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 follow these steps:

### Step 1: Understand the Formula for Nuclear Radius
The nuclear radius \( R \) of an isotope is given by the formula:
\[ R = R_0 A^{1/3} \]
where:
- \( R_0 \) is a constant approximately equal to \( 1.2 \times 10^{-15} \) meters,
- \( A \) is the atomic mass number of the isotope.

### Step 2: Identify the Atomic Mass Numbers
For the isotopes given:
- For gold \( _{79}^{197}Au \), the atomic mass \( A_1 = 197 \).
- For silver \( _{47}^{107}Ag \), the atomic mass \( A_2 = 107 \).

### Step 3: Write the Expressions for Nuclear Radii
Using the formula for nuclear radii, we can write:
- For gold:
\[ R_1 = R_0 A_1^{1/3} = R_0 (197)^{1/3} \]
- For silver:
\[ R_2 = R_0 A_2^{1/3} = R_0 (107)^{1/3} \]

### Step 4: Calculate the Ratio of Nuclear Radii
Now, we need to find the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{R_0 (197)^{1/3}}{R_0 (107)^{1/3}} \]
The \( R_0 \) cancels out:
\[ \frac{R_1}{R_2} = \frac{(197)^{1/3}}{(107)^{1/3}} = \left(\frac{197}{107}\right)^{1/3} \]

### Step 5: Calculate the Numerical Value
Now we compute the ratio:
\[ \frac{197}{107} \approx 1.8383 \]
Taking the cube root:
\[ \left(1.8383\right)^{1/3} \approx 1.225 \]

### Step 6: Conclusion
Thus, the ratio of the nuclear radii of the gold isotope \( _{79}^{197}Au \) and the silver isotope \( _{47}^{107}Ag \) is approximately:
\[ \frac{R_1}{R_2} \approx 1.225 \]

### Final Answer
The ratio of the nuclear radii of the gold isotope to the silver isotope is approximately \( 1.23 \).

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