(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

\[
P(z)=P(0) \exp \{-m g z / k T\}
\]

where the various symbols have their usual meanings. \({ }^{17}\)

(b) Derive the corresponding formula for an adiabatic atmosphere, that is, the one in which \(\left(P V^{\gamma}ight)\), rather than \((P V)\), stays constant. Also study the variation, with height, of the temperature \(T\) and the density \(n\) in such an atmosphere.