Please wait for the animation to completely load.
The equipartition of energy theorem says that the energy of an atom or particle is, on average, equally distributed between the different modes (different degrees of freedom) available. The way to count the modes, or degrees of freedom, is to count the number of quadratic terms in the energy expression. For example, for a monatomic gas (without any external forces), the energy of a particle is given as 1/2 mvx2 + 1/2 mvy2 + 1/2 mvz2. There are three terms that are quadratic. However, for a particle in a three-dimensional simple harmonic potential well, there are six different modes (degrees of freedom). The energy per particle has an average value of (f/2)kBT, where f is the number of degrees of freedom, kB is the Boltzmann constant and T is the temperature.
You can verify this result by taking the distribution function for a monatomic ideal gas and finding the total energy:5
n(ε)dε = 2πN/(π kBT)3/2 ε1/2 exp (−ε/kBT) dε ,
where ∫ n(ε)dε = E and E = (3/2) NkBT. For a harmonic oscillator, the distribution is different, but in the end it gives you E = 3kBT. Restart.
Try this animation of a diatomic gas with 20 particles. Notice that the graph shows the total kinetic energy of the diatomic particles and the kinetic energies of translation (motion in x and y directions) and rotation.
Now, try a mixture of 20 monatomic particles and 20 diatomic particles.
5This is a classical result because we are using the classical distribution (Section 15.5). If the temperature is very low and the density high, we will need to use a quantum mechanical distribution and these results do not hold (see Section 15.7 for an example of low temperature calculations in a solid).
Applet authored by Ernesto Martin and modified by Wolfgang Christian.