Pressure from molecules hitting the walls.

• Model: a gas is a very large number of molecules (of order Avogadro's number) in random motion. At ordinary conditions their spacing is ten or more times their size (2 Å), so between encounters they move in straight lines by Newton's first law.
• An encounter where two molecules come close enough for their forces to act is a collision. Molecules collide with each other and with the walls, and every collision is taken as elastic: total kinetic energy is conserved, and momentum is conserved as always.
• Take a cube of side l with axes along its edges. A molecule with velocity (v_x, v_y, v_z) hits the wall parallel to the yz-plane, area A = l². It bounces back as (−v_x, v_y, v_z): only the x-component flips.
• The molecule's momentum changes by −mv_x − mv_x = −2mv_x, so it hands the wall a momentum 2mv_x.
• In time Δt only molecules within v_xΔt of the wall can reach it, i.e. those in a volume Av_xΔt. On average half of them move towards the wall, so ½nAv_xΔt hit it, where n is the number density.
• Momentum delivered in Δt: Q = (2mv_x)(½nAv_xΔt). Pressure is force per unit area, Q/(AΔt) = nmv_x².
• Molecules have a spread of velocities, so each speed group adds its share and P = nm⟨v_x²⟩, with ⟨v_x²⟩ the average of v_x².
• No direction is special (the gas is isotropic), so ⟨v_x²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩ = ⅓⟨v²⟩. Hence P = ⅓nm⟨v²⟩.
• The cube shape does not matter: any small flat patch of any wall gives the same steps, and A and Δt drop out. By Pascal's law the pressure is the same throughout a gas in equilibrium.
• Collisions between molecules were ignored. In a steady state, any molecule knocked out of a velocity group is replaced by another knocked into it, so as long as collisions are brief compared with the time between them the result stands.

Teaching this idea? Put the simulation on your own page or LMS with this code.