JEE Main 2025 Jan 22 Shift 2, Physics Q8: Law of Equipartition of Energy

JEE Main2025Jan 22, Shift 2Physics

For a diatomic gas, if $γ_{1} = (\frac{C _{p}}{C _{v}})$ for rigid molecules and $γ_{2} = (\frac{C _{p}}{C _{v}})$ for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ?

( $C_{p}$ and $C_{v}$ are specific heats of the gas at constant pressure and volume)

A
$γ_{2} > γ_{1}$
B
$γ_{2} = γ_{1}$
C
$2 γ_{2} = γ_{1}$
D
$γ_{2} < γ_{1}$

Solution

By equipartition, the adiabatic index for a gas with $f$ degrees of freedom is $γ = 1 + \frac{2}{f}$ , which decreases as $f$ increases.

Rigid diatomic molecule: $f = 5$ (3 translational + 2 rotational), giving $γ_{1} = 1 + \frac{2}{5} = 1.4$ .

Diatomic molecule with vibrational modes also active: each vibrational mode contributes 2 to $f$ . Including one vibrational mode, $f = 7$ , giving $γ_{2} = 1 + \frac{2}{7} \approx 1.286$ .

Adding degrees of freedom lowers $γ$ , so $γ_{2} < γ_{1}$ .

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JEE Main 2025 Jan 22 Shift 2, Physics Q8: Law of Equipartition of Energy

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