Compare the radii of two nuclei with mass number 64 and 1 respectively.

Options

A. 4/1
B. 1/4
C. 2/4
D. 3/4

Explanation

To compare the radii of two nuclei with mass numbers 64 and 1, 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 \( 1.2 \, \text{fm} \) or \( 1.2 \times 10^{-15} \, \text{m} \)),
- \( A \) is the mass number of the nucleus.

### Step-by-step Solution:

1. **Identify the mass numbers**:
- For the first nucleus, \( A_1 = 64 \).
- For the second nucleus, \( A_2 = 1 \).

2. **Write the formula for the radii of the two nuclei**:
- The radius of the first nucleus (\( R_1 \)) is:
\[ R_1 = R_0 A_1^{1/3} = R_0 (64)^{1/3} \]
- The radius of the second nucleus (\( R_2 \)) is:
\[ R_2 = R_0 A_2^{1/3} = R_0 (1)^{1/3} \]

3. **Calculate the ratio of the radii**:
- To compare the two radii, we take the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{R_0 (64)^{1/3}}{R_0 (1)^{1/3}} \]
- The \( R_0 \) cancels out:
\[ \frac{R_1}{R_2} = \frac{(64)^{1/3}}{(1)^{1/3}} = (64)^{1/3} \]

4. **Calculate \( (64)^{1/3} \)**:
- Since \( 64 = 4^3 \), we have:
\[ (64)^{1/3} = 4 \]

5. **Conclusion**:
- Therefore, the ratio of the radii of the two nuclei is:
\[ \frac{R_1}{R_2} = 4 \]
- This means that the radius of the nucleus with mass number 64 is 4 times larger than that of the nucleus with mass number 1.