Question

Mass numbers of two nuclei are in the ratio of 4:3. Their nuclear densities will be in the ratio of:

• A. \( 4:3 \)
• B. \( \left( \frac{3}{4} \right)^{1/3} \)
• C. \( 1:1 \)
• D. \( \left( \frac{4}{3} \right)^{1/3} \)

Hint:

Nuclear density is independent of mass number. All nuclei have approximately the same density, regardless of their size.

Answer 1

The Correct Option is C

Step 1: Understanding Nuclear Density
The nuclear density \( \rho \) is given by: \[ \rho = \frac{\text{Mass of nucleus}}{\text{Volume of nucleus}} \] Since the mass of a nucleus is proportional to its mass number \( A \), and the volume of the nucleus is proportional to \( R^3 \), we can write: \[ \rho = \frac{m \times A}{\frac{4}{3} \pi R^3} \] where \( R \) is the nuclear radius.
Step 2: Relationship Between Radius and Mass Number
The nuclear radius is related to the mass number by the empirical relation: \[ R = R_0 A^{1/3} \] where \( R_0 \) is a constant.
Substituting this into the volume expression: \[ V \propto (A^{1/3})^3 = A \] Thus, the density of the nucleus becomes: \[ \rho \propto \frac{A}{A} = \text{constant} \]
Step 3: Ratio of Nuclear Densities
Since nuclear density is independent of mass number \( A \), the ratio of nuclear densities for nuclei with mass numbers in the ratio \( 4:3 \) is: \[ \frac{\rho_1}{\rho_2} = 1:1 \] Final Answer: The nuclear densities of the two nuclei will be in the ratio \( 1:1 \).