Question

A nucleus has mass number \(A_1\) and volume \(V_1\). Another nucleus has mass number \(A_2\) and volume \(V_2\). If the relation between mass numbers is \(A_2 = 4A_1\), then \(\frac{V_2}{V_1} =\) _______.

Answer 1

Correct Answer: 4

For a nucleus:

Volume: \[ V = \frac{4}{3} \pi R^3 \]

Relationship for the radius:

\[ R = R_0 \left( A \right)^{1/3} \]

Substitute \( R \) in the volume equation:

\[ V = \frac{4}{3} \pi \left( R_0 (A)^{1/3} \right)^3 A \]

Simplifying:

\[ \Rightarrow \frac{V_2}{V_1} = \frac{A_2}{A_1} = 4 \]

Answer 2

For a nucleus, the volume \( V \) is proportional to \( A \), the mass number, given by:

\[ V = \frac{4}{3}\pi R^3, \]

where the radius \( R \) of a nucleus is proportional to the cube root of its mass number \( A \):

\[ R = R_0 A^{1/3}. \]

Thus, the volume \( V \) of a nucleus can be expressed as:

\[ V \propto A. \]

Since \( A_2 = 4A_1 \), the ratio of volumes is:

\[ \frac{V_2}{V_1} = \frac{A_2}{A_1} = 4. \]