What is the pH of saturated solution of`Mg(OH)_(2) ? Ksp for Mg(OH)2=1.8*10^-11

Options

A. 10.1
B. 10.9
C. 10.5
D. 9.2

Answer

To find the pH of a saturated solution of magnesium hydroxide, \( \text{Mg(OH)}_2 \), we will use the solubility product constant \( K_{sp} \) provided in the question. Here are the steps to solve the problem:

### Step 1: Write the dissociation equation for \( \text{Mg(OH)}_2 \)

The dissociation of magnesium hydroxide in water can be represented as:
\[ \text{Mg(OH)}_2 (s) \rightleftharpoons \text{Mg}^{2+} (aq) + 2 \text{OH}^- (aq) \]

### Step 2: Write the expression for the solubility product \( K_{sp} \)

The solubility product \( K_{sp} \) for this equilibrium can be expressed as:
\[ K_{sp} = [\text{Mg}^{2+}][\text{OH}^-]^2 \]

### Step 3: Define solubility \( S \)

Let the solubility of \( \text{Mg(OH)}_2 \) in moles per liter be \( S \). At equilibrium:
- The concentration of \( \text{Mg}^{2+} \) will be \( S \)
- The concentration of \( \text{OH}^- \) will be \( 2S \) (since two hydroxide ions are produced for every formula unit of magnesium hydroxide that dissolves)

### Step 4: Substitute into the \( K_{sp} \) expression

Substituting these values into the \( K_{sp} \) expression gives:
\[ K_{sp} = S \cdot (2S)^2 = S \cdot 4S^2 = 4S^3 \]

### Step 5: Set up the equation with the given \( K_{sp} \)

Given that \( K_{sp} = 1.8 \times 10^{-11} \), we can set up the equation:
\[ 4S^3 = 1.8 \times 10^{-11} \]

### Step 6: Solve for \( S \)

Now, solve for \( S \):
\[ S^3 = \frac{1.8 \times 10^{-11}}{4} = 4.5 \times 10^{-12} \]
\[ S = \sqrt[3]{4.5 \times 10^{-12}} \approx 1.65 \times 10^{-4} \, \text{mol/L} \]

### Step 7: Calculate the concentration of \( \text{OH}^- \)

The concentration of hydroxide ions \( [\text{OH}^-] \) is:
\[ [\text{OH}^-] = 2S = 2 \times 1.65 \times 10^{-4} = 3.30 \times 10^{-4} \, \text{mol/L} \]

### Step 8: Calculate \( \text{pOH} \)

Using the concentration of hydroxide ions, we can find \( \text{pOH} \):
\[ \text{pOH} = -\log[\text{OH}^-] = -\log(3.30 \times 10^{-4}) \approx 3.48 \]

### Step 9: Calculate \( \text{pH} \)

Finally, we can find the pH using the relationship:
\[ \text{pH} + \text{pOH} = 14 \]
\[ \text{pH} = 14 - \text{pOH} = 14 - 3.48 \approx 10.52 \]

### Final Answer

Thus, the pH of the saturated solution of \( \text{Mg(OH)}_2 \) is approximately **10.5**.

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