Read the following statements :
(A) Volume of the nucleus is directly proportional to the mass number.
(B) Volume of the nucleus is independent of mass number.
(C) Density of the nucleus is directly proportional to the mass number.
(D) Density of the nucleus is directly proportional to the cube root of the mass number.
(E) Density of the nucleus is independent of the mass number.
Choose the correct option from the following options.

Solution

Physics sir · 4 steps

Start from R = R₀ A^(1/3): cube it for volume, then divide mass by volume and watch A cancel.

1. Nuclear radius. Experiments show the radius of a nucleus is R = R₀ A^(1/3), where R₀ ≈ 1.2 × 10⁻¹⁵ m and A is the mass number.
2. Volume. Cube the radius: V = 4/3π R³ = 4/3π R₀³ A. So V ∝ A, and statement (A) is true while (B) is false.

   ρ = m/V = (A mₚ)/(4/3π R₀³ A) = (3 mₚ)/(4π R₀³)

3. Density. The mass of the nucleus is about A mₚ, so A cancels between mass and volume. The density is the same for every nucleus, about 2.3 × 10¹⁷ kg m⁻³. So (E) is true, and (C) and (D) are false.
4. So. Only (A) and (E) are correct, which is option (b).

The idea to keep. Since R ∝ A^(1/3), volume grows exactly like mass (∝ A). Mass over volume is then constant, so every nucleus has the same density.

Answer from the NTA final answer key. Solution by PrepWiser.