Releasing air from a compressed air tank?

Question

A tank of compressed air of volume 1.0 m3 is pressurized to 50.0 atm at T = 273 K. A valve is opened and air is released until the pressure in the tank is 22.0 atm. How many air molecules were released?

Answer 1

Air is mostly diatonmic and has

gamma = Cp/Cv = 7/5

Assuming adiabatic decompression the formula is

PV^gamma = const

that is

T ~ P^(2/7)

and number of molecules is

N = PV/kT

ΔN = PV/kT =

V/k(Po/To - P/T) =

VPo/(kTo) (1 - (P/Po)^5/7)