The ratio of the nuclear radii of the gold isotope `._79^197Au` and silver isotope `._47^107Ag` is

Options

A. 1.23
B. 0.216
C. 2.13
D. 3.46

Explanation

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

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

where \( R_0 \) is a constant (approximately \( 1.2 \, \text{fm} \)) and \( A \) is the mass number of the nucleus.

### Step 1: Identify the mass numbers
- For gold \( _{79}^{197}Au \), the mass number \( A_{Au} = 197 \).
- For silver \( _{47}^{107}Ag \), the mass number \( A_{Ag} = 107 \).

### Step 2: Write the formula for the nuclear radii
The nuclear radius for gold can be expressed as:
\[ R_{Au} = R_0 \cdot (197)^{1/3} \]
The nuclear radius for silver can be expressed as:
\[ R_{Ag} = R_0 \cdot (107)^{1/3} \]

### Step 3: Calculate the ratio of the nuclear radii
The ratio of the nuclear radii \( \frac{R_{Au}}{R_{Ag}} \) is given by:
\[ \frac{R_{Au}}{R_{Ag}} = \frac{R_0 \cdot (197)^{1/3}}{R_0 \cdot (107)^{1/3}} = \frac{(197)^{1/3}}{(107)^{1/3}} \]

### Step 4: Simplify the ratio
This simplifies to:
\[ \frac{R_{Au}}{R_{Ag}} = \left( \frac{197}{107} \right)^{1/3} \]

### Step 5: Calculate the numerical value
Now we can calculate \( \frac{197}{107} \):
\[ \frac{197}{107} \approx 1.8383 \]
Taking the cube root:
\[ \left( 1.8383 \right)^{1/3} \approx 1.220 \]

### Conclusion
Thus, the ratio of the nuclear radii of the gold isotope to the silver isotope is approximately:
\[ \frac{R_{Au}}{R_{Ag}} \approx 1.22 \]

### Final Answer
The ratio of the nuclear radii of the gold isotope \( _{79}^{197}Au \) and silver isotope \( _{47}^{107}Ag \) is approximately \( 1.22 \).

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