Obtain approx. the ratio of the nuclear radii of the gold isotope `._79Au^(197)` and silver isotope `._47Ag^(107)`.

Options

A. `197:107`
B. `47:79`
C. `79:47`
D. `1:1`

Explanation

To find the ratio of the nuclear radii of the gold isotope \( _{79}^{197}\text{Au} \) and the silver isotope \( _{47}^{107}\text{Ag} \), we can use the formula for nuclear radius. The nuclear radius \( R \) can be approximated using the formula:

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

where:
- \( R_0 \) 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 gold \( _{79}^{197}\text{Au} \), the mass number \( A_{Au} = 197 \).
- For silver \( _{47}^{107}\text{Ag} \), the mass number \( A_{Ag} = 107 \).

2. **Calculate the nuclear radii**:
- For gold:
\[ R_{Au} = R_0 (197)^{1/3} \]
- For silver:
\[ R_{Ag} = R_0 (107)^{1/3} \]

3. **Find the ratio of the nuclear radii**:
\[ \frac{R_{Au}}{R_{Ag}} = \frac{R_0 (197)^{1/3}}{R_0 (107)^{1/3}} = \frac{(197)^{1/3}}{(107)^{1/3}} = \left(\frac{197}{107}\right)^{1/3} \]

4. **Calculate the numerical value**:
- First, calculate \( \frac{197}{107} \):
\[ \frac{197}{107} \approx 1.8383 \]
- Now take the cube root:
\[ \left(1.8383\right)^{1/3} \approx 1.220 \]

5. **Final Result**:
- The approximate ratio of the nuclear radii of gold to silver is:
\[ \frac{R_{Au}}{R_{Ag}} \approx 1.22 \]