Interpretation of Temperature in Kinetic Theory

Equation $P = \tfrac{1}{3}\tfrac{N}{V}m\overline{v^2}$ can be rewritten as

The quantity $\tfrac{1}{2}m\overline{v^2}$ is the average translational kinetic energy of a single molecule. In an ideal gas, since the molecules do not interact with one another (Section 3.3), there is no intermolecular potential energy term at all -- the internal energy of an ideal gas is therefore purely kinetic. The average total energy $E$ of the gas is thus

and the pressure relation above becomes $PV = \tfrac{2}{3}E$.

Distribution of speeds. Not every molecule moves at exactly $v_{rms}$ -- individual molecular speeds are spread over a whole range, from (in principle) zero to infinity, and it is natural to ask how many molecules have speeds in any given interval. This function -- the number of molecules as a function of speed -- is called the distribution of speeds, and its form at a fixed temperature $T$ is known as Maxwell's distribution of molecular speeds: a curve in which the number of molecules with speed between $v$ and $v + dv$ is proportional to a shaded strip of area $n_v\,dv$ under the curve. Once this distribution is known, average values of quantities like $\overline{v^2}$ can, in principle, be computed directly from it -- the kinetic theory derivation above effectively summarises that whole distribution into just its single mean-square value.

Relating energy to temperature. Combining $PV = \tfrac{2}{3}E$ with the ideal gas equation $PV = Nk_BT$ gives

so that the average energy per molecule is

$\dfrac{E}{N} = \dfrac{3}{2}k_BT$ …