A gaseous organic compound has a density of `2.5 kg//m^3` at 2 atm and at `273^@C`.The molecular formula of the compound can be :

Options

A. `C_3H_4O_2`
B. `C_4H_4`
C. `C_3H_4O`
D. `C_4H_8`

Explanation

To solve the problem, we need to determine the molecular formula of a gaseous organic compound given its density, pressure, and temperature. We will use the formula relating density, pressure, molecular mass, and temperature.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Density (ρ) = 2.5 kg/m³
- Pressure (P) = 2 atm
- Temperature (T) = 273°C

2. **Convert Density to Appropriate Units:**
- Since 1 m³ = 1000 L and 1 kg = 1000 g, we can convert the density:
\[ 2.5 \, \text{kg/m}^3 = 2.5 \, \text{g/L} \]

3. **Convert Temperature to Kelvin:**
- To convert Celsius to Kelvin, use the formula:
\[ T(K) = T(°C) + 273 \]
- Therefore:
\[ T = 273 + 273 = 546 \, K \]

4. **Use the Density Formula:**
- The relationship between density, pressure, molecular mass, and temperature is given by:
\[ \rho = \frac{PM}{RT} \]
- Rearranging this formula to find the molecular mass (M):
\[ M = \frac{\rho RT}{P} \]

5. **Substitute the Values:**
- Use the ideal gas constant \( R = 0.0821 \, \text{L atm/(K mol)} \):
\[ M = \frac{(2.5 \, \text{g/L}) \times (0.0821 \, \text{L atm/(K mol)}) \times (546 \, K)}{2 \, \text{atm}} \]

6. **Calculate the Molecular Mass:**
- Performing the calculation:
\[ M = \frac{2.5 \times 0.0821 \times 546}{2} \]
\[ M = \frac{113.1565}{2} \approx 56.578 \, \text{g/mol} \]
- Rounding gives \( M \approx 56 \, \text{g/mol} \).

7. **Determine Possible Molecular Formulas:**
- Now, we need to check the molecular masses of the given options:
- C₃H₄O₂: 72 g/mol
- C₄H₄: 52 g/mol
- C₃H₄O: 56 g/mol
- C₄H₈: 56 g/mol

8. **Identify the Correct Options:**
- The molecular mass we calculated is approximately 56 g/mol, which matches:
- C₃H₄O (56 g/mol)
- C₄H₈ (56 g/mol)

### Conclusion:
The molecular formula of the compound can be either **C₃H₄O** or **C₄H₈**.