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For a diatomic gas, if \({\gamma }_{1}=\left(\frac{Cp}{Cv}\right)\) for rigid molecules and \({\gamma }_{2}=\left(\frac{…

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For a diatomic gas, if \({\gamma }_{1}=\left(\frac{Cp}{Cv}\right)\) for rigid molecules and \({\gamma }_{2}=\left(\frac{Cp}{Cv}\right)\) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ? (Cp and Cv are specific heats of the gas at constant pressure and volume)

[JEE Main 2025, 22 Jan (Shift 2)]

a

\({\gamma }_{2}={\gamma }_{1}\)

b

\(2{\gamma }_{2}={\gamma }_{1}\)

c

\({\gamma }_{2}<{\gamma }_{1}\)

d

\({\gamma }_{2}>{\gamma }_{1}\)

✓ Correct answer: c)

\({\gamma }_{2}<{\gamma }_{1}\)

Explanation

For a diatomic gas, the adiabatic index y is given by:

\(\gamma =\frac{{C}_{p}}{{C}_{v}}=\frac{f+2}{f}\)

without vibration: f = 5

\({\gamma }_{1}=\frac{7}{5}=1.4\)


When vibrational modes are included, \({C}_{v}\) increases, reducing \({\gamma }_{2}\).
Thus, \({\gamma }_{2}<{\gamma }_{1}\).

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