20.5.2 The Equipartition Theorem

The particles of a monatomic gas are atoms. Their energy consists exclusively of their translational kinetic energy. A particle’s translational kinetic energy can be written

ϵ = \frac{1}{2} m v^{2} = \frac{1}{2} m v_{x}^{2} + \frac{1}{2} m v_{y}^{2} + \frac{1}{2} m v_{z}^{2} = ϵ_{x} + ϵ_{y} + ϵ_{z}

(20.32)

where we have written separately the energy associated with translational motion along the three axes. Because each axis in space is independent, we can think of $ϵ_{x}$ , $ϵ_{y}$ , and $ϵ_{z}$ as independent modes of storing energy within the system.

Other systems have additional modes of energy storage. For example,

Two atoms joined by a spring-like molecular bond can vibrate back and forth. Both kinetic and potential energy are associated with this vibration.
A diatomic molecule, in addition to translational kinetic energy, has rotational kinetic energy if it rotates end-over-end like a dumbbell.

We define the number of degrees of freedom as the number of distinct and independent modes of energy storage. A monatomic gas has three degrees of freedom. Systems that can vibrate or rotate have more degrees of freedom.

Collisions between molecules constantly move energy between one degree of freedom and another. For example, a collision could cause a diatomic molecule to rotate faster, which increases its rotational kinetic energy by decreasing its translational kinetic energy. The proof is beyond what we can do in this textbook, but it’s possible to show that the net result of the vast number of collisions is to cause the thermal energy to, on average, be shared equally among all the degrees of freedom. This conclusion is known as the equipartition theorem, meaning that the energy is equally divided.

A monatomic gas has three degrees of freedom and thus, as we found above, $E_{th} = \frac{3}{2} N k_{B} T$ .