Question

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

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

Hint:

The density of a nucleus is approximately constant across different nuclei, meaning it does not vary significantly with mass number \( A \).

Answer 1

The Correct Option is B

Step 1: Relate density, mass, and volume.
The mass density \( \rho \) of a nucleus is given by the formula: \[ \rho = \frac{\text{Mass}}{\text{Volume}} \] The mass of the nucleus is proportional to the mass number \( A \) (since mass is proportional to the number of nucleons), and the volume of the nucleus is proportional to \( R^3 \), where \( R \) is the radius of the nucleus.
Step 2: Use the empirical relationship for nuclear radii.
The radius \( R \) of a nucleus is approximately given by: \[ R = R_0 A^{1/3} \] where \( R_0 \) is a constant.
Step 3: Volume and mass relationship.
The volume of the nucleus is proportional to \( R^3 \), so: \[ \text{Volume} \propto A \] Thus, the mass of the nucleus is \( \propto A \), and the volume is also \( \propto A \).
Step 4: Conclusion.
Since both mass and volume are proportional to \( A \), the density \( \rho \) of the nucleus becomes independent of \( A \).
Hence, the correct answer is option (2), \( \rho \) is independent of \( A \).