The solubility product values of barium sulphate and barium carbonate are `1.0xx10^(-10)` and `5.0xx10^(-9)` respectively when the two salts are simultaneously equilibrated in pure water the ratio of `([SO_(4)^(2-)])/([CO_(3)^(2-)])` is

Options

A. 0.02
B. `0.10`
C. 0.05
D. 0.12

Answer

To solve the problem, we need to find the ratio of the concentrations of sulfate ions \([SO_4^{2-}]\) to carbonate ions \([CO_3^{2-}]\) when barium sulfate \((BaSO_4)\) and barium carbonate \((BaCO_3)\) are in equilibrium in pure water.

### Step-by-Step Solution:

1. **Write the Dissociation Equations:**
- For barium sulfate:
\[ BaSO_4 (s) \rightleftharpoons Ba^{2+} (aq) + SO_4^{2-} (aq) \]
- For barium carbonate:
\[ BaCO_3 (s) \rightleftharpoons Ba^{2+} (aq) + CO_3^{2-} (aq) \]

2. **Define the Solubility Products:**
- The solubility product \(K_{sp}\) for barium sulfate is given by:
\[ K_{sp1} = [Ba^{2+}][SO_4^{2-}] = S_1 \cdot S_1 = S_1^2 \]
- The solubility product \(K_{sp}\) for barium carbonate is given by:
\[ K_{sp2} = [Ba^{2+}][CO_3^{2-}] = S_2 \cdot S_2 = S_2^2 \]

3. **Express the Concentrations:**
- Let \(S_1\) be the solubility of barium sulfate and \(S_2\) be the solubility of barium carbonate.
- The total concentration of barium ions from both salts is:
\[ [Ba^{2+}] = S_1 + S_2 \]

4. **Set Up the Equations:**
- From the solubility products, we can write:
\[ K_{sp1} = (S_1 + S_2) \cdot S_1 \]
\[ K_{sp2} = (S_1 + S_2) \cdot S_2 \]

5. **Divide the Two Equations:**
- Dividing \(K_{sp1}\) by \(K_{sp2}\):
\[ \frac{K_{sp1}}{K_{sp2}} = \frac{(S_1 + S_2) \cdot S_1}{(S_1 + S_2) \cdot S_2} \]
- This simplifies to:
\[ \frac{K_{sp1}}{K_{sp2}} = \frac{S_1}{S_2} \]

6. **Substitute the Given Values:**
- Given \(K_{sp1} = 1.0 \times 10^{-10}\) and \(K_{sp2} = 5.0 \times 10^{-9}\):
\[ \frac{S_1}{S_2} = \frac{1.0 \times 10^{-10}}{5.0 \times 10^{-9}} = \frac{1}{50} = 0.02 \]

7. **Conclusion:**
- Therefore, the ratio of \([SO_4^{2-}]\) to \([CO_3^{2-}]\) is:
\[ \frac{[SO_4^{2-}]}{[CO_3^{2-}]} = 0.02 \]

### Final Answer:
The ratio of \([SO_4^{2-}]/[CO_3^{2-}]\) is **0.02**.