In which of the following cases is the solution of AgCl unsaturated?
(a) `[Ag^(+)] [Cl^(-)] lt K_(sp)`
(b) `[Ag^(+)] [Cl^(-)] gt K_(sp)`
(c ) `[Ag^(+)] [Cl^(-)]=K_(sp)`

Answer

To determine in which case the solution of AgCl is unsaturated, we need to understand the relationship between the ion concentrations and the solubility product constant (Ksp) for AgCl.

### Step-by-Step Solution:

1. **Understanding Saturation and Unsaturation**:
- A solution is said to be **unsaturated** when it can still dissolve more solute. This occurs when the product of the concentrations of the ions in solution is less than the Ksp value.
- Conversely, a solution is **saturated** when the product of the concentrations of the ions equals Ksp, and it is **supersaturated** when the product exceeds Ksp.

2. **Defining the Solubility Product (Ksp)**:
- For AgCl, the dissociation can be represented as:
\[ \text{AgCl (s)} \rightleftharpoons \text{Ag}^+ (aq) + \text{Cl}^- (aq) \]
- The Ksp expression for AgCl is:
\[ K_{sp} = [Ag^+][Cl^-] \]

3. **Analyzing the Given Options**:
- **Option (a)**: \([Ag^+][Cl^-] < K_{sp}\)
- This indicates that the product of the concentrations of Ag\(^+\) and Cl\(^-\) is less than Ksp. Therefore, the solution is unsaturated.

- **Option (b)**: \([Ag^+][Cl^-] > K_{sp}\)
- This indicates that the product of the concentrations is greater than Ksp. Thus, the solution is supersaturated.

- **Option (c)**: \([Ag^+][Cl^-] = K_{sp}\)
- This indicates that the product of the concentrations is equal to Ksp. Hence, the solution is saturated.

4. **Conclusion**:
- The solution of AgCl is unsaturated in **Option (a)**: \([Ag^+][Cl^-] < K_{sp}\).

### Final Answer:
The solution of AgCl is unsaturated in case (a) \([Ag^+][Cl^-] < K_{sp}\).