`Na_(2)SO_(4)` and `Na_(2)S` can be distinguished from each other by using:

Options

A. dilute `H_(2)SO_(4)`
B. acidified `KMnO_(4)` solution
C. sodium nitroprusside solution
D. cadmium acetate solution

Explanation

To distinguish between `Na2SO4` (sodium sulfate) and `Na2S` (sodium sulfide), we can use various reagents that will produce different reactions with each compound. Here’s a step-by-step breakdown of how each reagent can be used for this purpose:

### Step 1: Using Dilute H2SO4
1. **Reagent**: Dilute sulfuric acid (H2SO4).
2. **Reaction with Na2S**: When sodium sulfide (Na2S) reacts with dilute H2SO4, it produces hydrogen sulfide gas (H2S) and sodium ions (Na+). The reaction can be represented as:
\[ Na2S + H2SO4 \rightarrow H2S \uparrow + 2Na^+ \]
Here, H2S is released as a gas.
3. **Reaction with Na2SO4**: Sodium sulfate (Na2SO4) does not react with dilute H2SO4, and thus no gas is produced.

### Step 2: Using Acidified KMnO4
1. **Reagent**: Acidified potassium permanganate (KMnO4).
2. **Reaction with Na2S**: In an acidic medium, Na2S reacts with KMnO4, leading to the reduction of the pink color of KMnO4 to colorless Mn2+ ions. The reaction can be summarized as:
\[ Na2S + 5KMnO4 + 6H^+ \rightarrow 2Mn^{2+} + H2S + 5K^+ + 8H2O \]
3. **Reaction with Na2SO4**: Sodium sulfate does not react with KMnO4, and the pink color remains unchanged.

### Step 3: Using Sodium Nitroprusside Solution
1. **Reagent**: Sodium nitroprusside solution.
2. **Reaction with Na2S**: When Na2S is added to sodium nitroprusside, a purple-violet color develops due to the formation of a complex:
\[ S^{2-} + Na_2[Fe(CN)_5NO] \rightarrow [Fe(CN)_5NOS]^{-4} \quad (\text{purple-violet color}) \]
3. **Reaction with Na2SO4**: No color change occurs when sodium sulfate is added to sodium nitroprusside.

### Step 4: Using Cadmium Acetate Solution
1. **Reagent**: Cadmium acetate solution.
2. **Reaction with Na2S**: When Na2S reacts with cadmium ions (Cd2+), it forms a yellow precipitate of cadmium sulfide (CdS):
\[ Na2S + Cd^{2+} \rightarrow CdS \downarrow (\text{yellow precipitate}) + 2Na^+ \]
3. **Reaction with Na2SO4**: Sodium sulfate does not produce a precipitate with cadmium ions, as it remains soluble.

### Conclusion
All four reagents (dilute H2SO4, acidified KMnO4, sodium nitroprusside solution, and cadmium acetate solution) can be used to distinguish between sodium sulfate (Na2SO4) and sodium sulfide (Na2S) based on their different reactions.