How do you use the equipartition theorem to predict the average kinetic energy of one molecule?

The equipartition theorem states that at high-enough temperatures, the molecule completely occupies its energy states and can be said to have the full degrees of freedom associated with each kind of motion:

#K_(avg) = N/2k_BT = (N_(tr) + N_(rot) + N_(vib) + N_(el ec))/2k_BT#

where:

• #N# is the degrees of freedom for a certain kind of motion. These have corresponding translational, rotational, vibrational, and electronic energy states.
• #k_B# is the Boltzmann constant.
• #T# is the temperature in #"K"#.

In general:

• Translational energy states are very dense.
• Rotational energy states are less dense.
• Vibrational energy states are even less dense.
• Electronic energy states are extremely far apart usually.

Therefore,

• All molecules at most temperatures have #N_(tr) = 3# for translational motion (three dimensions).
• Many molecules at ordinary temperatures have #N_(rot) = 2# for rotational motion if they are linear molecules, or #N_(rot) = 3# if they are nonlinear molecules.
• Most molecules at ordinary temperatures have #N_(vib) <= 1# for vibrational motion, although it is usually best to ignore vibrational degrees of freedom at ordinary temperatures.
• Most molecules at ordinary temperatures have #N_(el ec)# #"<<"# #1# for electronic energy states, so we generally say #K_(el ec) ~~ 0#.

If we took #"NH"_3# at #"298 K"# as an example,

#K_(avg) ~~ (3 + 3 + 0 + 0)/2 k_BT#

#~~ 3k_BT#

Or if we took #"HCl"# at #"298 K"#:

#K_(avg) ~~ (3 + 2 + 0 + 0)/2 k_BT#

#~~ 5/2k_BT#