The solubility of mercurous chloride in water will be given as :

Options

A. `K_(sp)(Hg_(2)Cl_(2))`
B. `sqrt(K_(sp)(Hg_(2)Cl_(2))//4)`
C. `(K_(sp)(Hg_(2)Cl_(2)))/4]^(1/2)`
D. `K_(sp)(Hg_(2)Cl_(2))^(1/3)`

Explanation

To calculate the solubility of mercurous chloride (Hg2Cl2) in water, we will follow these steps:

### Step 1: Understand the Dimerization of Mercurous Chloride
Mercurous chloride exists as a dimer, represented as Hg2Cl2. This means that two mercury (I) ions (Hg^+) are bonded to two chloride ions (Cl^-).

### Step 2: Write the Dissociation Equation
When mercurous chloride dissolves in water, it dissociates as follows:
\[ \text{Hg}_2\text{Cl}_2 (s) \rightleftharpoons 2 \text{Hg}^+ (aq) + 2 \text{Cl}^- (aq) \]

### Step 3: Define the Molar Solubility
Let the molar solubility of mercurous chloride be \( x \) mol/L. According to the dissociation equation:
- For every 1 mole of Hg2Cl2 that dissolves, it produces 2 moles of Hg^+ and 2 moles of Cl^-.

Thus, at equilibrium:
- \([Hg^+] = 2x\)
- \([Cl^-] = 2x\)

### Step 4: Write the Expression for the Solubility Product (Ksp)
The solubility product (Ksp) expression for the dissociation of mercurous chloride is given by:
\[ K_{sp} = [Hg^+]^2 [Cl^-]^2 \]

Substituting the equilibrium concentrations:
\[ K_{sp} = (2x)^2 (2x)^2 = 4x^2 \cdot 4x^2 = 16x^4 \]

### Step 5: Solve for Molar Solubility
To find the molar solubility \( x \), we rearrange the Ksp expression:
\[ x = \sqrt[4]{\frac{K_{sp}}{16}} \]

### Step 6: Conclusion
Thus, the solubility of mercurous chloride in water can be expressed as:
\[ x = \frac{\sqrt{K_{sp}}}{4} \]