Editing Boltzmann constant (section)

=== Role in the equipartition of energy ===
{{main|Equipartition of energy}}
Given a [[thermodynamics|thermodynamic]] system at an [[thermodynamic temperature|absolute temperature]] {{mvar|T}}, the average thermal energy carried by each microscopic degree of freedom in the system is {{math|{{sfrac|1|2}}&nbsp;''kT''}} (i.e., about {{val|2.07|e=−21|u=J}}, or {{val|0.013|ul=eV}}, at room temperature). This is generally true only for classical systems with a [[Thermodynamic limit|large number of particles]].

In [[classical mechanics|classical]] [[statistical mechanics]], this average is predicted to hold exactly for homogeneous [[ideal gas]]es. Monatomic ideal gases (the six noble gases) possess three [[degrees of freedom (physics and chemistry)|degrees of freedom]] per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of {{math|{{sfrac|3|2}}&nbsp;''kT''}} per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate the [[root-mean-square speed]] of the atoms, which turns out to be inversely proportional to the square root of the [[atomic mass]]. The root mean square speeds found at room temperature accurately reflect this, ranging from {{val|1370|u=m/s}} for [[helium]], down to {{val|240|u=m/s}} for [[xenon]].

[[Kinetic theory of gases#Pressure|Kinetic theory]] gives the average pressure {{mvar|p}} for an ideal gas as
<math display="block"> p = \frac{1}{3}\frac{N}{V} m \overline{v^2}.</math>

Combination with the ideal gas law
<math display="block">p V = N k T</math>
shows that the average translational kinetic energy is
<math display="block"> \tfrac{1}{2}m \overline{v^2} = \tfrac{3}{2} k T.</math>

Considering that the translational motion velocity vector {{math|'''v'''}} has three degrees of freedom (one for each dimension) gives the average energy per degree of freedom equal to one third of that, i.e. {{math|{{sfrac|1|2}}&nbsp;''kT''}}.

The ideal gas equation is also obeyed closely by molecular gases; but the form for the heat capacity is more complicated, because the molecules possess additional internal degrees of freedom, as well as the three degrees of freedom for movement of the molecule as a whole. Diatomic gases, for example, possess a total of six degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and one vibrational). At lower temperatures, not all these degrees of freedom may fully participate in the gas heat capacity, due to quantum mechanical limits on the availability of excited states at the relevant thermal energy per molecule.