For a gas if ratio of specific heats at constant pressure and volume is g then value of degrees of freedom is

Options

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

Answer

To solve the problem, we need to find the value of degrees of freedom (F) for a gas given the ratio of specific heats at constant pressure and volume, denoted as gamma (γ).

### Step-by-Step Solution:

1. **Understanding the relationship between specific heats and degrees of freedom**:
The ratio of specific heats for a gas is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
where \(C_p\) is the specific heat at constant pressure and \(C_v\) is the specific heat at constant volume.

2. **Using the formula for specific heats**:
The specific heats can be related to the degrees of freedom (F) of the gas by the following equations:
\[ C_v = \frac{F}{2}R \]
\[ C_p = C_v + R = \frac{F}{2}R + R = \left(\frac{F}{2} + 1\right)R \]

3. **Substituting into the ratio**:
Now substituting \(C_p\) and \(C_v\) into the equation for gamma:
\[ \gamma = \frac{C_p}{C_v} = \frac{\left(\frac{F}{2} + 1\right)R}{\frac{F}{2}R} \]
Simplifying this gives:
\[ \gamma = \frac{\frac{F}{2} + 1}{\frac{F}{2}} = 1 + \frac{2}{F} \]

4. **Rearranging to find degrees of freedom**:
Rearranging the equation to isolate F:
\[ \gamma - 1 = \frac{2}{F} \]
\[ F = \frac{2}{\gamma - 1} \]

5. **Conclusion**:
The degrees of freedom (F) for the gas in terms of the specific heat ratio (γ) is given by:
\[ F = \frac{2}{\gamma - 1} \]

### Final Answer:
The value of degrees of freedom (F) is:
\[ F = \frac{2}{\gamma - 1} \]