\sect{Pressure of an ideal gas in the atmosphere}

%Baruch's  A19.

An ideal classical gas of ${N}$ particles of mass
${\mathsf{m}}$ is in a container of height ${L}$ which is in a
gravitational field of a constant acceleration ${g}$. The gas is
in uniform temperature ${T}$.

\begin{itemize}

\item [(a)]
Find the dependence ${P\left(h\right)}$ of the pressure on the height ${h}$.

\item [(b)]
Find the partition function and the internal energy. 
Examine the limits ${mgL \ll T}$ and ${mgL \gg T}$. 

\item [(c)]
Find ${P\left(h\right)}$ for an adiabatic atmosphere, i.e.
the atmosphere has been formed by a constant entropy process in
which ${T,\,\mu}$, are not equilibrated, but ${Pn^{-\gamma} =\const}$. 
The equilibrium is maintained within each atmospheric layer. 
Find $T(h)$ and $ n(h)$ at height $h$ in terms of 
the density $n_0$ and the temperature $T_0$ at $h=0$. 

\end{itemize}

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