The ratio of average translatory kinetic energy of He has molecules to `O_2` gas molecules is

Options

A. `25/21`
B. `21/25`
C. `3/2`
D. `1`

Answer

To find the ratio of the average translational kinetic energy of helium (He) molecules to that of oxygen (O₂) gas molecules, we can follow these steps:

### Step 1: Understand the formula for average translational kinetic energy
The average translational kinetic energy (KE) of a gas is given by the formula:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin.

### Step 2: Recognize that the kinetic energy depends on temperature
From the formula, we see that the average translational kinetic energy is directly proportional to the temperature of the gas. Therefore, if the temperature of both gases is the same, their average translational kinetic energies will also be the same.

### Step 3: Identify the gases and their degrees of freedom
- Helium (He) is a monatomic gas, which means it has 3 degrees of freedom (only translational).
- Oxygen (O₂) is a diatomic gas, which has 5 degrees of freedom (3 translational and 2 rotational).

### Step 4: Calculate the average translational kinetic energy for both gases
For both gases at the same temperature \( T \):
- For Helium:
\[ KE_{He} = \frac{3}{2} k T \]
- For Oxygen:
\[ KE_{O_2} = \frac{3}{2} k T \]

### Step 5: Find the ratio of average translational kinetic energy
Since both expressions for kinetic energy are the same, the ratio of the average translational kinetic energy of helium to that of oxygen is:
\[ \text{Ratio} = \frac{KE_{He}}{KE_{O_2}} = \frac{\frac{3}{2} k T}{\frac{3}{2} k T} = 1 \]

### Conclusion
The ratio of the average translational kinetic energy of He molecules to O₂ gas molecules is:
\[ \text{Ratio} = 1 \]