Solubility of AgCl in 0.2 M NaCl is x and that in 0.1 M `AgNO_(3)` is y. Then which of the following is correct?

Options

A. x=y
B. xgty
C. xlty
D. we cannot predict

Explanation

To solve the problem regarding the solubility of AgCl in different solutions, we need to analyze the solubility product (Ksp) of AgCl in the presence of NaCl and AgNO3.

### Step-by-Step Solution:

1. **Understanding the Dissociation of AgCl**:
AgCl dissociates in water as follows:
\[ \text{AgCl} \rightleftharpoons \text{Ag}^+ + \text{Cl}^- \]
The solubility product \( K_{sp} \) for AgCl can be expressed as:
\[ K_{sp} = [\text{Ag}^+][\text{Cl}^-] \]

2. **Solubility of AgCl in 0.2 M NaCl**:
In a solution containing 0.2 M NaCl, the concentration of Cl⁻ ions is already 0.2 M. Let the solubility of AgCl in this solution be \( x \) (which represents the concentration of Ag⁺ ions from the dissociation of AgCl).

Therefore, the equilibrium concentrations will be:
- \([\text{Ag}^+] = x\)
- \([\text{Cl}^-] = 0.2\)

Substituting these into the \( K_{sp} \) expression:
\[ K_{sp} = x \cdot 0.2 \]
Rearranging gives:
\[ x = \frac{K_{sp}}{0.2} \]

3. **Solubility of AgCl in 0.1 M AgNO3**:
AgNO3 dissociates as follows:
\[ \text{AgNO}_3 \rightleftharpoons \text{Ag}^+ + \text{NO}_3^- \]
In a solution containing 0.1 M AgNO3, the concentration of Ag⁺ ions is already 0.1 M. Let the solubility of AgCl in this solution be \( y \).

Therefore, the equilibrium concentrations will be:
- \([\text{Ag}^+] = 0.1 + y\)
- \([\text{Cl}^-] = y\)

Substituting these into the \( K_{sp} \) expression:
\[ K_{sp} = (0.1 + y) \cdot y \]
For small values of \( y \) compared to 0.1, we can approximate:
\[ K_{sp} \approx 0.1y \]
Rearranging gives:
\[ y = \frac{K_{sp}}{0.1} \]

4. **Comparing x and y**:
Now we have:
- \( x = \frac{K_{sp}}{0.2} \)
- \( y = \frac{K_{sp}}{0.1} \)

Since \( \frac{K_{sp}}{0.1} > \frac{K_{sp}}{0.2} \), it follows that:
\[ y > x \]

### Conclusion:
The solubility of AgCl in 0.1 M AgNO3 (y) is greater than its solubility in 0.2 M NaCl (x). Therefore, the correct statement is that \( y > x \).