The concentration of `Ag^(o+)` ions in a saturated solution of `Ag_(2)C_(2)O_(4)` is `2.2 xx 10^(-4)M`. Calculate the solubility product of `Ag_(2)C_(2)O_(4)`

Explanation

To calculate the solubility product (Ksp) of \( Ag_2C_2O_4 \) given the concentration of \( Ag^+ \) ions in a saturated solution, we will follow these steps:

### Step 1: Write the Dissociation Equation
The dissociation of \( Ag_2C_2O_4 \) in water can be represented as:
\[ Ag_2C_2O_4 (s) \rightleftharpoons 2Ag^+ (aq) + C_2O_4^{2-} (aq) \]

### Step 2: Define the Solubility
Let the solubility of \( Ag_2C_2O_4 \) be \( s \). From the dissociation equation, we can see that:
- For every 1 mole of \( Ag_2C_2O_4 \) that dissolves, it produces 2 moles of \( Ag^+ \) and 1 mole of \( C_2O_4^{2-} \).
- Therefore, at equilibrium:
- The concentration of \( Ag^+ \) ions = \( 2s \)
- The concentration of \( C_2O_4^{2-} \) ions = \( s \)

### Step 3: Use the Given Concentration
We are given that the concentration of \( Ag^+ \) ions in the saturated solution is:
\[ [Ag^+] = 2.2 \times 10^{-4} \, M \]
From our earlier definition, we have:
\[ 2s = 2.2 \times 10^{-4} \]
Thus, we can solve for \( s \):
\[ s = \frac{2.2 \times 10^{-4}}{2} = 1.1 \times 10^{-4} \, M \]

### Step 4: Calculate the Concentration of \( C_2O_4^{2-} \)
From the dissociation equation, we know:
\[ [C_2O_4^{2-}] = s = 1.1 \times 10^{-4} \, M \]

### Step 5: Write the Expression for Ksp
The solubility product \( Ksp \) is given by the expression:
\[ Ksp = [Ag^+]^2 \times [C_2O_4^{2-}] \]

### Step 6: Substitute the Values
Now substituting the values we have:
\[ Ksp = (2.2 \times 10^{-4})^2 \times (1.1 \times 10^{-4}) \]

### Step 7: Calculate Ksp
Calculating \( Ksp \):
\[ Ksp = (4.84 \times 10^{-8}) \times (1.1 \times 10^{-4}) = 5.324 \times 10^{-12} \]

### Final Answer
Thus, the solubility product \( Ksp \) of \( Ag_2C_2O_4 \) is approximately:
\[ Ksp \approx 5.3 \times 10^{-12} \]

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