Which of the following would increase the solubility of `Pb(OH)_(2)`?

Options

A. Add hydrochloric acid
B. Add a solution of `Pb(NO_(3))_(2)`
C. Add a solution of NaOH
D. None of the above-the solubility a compound is constant a constant temperature

Answer

To determine which of the following options would increase the solubility of \( \text{Pb(OH)}_2 \), we need to analyze how each option affects the equilibrium of the dissolution reaction. The dissolution of lead(II) hydroxide can be represented as follows:

\[ \text{Pb(OH)}_2 (s) \rightleftharpoons \text{Pb}^{2+} (aq) + 2 \text{OH}^- (aq) \]

### Step-by-Step Solution:

1. **Understand the Dissolution Reaction**:
The solid lead(II) hydroxide (\( \text{Pb(OH)}_2 \)) dissociates into lead ions (\( \text{Pb}^{2+} \)) and hydroxide ions (\( \text{OH}^- \)). The solubility product constant (\( K_{sp} \)) governs this equilibrium.

2. **Apply Le Chatelier's Principle**:
According to Le Chatelier's principle, if a system at equilibrium is disturbed, the system will shift in a direction that counteracts the disturbance. To increase the solubility of \( \text{Pb(OH)}_2 \), we want to shift the equilibrium to the right.

3. **Analyze Each Option**:
- **Option A: Add Hydrochloric Acid (HCl)**:
- HCl dissociates into \( \text{H}^+ \) and \( \text{Cl}^- \).
- The \( \text{H}^+ \) ions will react with \( \text{OH}^- \) ions to form water (\( \text{H}_2\text{O} \)).
- This decreases the concentration of \( \text{OH}^- \), shifting the equilibrium to the right to produce more \( \text{OH}^- \) and thus increasing solubility.

- **Option B: Add Lead(II) Nitrate (\( \text{Pb(NO}_3)_2 \))**:
- This will increase the concentration of \( \text{Pb}^{2+} \) ions in solution.
- According to Le Chatelier's principle, the equilibrium will shift to the left to counteract the increase in \( \text{Pb}^{2+} \), thus decreasing solubility.

- **Option C: Add Sodium Hydroxide (NaOH)**:
- NaOH dissociates into \( \text{Na}^+ \) and \( \text{OH}^- \).
- Increasing \( \text{OH}^- \) concentration will shift the equilibrium to the left, thus decreasing solubility.

4. **Conclusion**:
The only option that increases the solubility of \( \text{Pb(OH)}_2 \) is **Option A: Add Hydrochloric Acid (HCl)**.

### Final Answer:
**Option A: Add Hydrochloric Acid (HCl)** increases the solubility of \( \text{Pb(OH)}_2 \).