The solubility product constant `K_(sp)` of `Mg(OH)_(2)` is `9.0 xx 10^(-12)`. If a solution is 0.010 M with respect to `Mg^(2+)` ion, what is the maximum hydroxide ion concentration which could be present without causing the precipitation of `Mg(OH)_(2)`

Options

A. `1.5 xx 10^(-7) M`
B. `3.0 xx 10^(-7) M`
C. `1.5 xx 10^(-5) M`
D. `3.0 xx 10^(-5) M`

Explanation

To solve the problem, we need to determine the maximum concentration of hydroxide ions \([OH^-]\) that can be present in a solution containing \(0.010 \, M\) of \(Mg^{2+}\) without causing the precipitation of \(Mg(OH)_2\). The solubility product constant \(K_{sp}\) for \(Mg(OH)_2\) is given as \(9.0 \times 10^{-12}\).

### Step-by-Step Solution:

1. **Write the Dissolution Equation**:
The dissolution of magnesium hydroxide can be represented as:
\[ Mg(OH)_2 (s) \rightleftharpoons Mg^{2+} (aq) + 2OH^- (aq) \]

2. **Expression for \(K_{sp}\)**:
The solubility product constant \(K_{sp}\) is given by the equation:
\[ K_{sp} = [Mg^{2+}][OH^-]^2 \]

3. **Substituting Known Values**:
We know that the concentration of \(Mg^{2+}\) is \(0.010 \, M\). Let the concentration of hydroxide ions be \(s\). Thus, we can substitute these values into the \(K_{sp}\) expression:
\[ K_{sp} = (0.010)(s)^2 \]

4. **Setting Up the Equation**:
We can now set up the equation using the given \(K_{sp}\):
\[ 9.0 \times 10^{-12} = (0.010)(s)^2 \]

5. **Solving for \(s^2\)**:
Rearranging the equation to solve for \(s^2\):
\[ s^2 = \frac{9.0 \times 10^{-12}}{0.010} \]
\[ s^2 = 9.0 \times 10^{-10} \]

6. **Calculating \(s\)**:
Taking the square root of both sides to find \(s\):
\[ s = \sqrt{9.0 \times 10^{-10}} = 3.0 \times 10^{-5} \, M \]

7. **Conclusion**:
Therefore, the maximum concentration of hydroxide ions \([OH^-]\) that can be present without causing precipitation of \(Mg(OH)_2\) is:
\[ [OH^-] = 3.0 \times 10^{-5} \, M \]

### Final Answer:
The maximum hydroxide ion concentration which could be present without causing the precipitation of \(Mg(OH)_2\) is \(3.0 \times 10^{-5} \, M\).