Derive the kinetic theory equation pV=Nm/3(crms2) for an ideal gas.
Assume a point molecule in a container with dimensions lx by ly by lz 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 ly by lz 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=2lx/u1
>using Newton's second law, the force F1 on a molecule is given by F1=-2mu1/(2lx/u1)=-mu12/lx
>using Newton's third law, the force Fs on the face surface is given by Fs=mu12/lx
>the pressure p1 exerted by the molecule on the impact face is given by p1=Fs/area of impact face=mu12/lxlylz=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 +...+ mun2/V=m/V(u12 + u22 + u32+...+ un2)=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(crms2) where crms is the root mean square speed given by crms2=1/N(c12+c22+c32 +…+cn2)
