A very tall vertical cylinder contains a gas at a temperature true `T`. Assuming the gravitational field to be uniform, find the mean value of the potential energy of the gas molecules. Does this value depend on whether the gas consists of one kind of molecules or of serveral kinds ?

Explanation

`lt U gt = (int_0^oo mgz e^(-mgz//kT) dz)/(int_0^oo e^(-mgz//kT dz)) = kT (int_0^oo x e^(-x) dx)/(int_0^oo e^-x dx) = kT (Gamma (2))/(Gamma(1)) = kT`
When there many kinds of molecules. This formula holds for each kind and the average energy.
`lt U gt = (sum f_i kT)/(sum f_i) = kT`
Where `f_i alpha` fractional concentration of each kind at the ground level.