Question

A nucleus of mass number 189 splits into two nuclei having mass number 125 and 64. The ratio of radius of two daughter nuclei respectively is:

• A. 1:1
• B. 4:5
• C. 5:4
• D. 25:16:00

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The volume of a nucleus is directly proportional to the total number of nucleons (mass number \(A\)).
Assuming the nucleus to be spherical with radius \(R\), its volume is \(V = \frac{4}{3} \pi R^3\).
Therefore, \(R^3 \propto A\), which implies that the nuclear radius is proportional to the cube root of the mass number.
Key Formula or Approach:
The empirical relationship for nuclear radius is:
\[ R = R_0 A^{1/3} \]
where \(R_0\) is a constant (\(\approx 1.2\) to \(1.5\) fm).
For two different nuclei with mass numbers \(A_1\) and \(A_2\), the ratio of their radii is:
\[ \frac{R_1}{R_2} = \left( \frac{A_1}{A_2} \right)^{1/3} \]
Step 2: Detailed Explanation:
Given the mass numbers of the two daughter nuclei:
- \(A_1 = 125\)
- \(A_2 = 64\)
Substituting these values into the ratio formula:
\[ \frac{R_1}{R_2} = \left( \frac{125}{64} \right)^{1/3} \]
We recognize that \(125 = 5^3\) and \(64 = 4^3\).
Taking the cube root of the ratio:
\[ \frac{R_1}{R_2} = \frac{(5^3)^{1/3}}{(4^3)^{1/3}} = \frac{5}{4} \]
Thus, the ratio of the radii of the two nuclei is \(5 : 4\).
Step 3: Final Answer:
The ratio of the radius of the two daughter nuclei is \(5 : 4\).