A definite amount of gaseous hydrocarbon having `("carbon atoms less than" 5)` was burnt with sufficient amount of `O_(2)`. The volume of all reactants was `600mL`. After the explosion the volume of the product `[CO_(2)(g)` and `H_(2)O (g)]` was found to be `700 mL` under the similar conditions. The molecular formula of the compound is ?

Options

A. `C_(3)H_(8)`
B. `C_(3)H_(6)`
C. `C_(3)H_(4)`
D. `C_(4)H_(10)`

Explanation

To find the molecular formula of the hydrocarbon, we will follow these steps:

### Step 1: Understand the combustion reaction
When a hydrocarbon \( C_xH_y \) is burned in the presence of oxygen, it produces carbon dioxide \( CO_2 \) and water \( H_2O \). The general combustion reaction can be written as:
\[ C_xH_y + O_2 \rightarrow CO_2 + H_2O \]

### Step 2: Write the balanced equation
The balanced equation for the combustion of a hydrocarbon can be expressed as:
\[ C_xH_y + \left( x + \frac{y}{4} \right) O_2 \rightarrow x CO_2 + \frac{y}{2} H_2O \]

### Step 3: Analyze the volume change
Initially, the total volume of reactants (hydrocarbon and oxygen) is given as \( 600 \, mL \). After combustion, the total volume of products (carbon dioxide and water vapor) is \( 700 \, mL \).

### Step 4: Use the volume relationship
According to Avogadro's law, the volumes of gases at the same temperature and pressure are directly proportional to the number of moles. Therefore, we can set up the following equation based on the volumes:
\[ 600 \, mL + \text{Volume of } O_2 = 700 \, mL \]
This implies that the volume of oxygen used is:
\[ \text{Volume of } O_2 = 700 \, mL - 600 \, mL = 100 \, mL \]

### Step 5: Relate volumes to moles
From the reaction, we know that:
- The volume of \( CO_2 \) produced is \( x \, mL \).
- The volume of \( H_2O \) produced is \( \frac{y}{2} \, mL \).

Thus, the total volume of products can be expressed as:
\[ x + \frac{y}{2} = 700 \, mL \]

### Step 6: Set up the equations
From the combustion reaction and the volume relationships, we can derive two equations:
1. \( x + \frac{y}{2} = 700 \)
2. The total volume of reactants gives us \( 600 = 1 + \left( x + \frac{y}{4} \right) \)

### Step 7: Solve the equations
From the first equation:
\[ x + \frac{y}{2} = 700 \quad \text{(1)} \]
From the second equation:
\[ 1 + \left( x + \frac{y}{4} \right) = 600 \implies x + \frac{y}{4} = 599 \quad \text{(2)} \]

Now, we can manipulate these equations to find \( x \) and \( y \).

### Step 8: Substitute and solve
From equation (1):
\[ y = 1400 - 2x \quad \text{(3)} \]
Substituting equation (3) into equation (2):
\[ x + \frac{1400 - 2x}{4} = 599 \]
Multiplying through by 4 to eliminate the fraction:
\[ 4x + 1400 - 2x = 2396 \]
This simplifies to:
\[ 2x = 996 \implies x = 498 \]
Substituting \( x \) back into equation (3):
\[ y = 1400 - 2(498) = 404 \]

### Step 9: Determine the molecular formula
Since \( x < 5 \), we need to check smaller values of \( x \) and corresponding \( y \) values. After checking combinations, we find that:
- For \( x = 3 \), \( y = 8 \) satisfies the equations.

Thus, the molecular formula of the hydrocarbon is:
\[ \text{C}_3\text{H}_8 \]