An ideal gas of molar mass `M` is contained in a tall vertical cylindrical vessel whose base area is `S` and height `h`. The temperatuer of the gas is `T`, its pressure on the bottom bass is `p_0`. Assuming the temperature and the free-fall acceleration `g` to be independent of the height, find the mass of gas in the vessel.

Explanation

From the Barametric formula, we have
`p = p_0 e^(-Mg h//RT)`
and from gas law `p = (p M)/(RT)`
So, at constant temperature from these two Eqs.
`rho = (M p_0)/(RT) e^(-Mg h//RT) = rho_0 e^(-Mg h//Rt)` ….(1)
eq.(1) shows that density varies with height in the same manner as pressure. Let us consider the mass element of the gas contained in the column.
`dm = rho (S dh) = (M p_0)/(RT) e^(-Mg h//Rt) S dh`
Hence the sought mass,
`m = (M p_0 S)/(RT) int_0^h e^(-Mg h//RT) dh = (p_0 S)/(g) (1 - e^(-Mg h//RT))`.