Question

For a nucleus of mass number $A$ and radius $R$, mass density $\rho$. Then choose the correct option.

• A. \( \rho \propto A^{1/3} \)
• B. \( \rho \) is independent of \( A \)
• C. \( \rho \propto A \)
• D. \( \rho \propto A^3 \)

Hint:

The mass density of a nucleus is inversely proportional to the cube of the radius, and the radius is proportional to \( A^{1/3} \).

Answer 1

The Correct Option is A

The mass density \( \rho \) of a nucleus is defined as: \[ \rho = \frac{\text{Mass}}{\text{Volume}}. \] The mass of the nucleus is proportional to the mass number \( A \), so: \[ \text{Mass} \propto A. \] The volume of a nucleus is proportional to the cube of its radius, and the radius of the nucleus \( R \) is proportional to \( A^{1/3} \). Thus, the volume \( V \) of the nucleus is: \[ V \propto R^3 \propto A. \] Therefore, the mass density \( \rho \) is given by: \[ \rho = \frac{\text{Mass}}{\text{Volume}} \propto \frac{A}{A} = A^{1/3}. \] Thus, the correct answer is (1) \( \rho \propto A^{1/3} \).