The solubility of silver benzoate `(C_(6)H_(5)COOAg)` in `H_(2)O` and in a buffer solution of `pH = 2, 3`, and `4` are `S_(1), S_(2), S_(3)` and `S_(4)` respectively. The decreasing order of solubility is

Options

A. `S_(1) gt S_(2) gt S_(3) gt S_(4)`
B. `S_(4) gt S_(3) gt S_(2) gt S_(1)`
C. `S_(2) gt S_(3) gt S_(4) gt S_(1)`
D. `S_(3) gt S_(2) gt S_(4) gt S_(1)`

Explanation

To determine the decreasing order of solubility of silver benzoate `(C_(6)H_(5)COOAg)` in water and buffer solutions of different pH values, we can follow these steps:

### Step 1: Understand the Dissociation of Silver Benzoate
Silver benzoate dissociates in water to produce benzoate ions `(C_(6)H_(5)COO^-)` and silver ions `(Ag^+)`. The equilibrium can be represented as:
\[ C_{6}H_{5}COOAg (s) \rightleftharpoons C_{6}H_{5}COO^- (aq) + Ag^+ (aq) \]

### Step 2: Write the Expression for Solubility Product (Ksp)
The solubility product constant (Ksp) for this dissociation can be expressed as:
\[ K_{sp} = [C_{6}H_{5}COO^-][Ag^+] \]
If the solubility of silver benzoate is `S`, then at equilibrium:
\[ [C_{6}H_{5}COO^-] = S \]
\[ [Ag^+] = S \]
Thus, we can write:
\[ K_{sp} = S^2 \]
From this, we can find the solubility `S`:
\[ S = \sqrt{K_{sp}} \]

### Step 3: Effect of pH on Solubility
In acidic solutions (lower pH), the concentration of hydrogen ions `(H^+)` increases. The benzoate ion `(C_{6}H_{5}COO^-)` can react with `H^+` to form benzoic acid `(C_{6}H_{5}COOH)`, which shifts the equilibrium to the right, increasing the solubility of silver benzoate.

The relationship can be expressed as:
\[ S_{buffer} = \sqrt{K_{sp}} \left(1 + \frac{[H^+]}{K_a}\right)^{1/2} \]
Where `K_a` is the acid dissociation constant for benzoic acid.

### Step 4: Analyze the Given pH Values
- At **pH 2**, the concentration of `H^+` is highest.
- At **pH 3**, the concentration of `H^+` is lower than at pH 2.
- At **pH 4**, the concentration of `H^+` is lower than at pH 3.
- At **pH 7** (pure water), the concentration of `H^+` is the lowest.

### Step 5: Determine the Order of Solubility
Given the relationship that higher `H^+` concentration leads to higher solubility, we can conclude:
- **S2** (pH 2) > **S3** (pH 3) > **S4** (pH 4) > **S1** (water, pH 7)

### Final Order of Solubility
Thus, the decreasing order of solubility is:
\[ S_2 > S_3 > S_4 > S_1 \]