The pH of aqueous solution of magnesium hydroxide is 9.0 . If `K_(sp)` of magnesium hydroxide is `1.0xx10^(-11) M^(3)` , then the concentration of magnesium ions in the solution is

Options

A. `10^(-7) M`
B. `10^(-2)M`
C. `0.1 M`
D. `10^(-6)M`

Answer

To find the concentration of magnesium ions in the solution, we can follow these steps:

### Step 1: Calculate the concentration of hydroxide ions (OH⁻)
Given that the pH of the solution is 9.0, we can find the concentration of hydrogen ions (H⁺) using the formula:

\[ \text{pH} = -\log[H^+] \]

To find [H⁺]:

\[ [H^+] = 10^{-\text{pH}} = 10^{-9} \, \text{M} \]

### Step 2: Calculate the concentration of hydroxide ions (OH⁻)
Using the ion product of water at 25°C, where \(K_w = [H^+][OH^-] = 1.0 \times 10^{-14}\):

\[ [OH^-] = \frac{K_w}{[H^+]} = \frac{1.0 \times 10^{-14}}{10^{-9}} = 1.0 \times 10^{-5} \, \text{M} \]

### Step 3: Write the dissociation equation for magnesium hydroxide
The dissociation of magnesium hydroxide (Mg(OH)₂) in water can be represented as:

\[ \text{Mg(OH)}_2 \rightleftharpoons \text{Mg}^{2+} + 2\text{OH}^- \]

Let the concentration of magnesium ions, [Mg²⁺], be \(x\). Then, the concentration of hydroxide ions will be \(2x\).

### Step 4: Set up the expression for \(K_{sp}\)
The solubility product constant (\(K_{sp}\)) for magnesium hydroxide is given by:

\[ K_{sp} = [Mg^{2+}][OH^-]^2 \]

Substituting the values we have:

\[ K_{sp} = x \cdot (1.0 \times 10^{-5})^2 \]

### Step 5: Substitute the value of \(K_{sp}\)
Given that \(K_{sp} = 1.0 \times 10^{-11}\):

\[ 1.0 \times 10^{-11} = x \cdot (1.0 \times 10^{-5})^2 \]

### Step 6: Solve for \(x\)
Calculating \( (1.0 \times 10^{-5})^2 \):

\[ (1.0 \times 10^{-5})^2 = 1.0 \times 10^{-10} \]

Now substituting back:

\[ 1.0 \times 10^{-11} = x \cdot 1.0 \times 10^{-10} \]

Solving for \(x\):

\[ x = \frac{1.0 \times 10^{-11}}{1.0 \times 10^{-10}} = 0.1 \, \text{M} \]

### Conclusion
The concentration of magnesium ions \([Mg^{2+}]\) in the solution is:

\[ \text{[Mg}^{2+}] = 0.1 \, \text{M} \]

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