\sect{Gas in gravitation confined between adhesive plates}

{\em Based on Exam 2004/2012.} \\

A classical ideal gas that consists of $N$ mass $\mass$ particles 
is confined between two horizontal plates that have each area~$\mathsf{A}$, 
while the vertical distance between them is~$L$.
The gravitational force is $f$ oriented towards the lower plate.
In the calculation below fix the center of the box 
as the reference point of the potential.  
 
\Dn

The particles can be adsorbed by the plates. The adsorption energy is $-\epsilon$.
The adsorbed particles can move along the plates freely forming a two dimensional classical gas.
The system is in thermal equilibrium, the temperature is~$T$.   

\begin{enumerate}

\item
Calculate the one particle partition function $Z(\beta,\mathsf{A},L,f)$ of the whole system. \\
Tip: express the answer using $\sinh$ and $\cosh$ functions. 

\item
Find the ratio $N_A/N_V$, where $N_A$ and $N_V$ are the number 
of adsorbed and non-adsorbed particles.

\item
What is the value of this ratio at high temperatures. \\
Express the result using the thermal wavelength~$\lambda_T$.

\item
Find an expression for $F_V$ in the formula $dW = (N_V F_V + N_A F_A) dL $. \\
Tip: the expression is quite simple (a single term).

\item
Find a high temperature approximation for~$F_V$. \\
Tip: it is possible to guess the result without any computation.

\item
Find a zero temperature approximation for~$dW$. \\
Tip: it is possible to guess the result without any computation.

\end{enumerate}