To find out degree of freedom, the correct expression is:

Options

A. `f = (2)/(gamma - 1)`
B. `f = (gamma + 1)/(2)`
C. `f = (2)/(gamma + 1)`
D. `f = (1)/(gamma + 1)`

Explanation

To find the degree of freedom (f) in terms of the adiabatic constant (γ), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- The degree of freedom (f) is a measure of the number of independent ways in which a system can move.
- The adiabatic constant (γ) is defined as the ratio of the specific heat at constant pressure (Cₚ) to the specific heat at constant volume (Cᵥ):
\[ \gamma = \frac{C_p}{C_v} \]

2. **Expressing Cₚ and Cᵥ**:
- For an ideal gas, the specific heats can be expressed in terms of the degree of freedom:
\[ C_p = \left(\frac{f}{2} + 1\right) nR\Delta T \]
\[ C_v = \frac{f}{2} nR\Delta T \]
where \( n \) is the number of moles, \( R \) is the gas constant, and \( \Delta T \) is the change in temperature.

3. **Substituting Cₚ and Cᵥ into the expression for γ**:
- Substitute the expressions for Cₚ and Cᵥ into the equation for γ:
\[ \gamma = \frac{C_p}{C_v} = \frac{\left(\frac{f}{2} + 1\right) nR\Delta T}{\frac{f}{2} nR\Delta T} \]

4. **Simplifying the Expression**:
- The \( nR\Delta T \) terms cancel out:
\[ \gamma = \frac{\frac{f}{2} + 1}{\frac{f}{2}} = \frac{f/2 + 1}{f/2} \]
- This simplifies to:
\[ \gamma = 1 + \frac{2}{f} \]

5. **Rearranging to Find f**:
- Rearranging the equation to express f in terms of γ:
\[ \gamma - 1 = \frac{2}{f} \]
- Taking the reciprocal gives:
\[ f = \frac{2}{\gamma - 1} \]

6. **Final Expression**:
- Thus, the degree of freedom (f) in terms of the adiabatic constant (γ) is:
\[ f = \frac{2}{\gamma - 1} \]

### Conclusion:
The correct expression for the degree of freedom in terms of the adiabatic constant is:
\[ f = \frac{2}{\gamma - 1} \]