If the solubility of `Ag_(2)SO_(4)` in `10^(-2)M Na_(2)SO_(4)` solution be `2xx10^(-8)M` then `K_(sp)` of `Ag_(2)SO_(4)` will be

Options

A. `32xx10^(-24)`
B. `16xx10^(-18)`
C. `32xx10^(-18)`
D. `16xx10^(-24)`

Explanation

To find the solubility product constant (Ksp) of `Ag2SO4` given its solubility in a `10^(-2) M Na2SO4` solution, we can follow these steps:

### Step 1: Write the dissociation equation for `Ag2SO4`
`Ag2SO4` dissociates in water according to the following equation:
\[ Ag_2SO_4 (s) \rightleftharpoons 2Ag^+ (aq) + SO_4^{2-} (aq) \]

### Step 2: Define the solubility
Let the solubility of `Ag2SO4` in the solution be \( x \). This means:
- The concentration of `Ag^+` ions produced will be \( 2x \).
- The concentration of `SO4^{2-}` ions produced will be \( x \).

### Step 3: Consider the contribution of sulfate ions from `Na2SO4`
Since we have a `10^(-2) M Na2SO4` solution, it contributes:
- \( 10^{-2} M \) of `SO4^{2-}` ions.

Thus, the total concentration of `SO4^{2-}` ions in the solution will be:
\[ [SO_4^{2-}] = x + 10^{-2} \]

### Step 4: Substitute the given solubility
We are given that the solubility \( x \) of `Ag2SO4` is \( 2 \times 10^{-8} M \).

### Step 5: Calculate the total concentration of sulfate ions
Substituting \( x \):
\[ [SO_4^{2-}] = 2 \times 10^{-8} + 10^{-2} \]
Since \( 10^{-2} \) is much larger than \( 2 \times 10^{-8} \), we can approximate:
\[ [SO_4^{2-}] \approx 10^{-2} \]

### Step 6: Write the expression for Ksp
The solubility product constant \( K_{sp} \) for `Ag2SO4` is given by:
\[ K_{sp} = [Ag^+]^2 \cdot [SO_4^{2-}] \]
Substituting the concentrations:
\[ K_{sp} = (2x)^2 \cdot (x + 10^{-2}) \]

### Step 7: Substitute the values
Now, substituting \( x = 2 \times 10^{-8} \):
\[ K_{sp} = (2(2 \times 10^{-8}))^2 \cdot (10^{-2}) \]
\[ K_{sp} = (4 \times 10^{-8})^2 \cdot (10^{-2}) \]
\[ K_{sp} = 16 \times 10^{-16} \cdot 10^{-2} \]
\[ K_{sp} = 16 \times 10^{-18} \]

### Final Answer
Thus, the solubility product \( K_{sp} \) of `Ag2SO4` is:
\[ K_{sp} = 16 \times 10^{-18} \]

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