n molecules of a gas are enclosed in a cubical vessel of edge length $\ell .$ then pressure exerted by the gas, if each molecule has mass m and rms speed V, is

The momentum imparted per unit time to the wall by molecule will be
$\mathrm{\Delta F}=\mathrm{n\Delta p}$

$$\begin{gathered} \text{ so change in momentum will be }\mathrm{\Delta p}=2{\mathrm{mv}}_{\mathrm{x}} \\ \text{ and molecule will collide form the two surfaces of wall so total time }\mathrm{\Delta t}=\frac{2\mathrm{L}}{{\mathrm{v}}_{\mathrm{x}}} \\ \text{ so number of collision by molecule in unit time will be }\mathrm{n}=\frac{1}{\mathrm{\Delta t}}=\frac{{\mathrm{v}}_{\mathrm{x}}}{2\mathrm{L}} \\ \text{ so }\mathrm{F}=\frac{{\mathrm{mv}}_{\mathrm{x}}^2}{\mathrm{L}} \\ \mathrm{now} \mathrm{for} \mathrm{cube} \mathrm{all} \mathrm{the} \mathrm{direction} \mathrm{are} \mathrm{equivalent},\mathrm{we} \mathrm{have} \\ \mathrm{F}=\frac{{\mathrm{mnv}}^2}{3\ln }\text{ (i.e }\frac{{\mathrm{v}}^2}{\mathrm{n}}={\mathrm{v}}^2\text{ it is the sum of three velocity in }\mathrm{x},\mathrm{y},\mathrm{z}\text{ direction) } \\ \text{ so pressure }\mathrm{p}=\frac{\mathrm{F}}{{\mathrm{l}}^2} \\ \mathrm{p}=\frac{{\mathrm{mnv}}^2}{3{\mathrm{l}}^3} \end{gathered}$$