Degrees of freedom

The formula $E_{\text{int}} = \frac{f}{2}nRT$ contains the number $f$ of degrees of freedom: the independent ways in which a molecule can move and store energy. The equipartition principle assigns to each degree of freedom an average energy of $\frac{1}{2}k_B T$ per molecule: counting degrees of freedom therefore means counting the “drawers” in which thermal energy can be stored.

• Monatomic gas ($f = 3$): the molecule is a single atom, which can only translate along the three axes $x$, $y$, $z$. Three translations, so $f = 3$ and $E_\text{int} = \frac{3}{2}nRT$.
• Diatomic gas ($f = 5$): the molecule is like a small dumbbell. Besides the three translations, it can rotate about two axes perpendicular to the molecular axis (rotation about the axis itself has negligible moment of inertia and does not count). Three translations plus two rotations, so $f = 5$ and $E_\text{int} = \frac{5}{2}nRT$.

Further detail — Degrees of freedom "switch on"

At very high temperatures (above a thousand kelvin or so) the vibrational degrees of freedom of the two atoms along the molecular axis also become active, and $f$ increases further. But at ordinary temperatures quantum mechanics “freezes” them, and $f = 3$ (monatomic) or $f = 5$ (diatomic) is enough. The fact that the vibrational modes remain switched off at room temperature was one of the first historical clues that classical physics is not enough to explain specific heats.

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Topics: Teoria cinetica dei gas Concepts: Energia interna Skills: Interpretazione micro-macro

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