Question

The mass number of nucleus having radius equal to half of the radius of nucleus with mass number 192 is:

• A. 24
• B. 32
• C. 40
• D. 20

Answer 1

The Correct Option is A

The radius \( R \) of a nucleus is proportional to the cube root of its mass number \( A \), described by the formula \( R = R_0 A^{1/3} \), where \( R_0 \) is a constant. We are given a nucleus with mass number \( A_2 = 192 \) and another nucleus with mass number \( A_1 \) such that its radius \( R_1 \) is half the radius \( R_2 \) of the nucleus with mass number 192, i.e., \( R_1 = \frac{1}{2} R_2 \).

1. The radius of the nucleus with mass number 192 is \( R_2 = R_0 (192)^{1/3} \).
2. The radius of the nucleus with mass number \( A_1 \) is \( R_1 = R_0 (A_1)^{1/3} \).
3. The given relationship is \( R_0 (A_1)^{1/3} = \frac{1}{2} R_0 (192)^{1/3} \).
4. Simplifying by canceling \( R_0 \), we get \( (A_1)^{1/3} = \frac{1}{2} (192)^{1/3} \).
5. Cubing both sides to solve for \( A_1 \): \( A_1 = \left( \frac{1}{2} (192)^{1/3} \right)^3 \).
6. Approximating \( (192)^{1/3} \approx 5.82 \).
7. Substituting this value: \( A_1 = \left( \frac{5.82}{2} \right)^3 = (2.91)^3 \approx 24.68 \).

Rounding to the nearest whole number, the mass number of the nucleus is 24.