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 solve the problem of finding 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
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 Temperature to Kelvin
The temperature given in the problem is \( 27^\circ C \). To convert this to Kelvin:
\[ T(K) = T(°C) + 273.15 = 27 + 273.15 = 300.15 \, K \]

### Step 3: Identify the Constants
In the formula for kinetic energy per molecule, \( k \) and \( T \) are constants for a given temperature. Since we are comparing the kinetic energies of different gases at the same temperature, \( k \) and \( T \) will be the same for all gases.

### Step 4: Calculate Kinetic Energy for Each Gas
Since the formula for kinetic energy per molecule does not depend on the type of gas but only on the temperature and Boltzmann constant, we can conclude that:
\[ KE_{CO} = KE_{N2} = KE_{C2H4} \]

### Step 5: Establish the Ratio
Since the kinetic energy per molecule is the same for all three gases, the ratio of kinetic energy per molecule of CO, N2, and C2H4 is:
\[ \text{Ratio} = KE_{CO} : KE_{N2} : KE_{C2H4} = 1 : 1 : 1 \]

### Conclusion
Thus, the ratio of kinetic energy per molecule of CO, N2, and C2H4 is \( 1 : 1 : 1 \).

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