Compare the radii of two nuclei with mass number 125 and 64 respectively.

Options

A. 5/2
B. 5/3
C. 5/4
D. 3/5

Explanation

To compare the radii of two nuclei with mass numbers 125 and 64, we can use the 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 \( 1.2 \, \text{fm} \)),
- \( A \) is the mass number of the nucleus.

### Step-by-Step Solution:

1. **Identify the mass numbers**:
- Let \( A_1 = 125 \) (mass number of the first nucleus).
- Let \( A_2 = 64 \) (mass number of the second nucleus).

2. **Write the formula for the radii**:
- The radius of the first nucleus \( r_1 \) is given by:
\[ r_1 = r_0 A_1^{1/3} \]
- The radius of the second nucleus \( r_2 \) is given by:
\[ r_2 = r_0 A_2^{1/3} \]

3. **Set up the ratio of the radii**:
- To compare the two radii, we can find the ratio \( \frac{r_1}{r_2} \):
\[ \frac{r_1}{r_2} = \frac{r_0 A_1^{1/3}}{r_0 A_2^{1/3}} = \frac{A_1^{1/3}}{A_2^{1/3}} \]
- Here, the \( r_0 \) cancels out.

4. **Substitute the values of \( A_1 \) and \( A_2 \)**:
- Substitute \( A_1 = 125 \) and \( A_2 = 64 \):
\[ \frac{r_1}{r_2} = \frac{125^{1/3}}{64^{1/3}} = \left(\frac{125}{64}\right)^{1/3} \]

5. **Calculate \( \frac{125}{64} \)**:
- Calculate \( \frac{125}{64} = \frac{5^3}{4^3} = \left(\frac{5}{4}\right)^3 \).

6. **Take the cube root**:
- Now take the cube root:
\[ \left(\frac{5}{4}\right)^{1} = \frac{5}{4} \]

7. **Conclusion**:
- Thus, the ratio of the radii of the two nuclei is:
\[ \frac{r_1}{r_2} = \frac{5}{4} \]
- This means that the radius of the nucleus with mass number 125 is \( \frac{5}{4} \) times the radius of the nucleus with mass number 64.