A polyatomic gas with n degrees of freedom has a mean energy per molecules given by

Options

A. `(n k T )/(N)`
B. `(n k T )/(2 N)`
C. `(n k T)/(2 )`
D. `(3 k T)/(2)`

Explanation

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

### Step 1: Understand the concept of degrees of freedom
Degrees of freedom refer to the number of independent ways in which a molecule can store energy. For a polyatomic gas, these can include translational, rotational, and vibrational motions.

### Step 2: Recall the formula for mean energy
The mean energy per molecule for a gas can be expressed in terms of its degrees of freedom. The formula for the average energy associated with each degree of freedom is given by:
\[ E = \frac{1}{2} k T \]
where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

### Step 3: Calculate the total mean energy for \( n \) degrees of freedom
For \( n \) degrees of freedom, the total mean energy per molecule can be calculated by multiplying the energy per degree of freedom by the number of degrees of freedom:
\[ E_{\text{total}} = n \times \frac{1}{2} k T = \frac{n}{2} k T \]

### Step 4: Express the mean energy in terms of the gas constant \( R \)
For one mole of gas, the relationship between the Boltzmann constant \( k \) and the gas constant \( R \) is given by:
\[ R = N_A k \]
where \( N_A \) is Avogadro's number. Therefore, the mean energy per mole can be expressed as:
\[ E_{\text{mole}} = \frac{n}{2} R T \]

### Step 5: Finalize the mean energy per molecule
To find the mean energy per molecule, we divide the mean energy per mole by Avogadro's number:
\[ E_{\text{molecule}} = \frac{E_{\text{mole}}}{N_A} = \frac{n}{2} k T \]

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