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

Options

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

Explanation

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. **Understand the Kinetic Energy per Molecule**:
The mean kinetic energy (K.E.) per molecule of a gas is given by the formula:
\[ \text{K.E. per molecule} = \frac{F}{2} kT \]
where \( F \) is the degree of freedom, \( k \) is the Boltzmann constant, and \( T \) is the absolute temperature in Kelvin.

2. **Calculate the Total Kinetic Energy for 1 Mole**:
Since 1 mole of gas contains \( N_a \) (Avogadro's number) of molecules, the total kinetic energy for 1 mole of gas can be calculated as:
\[ \text{Total K.E.} = \text{K.E. per molecule} \times N_a = \left(\frac{F}{2} kT\right) \times N_a \]
This simplifies to:
\[ \text{Total K.E.} = \frac{F}{2} kT N_a \]

3. **Relate Boltzmann Constant to Gas Constant**:
We know that \( k \times N_a = R \), where \( R \) is the universal gas constant. Thus, we can substitute \( k N_a \) with \( R \):
\[ \text{Total K.E.} = \frac{F}{2} R T \]

4. **Mean Kinetic Energy per Degree of Freedom**:
To find the mean kinetic energy per degree of freedom, we need to divide the total kinetic energy by the number of degrees of freedom \( F \):
\[ \text{Mean K.E. per degree of freedom} = \frac{\text{Total K.E.}}{F} = \frac{\frac{F}{2} R T}{F} \]
This simplifies to:
\[ \text{Mean K.E. per degree of freedom} = \frac{1}{2} R T \]

5. **Final Result**:
Thus, the mean kinetic energy of 1 mole per degree of freedom is:
\[ \frac{1}{2} R T \]

### Conclusion:
The mean kinetic energy of 1 mole of gas per degree of freedom is \( \frac{1}{2} R T \).