A polyatomic gas with (n) degress of freedom has a mean energy per molecule given by.

Options

A. `n/2 R T`
B. `1/2 R T`
C. `n/2 k T`
D. `1/2 k T`

Answer

To find the mean energy per molecule of a polyatomic gas with \( n \) degrees of freedom, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Mean Energy**:
The mean energy per molecule of a gas can be derived from the kinetic theory of gases. It is related to the degrees of freedom of the gas molecules.

2. **Identify the Formula**:
The mean energy per molecule \( u \) is given by the formula:
\[ u = \frac{f}{2} k T \]
where:
- \( u \) = mean energy per molecule
- \( f \) = degrees of freedom
- \( k \) = Boltzmann constant
- \( T \) = absolute temperature in Kelvin

3. **Substitute the Degrees of Freedom**:
For a polyatomic gas, the degrees of freedom \( f \) is represented as \( n \). Therefore, we can substitute \( n \) into the formula:
\[ u = \frac{n}{2} k T \]

4. **Final Expression**:
The final expression for the mean energy per molecule of a polyatomic gas is:
\[ u = \frac{n}{2} k T \]

5. **Select the Correct Option**:
Based on the derived formula, we can identify the correct option from the given choices in the question.

### Conclusion:
The mean energy per molecule of a polyatomic gas with \( n \) degrees of freedom is given by:
\[ u = \frac{n}{2} k T \]