On the basis of kinetic theory of gases, the mean `K.E.` of 1 mole of gas per degree of freedom is

Options

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

Answer

To find the mean kinetic energy of 1 mole of gas per degree of freedom based on the kinetic theory of gases, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Kinetic Energy Formula**:
The mean kinetic energy (K.E.) of a gas is given by the formula:
\[ \text{K.E.} = \frac{F}{2} \cdot k \cdot N_a \cdot T \]
where:
- \( F \) = degrees of freedom
- \( k \) = Boltzmann constant
- \( N_a \) = Avogadro's number
- \( T \) = absolute temperature in Kelvin

2. **Identifying Degrees of Freedom**:
For a monoatomic gas, the degrees of freedom \( F \) is 3 (movement in x, y, and z directions). However, since we want the mean K.E. per degree of freedom, we will consider \( F = 1 \) for this calculation.

3. **Substituting Values**:
We substitute \( F = 1 \) into the kinetic energy formula:
\[ \text{K.E.} = \frac{1}{2} \cdot k \cdot N_a \cdot T \]

4. **Relating to Universal Gas Constant**:
We know that \( k \cdot N_a = R \) (where \( R \) is the universal gas constant). Thus, we can rewrite the equation as:
\[ \text{K.E.} = \frac{1}{2} \cdot R \cdot T \]

5. **Final Result**:
Therefore, the mean kinetic energy of 1 mole of gas per degree of freedom is:
\[ \text{K.E. per degree of freedom} = \frac{1}{2} R T \]

### Conclusion:
The answer is \( \frac{1}{2} R T \).