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 expressed as:\[\rho = \frac{\text{Mass}}{\text{Volume}}.\]Since the nuclear mass is directly proportional to the mass number \( A \):\[\text{Mass} \propto A.\]The nuclear volume \( V \) is proportional to the cube of its radius \( R \), and \( R \) is proportional to \( A^{1/3} \). Consequently, the volume is:\[V \propto R^3 \propto A.\]Thus, the mass density \( \rho \) is determined by:\[\rho = \frac{\text{Mass}}{\text{Volume}} \propto \frac{A}{A} = A^{1/3}.\]Therefore, the correct relationship is \( \rho \propto A^{1/3} \). (1)