The ratio of average translatory kinetic energy of He gas 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 translatory kinetic energy of helium (He) gas molecules to oxygen (O₂) gas molecules, we can follow these steps:

### Step 1: Understand the Degrees of Freedom
- Helium (He) is a monatomic gas, which means it has 3 translational degrees of freedom.
- Oxygen (O₂) is a diatomic gas, which also has 3 translational degrees of freedom.

### Step 2: Use the Equipartition Theorem
According to the equipartition theorem, the average translational kinetic energy (U) of a gas can be expressed as:
\[ U = \frac{F}{2} RT \]
where \( F \) is the number of degrees of freedom, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

### Step 3: Calculate the Average Translatory Kinetic Energy for Helium
For helium (He):
- Degrees of freedom \( F_1 = 3 \)
- Therefore, the average translatory kinetic energy \( U_1 \) is:
\[ U_1 = \frac{3}{2} RT \]

### Step 4: Calculate the Average Translatory Kinetic Energy for Oxygen
For oxygen (O₂):
- Degrees of freedom \( F_2 = 3 \) (considering only translational motion)
- Therefore, the average translatory kinetic energy \( U_2 \) is:
\[ U_2 = \frac{3}{2} RT \]

### Step 5: Find the Ratio of Average Kinetic Energies
Now, we need to find the ratio of \( U_1 \) to \( U_2 \):
\[ \text{Ratio} = \frac{U_1}{U_2} = \frac{\frac{3}{2} RT}{\frac{3}{2} RT} \]

### Step 6: Simplify the Ratio
The \( RT \) terms cancel out:
\[ \text{Ratio} = \frac{3/2}{3/2} = 1 \]

### Conclusion
Thus, the ratio of the average translatory kinetic energy of helium gas molecules to oxygen gas molecules is:
\[ \text{Ratio} = 1:1 \]