Kinetic Interpretation of Temperature

Comparing with the ideal gas equation. Equation (9.6) can be rewritten as $PV = \tfrac13 Nm\overline{v^2}$ (9.7). Comparing this directly with the macroscopic ideal gas equation $PV = NkT$ (where $k$ is Boltzmann's constant) gives

Multiplying both sides by $3/2$,

The right-hand side, $\tfrac12 m\overline{v^2}$, is exactly the average translational kinetic energy of a single molecule, usually written $\overline{KE}=\epsilon$. So

The physical meaning. Equation (9.10) is one of the most important results in kinetic theory: it says the temperature of a gas is nothing more than a direct measure of the average translational kinetic energy carried by each of its molecules. Two consequences follow immediately: (i) average kinetic energy per molecule is directly proportional to the absolute temperature -- this single equation is the bridge connecting the macroscopic, easily-measured world of temperature to the microscopic, unobservable world of molecular motion; and (ii) average kinetic energy per molecule depends only on temperature, never on the mass of the molecule -- so if the temperature of an ideal gas is known from a thermometer, the average kinetic energy of every molecule in it is already known, entirely without needing to identify what kind of molecule it is.

Internal energy of an ideal gas. Multiplying the average kinetic energy of one molecule by the total number of molecules $N$ gives the total internal energy of the gas:

Since $Nk=\mu R$ for $\mu$ moles of gas (where $R$ is the universal gas constant), this is often written $U=\tfrac32\mu RT$. Crucially, this shows the internal energy of an ideal gas depends only on its absolute temperature -- it is completely independent of the gas's pressure or volume. …