Specific heats of monoatomic and diatomic gases are same and satisfy the relation which is

Options

A. `C_p"(mono)"=C_p"(dia)"`
B. `C_p"(mono)"=C_v"(dia)"`
C. `C_v"(mono)"=C_v"(dia)"`
D. `C_v"(mono)"=C_p"(dia)"`

Explanation

To solve the problem regarding the specific heats of monoatomic and diatomic gases, we need to understand the definitions and relationships between the specific heats and the degrees of freedom of these gases.

### Step-by-Step Solution:

1. **Understanding Specific Heats**:
- The specific heat at constant volume (Cv) and at constant pressure (Cp) for a gas are related to its degrees of freedom (F).
- The relationships are given by:
- For a monoatomic gas:
\[ C_v = \frac{3}{2} R \]
\[ C_p = C_v + R = \frac{3}{2} R + R = \frac{5}{2} R \]
- For a diatomic gas:
\[ C_v = \frac{5}{2} R \]
\[ C_p = C_v + R = \frac{5}{2} R + R = \frac{7}{2} R \]

2. **Setting Up the Equation**:
- We are given that the specific heats of monoatomic and diatomic gases are the same. This means:
\[ C_{p, \text{monoatomic}} = C_{p, \text{diatomic}} \]
- Substituting the values we derived:
\[ \frac{5}{2} R = \frac{7}{2} R \]
- This equation does not hold true, indicating that the specific heats cannot be equal under normal conditions.

3. **Conclusion**:
- Since the specific heats of monoatomic and diatomic gases cannot be equal, we can conclude that the only valid relation is:
\[ C_{p, \text{monoatomic}} \neq C_{p, \text{diatomic}} \]
- Therefore, the specific heats of monoatomic and diatomic gases do not satisfy the relation of being equal.