Ideal Gas Law Derivation

The Ideal Gas Law can be derived for point particles in a box with an isotropic velocity distribution that only undergo elastic collisions with the walls. For N such particles of mass m, with temperature T and pressure P, in a box of cross sectional area A and length l such that the volume V is Al:

• The time between collisions for a particle with a wall of area A is 2l / v_⊥ where v_⊥ is the magnitude of the perpendicular velocity component.
• The corresponding collision impulse is 2mv_⊥.
• The corresponding average force on the wall is mv_⊥² / l.
• The corresponding average pressure on the wall is mv_⊥² / V.
• P = Nmv_⊥² / V where v_⊥² is the average square perpendicular velocity component.
• v² = 3v_⊥² where v² is the average square velocity.
• PV = Nmv² / 3
• T ∝ mv² / 2
• PV = NkT for some constant k.