JEE Physics Nuclei MCQs Set 06

Answer: (c) decreases with mass number at high mass numbers

Explanation:
1. For heavy nuclei B_N falls as A increases, so splitting a heavy nucleus into two medium-mass nuclei raises B_N and releases energy.

Teacher's Note:
1. At low A the curve rises, which makes fusion possible.
2. Fission and fusion each move toward the peak at iron.

Answer: (a) decrease the energy of neutrons

Explanation:
1. A moderator (heavy water, graphite) slows fast fission neutrons to thermal energies by elastic collisions so that they can cause further fission of U-235.

Teacher's Note:
1. Control rods absorb neutrons; the coolant removes heat.
2. Moderator atoms should be light so that neutrons lose maximum energy per collision.

Answer: (b) 1/e

Explanation:
1. N = N₀e^(−λt) with t = 1/λ gives N/N₀ = e⁻¹
2. The survival probability is 1/e ≈ 0.37.

Teacher's Note:
1. About 37% of nuclei survive one mean life and 63% decay.
2. Mean life τ = 1/λ = 1.44 T½.

Answer: (b) 1 : 12

Explanation:
1. First sample: 6 h is 6 half-lives, N₁ = 2 × (1/2)⁶ = 2/64
2. Second sample: 6 h is 3 half-lives, N₂ = 3 × (1/2)³ = 24/64
3. N₁ : N₂ = 2 : 24 = 1 : 12.

Teacher's Note:
1. Find the number of half-lives separately for each sample.
2. Do not forget the initial ratio when computing the final ratio.

Answer: (b) \(\left(\frac{N_A}{M_w}\right) m\lambda\)

Explanation:
1. Number of atoms in m gram = (m/M_w)N_A
2. Activity = λN = (N_A/M_w) m λ.

Teacher's Note:
1. Activity R = λN, where N counts the active atoms present.
2. N = (mass / molar mass) × N_A.

Answer: (b) β-rays

Explanation:
1. β-particles are emitted with all energies from zero up to a maximum because the energy is shared with a neutrino
2. α and γ spectra are line (discrete) spectra.

Teacher's Note:
1. The continuous β spectrum led Pauli to propose the neutrino.
2. Line spectra reflect discrete nuclear energy levels.

Answer: (a) speed of α-particles is a characteristic property

Explanation:
1. α-particles from a given nuclide have a definite, characteristic speed
2. β-particles have a range of speeds and all γ-photons travel at c, so neither speed is a characteristic.

Teacher's Note:
1. Each α-emitter gives α-particles of fixed energy.
2. γ-photons are identified by their energy, not their speed.

Answer: (c) γ-rays

Explanation:
1. γ-rays are electromagnetic waves and travel at the speed of light (3 × 10⁸ m/s)
2. α, β and cathode-ray particles are slower.

Teacher's Note:
1. β-particles can reach 99% of c but never equal it.
2. α-particles typically move at about 10⁷ m/s.

Answer: (c) after the next 1/λ time interval, the active nuclei in the sample will be N(1 − 1/e)

Explanation:
1. After a time 1/λ the number of active nuclei is N/e, not N(1 − 1/e); N(1 − 1/e) is the number that has decayed
2. The other three statements are correct.

Teacher's Note:
1. Remaining = N e^(−λt); decayed = N(1 − e^(−λt)).
2. At t = 1/λ these are N/e and N(1 − 1/e).

Answer: (b) decrease

Explanation:
1. In β⁻ emission a neutron becomes a proton, so n decreases by 1 and p increases by 1
2. Hence n/p decreases.

Teacher's Note:
1. β⁻ emitters have too high n/p ratio; the decay moves them toward stability.
2. β⁺ decay and K-capture increase n/p.

Answer: (c) C, A, B

Explanation:
1. The left plate is negative and the right plate positive
2. α-particles are positive and bend toward the negative plate: path C
3. β-particles are negative and bend toward the positive plate: path A
4. γ-rays are neutral and go straight: path B. So the order is C, A, B.

Teacher's Note:
1. Opposite charges attract: positive α toward the negative plate.
2. β bends more than α because of its much smaller mass.

Answer: (d) it is not possible

Explanation:
1. Radioactive decay is a nuclear process, independent of temperature, pressure or chemical state
2. The decay constant λ cannot be changed by any physical or chemical means.

Teacher's Note:
1. Chemical reactions involve only outer electrons, not the nucleus.
2. λ is a constant of the nuclide.

Answer: (d) photon

Explanation:
1. Emission of a proton, neutron or electron changes A or Z
2. A γ-photon carries energy only, so A and Z stay the same.

