Nine volumes of gaseous mixture consisting of gaseous organic compound A and just sufficient amount of oxygen required for complete combustion yielded on burning 4 volumes of `CO_(2)`, 6 volumes of water vapours, and 2 volumes of `N_(2)`, at all volumes measured at the same temperature and pressure. If the compounds contains `C, H`, and `N` only, the molecular formula of compound `A` is

Options

A. `C_(2)H_(3)N_(2)`
B. `C_(2)H_(6)N_(2)`
C. `C_(3)H_(6)N_(2)`
D. `C_(3)H_(6)N`

Explanation

To determine the molecular formula of the gaseous organic compound A, we can follow these steps:

### Step 1: Write the combustion reaction
The combustion of the organic compound A (which contains carbon (C), hydrogen (H), and nitrogen (N)) can be represented as follows:

\[ \text{C}_x\text{H}_y\text{N}_z + O_2 \rightarrow CO_2 + H_2O + N_2 \]

### Step 2: Analyze the products of combustion
From the problem, we know that the combustion of the mixture produced:
- 4 volumes of \( CO_2 \)
- 6 volumes of \( H_2O \)
- 2 volumes of \( N_2 \)

### Step 3: Relate the volumes of products to the elements in the compound
From the stoichiometry of the combustion reaction:
- Each mole of \( CO_2 \) produced corresponds to one mole of carbon in the compound.
- Each mole of \( H_2O \) produced corresponds to two moles of hydrogen in the compound.
- Each mole of \( N_2 \) produced corresponds to one mole of nitrogen in the compound.

### Step 4: Set up equations based on the volumes
1. For carbon:
\[ x = \text{moles of C} = \frac{4 \text{ (from } CO_2\text{)}}{1} = 4 \]
2. For hydrogen:
\[ y = \text{moles of H} = 2 \times \text{moles of } H_2O = 2 \times 6 = 12 \]
3. For nitrogen:
\[ z = \text{moles of N} = \frac{2 \text{ (from } N_2\text{)}}{1} = 2 \]

### Step 5: Compile the results
From the above calculations, we have:
- \( x = 4 \)
- \( y = 12 \)
- \( z = 2 \)

### Step 6: Write the molecular formula
The molecular formula of compound A can be written as:
\[ \text{C}_4\text{H}_{12}\text{N}_2 \]

### Step 7: Simplify if necessary
The simplest form of the molecular formula is:
\[ \text{C}_2\text{H}_6\text{N} \]

### Conclusion
Thus, the molecular formula of compound A is:
\[ \text{C}_2\text{H}_6\text{N}_2 \]