What will be average value of energy along one degree of freedom for an ideal gas in thermal equilibrium at temp. T

Answer

To find the average value of energy along one degree of freedom for an ideal gas in thermal equilibrium at temperature \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Degrees of Freedom**:
For an ideal gas, the degrees of freedom \( F \) corresponds to the number of independent ways in which a molecule can store energy. In three-dimensional space, each molecule has three translational degrees of freedom (movement in x, y, and z directions). However, since the question asks for energy along one degree of freedom, we will consider \( F = 1 \).

2. **Using the Equipartition Theorem**:
According to the equipartition theorem, each degree of freedom contributes an average energy of \( \frac{1}{2} k_B T \) to the total energy of the system, where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature.

3. **Calculating Average Energy for One Degree of Freedom**:
Since we are interested in only one degree of freedom, we can calculate the average energy \( E \) as follows:
\[ E = \frac{1}{2} k_B T \]

4. **Final Result**:
Therefore, the average value of energy along one degree of freedom for an ideal gas in thermal equilibrium at temperature \( T \) is:
\[ E = \frac{1}{2} k_B T \]

### Summary:
The average value of energy along one degree of freedom for an ideal gas in thermal equilibrium at temperature \( T \) is given by:
\[ E = \frac{1}{2} k_B T \]