8.4 ml of a gaseous hydrocarbon was burnt with 50 ml of `O_2` in a eudiometer tube. The volume of the products formed after cooling to room temperature was 37.4 ml, when reacted with NaOH, the volume contracted to 3.8 ml. The molecular formula of the hydrocarbon is

Options

A. `C_4H_6`
B. `C_2H_2`
C. `C_2H_8`
D. `C_5H_10`

Answer

To find the molecular formula of the hydrocarbon, we will follow a systematic approach based on the information given in the question.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Volume of hydrocarbon (CₓHᵧ) = 8.4 ml
- Volume of O₂ = 50 ml
- Volume of products after cooling = 37.4 ml
- Volume after reacting with NaOH = 3.8 ml

2. **Calculate the Volume of CO₂ Produced:**
- The volume of gas after cooling (37.4 ml) includes unreacted O₂ and the products CO₂ and H₂O.
- The volume of gas that reacted with NaOH (3.8 ml) indicates the volume of CO₂ produced, as NaOH absorbs CO₂.
- Therefore, the volume of CO₂ produced = 37.4 ml - 3.8 ml = 33.6 ml.

3. **Determine the Volume of O₂ Used:**
- The total volume of O₂ initially was 50 ml.
- The volume of O₂ remaining after the reaction can be calculated as follows:
- Volume of O₂ consumed = Initial O₂ - Remaining O₂
- Let’s denote the volume of O₂ consumed as V_O₂.
- Thus, V_O₂ = 50 ml - (Volume of gas after cooling - Volume of CO₂ produced)
- V_O₂ = 50 ml - 37.4 ml = 12.6 ml.

4. **Write the Balanced Combustion Reaction:**
- The general combustion reaction for the hydrocarbon can be written as:
\[ CₓHᵧ + O₂ \rightarrow CO₂ + H₂O \]
- From the stoichiometry of the reaction:
- For every x moles of CO₂ produced, x moles of carbon are consumed.
- For every y/2 moles of H₂O produced, y moles of hydrogen are consumed.
- The amount of O₂ consumed can be expressed as:
\[ \text{O}_2 \text{ required} = \frac{x + \frac{y}{4}}{2} \]

5. **Relate the Volumes to Moles:**
- Using the volumes of gases, we can relate the volumes to moles (since at the same temperature and pressure, volumes are directly proportional to moles).
- The volume of CO₂ produced (33.6 ml) corresponds to x moles.
- The volume of H₂O produced (which is not directly measured but can be inferred) corresponds to y/2 moles.
- The volume of O₂ consumed corresponds to \(\frac{x + \frac{y}{4}}{2}\) moles.

6. **Set Up the Equations:**
- From the volume of CO₂:
\[ x = \frac{33.6}{8.4} = 4 \]
- From the volume of O₂:
\[ 12.6 = \frac{4 + \frac{y}{4}}{2} \implies 25.2 = 4 + \frac{y}{4} \implies \frac{y}{4} = 21.2 \implies y = 84.8 \]

7. **Determine the Molecular Formula:**
- Now we have x = 4 and y = 6.
- Therefore, the molecular formula of the hydrocarbon is C₄H₆.

### Final Answer:
The molecular formula of the hydrocarbon is **C₄H₆**.