Obtain the approximate value of the radius of a nucleus `"nucleus"_92 U^238, R_0` is `1.2xx10^(-15) m`.

Options

A. `3.143xx10^(-3) m `
B. `5.321xx10^(-6) m `
C. `7.437xx10^(-15) m `
D. `8.351xx10^(-12) m `

Answer

To find the approximate radius of the nucleus of Uranium-238 (\( _{92}^{238}\text{U} \)), we can use the empirical formula for the radius of a nucleus, which is given by:

\[ R = R_0 A^{1/3} \]

where:
- \( R \) is the radius of the nucleus,
- \( R_0 \) is a constant approximately equal to \( 1.2 \times 10^{-15} \, \text{m} \),
- \( A \) is the mass number of the nucleus.

### Step-by-Step Solution:

1. **Identify the mass number (A)**:
For Uranium-238, the mass number \( A \) is 238.

2. **Use the formula for the radius**:
Substitute \( R_0 \) and \( A \) into the formula:
\[ R = R_0 A^{1/3} \]

3. **Calculate \( A^{1/3} \)**:
Calculate the cube root of the mass number:
\[ A^{1/3} = 238^{1/3} \]
Using a calculator, we find:
\[ 238^{1/3} \approx 6.2 \]

4. **Substitute \( R_0 \) and \( A^{1/3} \) into the radius formula**:
Now substitute \( R_0 = 1.2 \times 10^{-15} \, \text{m} \) and \( A^{1/3} \approx 6.2 \):
\[ R \approx 1.2 \times 10^{-15} \times 6.2 \]

5. **Perform the multiplication**:
Calculate the product:
\[ R \approx 7.44 \times 10^{-15} \, \text{m} \]

### Final Result:
The approximate radius of the nucleus of Uranium-238 is:
\[ R \approx 7.44 \times 10^{-15} \, \text{m} \]