The average value of rotational kinetic energy of one mole of oxygen gas at temperature T will be

Options

A. RT
B. `3/2` RT
C. `5/2` RT
D. `1/2` RT

Answer

To find the average value of rotational kinetic energy of one mole of oxygen gas at temperature T, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Degrees of Freedom**:
- For a diatomic gas like oxygen (O2), the degrees of freedom (f) are divided into translational and rotational.
- The translational degrees of freedom are 3 (movement in x, y, and z directions).
- The rotational degrees of freedom for a diatomic molecule are 2 (rotation about two axes perpendicular to the bond).

2. **Total Degrees of Freedom**:
- The total degrees of freedom for one mole of diatomic gas is given by:
\[ f = f_{\text{translational}} + f_{\text{rotational}} = 3 + 2 = 5 \]

3. **Average Kinetic Energy**:
- The average kinetic energy per degree of freedom is given by:
\[ \text{Average K.E.} = \frac{1}{2} k T \]
- For one mole of gas, we use the ideal gas constant \( R \) instead of Boltzmann's constant \( k \):
\[ \text{Average K.E. per mole} = \frac{1}{2} f R T \]

4. **Calculate Rotational Kinetic Energy**:
- Since we are interested in the average rotational kinetic energy, we only consider the rotational degrees of freedom (f = 2):
\[ \text{Average Rotational K.E.} = \frac{1}{2} \cdot 2 \cdot R T \]
- Simplifying this gives:
\[ \text{Average Rotational K.E.} = R T \]

5. **Final Answer**:
- Therefore, the average value of the rotational kinetic energy of one mole of oxygen gas at temperature T is:
\[ \text{Average Rotational K.E.} = R T \]

### Conclusion:
The average value of rotational kinetic energy of one mole of oxygen gas at temperature T is \( R T \).