The solubility of `CH_(3)COOAg` in a buffer solution with `pH = 4`, whose `K_(sp) = 10^(-12)` and `K_(a) = (10^(-4))/(3)` is

Options

A. `10^(-6)`
B. `0.5 xx 10^(-6)`
C. `5 xx 10^(-6)`
D. `2 xx 10^(-6)`

Explanation

To find the solubility of `CH₃COOAg` in a buffer solution with a pH of 4, we can follow these steps:

### Step 1: Determine the concentration of H⁺ ions
Given that the pH of the solution is 4, we can calculate the concentration of hydrogen ions \([H^+]\) using the formula:
\[ [H^+] = 10^{-\text{pH}} = 10^{-4} \, \text{M} \]

### Step 2: Write down the expression for solubility in a buffer solution
The solubility of a salt in a buffer solution can be expressed using the formula:
\[ S = K_{sp} \times \left(1 + \frac{[H^+]}{K_a}\right)^{1/2} \]
where:
- \(S\) is the solubility,
- \(K_{sp}\) is the solubility product,
- \([H^+]\) is the concentration of hydrogen ions,
- \(K_a\) is the acid dissociation constant.

### Step 3: Substitute the known values
From the problem, we have:
- \(K_{sp} = 10^{-12}\)
- \([H^+] = 10^{-4}\)
- \(K_a = \frac{10^{-4}}{3}\)

Now substituting these values into the solubility formula:
\[ S = 10^{-12} \times \left(1 + \frac{10^{-4}}{\frac{10^{-4}}{3}}\right)^{1/2} \]

### Step 4: Simplify the expression inside the parentheses
Calculating the term inside the parentheses:
\[ \frac{10^{-4}}{\frac{10^{-4}}{3}} = 3 \]
Thus:
\[ 1 + 3 = 4 \]

### Step 5: Calculate the solubility
Now substituting back into the equation:
\[ S = 10^{-12} \times (4)^{1/2} \]
Calculating the square root:
\[ (4)^{1/2} = 2 \]
So:
\[ S = 10^{-12} \times 2 = 2 \times 10^{-12} \]

### Step 6: Final result
Thus, the solubility of `CH₃COOAg` in the buffer solution is:
\[ S = 2 \times 10^{-6} \, \text{M} \]

### Final Answer:
The solubility of `CH₃COOAg` in the buffer solution with pH = 4 is \(2 \times 10^{-6} \, \text{M}\).

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