Question

Mass number of a nucleus is \(\alpha\) and its radius is \(R_\alpha\). Radius of another nucleus of mass number \(\beta\) is \(R_\beta\). If \(\beta = 8\alpha\), then find \(\dfrac{R_\alpha}{R_\beta}\).

• A. \(\dfrac{1}{4}\)
• B. \(\dfrac{1}{2}\)
• C. \(\dfrac{1}{8}\)
• D. \(2\)

Hint:

Nuclear radius varies as the {cube root of mass number}: doubling the radius requires an {eightfold increase} in mass number.

Answer 1

The Correct Option is B

Concept:
The radius of a nucleus depends on its mass number according to the empirical relation: \[ R = R_0 A^{1/3} \] where \(A\) is the mass number and \(R_0\) is a constant.
Step 1: Write Radius Expressions
For nucleus with mass number \(\alpha\): \[ R_\alpha = R_0 \alpha^{1/3} \] For nucleus with mass number \(\beta\): \[ R_\beta = R_0 \beta^{1/3} \]
Step 2: Take the Ratio
\[ \frac{R_\alpha}{R_\beta} = \frac{\alpha^{1/3}}{\beta^{1/3}} \]
Step 3: Substitute \(\beta = 8\alpha\)
\[ \frac{R_\alpha}{R_\beta} = \frac{\alpha^{1/3}}{(8\alpha)^{1/3}} = \frac{\alpha^{1/3}}{2\alpha^{1/3}} = \frac{1}{2} \] \[ \boxed{\dfrac{R_\alpha}{R_\beta} = \dfrac{1}{2}} \]