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 simultaneously equilibrated in pure water.

### Step-by-Step Solution:

1. **Identify the Solubility Products**:
- The solubility product \(K_{sp}\) for barium sulfate \((BaSO_4)\) is given as \(1.0 \times 10^{-10}\).
- The solubility product \(K_{sp}\) for barium carbonate \((BaCO_3)\) is given as \(5.0 \times 10^{-9}\).

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

3. **Define the Solubility**:
- Let \(S_1\) be the solubility of \(BaSO_4\) in mol/L, and \(S_2\) be the solubility of \(BaCO_3\) in mol/L.
- From the equilibrium expressions:
\[ K_{sp1} = [Ba^{2+}][SO_4^{2-}] = S_1 \cdot [SO_4^{2-}] \]
\[ K_{sp2} = [Ba^{2+}][CO_3^{2-}] = S_2 \cdot [CO_3^{2-}] \]

4. **Express the Concentrations**:
- Since both salts are present in the same solution, the concentration of \(Ba^{2+}\) will be the same for both salts. Let’s denote it as \(x\).
- Thus:
\[ K_{sp1} = x \cdot [SO_4^{2-}] \]
\[ K_{sp2} = x \cdot [CO_3^{2-}] \]

5. **Substituting the Values**:
- From the solubility product expressions:
\[ [SO_4^{2-}] = \frac{K_{sp1}}{x} = \frac{1.0 \times 10^{-10}}{x} \]
\[ [CO_3^{2-}] = \frac{K_{sp2}}{x} = \frac{5.0 \times 10^{-9}}{x} \]

6. **Finding the Ratio**:
- The ratio of \([SO_4^{2-}]\) to \([CO_3^{2-}]\) is:
\[ \frac{[SO_4^{2-}]}{[CO_3^{2-}]} = \frac{\frac{1.0 \times 10^{-10}}{x}}{\frac{5.0 \times 10^{-9}}{x}} = \frac{1.0 \times 10^{-10}}{5.0 \times 10^{-9}} = \frac{1.0}{5.0} \times 10^{-1} = \frac{1}{50} \]

7. **Final Answer**:
- Therefore, the ratio \(\frac{[SO_4^{2-}]}{[CO_3^{2-}]}\) is \(\frac{1}{50}\).