If `K_(sp)` of `Ag_(2)CO_(3)` is `8 xx 10^(-12)`, the molar solubility of `Ag_(2)CO_(3)` in 0.1 M `AgNO_(3)` is `x xx 10^(-10) M`. The numerical value of x is ________.

Explanation

To solve the problem, we need to find the molar solubility of \( \text{Ag}_2\text{CO}_3 \) in a solution that already contains \( 0.1 \, \text{M} \) of \( \text{AgNO}_3 \). We are given the \( K_{sp} \) value for \( \text{Ag}_2\text{CO}_3 \) as \( 8 \times 10^{-12} \).

### Step-by-Step Solution:

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

2. **Define the Solubility**:
Let the molar solubility of \( \text{Ag}_2\text{CO}_3 \) be \( s \). Therefore, the concentration of \( \text{Ag}^+ \) ions from the dissolution will be \( 2s \) and the concentration of \( \text{CO}_3^{2-} \) ions will be \( s \).

3. **Account for the Common Ion Effect**:
Since we have \( 0.1 \, \text{M} \) of \( \text{AgNO}_3 \) in the solution, it dissociates completely to give \( 0.1 \, \text{M} \) of \( \text{Ag}^+ \) ions. Thus, the total concentration of \( \text{Ag}^+ \) ions in the solution becomes:
\[ [\text{Ag}^+] = 0.1 + 2s \approx 0.1 \quad (\text{since } s \text{ is very small compared to } 0.1) \]

4. **Set Up the \( K_{sp} \) Expression**:
The solubility product constant \( K_{sp} \) for \( \text{Ag}_2\text{CO}_3 \) is given by:
\[ K_{sp} = [\text{Ag}^+]^2 [\text{CO}_3^{2-}] \]
Substituting the concentrations:
\[ K_{sp} = (0.1)^2 (s) \]

5. **Substitute the Given \( K_{sp} \) Value**:
We know \( K_{sp} = 8 \times 10^{-12} \), so we can write:
\[ 8 \times 10^{-12} = (0.1)^2 (s) \]
Simplifying this gives:
\[ 8 \times 10^{-12} = 0.01s \]

6. **Solve for \( s \)**:
Rearranging the equation to solve for \( s \):
\[ s = \frac{8 \times 10^{-12}}{0.01} = 8 \times 10^{-10} \, \text{M} \]

7. **Find the Value of \( x \)**:
The problem states that the molar solubility is \( x \times 10^{-10} \, \text{M} \). From our calculation, we found \( s = 8 \times 10^{-10} \, \text{M} \), which means:
\[ x = 8 \]

### Final Answer:
The numerical value of \( x \) is **8**.