Question

Two nuclei have mass numbers \(64\) and \(27\). The ratio of their nuclear densities is:

• A. \(4:3\)
• B. \(3:4\)
• C. \(1:1\)
• D. \(16:9\)

Hint:

Nuclear density is independent of the size of the nucleus because both nuclear mass and nuclear volume are directly proportional to mass number \(A\).

Answer 1

The Correct Option is C

Concept: The density of atomic nuclei is approximately constant for all elements irrespective of their mass numbers. This happens because the radius of a nucleus depends on the mass number according to the relation: \[ R = R_0 A^{1/3} \] where:

• \(R\) = radius of nucleus
• \(R_0\) = nuclear constant
• \(A\) = mass number

Since volume of a nucleus is proportional to \(R^3\), \[ V \propto (A^{1/3})^3 \] \[ V \propto A \] Also, mass of nucleus is directly proportional to mass number \(A\). Thus, \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] \[ \rho \propto \frac{A}{A} \] \[ \rho = \text{constant} \] Therefore, all nuclei have nearly the same density.

Step 1: Write the relation for nuclear density.
Density is given by: \[ \rho = \frac{\text{Mass of nucleus}}{\text{Volume of nucleus}} \] Mass of nucleus: \[ m \propto A \] Radius of nucleus: \[ R = R_0 A^{1/3} \] Volume of nucleus: \[ V = \frac{4}{3}\pi R^3 \] Substituting the radius relation: \[ V \propto (A^{1/3})^3 \] \[ V \propto A \]

Step 2: Find the density dependence.
Since, \[ \rho \propto \frac{A}{A} \] \[ \rho = \text{constant} \] Thus nuclear density does not depend on mass number.

Step 3: Compare the two nuclei.
Given mass numbers: \[ A_1 = 64 \] \[ A_2 = 27 \] Since nuclear density is constant for all nuclei: \[ \rho_1 = \rho_2 \] Therefore, \[ \rho_1 : \rho_2 = 1:1 \] Hence, the correct answer is: \[ \boxed{1:1} \]