Kinetic theory

Molecular impacts create pressure

A gas exerts pressure because molecules transfer momentum to walls. Faster molecular motion raises the momentum delivered per collision, while more molecules per volume increases collision rate.

Average the momentum flux

For an ideal gas with isotropic molecular velocities, each Cartesian component contributes one third of the mean squared speed. The kinetic-theory result is p = (1/3)ρ⟨v²⟩, where ρ is mass density and ⟨v²⟩ is the average squared molecular speed. Combining with pV = Nk_BT gives ⟨K_trans⟩ = 3k_BT/2 per molecule.

At 300 K, the average translational kinetic energy per molecule is 1.5(1.381×10⁻²³)(300) = 6.21×10⁻²¹ J. Temperature sets this energy scale independent of the gas amount.

Average speed is not the same as rms speed

The equation uses ⟨v²⟩, so v_rms = √⟨v²⟩, not the ordinary average speed. Real gases also have a distribution of speeds, not one common molecular velocity.

Work it: For nitrogen at 300 K with molar mass 0.028 kg/mol, estimate v_rms = √(3RT/M). Check whether the result is plausible compared with the speed of sound.