The mean kinetic energy of one mole of gas per degree of

Options

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

Explanation

To solve the question regarding the mean kinetic energy of one mole of gas per degree of freedom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Degrees of Freedom**:
- In the kinetic theory of gases, each molecule of a gas can move in three dimensions (x, y, z), which gives it three translational degrees of freedom. Additionally, depending on the type of gas (monoatomic, diatomic, etc.), there can be rotational and vibrational degrees of freedom.

2. **Mean Kinetic Energy Formula**:
- The average kinetic energy (KE) of one mole of gas is given by the formula:
\[ KE = \frac{3}{2}RT \]
- Here, \(R\) is the universal gas constant, and \(T\) is the absolute temperature in Kelvin.

3. **Kinetic Energy per Degree of Freedom**:
- The average kinetic energy associated with each degree of freedom is given by:
\[ KE_{per \ degree \ of \ freedom} = \frac{1}{2}RT \]
- This is derived from the total kinetic energy divided by the number of degrees of freedom. For a monoatomic gas, there are 3 translational degrees of freedom.

4. **Conclusion**:
- Since the question asks for the mean kinetic energy of one mole of gas per degree of freedom, we conclude that:
\[ KE_{per \ degree \ of \ freedom} = \frac{1}{2}RT \]
- Therefore, the correct answer is option 4: \(\frac{1}{2}RT\).

### Final Answer:
The mean kinetic energy of one mole of gas per degree of freedom is \(\frac{1}{2}RT\).

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