Question

If the ratio of the radius of a nucleus with 61 neutrons to that of helium nucleus is 3, the atomic number of this nucleus is

• A. $27$
• B. $47$
• C. $51$
• D. $61$
• E. $108$

Hint:

Always cube the radius ratio → then use $A = Z + N$ to find atomic number.

Answer 1

The Correct Option is B

Concept: Nuclear radius relation
The radius of a nucleus depends on its mass number $A$ as: \[ R = R_0 A^{1/3} \] Thus, radius is proportional to the cube root of mass number: \[ R \propto A^{1/3} \] ---

Step 1: Form ratio of radii
Given: \[ \frac{R_{\text{unknown}}}{R_{\text{He}}} = 3 \] Using proportionality: \[ \frac{A^{1/3}}{4^{1/3}} = 3 \] ---

Step 2: Cube both sides to remove root \[ \frac{A}{4} = 27 \] \[ A = 108 \] ---

Step 3: Use relation between $A$, $Z$, and $N$
We know: \[ A = Z + N \] Given: \[ N = 61 \] So: \[ 108 = Z + 61 \] ---

Step 4: Solve for atomic number \[ Z = 108 - 61 = 47 \] ---

Step 5: Physical Interpretation

• Nuclear size depends only on total nucleons ($A$), not separately on protons or neutrons
• Even though neutrons = 61, total nucleons determine radius
• Atomic number tells number of protons in nucleus --- Final Answer: \[ \boxed{47} \]