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 solve the question regarding which formula is correct, we will start by recalling some fundamental relationships in thermodynamics related to specific heats.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- \( C_p \) is the specific heat at constant pressure.
- \( C_v \) is the specific heat at constant volume.
- \( R \) is the universal gas constant.
- \( \gamma \) (gamma) is the ratio of specific heats, defined as \( \gamma = \frac{C_p}{C_v} \).

2. **Use the Relationship Between Specific Heats**:
- We know from thermodynamics that:
\[ C_p - C_v = R \quad \text{(Equation 1)} \]
- Also, we have:
\[ \frac{C_p}{C_v} = \gamma \quad \text{(Equation 2)} \]

3. **Express \( C_p \) in Terms of \( C_v \)**:
- From Equation 2, we can express \( C_p \) as:
\[ C_p = \gamma C_v \quad \text{(Equation 3)} \]

4. **Substitute Equation 3 into Equation 1**:
- Substituting \( C_p \) from Equation 3 into Equation 1 gives:
\[ \gamma C_v - C_v = R \]
- This simplifies to:
\[ C_v (\gamma - 1) = R \]
- Therefore, we can express \( C_v \) as:
\[ C_v = \frac{R}{\gamma - 1} \quad \text{(Equation 4)} \]

5. **Find \( C_p \) Using Equation 4**:
- Now, substituting Equation 4 back into Equation 3 to find \( C_p \):
\[ C_p = \gamma C_v = \gamma \left(\frac{R}{\gamma - 1}\right) \]
- This simplifies to:
\[ C_p = \frac{\gamma R}{\gamma - 1} \quad \text{(Equation 5)} \]

6. **Evaluate the Options**:
- Now, we can evaluate the options given in the question:
- From our derived equations, we have:
- \( C_v = \frac{R}{\gamma - 1} \)
- \( C_p = \frac{\gamma R}{\gamma - 1} \)
- Therefore, we can conclude that all the options provided in the question are correct.

7. **Final Answer**:
- Hence, the correct answer is **Option D: All of the above**.