Which of the following formula is correct ? [ Where symbols have their usual meaning ]

Options

A. `C_V = R/(gamma - 1)`
B. `C_p =(gammaR)/(gamma - 1)`
C. `C_p /C_V = gamma `
D. All of these

Answer

To determine which of the given formulas is correct, we need to analyze the options based on the definitions and relationships involving the specific heats at constant pressure (Cp) and constant volume (Cv), as well as the adiabatic constant (gamma).

### Step-by-Step Solution:

1. **Understanding Gamma (γ)**:
- Gamma (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**:
- The molar specific heat at constant volume (Cv) can be expressed in terms of the degrees of freedom (F) of the gas:
\[ C_v = \frac{F}{2} R \]
- The molar specific heat at constant pressure (Cp) can be expressed as:
\[ C_p = C_v + R = \frac{F}{2} R + R = \left(\frac{F}{2} + 1\right) R \]

3. **Finding the Relationship Between Cp, Cv, and Gamma**:
- From the definition of gamma:
\[ \gamma = \frac{C_p}{C_v} = \frac{\left(\frac{F}{2} + 1\right) R}{\frac{F}{2} R} = \frac{\frac{F}{2} + 1}{\frac{F}{2}} = 1 + \frac{2}{F} \]

4. **Rearranging to Find Degrees of Freedom**:
- Rearranging the equation for degrees of freedom (F):
\[ F = \frac{2}{\gamma - 1} \]

5. **Expressing Cv in Terms of Gamma**:
- Substituting the expression for F back into the equation for Cv:
\[ C_v = \frac{F}{2} R = \frac{1}{2} \cdot \frac{2}{\gamma - 1} R = \frac{R}{\gamma - 1} \]

6. **Expressing Cp in Terms of Gamma**:
- Using the relationship \(C_p = \gamma C_v\):
\[ C_p = \gamma \cdot \frac{R}{\gamma - 1} \]

7. **Conclusion**:
- After analyzing the relationships, we find that:
- \(C_p = \gamma C_v\)
- \(C_v = \frac{R}{\gamma - 1}\)
- \(C_p = \frac{\gamma R}{\gamma - 1}\)
- Therefore, all options can be correct based on the relationships derived.

### Final Answer:
All options are correct, and the correct formula is:
\[ \gamma = \frac{C_p}{C_v}, \quad C_v = \frac{R}{\gamma - 1}, \quad C_p = \gamma C_v \]