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

Options

A. `(n kT)/(N)`
B. `(n kT)/(2N)`
C. `(nkT)/(2)`
D. `(3kT)/(2)`

Answer

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

### Step-by-Step Solution:

1. **Understand the Concept of Degrees of Freedom**:
- The degrees of freedom of a gas molecule refer to the number of independent ways in which the molecule can store energy. For a polyatomic gas, this includes translational, rotational, and vibrational motions.

2. **Mean Energy per Degree of Freedom**:
- According to the equipartition theorem, each degree of freedom contributes an energy of \( \frac{1}{2} k T \) to the total energy of the system, where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin.

3. **Calculate Total Mean Energy**:
- For a gas with \( n \) degrees of freedom, the total mean energy per molecule can be expressed as:
\[ E = \frac{n}{2} k T \]
- This equation indicates that the mean energy is directly proportional to the number of degrees of freedom and the temperature.

4. **Final Expression**:
- Therefore, the mean energy per molecule of the polyatomic gas with \( n \) degrees of freedom is:
\[ E = \frac{n}{2} k T \]

5. **Identify the Correct Option**:
- From the options given in the question, the correct expression for the mean energy per molecule of the polyatomic gas is:
\[ \frac{n}{2} k T \]