What is `[Ag^(+)]` in a solution made by dissolving both `Ag_(2)CrO_(4)` and `Ag_(2)C_(2)O_(4)` until saturation is reached with respect to both salts ?
`"["K_(sp)(Ag_(2)C_(2)O_(4))=2xx10^(-11)," " K_(sp)(Ag_(2)CrO_(4))=2xx10^(-12)"]"`

Options

A. `2.80xx10^(-4)`
B. `7.6xx10^(-5)`
C. `6.63xx10^(-6)`
D. `3.52xx10^(-4)`

Answer

To find the concentration of \([Ag^+]\) in a solution made by dissolving both \(Ag_2CrO_4\) and \(Ag_2C_2O_4\) until saturation is reached, we can follow these steps:

### Step 1: Write the Dissociation Reactions
For \(Ag_2CrO_4\):
\[ Ag_2CrO_4 (s) \rightleftharpoons 2Ag^+ (aq) + CrO_4^{2-} (aq) \]
For \(Ag_2C_2O_4\):
\[ Ag_2C_2O_4 (s) \rightleftharpoons 2Ag^+ (aq) + C_2O_4^{2-} (aq) \]

### Step 2: Write the Expression for \(K_{sp}\)
The solubility product expressions for both salts are:
1. For \(Ag_2CrO_4\):
\[ K_{sp}(Ag_2CrO_4) = [Ag^+]^2 [CrO_4^{2-}] = 2 \times 10^{-12} \]
2. For \(Ag_2C_2O_4\):
\[ K_{sp}(Ag_2C_2O_4) = [Ag^+]^2 [C_2O_4^{2-}] = 2 \times 10^{-11} \]

### Step 3: Set Up the Charge Balance Equation
Since the solution is electrically neutral, the total positive charge must equal the total negative charge:
\[ [Ag^+] = 2[CrO_4^{2-}] + 2[C_2O_4^{2-}] \]

### Step 4: Substitute the Concentrations from \(K_{sp}\)
From the \(K_{sp}\) expressions, we can express \([CrO_4^{2-}]\) and \([C_2O_4^{2-}]\) in terms of \([Ag^+]\):
1. From \(Ag_2CrO_4\):
\[ [CrO_4^{2-}] = \frac{K_{sp}}{[Ag^+]^2} = \frac{2 \times 10^{-12}}{[Ag^+]^2} \]
2. From \(Ag_2C_2O_4\):
\[ [C_2O_4^{2-}] = \frac{K_{sp}}{[Ag^+]^2} = \frac{2 \times 10^{-11}}{[Ag^+]^2} \]

### Step 5: Substitute Back into the Charge Balance Equation
Substituting these expressions into the charge balance equation gives:
\[ [Ag^+] = 2\left(\frac{2 \times 10^{-12}}{[Ag^+]^2}\right) + 2\left(\frac{2 \times 10^{-11}}{[Ag^+]^2}\right) \]
\[ [Ag^+] = \frac{4 \times 10^{-12}}{[Ag^+]^2} + \frac{4 \times 10^{-11}}{[Ag^+]^2} \]
Combining the terms:
\[ [Ag^+] = \frac{4 \times 10^{-12} + 4 \times 10^{-11}}{[Ag^+]^2} \]
\[ [Ag^+] = \frac{4.4 \times 10^{-11}}{[Ag^+]^2} \]

### Step 6: Rearranging and Solving for \([Ag^+]\)
Multiplying both sides by \([Ag^+]^2\):
\[ [Ag^+]^3 = 4.4 \times 10^{-11} \]
Taking the cube root:
\[ [Ag^+] = (4.4 \times 10^{-11})^{1/3} \]
Calculating this gives:
\[ [Ag^+] \approx 3.52 \times 10^{-4} \, \text{M} \]

### Final Answer
Thus, the concentration of \([Ag^+]\) in the solution is approximately \(3.52 \times 10^{-4} \, \text{M}\).

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