The solubility product of `MgF_(2)` is `7.4xx10^(-11)`. Calculate the solubility of `MgF_(2)` in `0.1`M NaF solution

Options

A. `7.4xx10^(-9)`
B. `3.7xx10^(-9)`
C. `3.7xx10^(-11)`
D. `7.4xx10^(-11)`

Answer

To calculate the solubility of \( MgF_2 \) in a \( 0.1 \, M \) NaF solution, we can follow these steps:

### Step 1: Write the dissociation equation for \( MgF_2 \)
The dissociation of \( MgF_2 \) in water can be represented as:
\[ MgF_2 (s) \rightleftharpoons Mg^{2+} (aq) + 2F^{-} (aq) \]

### Step 2: Define the solubility product expression
The solubility product \( K_{sp} \) for \( MgF_2 \) is given by:
\[ K_{sp} = [Mg^{2+}][F^{-}]^2 \]
Given \( K_{sp} = 7.4 \times 10^{-11} \).

### Step 3: Set up the equilibrium concentrations
Let the solubility of \( MgF_2 \) in the \( 0.1 \, M \) NaF solution be \( S \). The concentration of \( Mg^{2+} \) will be \( S \) and the concentration of \( F^{-} \) will be \( 2S \) from the dissociation of \( MgF_2 \) plus \( 0.1 \, M \) from the NaF solution:
\[ [F^{-}] = 2S + 0.1 \]

### Step 4: Substitute into the \( K_{sp} \) expression
Substituting the equilibrium concentrations into the \( K_{sp} \) expression, we get:
\[ K_{sp} = [Mg^{2+}][F^{-}]^2 = S(2S + 0.1)^2 \]
Setting this equal to the given \( K_{sp} \):
\[ 7.4 \times 10^{-11} = S(2S + 0.1)^2 \]

### Step 5: Expand and simplify the equation
Expanding \( (2S + 0.1)^2 \):
\[ (2S + 0.1)^2 = 4S^2 + 0.4S + 0.01 \]
Thus, the equation becomes:
\[ 7.4 \times 10^{-11} = S(4S^2 + 0.4S + 0.01) \]
This simplifies to:
\[ 7.4 \times 10^{-11} = 4S^3 + 0.4S^2 + 0.01S \]

### Step 6: Solve the cubic equation
This is a cubic equation in \( S \). However, since \( S \) is expected to be small, we can neglect the higher order terms. Assuming \( S \) is small compared to \( 0.1 \):
\[ 7.4 \times 10^{-11} \approx 0.01S \]
Solving for \( S \):
\[ S \approx \frac{7.4 \times 10^{-11}}{0.01} = 7.4 \times 10^{-9} \, M \]

### Conclusion
The solubility of \( MgF_2 \) in \( 0.1 \, M \) NaF solution is approximately:
\[ \boxed{7.4 \times 10^{-9} \, M} \]