4 moles CO, 5 moles `N_2` and 2 mole `C_2H_4` are placed in a 5 litre vessel at `27^@C` . The ratio of kinetic energy per molecule of `CO_2, He` and `NH_3` is

Options

A. `4:5:2`
B. `5:5:4`
C. `2:4:5 `
D. `1:1:1`

Answer

To find the ratio of kinetic energy per molecule of CO, N₂, and C₂H₄, we can follow these steps:

### Step 1: Understand the formula for kinetic energy per molecule
The kinetic energy (KE) per molecule of an ideal gas is given by the formula:
\[ KE = \frac{3}{2} k T \]
where:
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin.

### Step 2: Convert the temperature to Kelvin
The temperature given is \( 27^\circ C \). To convert this to Kelvin, we use the formula:
\[ T(K) = T(°C) + 273.15 \]
So,
\[ T = 27 + 273.15 = 300.15 \, K \]

### Step 3: Identify that the kinetic energy formula does not depend on the number of moles
The formula for kinetic energy per molecule does not involve the number of moles. This means that regardless of how many moles of each gas we have, the kinetic energy per molecule will be the same for all gases at the same temperature.

### Step 4: Calculate the ratio of kinetic energy per molecule
Since the kinetic energy per molecule for all three gases (CO, N₂, and C₂H₄) is given by the same formula and depends only on temperature, we can conclude that:
\[ KE_{CO} : KE_{N_2} : KE_{C_2H_4} = 1 : 1 : 1 \]

### Conclusion
The ratio of kinetic energy per molecule of CO, N₂, and C₂H₄ is:
\[ \text{Ratio} = 1 : 1 : 1 \]