Kinetic Theory — Degrees of Freedom & Specific Heats (Medium)

Quick Recap — Degrees of Freedom & Specific Heats

• Degrees of freedom $f$: monatomic 3 (translational); diatomic 5 (3 trans + 2 rot); at high $T$ vibrational modes add.
• Equipartition: each degree of freedom contributes $\dfrac12 k_BT$ per molecule, so internal energy $U=\dfrac{f}{2}nRT$.
• Molar specific heats: $C_V=\dfrac{f}{2}R$, $C_P=C_V+R$, and $\gamma=\dfrac{C_P}{C_V}=1+\dfrac{2}{f}$. So monatomic $\gamma=\tfrac53$, diatomic $\gamma=\tfrac75$.
• Mean free path: $\lambda=\dfrac{1}{\sqrt2\,\pi d^2 n}\propto\dfrac{T}{P}$ — it grows at low pressure (fewer molecules per volume).

Worked mini-example. A diatomic gas has $f=5$: $C_V=\tfrac52 R$, $C_P=\tfrac72 R$, and $\gamma=1+\dfrac25=\dfrac75=1.4$.