The average energy per molecule of a triatomic gas at room temperature T is

Options

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

Explanation

To find the average energy per molecule of a triatomic gas at room temperature \( T \), we can follow these steps:

### Step 1: Understand Degrees of Freedom
A triatomic gas can be either linear or nonlinear. For a nonlinear triatomic gas, it has:
- 3 translational degrees of freedom (movement in x, y, and z directions)
- 3 rotational degrees of freedom (rotation about three perpendicular axes)

Thus, the total degrees of freedom \( f \) for a nonlinear triatomic gas is:
\[ f = 3 \text{ (translational)} + 3 \text{ (rotational)} = 6 \]

### Step 2: Apply the Equipartition Theorem
According to the equipartition theorem, the average energy per degree of freedom is given by:
\[ \text{Average energy per molecule} = \frac{1}{2} f k T \]
where:
- \( k \) is the Boltzmann constant
- \( T \) is the absolute temperature

### Step 3: Calculate the Average Energy
Substituting the value of \( f \) into the equation:
\[ \text{Average energy per molecule} = \frac{1}{2} \times 6 \times k T \]
This simplifies to:
\[ \text{Average energy per molecule} = 3 k T \]

### Conclusion
Thus, the average energy per molecule of a triatomic gas at room temperature \( T \) is:
\[ \text{Average energy} = 3 k T \]