If the radii of nuclei of `._(13)Al^(27) and ._(30)Zn^(64)` are `R_(1) and R_(2)` respectively, then `(R_(1))/(R_(2))=`

Options

A. `(64)/(27)`
B. `(3)/(4)`
C. `(27)/(64)`
D. `(4)/(3)`

Answer

To find the ratio of the radii of the nuclei of Aluminium (Al) and Zinc (Zn), 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 \( 1.2 \, \text{fm} \) for nuclear radii),
- \( A \) is the mass number (number of nucleons) of the nucleus.

### Step-by-Step Solution:

1. **Identify the mass numbers**:
- For Aluminium (\( _{13}^{27}Al \)), the mass number \( A_1 = 27 \).
- For Zinc (\( _{30}^{64}Zn \)), the mass number \( A_2 = 64 \).

2. **Write the formula for the radius of each nucleus**:
- For Aluminium:
\[ R_1 = R_0 (27)^{1/3} \]
- For Zinc:
\[ R_2 = R_0 (64)^{1/3} \]

3. **Calculate the ratio of the radii**:
\[ \frac{R_1}{R_2} = \frac{R_0 (27)^{1/3}}{R_0 (64)^{1/3}} \]
- The \( R_0 \) terms cancel out:
\[ \frac{R_1}{R_2} = \frac{(27)^{1/3}}{(64)^{1/3}} \]

4. **Simplify the ratio**:
- Calculate \( (27)^{1/3} \) and \( (64)^{1/3} \):
- \( (27)^{1/3} = 3 \) (since \( 3^3 = 27 \))
- \( (64)^{1/3} = 4 \) (since \( 4^3 = 64 \))
- Thus, the ratio becomes:
\[ \frac{R_1}{R_2} = \frac{3}{4} \]

5. **Conclusion**:
- The final answer is:
\[ \frac{R_1}{R_2} = \frac{3}{4} \]