Kinetic Theory and r.m.s. Speed

Kinetic theory derives the pressure of a gas from nothing but Newton's laws applied to particles bouncing off the walls.

pV = ⅓ N m ⟨c²⟩

N is the number of particles, m the mass of one particle, and ⟨c²⟩ the mean square speed.

Why a mean square speed, not a mean speed — the particles move in all directions, so the mean velocity is zero: for every particle heading one way there is, on average, one heading the other. Squaring makes every value positive, so the average is not destroyed by the symmetry. The factor of one third arises because the motion is shared equally between the three perpendicular directions, and only the component along one axis contributes to the pressure on a given pair of walls.

Root mean square speed — the square root of the mean square speed, c_rms = √⟨c²⟩. It is a typical particle speed, and it has the units of speed, which ⟨c²⟩ does not.

Linking to temperature — comparing pV = ⅓Nm⟨c²⟩ with pV = NkT gives the mean kinetic energy of one particle:

½ m ⟨c²⟩ = ³⁄₂ k T

This is the precise version of the statement that temperature measures mean kinetic energy. Notice what it does not contain: mass. At a given temperature every gas has the same mean kinetic energy per particle, whatever the particles are.

Example: at 300 K the mean kinetic energy of a particle is ³⁄₂ × 1.38 × 10⁻²³ × 300 = 6.21 × 10⁻²¹ J — the same for helium and for carbon dioxide.

Lighter particles move faster — since the kinetic energies match but the masses do not, ⟨c²⟩ must be larger for lighter particles. This is why hydrogen and helium escape from the atmosphere over geological time while nitrogen and oxygen do not: at the same temperature the lighter molecules are moving several times faster.