding energy per nucleon curve, nuclear fission of a heavy nucleus (e.g., ²³⁵U) into two lighter nuclei releases energy because:
- A) The binding energy per nucleon of the fragments is greater than that of the parent nucleus.
- B) The binding energy per nucleon of the parent is greater than that of the fragments.
- C) The mass defect of the parent nucleus is zero.
- D) The total mass of the fragments is greater than the parent mass.
Answer: A) The binding energy per nucleon of the fragments is greater than that of the parent nucleus.
Heavy nuclei have lower BE per nucleon; when they split into lighter fragments with higher BE per nucleon, the total binding energy increases. The lost mass appears as released energy.
7. Assuming the nucleus to be spherical and of constant density, the nuclear density is approximately:
- A) 2.3 × 10¹⁷ kg/m³
- B) 2.3 × 10¹⁵ kg/m³
- C) 2.3 × 10¹³ kg/m³
- D) 2.3 × 10¹¹ kg/m³
Answer: A) 2.3 × 10¹⁷ kg/m³
Using the formula density = mass/volume with R = R₀A^{1/3}, the mass is proportional to A and volume to A, so density is constant at about 2.3 × 10¹⁷ kg/m³, independent of mass number.
8. For a radioactive substance, which of the following correctly relates the decay constant (λ) to the half-life (T₁/₂) and the mean life (τ)?
- \(T_{1/2} = \dfrac{\ln 2}{\lambda}\) and \(\tau = \dfrac{1}{\lambda}\)
- \(T_{1/2} = \dfrac{\lambda}{\ln 2}\) and \(\tau = \dfrac{1}{\lambda}\)
- \(T_{1/2} = \ln 2 \cdot \lambda\) and \(\tau = \lambda\)
- \(T_{1/2} = \dfrac{1}{\lambda}\) and \(\tau = \dfrac{\ln 2}{\lambda}\)
Answer: \(T_{1/2} = \dfrac{\ln 2}{\lambda}\) and \(\tau = \dfrac{1}{\lambda}\)
The half-life is the time for half the nuclei to decay: N₀/2 = N₀ e^{-λT₁/₂} → λT₁/₂ = ln2 → T₁/₂ = ln2/λ. The mean life is the average lifetime of a nucleus and equals 1/λ.
9. The radioactive decay law states that the rate of decay is proportional to the number of undecayed nuclei. Which of the following equations correctly represents the exponential form of this law?
- \(N = N_0 e^{-\lambda t}\)
- \(N = N_0 e^{\lambda t}\)
- \(N = N_0 (1 - e^{-\lambda t})\)
- \(N = N_0 \lambda t\)
Answer: \(N = N_0 e^{-\lambda t}\)
The differential form is dN/dt = -λN, which integrates to N = N₀ e^{-λt}. This shows an exponential decrease of undecayed nuclei with time, where λ is the decay constant.
10. Which of the following is an example of isotopes?
- A) ¹H¹ and ²H¹
- B) ¹⁴C⁶ and ¹⁴N⁷
- C) ¹²C⁶ and ¹⁴N⁷
- D) ²³Na¹¹ and ²⁴Mg¹²
Answer: A) ¹H¹ and ²H¹
Isotopes have the same atomic number but different mass numbers. ¹H and ²H both have Z=1 (hydrogen), but mass numbers 1 and 2 respectively, making them isotopes.
11. The binding energy per nucleon (BE/n) curve shows that the most stable nucleus is:
- A) ²³⁵U (uranium)
- B) ⁵⁶Fe (iron)
- C) ⁴He (helium)
- D) ²H (deuterium)
Answer: B) ⁵⁶Fe (iron)
The BE per nucleon curve peaks at iron-56, indicating it has the maximum binding energy per nucleon and is therefore the most stable nucleus.