Teacher's Note:
1. γ-decay takes the nucleus from an excited level to a lower one.
2. It usually follows α or β decay.

Answer: (a) 1 : 1

Explanation:
1. Two days = 48 h
2. First: 48/12 = 4 half-lives, 2 × (1/2)⁴ = 1/8
3. Second: 48/16 = 3 half-lives, 1 × (1/2)³ = 1/8
4. The ratio is 1 : 1.

Teacher's Note:
1. Compute each substance's remaining fraction separately.
2. The substance with the longer half-life decays more slowly.

Answer: (b) Co (60)

Explanation:
1. Cobalt-60 emits penetrating γ-rays that destroy cancer cells (cobalt therapy); I-131 is used for the thyroid.

Teacher's Note:
1. Cobalt-60 has a half-life of 5.3 years.
2. Radioactive iodine is used for thyroid diagnosis and treatment.

Answer: (d) 0.8 × 10¹⁰

Explanation:
1. 12 days = 3 half-lives
2. N = 6.4 × 10¹⁰ × (1/2)³ = 0.8 × 10¹⁰.

Teacher's Note:
1. Divide the number of atoms by 2 for each half-life.
2. Be careful to use the half-life in the same unit as the time.

Answer: (b) chargeless and has spin

Explanation:
1. A neutrino has no charge, a nearly zero rest mass and spin 1/2, introduced by Pauli to save energy, momentum and angular momentum conservation in β-decay.

Teacher's Note:
1. The neutrino interacts extremely weakly with matter.
2. The anti-neutrino accompanies β⁻ decay.

Answer: (d) 1

Explanation:
1. α-particles = (238 − 222)/4 = 4
2. Charge after 4 α-decays = 90 − 8 = 82
3. To reach 83, one β⁻ particle must be emitted.

Teacher's Note:
1. Always find α first from the mass number.
2. Compare the resulting Z with the final Z to get β.

Answer: (a) T_1/2 = ln2 / λ

Explanation:
1. At t = T_1/2, N = N₀/2, so e^(−λT) = 1/2
2. Taking logarithms, λT_1/2 = ln2, i.e. T_1/2 = 0.693/λ.

Teacher's Note:
1. The mean life is 1/λ and the half-life is 0.693 times the mean life.
2. (λ + T) is not a meaningful sum; units differ.

Answer: (d) 0.75

Explanation:
1. 10 years = 2 half-lives, so the survival probability is (1/2)² = 0.25
2. Probability of disintegration = 1 − 0.25 = 0.75.

Teacher's Note:
1. Survival and disintegration probabilities add up to 1.
2. Fraction decayed after n half-lives = 1 − (1/2)ⁿ.

Answer: (b) 0.32 g

Explanation:
1. 19/3.8 = 5 half-lives
2. Remaining = 10.24/2⁵ = 10.24/32 = 0.32 g.

Teacher's Note:
1. Count the half-lives first: n = t/T.
2. Each half-life halves the remaining mass.

Answer: (d) 6 β, 8 α

Explanation:
1. α-particles = (238 − 206)/4 = 8
2. Z after 8 α-decays = 92 − 16 = 76, but the final Z = 82, so 6 β⁻ particles are emitted.

Teacher's Note:
1. This is the uranium series, ending at stable Pb-206.
2. Mass number change is due to α-decay only.

Answer: (a) α-rays

Explanation:
1. α-rays are helium nuclei (charge +2e); β-rays are electrons, while γ and X-rays are neutral photons.

Teacher's Note:
1. α-particles deflect toward the negative plate.
2. γ and X-rays are electromagnetic waves.

Answer: (b) 100 hours

Explanation:
1. 256 g → 1 g is a factor of 256 = 2⁸, so 8 half-lives are needed
2. t = 8 × 12.5 = 100 hours.

Teacher's Note:
1. Write the ratio as a power of 2 to read off the number of half-lives.
2. Multiply by the half-life to get the time.

Answer: (a) graph (a)

Explanation:
1. Activity A = λN, and λ is a constant
2. So A is directly proportional to N and the graph of N against A is a straight line through the origin: graph (a).

Teacher's Note:
1. A and N both decay exponentially with the same decay constant.
2. Hence their ratio is the constant λ.

Answer: (c) 3.5 × 10¹⁶

Explanation:
1. 30 days = 3 half-lives, so remaining = 4 × 10¹⁶/8 = 0.5 × 10¹⁶
2. Decayed = 4 × 10¹⁶ − 0.5 × 10¹⁶ = 3.5 × 10¹⁶.

Teacher's Note:
1. The question asks for the decayed number, not the remaining number.
2. Fraction decayed after 3 half-lives = 7/8.