Degrees of freedom and equipartition of energy

A molecule's translational kinetic energy is ε_t = ½mv_x² + ½mv_y² + ½mv_z². At temperature T its average is (3/2)k_BT, and with no preferred direction each term averages ½k_BT.

Degrees of freedom count the independent coordinates needed to locate something: one for motion along a line, two in a plane, three in space. Moving freely in three dimensions, a molecule therefore has three translational degrees of freedom, each contributing one squared term.

A monatomic gas such as argon has only these three.

A diatomic molecule such as O₂ or N₂ can also rotate about two axes perpendicular to the line joining its atoms (Fig. 12.6), adding ½I₁ω₁² + ½I₂ω₂². Spin about the bond line itself has almost no moment of inertia and, for quantum-mechanical reasons, stores no energy here.

Treating O₂ as a rigid rotator (no vibration) works at moderate temperatures. Molecules such as CO vibrate even then: the atoms oscillate along their axis and add ε_v = ½m(dy/dt)² + ½ky², where k is the force constant and y the vibrational coordinate.

Each translational or rotational degree of freedom adds one squared term; a vibrational mode adds two, one kinetic and one potential.

Law of equipartition of energy (first proved by Maxwell): in thermal equilibrium the energy is shared equally among all the squared terms, each averaging ½k_BT.

So each translational and each rotational degree of freedom gets ½k_BT, and each vibrational frequency gets 2 × ½k_BT = k_BT.

The proof is beyond this book; the law is used here to predict specific heats.