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: Establish the relationship between density, mass, and volume.
The mass density \( \rho \) of a nucleus is defined as:\[\rho = \frac{\text{Mass}}{\text{Volume}}\]Nuclear mass is directly proportional to the mass number \( A \) (as mass correlates with the number of nucleons), and nuclear volume is proportional to \( R^3 \), with \( R \) representing the nuclear radius.
Step 2: Apply the empirical formula for nuclear radii.
The radius \( R \) of a nucleus can be approximated by:\[R = R_0 A^{1/3}\]where \( R_0 \) is a constant.
Step 3: Determine the volume-mass relationship.
Given that nuclear volume is proportional to \( R^3 \), we have:\[\text{Volume} \propto A\]Consequently, both the mass (\( \propto A \)) and the volume (\( \propto A \)) of the nucleus are proportional to \( A \).
Step 4: Draw a conclusion.
As both mass and volume are proportional to \( A \), the nuclear density \( \rho \) is independent of \( A \).
Therefore, the correct conclusion is that \( \rho \) is independent of \( A \). The corresponding option is (2).