How do we differentiate between `CO_(3)^(2-)` and `SO_(3)^(2-)` in dilute `H_(2)OSO_(4)` group ? ( Note `:` These are sodium salts ?

Options

A. By passing gas liberated with dilute `H_(2)SO_(4)` through lime water.
B. By passing gas liberated with dilute `H_(2)SO_(4)` through acidified `K_(2)Cr_(2)O_(7)` solution.
C. By the addidtion of lead acetate solution in their aqueous solutions .
D. By the addition of silver nitrate solution in their aqueous solutions.

Explanation

To differentiate between the carbonate ion \((CO_3^{2-})\) and the sulfate ion \((SO_3^{2-})\) in dilute \(H_2SO_4\), we can follow these steps:

### Step-by-Step Solution:

1. **Preparation of Sodium Salts**: Start with the sodium salts of carbonate and sulfate, which are \(Na_2CO_3\) (sodium carbonate) and \(Na_2SO_3\) (sodium sulfite).

2. **Reaction with Dilute \(H_2SO_4\)**: Add dilute sulfuric acid (\(H_2SO_4\)) to both sodium salts. The reactions are as follows:
- For sodium carbonate:
\[ Na_2CO_3 + H_2SO_4 \rightarrow Na_2SO_4 + H_2O + CO_2 \uparrow \]
Here, carbon dioxide gas \((CO_2)\) is released.

- For sodium sulfite:
\[ Na_2SO_3 + H_2SO_4 \rightarrow Na_2SO_4 + H_2O + SO_2 \uparrow \]
Here, sulfur dioxide gas \((SO_2)\) is released.

3. **Observation of Gases**: Collect the gases released from both reactions:
- The gas from sodium carbonate is \(CO_2\).
- The gas from sodium sulfite is \(SO_2\).

4. **Testing with Lime Water**: Pass the gases through lime water (\(Ca(OH)_2\)):
- \(CO_2\) will react with lime water to form calcium carbonate, which makes the solution milky:
\[ CO_2 + Ca(OH)_2 \rightarrow CaCO_3 \downarrow \text{ (milky)} \]
- \(SO_2\) does not produce any significant reaction with lime water.

5. **Testing with Acidified Potassium Dichromate**: To further differentiate, pass the gas through acidified potassium dichromate \((K_2Cr_2O_7)\):
- The \(SO_2\) gas will reduce the orange dichromate solution to green chromium(III) ions:
\[ SO_2 + K_2Cr_2O_7 \rightarrow Cr^{3+} \text{ (green)} \]
- The \(CO_2\) gas will not affect the color of the dichromate solution.

### Conclusion:
By observing the reactions and the color changes, we can differentiate between the carbonate ion and the sulfate ion. The presence of a milky solution with lime water indicates \(CO_2\) from carbonate, while the color change from orange to green in the dichromate solution indicates \(SO_2\) from sulfate.

### Final Answer:
The correct method to differentiate between \(CO_3^{2-}\) and \(SO_3^{2-}\) is by passing the gas liberated with dilute \(H_2SO_4\) through acidified \(K_2Cr_2O_7\) solution.