JEE Physics Nuclei MCQs Set 32

Answer: (d) (i), (ii), (iii) and (iv)

Explanation:
1. α-decay: \( Z \) falls by 2 and \( A \) by 4 (i true).
2. β⁺: a proton becomes a neutron, so \( Z \) falls by 1 and \( A \) is unchanged (ii true).
3. β⁻: a neutron becomes a proton, so \( Z \) rises by 1 (iii true).
4. γ-emission only lowers the nuclear energy (iv true).

Teacher's Note:
1. Only α-decay changes the mass number.

Answer: (b) (i) and (iv) only

Explanation:
1. α-particles are emitted with discrete energies, so they have almost the same speed.
2. β-particles share the decay energy with the antineutrino, so their energies form a continuous spectrum and the speeds vary widely.

Teacher's Note:
1. The antineutrino carries off a variable part of the energy in β-decay.

Answer: (b) 81%

Explanation:
1. The same fraction remains in each equal interval: after one day 0.90 remains.
2. After two days, \( 0.90 \times 0.90 = 0.81 \), that is 81%.

Teacher's Note:
1. Exponential decay multiplies by a fixed factor per day.

Answer: (c) 63%

Explanation:
1. After time \( T \), \( N = N_0e^{-1} = 0.368N_0 \) remain.
2. Decayed fraction = 1 – \( 0.368 = 0.632 \), that is 63%.

Teacher's Note:
1. \( e^{-1} = 0.368 \).

Answer: (c) only for A = 12

Explanation:
1. The atomic mass unit is defined as 1/12 of the mass of a carbon-12 atom, so exactly \( A u \) holds only for \( A = 12 \).
2. Other atoms differ slightly because of binding energy.

Teacher's Note:
1. \( 1 u = (1/12) \) of the mass of a \( C-12 \) atom.

Answer: (d) 80

Explanation:
1. Count rate is proportional to the volume kept, so initially the remaining volume \( V \) has count rate cV/100.
2. After three half-lives it falls to 1/8: (cV/100)/8 = \( c/10 \), so \( V = 80 \) cm³.

Teacher's Note:
1. Activity is proportional to the amount of substance.

Answer: (b) \( (mN_A\lambda/M)e^{-\lambdat} \)

Explanation:
1. The number of nuclei left is \( N = (mN_A/M)e^{-\lambda t} \).
2. Activity = \( \lambdaN \) = (mN_Aλ/M)e^–λt.

Teacher's Note:
1. Activity decays exponentially at the same rate as \( N \).

Answer: (a) λ = λ₁ + λ₂

Explanation:
1. Both processes remove nuclei independently, so the decay rates add: \( dN/dt = -(\lambda_1+\lambda_2)N \).

Teacher's Note:
1. Parallel decay channels add their decay constants.

Answer: (c) λm N_A/M

Explanation:
1. Number of nuclei \( N = mN_A/M \).
2. Activity = \( \lambdaN \) = λmN_A/M.

Teacher's Note:
1. Activity = decay constant × number of undecayed nuclei.

Answer: (c) γ decay and K-capture only

Explanation:
1. α-decay lowers \( Z \) by 2 but leaves all electrons, so the atom becomes a negative ion; β⁻ raises \( Z \) by 1, so it becomes a positive ion.
2. γ-decay does not change \( Z \).
3. In K-capture a nuclear proton captures an inner electron, so \( Z \) and the electron count both fall by 1, and the atom stays neutral.

Teacher's Note:
1. A neutral atom stays neutral only if \( Z \) and the number of electrons change together.

Answer: (d) 3/8 s

Explanation:
1. When 3/4 decays, 1/4 remains, so two half-lives have passed in \( 3/4 s \).
2. Half-life = \( (3/4)/2 = 3/8 s \).

Teacher's Note:
1. \( 1/4 = (1/2)^{2} \).

Answer: (b) γ, β, α

Explanation:
1. γ-rays are the most penetrating, β-particles come next, and α-particles are stopped even by a sheet of paper.

Teacher's Note:
1. Penetrating power is the reverse of ionising power.

Answer: (a) f₁ > f₂

Explanation:
1. In one mean life, the fraction decayed is 1 – \( e^{-1} = 0.632 \).
2. In one half-life it is 0.5.
3. So f₁ > f₂, whatever the sample.

Teacher's Note:
1. Mean life is longer than half-life, by the factor 1/0.693.

Answer: (c) n₁ = 8, n₂ = 6

Explanation:
1. The mass number falls by 32, and each α-decay reduces it by 4, so n₁ = 8.
2. The atomic number would fall by 16 to 76, but it is 82, so 6 β-decays (each \( +1 \)) are needed: n₂ = 6.

Teacher's Note:
1. Mass number changes only in α-decay.
2. Each β⁻ decay raises \( Z \) by 1.

Answer: (b) T_{½} = ln 2/λ, T = 1/λ

Explanation:
1. From \( N = N_0e^{-\lambda t} \), the half-life is \( \ln 2/\lambda \) and the mean life is 1/λ.

Teacher's Note:
1. T_½ = \( 0.693 T \).

Answer: (c) \( A_2 = A_1 e^{(t_1 - t_2)/T} \)

Explanation:
1. Activity decays as \( A = A_0e^{-t/T} \), where \( T = 1/\lambda \) is the mean life.
2. So \( A_2/A_1 = e^{-(t_2-t_1)/T} \), which is A₁e^{(t₁–t₂)/T}.

Teacher's Note:
1. The ratio of activities depends only on the time difference.

Answer: (d) 2(ln 2)T

Explanation:
1. If 75% decays, 25% = 1/4 remains, so two half-lives have passed.
2. Half-life = \( (\ln 2)T \), so the time is \( 2(\ln 2)T \).

Teacher's Note:
1. Half-life = 0.693 × mean life.

Answer: (c) F₁ = F₃ > F₂

Explanation:
1. The nuclear force is charge-independent, so it is the same for \( n-n, p-p \) and \( n-p \) pairs.
2. Two protons also repel one another electrically, which reduces their net attraction, so F₂ is smaller.

Teacher's Note:
1. Nuclear force is the same for all pairs of nucleons.
2. Coulomb repulsion acts only between protons.

Answer: (c) 1/√2

Explanation:
1. After half a half-life, \( N/N_0 = (1/2)^{1/2} = 1/\sqrt2 \).

Teacher's Note:
1. \( N \) = N₀(1/2)^t/T_½.

Answer: (b) (ii) and (iii) only

Explanation:
1. A free neutron is unstable (half-life about \( 10 \min \)) and decays into a proton, so (iii) is true and \( (i) \) is false.
2. A neutron is heavier than a proton, so an isolated proton cannot change into a neutron; inside a nucleus, binding energy makes \( p \to n \) possible: (ii) true, (iv) false.

Teacher's Note:
1. Free neutron decay is the source of β⁻ particles.
2. Free protons are stable.

Answer: (c) β, α, γ

Explanation:
1. Deflection in a transverse field is proportional to \( q/m \).
2. β-particles (electrons) have a very large \( q/m \), α-particles a much smaller one, and γ-rays, being neutral, are not deflected.
3. Order: β, α, γ.

Teacher's Note:
1. Deflection is proportional to the charge-to-mass ratio.

Answer: (c) α, β, γ

Explanation:
1. Ionisation depends on charge and on the time spent near gas atoms: α (charge \( +2e \), slow) ionises most, β less, and γ (no charge) the least.

Teacher's Note:
1. Ionising power: \( \alpha \gt \beta \gt \gamma \).