Kinetic Interpretation of Temperature

Rearranging the result found while deriving $v_{rms}$ in Section 9.4, $\tfrac{1}{3}Nm\overline{v^2} = nRT$, and recognising that $\tfrac{1}{2}m\overline{v^2}$ is precisely the average translational kinetic energy of a single molecule, gives, for one mole ($N=N_A$, so $n=1$):

using $k_B = R/N_A$, Boltzmann's constant. So the average translational kinetic energy of a SINGLE gas molecule is

and for one mole, the total translational kinetic energy of all $N_A$ molecules is $\tfrac{3}{2}N_Ak_BT = \tfrac{3}{2}RT$.

This result is genuinely remarkable, and is the real payoff of the whole kinetic-theory programme: it shows that the absolute temperature $T$ of a gas is, up to the fixed constant $\tfrac{3}{2}k_B$, DIRECTLY and ENTIRELY a measure of the average translational kinetic energy of its molecules. Temperature stops being merely a number read off a thermometer scale and becomes instead a genuinely mechanical quantity -- a measure of how energetically, on average, the molecules of a substance are moving. Two gases at the same temperature, whatever their chemical identity, always have molecules with exactly the same average translational kinetic energy $\tfrac{3}{2}k_BT$ per molecule (though, since their molar masses differ, they will generally have different RMS SPEEDS, per Section 9.4, since $v_{rms}=\sqrt{3k_BT/m}$ still depends on $m$). …