At `30^(@) C` the solubility of `Ag_(2)CO_(3) (K_(SP)=8xx10^(-12))` would be gretest in one litre of:

Options

A. `0.05 M Na_(2)CO_(3)`
B. `0.05 M AgNO_(3)`
C. pure water
D. `0.05 M K_(2)CO_(3)`

Explanation

To determine the greatest solubility of \( \text{Ag}_2\text{CO}_3 \) (silver carbonate) at \( 30^\circ C \) in one litre of different solutions, we need to consider the common ion effect. The solubility product constant (\( K_{sp} \)) for \( \text{Ag}_2\text{CO}_3 \) is given as \( 8 \times 10^{-12} \).

### Step-by-Step Solution:

1. **Identify the dissociation of \( \text{Ag}_2\text{CO}_3 \)**:
\[ \text{Ag}_2\text{CO}_3 (s) \rightleftharpoons 2 \text{Ag}^+ (aq) + \text{CO}_3^{2-} (aq) \]
From this equation, we can see that one mole of \( \text{Ag}_2\text{CO}_3 \) produces 2 moles of \( \text{Ag}^+ \) ions and 1 mole of \( \text{CO}_3^{2-} \) ions.

2. **Write the expression for \( K_{sp} \)**:
\[ K_{sp} = [\text{Ag}^+]^2 [\text{CO}_3^{2-}] \]
Substituting the solubility \( s \) for \( \text{Ag}_2\text{CO}_3 \):
\[ K_{sp} = (2s)^2 (s) = 4s^3 \]

3. **Set up the equation using \( K_{sp} \)**:
\[ 4s^3 = 8 \times 10^{-12} \]
Solving for \( s \):
\[ s^3 = \frac{8 \times 10^{-12}}{4} = 2 \times 10^{-12} \]
\[ s = \sqrt[3]{2 \times 10^{-12}} \approx 1.26 \times 10^{-4} \text{ mol/L} \]

4. **Consider the effect of common ions**:
- In solutions like sodium carbonate (\( \text{Na}_2\text{CO}_3 \)) or potassium carbonate (\( \text{K}_2\text{CO}_3 \)), there are carbonate ions (\( \text{CO}_3^{2-} \)) present. The presence of these ions will shift the equilibrium to the left, decreasing the solubility of \( \text{Ag}_2\text{CO}_3 \).
- In the case of silver nitrate (\( \text{AgNO}_3 \)), silver ions (\( \text{Ag}^+ \)) are present, which will also decrease the solubility of \( \text{Ag}_2\text{CO}_3 \).

5. **Conclusion**:
The solubility of \( \text{Ag}_2\text{CO}_3 \) will be greatest in pure water since there are no common ions present to affect the solubility.

### Final Answer:
The greatest solubility of \( \text{Ag}_2\text{CO}_3 \) at \( 30^\circ C \) would be in pure water.