\sect{Baruch's  D51.}

Consider a fluid in two compartments connected with a small hole.
Although particles can pass easily through the hole, it is small
enough so that within each compartment the fluid is in
thermodynamic equilibrium. The compartments have pressure,
temperature, volume and particle number $P_1$, $T_1$, $V_1$, $N_1$
and $P_2$, $T_2$, $V_2$, $N_2$, respectively. There is an energy
transfer rate $dE/dt$ and particle transfer rate $dN/dt$ through
the hole.
\begin{itemize}
\item[(a)] Identify the kinetic
coefficients for $dE/dt$ and $dN/dt$ driven by temperature and
chemical potential differences. Rewrite the equations in terms of
$\Delta T=T_1-T_2$ and  $\Delta P=P_1-P_2$ to first order in
$\Delta T$ and  $\Delta P$.
\item[(b)]  If  $\Delta T=0$ one measures  $\epsilon_1= (dE/dt)/(dN/dt)$. One can also
adjust the ratio $\epsilon_2=\Delta P/\Delta T$ so that $dN/dt=0$.
Show the relation
\[\epsilon_2=\frac{1}{T}[\frac{E}{V}+P-\frac{N}{V}\epsilon_1]\]
($E/V$ or $P$ for either compartment).
\item[(c)]  Assume that
the work done during the transfer by the pressure is via reducing
the effective volume to zero within the hole.  Evaluate
$\epsilon_1$ and show that  $\epsilon_2=0$.\\
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%