Question

For a diatomic gas, if \( \gamma_1 = \frac{C_P}{C_V} \) for rigid molecules and \( \gamma_2 = \frac{C_P}{C_V} \) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct? (where \( C_P \) and \( C_V \) are specific heats of the gas at constant pressure and volume)

• A. \( \gamma_2 = \gamma_1 \)
• B. \( \gamma_2>\gamma_1 \)
• C. \( 2 \gamma_2 = \gamma_1 \)
• D. \( \gamma_2<\gamma_1 \)

Hint:

For diatomic gases with vibrational modes, the specific heat ratio \( \gamma \) will decrease compared to rigid molecules, as vibrational modes add additional degrees of freedom, which reduces the overall energy increase per unit temperature.

Answer 1

The Correct Option is D

Regarding diatomic molecules:
- For rigid molecules, the specific heat ratio \( \gamma_1 = \frac{C_P}{C_V} \) is generally 5/3 for a monatomic gas. 
- For diatomic gases, when vibrational modes are considered, the specific heat ratio \( \gamma_2 \) decreases. This is because vibrational modes introduce additional degrees of freedom, which in turn reduce the specific heat ratio. 

Consequently, \( \gamma_2 \) is less than \( \gamma_1 \). This occurs because vibrational modes increase internal energy without a proportional increase in temperature. Therefore, the correct conclusion is \( \boxed{\gamma_2 < \gamma_1} \).