(a) Show that, if the temperature is uniform, the pressure of a classical gas in a uniform gravitational field decreases with height according to the barometric formula
P(z)=P(0)exp{−mgz/kT},
where the various symbols have their usual meanings.
(b) Derive the corresponding formula for an adiabatic atmosphere, i.e. the one in which (PVγ), rather than (PV), stays constant. Also study the variation, with height, of the temperature T and the density n of the atmosphere.