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\sect{A divided box with a hole in one side}

A cylinder of length ${L}$ and cross section ${A}$ is divided into
two compartments by a piston. The piston has mass ${M}$ and it is
free to move without friction. Its distance from the left basis of
the cylinder is denoted by ${x}$. In the left side of the piston
there is an ideal Bose gas of ${N_{a}}$ particles with mass
${\mathsf{m}_{a}}$. In the right side of the piston there is an
ideal Bose gas of ${N_{b}}$ particles with mass ${\mathsf{m}_{b}}$.
The temperature of the system is ${T}$. 
Assume that the left gas can be treated within the framework of the Boltzmann approximation. 
Assume that the right gas is in condensation. In items (3-5) consider separately two cases: \\
 
(a) A small hole is drilled in the left wall of the box.

(b) A small hole is drilled in the right wall of the box. \\

The area of the hole is ${\delta A}$.

\begin {itemize}
\item[(1)]
Find the equilibrium position of the piston.

\item[(2)]
What is the frequency of small oscillations of the piston.

\item[(3)]
What is the velocity distribution ${N(v)}$ of the emitted particles?

\item[(4)]
What is the flux (particles per unit time) of the emitted particles?

\item[(5)]
Is the piston going to move? If yes write an expression for its velocity. 

\end {itemize}

In item (3) use normalization that makes sense for the calculation in item (4).
In item (5) assume that the process is quasi-static, 
such that at any moment the system is at equilibrium.
Express your answers using ${L, A, \delta A,  N_{a}, N_{b}, \mathsf{m}_{a},\mathsf{m}_{b}, T, M}$.

\[\int_{0}^{\infty}\frac{xdx}{e^{x}-1}=\frac{\pi^2}{6}\]


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