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)
Show Hint
- \( \gamma_2 = \gamma_1 \)
- \( \gamma_2>\gamma_1 \)
- \( 2 \gamma_2 = \gamma_1 \)
- \( \gamma_2<\gamma_1 \)
The Correct Option is D
Solution and Explanation
For diatomic molecules:
- For rigid molecules, the specific heat ratio \( \gamma_1 = \frac{C_P}{C_V} \) is typically 5/3 for a monoatomic gas.
- For diatomic gases with vibrational modes included, the value of \( \gamma_2 \) will be lower, since vibrational modes contribute more degrees of freedom which lower the specific heat ratio.
Thus, \( \gamma_2 \) is smaller than \( \gamma_1 \), as vibrational modes lead to higher internal energy without increasing the temperature as much. Therefore, the correct answer is \( \boxed{\gamma_2 < \gamma_1} \).
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