A solution is saturated with respect to `SrCO_(3)` and `SrF_(2)`. The `[CO_(3)^(2-)]` was found to be `1.2 xx 10^(-3)M`. The concnetration of `F^(Theta)` in the solution would be
Given `K_(sp)` of `SrCO_(3) = 7.0 xx10^(-10)M^(2)`,
`K_(sp) "of" SrF_(2) = 7.9 xx 10^(-10) M^(3)`,

Options

A. `1.3 xx 10^(-3)M`
B. `2.6 xx 10^(-2)M`
C. `3.7 xx 10^(-2)M`
D. `5.8 xx 10^(-7)M`

Explanation

To solve the problem, we need to find the concentration of fluoride ions \([F^-]\) in a saturated solution of strontium carbonate \((SrCO_3)\) and strontium fluoride \((SrF_2)\). We are given the concentration of carbonate ions \([CO_3^{2-}] = 1.2 \times 10^{-3} \, M\) and the solubility product constants \(K_{sp}\) for both salts.

### Step-by-Step Solution:

1. **Write the Dissociation Reactions:**
- For strontium carbonate:
\[ SrCO_3 (s) \rightleftharpoons Sr^{2+} (aq) + CO_3^{2-} (aq) \]
- For strontium fluoride:
\[ SrF_2 (s) \rightleftharpoons Sr^{2+} (aq) + 2F^{-} (aq) \]

2. **Write the Expressions for \(K_{sp}\):**
- For \(SrCO_3\):
\[ K_{sp} = [Sr^{2+}][CO_3^{2-}] \]
- For \(SrF_2\):
\[ K_{sp} = [Sr^{2+}][F^{-}]^2 \]

3. **Substitute Known Values:**
- Given \(K_{sp}\) values:
\[ K_{sp} (SrCO_3) = 7.0 \times 10^{-10} \, M^2 \]
\[ K_{sp} (SrF_2) = 7.9 \times 10^{-10} \, M^3 \]
- We know \([CO_3^{2-}] = 1.2 \times 10^{-3} \, M\).

4. **Express the Concentration of Strontium Ion:**
- From the \(K_{sp}\) expression for \(SrCO_3\):
\[ [Sr^{2+}] = \frac{K_{sp}}{[CO_3^{2-}]} = \frac{7.0 \times 10^{-10}}{1.2 \times 10^{-3}} \]

5. **Calculate \([Sr^{2+}]\):**
\[ [Sr^{2+}] = \frac{7.0 \times 10^{-10}}{1.2 \times 10^{-3}} \approx 5.83 \times 10^{-7} \, M \]

6. **Substitute \([Sr^{2+}]\) into the \(K_{sp}\) Expression for \(SrF_2\):**
- From the \(K_{sp}\) expression for \(SrF_2\):
\[ K_{sp} = [Sr^{2+}][F^{-}]^2 \]
- Rearranging gives:
\[ [F^{-}]^2 = \frac{K_{sp}}{[Sr^{2+}]} \]

7. **Calculate \([F^{-}]\):**
\[ [F^{-}]^2 = \frac{7.9 \times 10^{-10}}{5.83 \times 10^{-7}} \]
\[ [F^{-}]^2 \approx 1.35 \times 10^{-3} \]
\[ [F^{-}] \approx \sqrt{1.35 \times 10^{-3}} \approx 3.67 \times 10^{-2} \, M \]

8. **Final Result:**
The concentration of fluoride ions \([F^-]\) in the solution is approximately:
\[ [F^-] \approx 3.7 \times 10^{-2} \, M \]