Let the solubility of an aqueous solution of `Mg(OH)_(2)` be x then its `K_(sp)` is

Options

A. `4 x^(3)`
B. `108 x^(5)`
C. `27 x^(4)`
D. 9x

Explanation

To find the solubility product constant (Ksp) of magnesium hydroxide, Mg(OH)₂, given its solubility in an aqueous solution is x, we can follow these steps:

### Step 1: Write the dissociation equation
When magnesium hydroxide dissolves in water, it dissociates into magnesium ions (Mg²⁺) and hydroxide ions (OH⁻):
\[ \text{Mg(OH)}_2 (s) \rightleftharpoons \text{Mg}^{2+} (aq) + 2 \text{OH}^- (aq) \]

### Step 2: Define the solubility
Let the solubility of Mg(OH)₂ be x. This means that in a saturated solution:
- The concentration of Mg²⁺ ions will be x (from the dissociation).
- The concentration of OH⁻ ions will be 2x (since two hydroxide ions are produced for each formula unit of Mg(OH)₂ that dissolves).

### Step 3: Write the expression for Ksp
The solubility product constant (Ksp) is given by the product of the concentrations of the ions, each raised to the power of their coefficients in the balanced equation:
\[ K_{sp} = [\text{Mg}^{2+}][\text{OH}^-]^2 \]

Substituting the concentrations from our earlier step:
\[ K_{sp} = (x)(2x)^2 \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ K_{sp} = x \cdot (2x)^2 \]
\[ K_{sp} = x \cdot 4x^2 \]
\[ K_{sp} = 4x^3 \]

### Conclusion
Thus, the solubility product constant \( K_{sp} \) for magnesium hydroxide is:
\[ K_{sp} = 4x^3 \]