A nucleus with `A = 235` splits into two nuclei whose mass numbers are in the ratio `2:1`. IF `R_(0) = 1.4` fm, find the radii of the new nuclei.

Answer

To solve the problem, we need to find the radii of the two new nuclei formed when a nucleus with mass number \( A = 235 \) splits into two nuclei whose mass numbers are in the ratio \( 2:1 \).

### Step-by-Step Solution:

1. **Determine the Mass Numbers of the New Nuclei:**
Let the mass numbers of the two new nuclei be \( A_1 \) and \( A_2 \). Given the ratio \( A_1 : A_2 = 2 : 1 \), we can express this as:
\[ A_1 = 2x \quad \text{and} \quad A_2 = x \]
Since the total mass number before the split is \( A = 235 \), we have:
\[ A_1 + A_2 = 235 \implies 2x + x = 235 \implies 3x = 235 \implies x = \frac{235}{3} \approx 78.33 \]
Therefore:
\[ A_1 = 2x \approx 156.67 \quad \text{and} \quad A_2 = x \approx 78.33 \]
For practical purposes, we can round these to the nearest whole numbers:
\[ A_1 \approx 157 \quad \text{and} \quad A_2 \approx 78 \]

2. **Calculate the Radii of the New Nuclei:**
The radius \( R \) of a nucleus is given by the formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 = 1.4 \) fm.

- For nucleus with mass number \( A_1 = 157 \):
\[ R_1 = R_0 A_1^{1/3} = 1.4 \, \text{fm} \times (157)^{1/3} \]
- For nucleus with mass number \( A_2 = 78 \):
\[ R_2 = R_0 A_2^{1/3} = 1.4 \, \text{fm} \times (78)^{1/3} \]

3. **Calculate the Values:**
- First, calculate \( (157)^{1/3} \) and \( (78)^{1/3} \):
- \( (157)^{1/3} \approx 5.368 \)
- \( (78)^{1/3} \approx 4.273 \)

- Now substitute these values into the radius formulas:
\[ R_1 = 1.4 \, \text{fm} \times 5.368 \approx 7.515 \, \text{fm} \]
\[ R_2 = 1.4 \, \text{fm} \times 4.273 \approx 5.979 \, \text{fm} \]

### Final Results:
- The radius of the first nucleus \( R_1 \approx 7.52 \, \text{fm} \)
- The radius of the second nucleus \( R_2 \approx 5.98 \, \text{fm} \)