Show that the average K.E. per molecule does not depends on pressure, volume or nature of gas staring from the expression of pressure of an ideal gas.

Explanation

We know that the pressure exerted by an ideal gas `p=(1)/(3)mbarv^(2)`
i.e. `PV=((1)/(3)mbarv^(2))V` where n = no density of molecules.
but n V = N = total number of molecules in a given volume of a container.
Thus `PV=(1)/(3)Nbarv^(2)`
i.e. `(PV)=((2)/(3)N)xx((1)/(2)mbarv^(2))`
i.e. `(PV)/(N)=(2)/(3)((1)/(2)mbarv^(2))`
i.e. `((1)/(2)mbarv^(2))N=(3)/(2)PV=E`
where `PV = muRT`
and E = internal energy of ideal gas which is purely kinetic.
i.e. `(1)/(2)mbarv^(2)=(3)/(2)((muR)/(N))TpropT.`
`therefore KE prop T` and is independent of pressure, volume and nature of gas.