Derive the kinetic theory equation pV=Nm/3(crms2) for an ideal gas.

Assume a point molecule in a container with dimensions l_x by l_y by l_z travelling at velocity c₁ given by c₁²=u₁²+v₁²+w₁²​​​​​​ where u, v, w are the x, y, and z velocity components.

>each impact of the molecule with the l_y by l_z container face reverses the x component of the velocity, so the change in momentum dp is given by dp=(-mu₁)-mu₁=-2mu₁

>the time between successive impacts t is given by t=2l_x/u₁

>using Newton's second law, the force F₁ on a molecule is given by F₁=-2mu₁/(2l_x/u₁)=-mu₁²/l_x

>using Newton's third law, the force F_s on the face surface is given by F_s=mu₁²/l_x

>the pressure p₁ exerted by the molecule on the impact face is given by p₁=F_s/area of impact face=mu₁²/l_xl_yl_z=mu₁²/V where V is the container volume

>the total pressure p exerted by N molecules on the impact face is given by p=mu₁²/V + mu₂²/V + mu₃²/V +...+ mu_n²/V=m/V(u₁² + u₂² + u₃²+...+ u_n²)=Nm/V(u²) where u is the mean u component speed

>the motion of molecules is random meaning there is no preferred direction so p=Nm/3V(u²+v²+w²)=Nm/3V(c_rms²) where c_rms is the root mean square speed given by c_rms²=1/N(c₁²+c₂²​​​​​​​+c₃² +…+c_n²)