Ratio Of Specific Heats

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An ideal diatomic gas with rigid molecules undergoes no vibrational excitation at moderate temperatures; what is the ratio of its specific heats Cp/Cv?

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Fixed entirely by its degrees of freedom $f$, the adiabatic index of an ideal gas is $\gamma=C_p/C_v=1+\dfrac{2}{f}$. A rigid diatomic molecule has $f=5$ (three translational plus two rotational modes), so $\gamma=1+\dfrac{2}{5}=1.4$. Equivalently, $C_v=\frac{5}{2}R$ and $C_p=\frac{7}{2}R$, whose ratio is $7/5=1.4$. The value of $\gamma$ governs how steeply pressure falls during an adiabatic expansion and also sets the speed of sound in the gas through $v=\sqrt{\gamma P/\rho}$. The value 1.67 corresponds to a monatomic gas with $f=3$, where $\gamma=5/3$. The value 1.33 belongs to a polyatomic gas with $f=6$. The value 1.5 has no standard molecular interpretation and would require a non-integer degree-of-freedom count. This is the NCERT result connecting molecular structure to thermodynamic behaviour, central to analysing adiabatic processes, heat-engine cycles, and sound propagation. A larger number of degrees of freedom lowers $\gamma$ because the same added heat is shared among more storage modes, leaving relatively less to raise the pressure. As a plausibility check, $\gamma$ for real diatomic gases such as nitrogen and oxygen is measured to be very close to 1.4 at room temperature, which strongly validates the rigid-rotor diatomic model used here and the value obtained.

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