Diatomic Molecules

A monatomic gas such as helium consists of single atoms, and a He atom -- being, for these purposes, effectively a point mass -- has only 3 translational degrees of freedom, and no others.

A diatomic molecule such as O$_2$ or N$_2$, with its two atoms lying along (say) the $x$-axis, also has these same 3 translational degrees of freedom for the motion of its centre of mass. But it can, in addition, rotate: about the $z$-axis and about the $y$-axis, both of which pass through the centre of mass perpendicular to the molecular axis (Fig. 3.3). Rotation about the molecule's own axis (the $x$-axis, along the bond) is not counted, because spinning the two point-like atoms about the line joining them produces no change in either atom's position -- there is no meaningful rotational motion to speak of about that axis. So a diatomic molecule, treated as a rigid rotator, has 2 additional rotational degrees of freedom, for a total of 5 (3 translational + 2 rotational).

If $I_z$ and $I_y$ are the moments of inertia about the $z$- and $y$-axes, and $\omega_z$, $\omega_y$ the corresponding angular speeds, the two rotational kinetic energy terms are $\tfrac{1}{2}I_z\omega_z^2$ and $\tfrac{1}{2}I_y\omega_y^2$. So the total energy of a rigid diatomic molecule is

Each of these five quadratic terms contributes $\tfrac{1}{2}k_BT$ by the law of equipartition, for a total average energy of $\tfrac{5}{2}k_BT$ per molecule.

Vibration in non-rigid molecules. The "rigid rotator" assumption is itself an idealisation. Real diatomic molecules such as O$_2$, N$_2$ and CO contain covalent bonds and can additionally vibrate, with the atoms oscillating about their mean separation along the internuclear axis, behaving like a one-dimensional harmonic oscillator:

$E_{\text{vibrational}} = \tfrac{1}{2}m u^2 + \tfrac{1}{2}kr^2$ …