Question

A nucleus has mass number \(\alpha\) and radius \(R_\alpha\). Another nucleus has mass number \(\beta\) and radius \(R_\beta\). If \(\beta=8\alpha\), then \(R_\alpha/R_\beta\) is:

• A. \(1\)
• B. \(8\)
• C. \(0.5\)
• D. \(2\)

Hint:

Nuclear radius scales as the cube root of mass number: doubling radius requires eight times mass.

Answer 1

The Correct Option is C

Concept: The radius of a nucleus is related to its mass number \(A\) by: \[ R=R_0A^{1/3} \]
Step 1: Write expressions for both radii \[ R_\alpha=R_0\alpha^{1/3},\quad R_\beta=R_0\beta^{1/3} \]
Step 2: Use the given relation \[ \beta=8\alpha \Rightarrow R_\beta=R_0(8\alpha)^{1/3} =2R_0\alpha^{1/3} \]
Step 3: Take ratio \[ \frac{R_\alpha}{R_\beta} =\frac{R_0\alpha^{1/3}}{2R_0\alpha^{1/3}} =\frac12 \] Final Answer: \[ \boxed{0.5} \]