Determine the molar solubility of `MgF_(2)` from its solubility product `K_(sp)=4xx10^(-9)` :

Options

A. `10^(-3)`
B. `6.32 xx10^(-5)`
C. `2xx10^(-5)`
D. None of these

Answer

To determine the molar solubility of \( \text{MgF}_2 \) from its solubility product \( K_{sp} = 4 \times 10^{-9} \), we can follow these steps:

### Step 1: Write the Dissociation Equation
When \( \text{MgF}_2 \) dissolves in water, it dissociates into magnesium ions and fluoride ions:
\[ \text{MgF}_2 (s) \rightleftharpoons \text{Mg}^{2+} (aq) + 2 \text{F}^- (aq) \]

### Step 2: Define Molar Solubility
Let the molar solubility of \( \text{MgF}_2 \) be \( S \). This means that when \( S \) moles of \( \text{MgF}_2 \) dissolve, the concentrations of the ions in solution will be:
- \( [\text{Mg}^{2+}] = S \)
- \( [\text{F}^-] = 2S \)

### Step 3: Write the Expression for \( K_{sp} \)
The solubility product \( K_{sp} \) is defined as:
\[ K_{sp} = [\text{Mg}^{2+}][\text{F}^-]^2 \]
Substituting the concentrations from Step 2 into the \( K_{sp} \) expression gives:
\[ K_{sp} = (S)(2S)^2 = S \cdot 4S^2 = 4S^3 \]

### Step 4: Substitute the Given \( K_{sp} \) Value
We know from the problem that \( K_{sp} = 4 \times 10^{-9} \). Therefore, we can set up the equation:
\[ 4S^3 = 4 \times 10^{-9} \]

### Step 5: Solve for \( S \)
Dividing both sides by 4:
\[ S^3 = 10^{-9} \]
Now, taking the cube root of both sides:
\[ S = \sqrt[3]{10^{-9}} = 10^{-3} \]

### Step 6: Conclusion
Thus, the molar solubility of \( \text{MgF}_2 \) is:
\[ S = 10^{-3} \, \text{mol/L} \]

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