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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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