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

To solve this problem, we need to apply the formula relating the radius of a nucleus to its mass number. The radius \( R \) of a nucleus is given by the formula:

\(R = R_0 A^{1/3}\)

where \( R_0 \) is a constant (~1.2-1.3 femtometers) and \( A \) is the mass number.

We are given that the radius of the first nucleus is half of the radius of a nucleus with mass number 192, denoted as \( R_1 = \frac{1}{2} R_2 \). Let \( A_1 \) be the mass number of the first nucleus, and \( A_2 = 192 \).

1. For the nucleus with mass number 192, its radius \( R_2 \) is: \(R_2 = R_0 (192)^{1/3}\)
2. For the nucleus with mass number \( A_1 \), its radius \( R_1 \) is: \(R_1 = R_0 (A_1)^{1/3}\)
3. According to the problem, the relation given is: \(R_0 (A_1)^{1/3} = \frac{1}{2} R_0 (192)^{1/3}\)
4. Canceling \( R_0 \) from both sides: \((A_1)^{1/3} = \frac{1}{2} (192)^{1/3}\)
5. Cubing both sides to eliminate the cube root gives: \(A_1 = \left( \frac{1}{2} (192)^{1/3} \right)^3\)
6. Calculate \( (192)^{1/3} \): \((192)^{1/3} = 5.82\) (approximately)
7. Thus: \(A_1 = \left( \frac{5.82}{2} \right)^3 = \left( 2.91 \right)^3 \approx 24.68\)

Rounding 24.68 to the nearest whole number gives 24. Therefore, the mass number of the nucleus having a radius equal to half of the radius of the nucleus with mass number 192 is 24.

Answer 2

The radius \( R \) of a nucleus is proportional to the cube root of its mass number \( A \):

\[ R \propto A^{1/3}. \]

Let \( R_1 \) and \( R_2 \) be the radii of two nuclei with mass numbers \( A_1 \) and \( A_2 \), respectively. Given:

\[ R_1 = \frac{1}{2} R_2 \quad \text{and} \quad A_2 = 192. \]

Using the proportionality,

\[ \frac{R_1}{R_2} = \left( \frac{A_1}{A_2} \right)^{1/3}. \]

Substitute \( R_1 = \frac{1}{2} R_2 \):

\[ \frac{1}{2} = \left( \frac{A_1}{192} \right)^{1/3}. \]

Cubing both sides:

\[ \frac{1}{8} = \frac{A_1}{192}. \]

Solving for \( A_1 \):

\[ A_1 = 192 \times \frac{1}{8} = 24. \]

Thus, the answer is:

\[ 24. \]