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 the mass numbers is \(A_2 = 3 A_1\), find \(\frac{V_1}{V_2}\).

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

Hint:

For nuclei, \(V \propto A\) since \(V \propto R^3\) and \(R \propto A^{1/3}\). Volume ratios follow mass number ratios.

Answer 1

The Correct Option is C

Step 1: Start from the nuclear radius law.
\[ R = r_0 A^{1/3} \]
Step 2: Cube it to get the volume.
\[ V = \frac43\pi R^3 = \frac43\pi r_0^3 A \]
so the volume is directly proportional to the mass number, with the same constant for every nucleus.
Step 3: Take the ratio, the constants cancel.
\[ \frac{V_1}{V_2} = \frac{A_1}{A_2} \]
Step 4: Substitute $A_2 = 3A_1$.
\[ \frac{V_1}{V_2} = \frac{A_1}{3A_1} = \frac13 \]
\[ \boxed{\frac{V_1}{V_2}=\frac13} \]