The relation `C_p-C_v=R(C_p,C_v)`: Molar specific heats at constant pressure, volume is exactly true for

Options

A. An ideal mono-atomic gas
B. Any ideal gas, whether mono, di-or-poly-atomic
C. Any real gas above its critical temperature
D. All real gases

Answer

To solve the question regarding the relation \( C_p - C_v = R \) and its validity for different types of gases, we can follow these steps:

### Step 1: Understand the Terms
- **\( C_p \)**: Molar specific heat at constant pressure.
- **\( C_v \)**: Molar specific heat at constant volume.
- **\( R \)**: Universal gas constant.

### Step 2: Recall Mayer's Relation
Mayer's relation states that for an ideal gas, the difference between the molar specific heats at constant pressure and constant volume is equal to the universal gas constant:
\[ C_p - C_v = R \]

### Step 3: Identify Ideal Gases
- Ideal gases are defined as gases that follow the ideal gas law perfectly, which means they have no intermolecular forces and occupy no volume.
- The relation \( C_p - C_v = R \) holds true for all ideal gases, regardless of whether they are monoatomic, diatomic, or polyatomic.

### Step 4: Consider Real Gases
- For real gases, especially at conditions close to their critical temperature or under high pressure, the behavior deviates from the ideal gas law.
- Therefore, Mayer's relation does not hold for real gases under all conditions.

### Step 5: Conclusion
The relation \( C_p - C_v = R \) is exactly true for:
- All ideal gases (monoatomic, diatomic, and polyatomic).

Thus, the correct answer is that the relation is valid for **all ideal gases**.

### Final Answer
The relation \( C_p - C_v = R \) is exactly true for **all ideal gases**.

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