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\sect{Classical gas in volume-surface phases equilibrium}

An ideal gas composed of point particles with
mass ${\mathsf{m}}$, moves between parallel boards of a capacitor. The
surface of each one of them is ${A}$ and the distance between them
is ${L}$, as described in the figure.

Force ${\vec{f}}$ operates
on the particles, in  vertical direction to the boards, which pushes
the particles to the lower board.  particles can be adsorbed to the
boards. the adsorbed particles move over them freely,
and adsorbed potential ${-E}$ operates on them (when ${E>0}$)in
addition to force ${\vec{f}}$.

The system is in balance, in temperature ${T}$. Moreover, It's
given that the average number of the particles that move between
the boards and are not adsorbed over them is ${N}$, and their
average density is ${\bar{n}}$.

Assume that the gas particles maintain Maxwell-Boltzman
statistics and therefore it's possible to  carry out the calculations in the
classical statistical mechanics frame.

Express all of your answers with
${E,L,\bar{n},T,f=|\vec{f}|,\mathsf{m}}$ and through physical and
mathematical constants only.

\begin {itemize}
\item[(a)]
Calculate ${n\left(x,y,z\right)}$, The density of the particles per
volume unit in some point between the boards. Define the coordinate
system you use.

\item[(b)]
Calculate the ratio ${\frac {\Phi_{+}}{\Phi_{-}}}$ between the flow
that hits the upper board and the flow that hits the lower board.

\item[(c)]
Calculate ${}$ and ${}$ which are the densities of the particles
adsorbed over the upper board and the lower board respectively.
Moreover,calculate the ratio ${\frac{}{}}$.

Guideline: It's possible to make the calculation through the
chemical potentials of the gas between the boards and over them.

\end {itemize}



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