Question

Mass numbers of two nuclei are in the ratio 2:3. The ratio of the nuclear densities would be:

• A. \( 2:3^{1/3} \)
• B. \( 3^{1/3}:2 \)
• C. \( 2:3 \)
• D. \( 3:2 \)
• E. \( 1:1 \)

Hint:

The density of a nucleus is independent of its mass number because the volume of a nucleus scales as \( A^{1/3} \), and mass scales directly with \( A \).

Answer 1

Answer: (E)

Step 1: The nuclear density \( \rho \) is given by: \[ \rho = \frac{{Mass}}{{Volume}} = \frac{A}{\frac{4}{3} \pi R^3} \] where \( A \) is the mass number, and \( R \) is the radius of the nucleus. 
The radius of the nucleus is related to the mass number \( A \) by the empirical relation: \[ R \propto A^{1/3} \] 
Step 2: Therefore, the density is given by: \[ \rho \propto \frac{A}{R^3} \propto \frac{A}{(A^{1/3})^3} = \frac{A}{A} = 1 \] 
Step 3: Hence, the ratio of the nuclear densities of the two nuclei will be \( 1:1 \).