Deriving pV = nRT from molecular collisions

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

F = \frac{\Delta p_{\text{total}}}{\Delta t} = \frac{Nm\overline{v_x^2}}{L} \ \Rightarrow\ p = \frac{F}{L^2} = \frac{Nm\overline{v_x^2}}{L^3} = \frac{Nm\overline{v_x^2}}{V}

problems.proof.analysis

Multiplying the momentum per collision ( $2mv_x$ ) by the collision frequency ( $Nv_x/(2L)$ ) gives the average force on the wall. Dividing by the wall area $L^2$ and volume $V=L^3$ gives the pressure in terms of $\overline{v_x^2}$ , the mean square velocity along one direction.