Mean kinetic energy per degree of freedom for gas molecule is

Options

A. `(3KT)/(2)`
B. `KT`
C. `(1)/(2)KT`
D. `(5KT)/(2)`

Explanation

To find the mean kinetic energy per degree of freedom for a gas molecule, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Degrees of Freedom**:
- In kinetic theory, the degrees of freedom refer to the number of independent ways in which a molecule can move or store energy. For a monatomic gas, there are 3 translational degrees of freedom (movement in x, y, and z directions).

2. **Apply the Law of Equipartition of Energy**:
- According to the law of equipartition of energy, the total energy of a system in thermal equilibrium is distributed equally among all degrees of freedom. Each degree of freedom contributes an amount of energy.

3. **Energy Contribution per Degree of Freedom**:
- The energy associated with each degree of freedom is given by the formula:
\[ \text{Energy per degree of freedom} = \frac{1}{2} k_B T \]
- Here, \( k_B \) is the Boltzmann constant, and \( T \) is the absolute temperature in Kelvin.

4. **Mean Kinetic Energy Calculation**:
- Since we are interested in the mean kinetic energy per degree of freedom, we can directly state:
\[ \text{Mean kinetic energy per degree of freedom} = \frac{1}{2} k_B T \]

5. **Conclusion**:
- Therefore, the mean kinetic energy per degree of freedom for a gas molecule is:
\[ \frac{1}{2} k_B T \]

### Final Answer:
The mean kinetic energy per degree of freedom for a gas molecule is \(\frac{1}{2} k_B T\).

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