If a saturated solution prepared by dissolving `Ag_(2)CO_(3)` in water has `[Ag^(+)] = 2.5 xx 10^(-)`. What is the value of `K_(sp)` for `Ag_(2)CO_(3)` ?

Options

A. `78.1 xx 10^(-12)`
B. `7.81 xx 10^(-12)`
C. `93.8 xx 10^(-12)`
D. `9.38 xx 10^(-12)`

Answer

To find the solubility product constant (Ksp) for silver carbonate (Ag₂CO₃), we can follow these steps:

### Step 1: Write the Dissociation Equation
The dissociation of silver carbonate in water can be represented as:
\[ \text{Ag}_2\text{CO}_3 (s) \rightleftharpoons 2 \text{Ag}^+ (aq) + \text{CO}_3^{2-} (aq) \]

### Step 2: Determine Ion Concentrations
From the problem, we know that the concentration of silver ions, \([Ag^+]\), in the saturated solution is given as:
\[ [Ag^+] = 2.5 \times 10^{-4} \, \text{M} \]

Since the stoichiometry of the dissociation shows that 2 moles of \(\text{Ag}^+\) are produced for every mole of \(\text{Ag}_2\text{CO}_3\), we can find the concentration of carbonate ions \([CO_3^{2-}]\):
\[ [CO_3^{2-}] = \frac{1}{2} [Ag^+] = \frac{1}{2} (2.5 \times 10^{-4}) = 1.25 \times 10^{-4} \, \text{M} \]

### Step 3: Write the Expression for Ksp
The solubility product constant \(K_{sp}\) is defined as:
\[ K_{sp} = [Ag^+]^2 [CO_3^{2-}] \]

### Step 4: Substitute the Concentrations into the Ksp Expression
Now we can substitute the values we found into the \(K_{sp}\) expression:
\[ K_{sp} = (2.5 \times 10^{-4})^2 \times (1.25 \times 10^{-4}) \]

### Step 5: Calculate Ksp
Calculating each part:
1. Calculate \((2.5 \times 10^{-4})^2\):
\[ (2.5 \times 10^{-4})^2 = 6.25 \times 10^{-8} \]

2. Now multiply by \((1.25 \times 10^{-4})\):
\[ K_{sp} = 6.25 \times 10^{-8} \times 1.25 \times 10^{-4} = 7.8125 \times 10^{-12} \]

### Final Answer
Thus, the value of \(K_{sp}\) for \(Ag_2CO_3\) is:
\[ K_{sp} = 7.8125 \times 10^{-12} \]

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