Kinetic interpretation of temperature

Rewrite P = ⅓nm⟨v²⟩ with N = nV molecules: PV = ⅔N(½m⟨v²⟩). The bracket is the average translational kinetic energy of one molecule.

For an ideal gas the internal energy is purely kinetic: E = N × ½m⟨v²⟩, so PV = ⅔E. (E here is the translational part; other degrees of freedom come in §12.5.)

Compare with PV = Nk_BT: E = (3/2)Nk_BT, i.e. ½m⟨v²⟩ = (3/2)k_BT.

So a molecule's mean kinetic energy rises in direct proportion to the kelvin temperature, whatever the pressure, the volume or the kind of ideal gas. The Boltzmann constant is the bridge between a measured temperature and a molecular energy.

Side result: the internal energy of an ideal gas depends only on T, not on P or V. With this reading of temperature, kinetic theory agrees fully with the ideal-gas equation and the gas laws.

Mixtures: P = ⅓[n₁m₁⟨v₁²⟩ + n₂m₂⟨v₂²⟩ + …]. In equilibrium every species has the same average kinetic energy (3/2)k_BT, so P = (n₁ + n₂ + …)k_BT, which is Dalton's law again.

Root mean square speed: v_rms = √⟨v²⟩ = √(3k_BT/m).

Nitrogen at 300 K: m = M/N_A = 28/(6.02 × 10²⁶) = 4.65 × 10⁻²⁶ kg, so ⟨v²⟩ = 3k_BT/m = (516)² m² s⁻² and v_rms = 516 m s⁻¹, comparable to how fast sound travels in air.

At the same temperature, lighter molecules have larger rms speeds.