\sect{Hard spheres in a box}
% Baruch's A17.
Consider a gas of $N$ hard spheres in a box. A single sphere
occupies a volume $\omega$ , while its center of mass can explore
a volume $V$ (if the rest of the space is empty). There are no
other interactions between the spheres except for the constraints
of hard core exclusion.
\begin{itemize}
\item[(a)] Calculate the partition function for this gas. You will need
to integrate over the spatial part of phase space. Use the
approximation
\[\int_{|x_i-x_j|>a} d^3x_1d^3x_2...d^3x_N\approx
V(V-\omega)(V-2\omega)...(V-(N-1)\omega)\]
and explain under which physical circumstances this approximation is valid.
\item[(b)] Calculate the entropy. Use the approximation
\[(V-a\omega)(V-(N-a)\omega)\approx (V-N\omega/2)^2\]
where $a$ is an integer. When is this approximation valid?
\item[(c)] Find the equation of state.
\item[(d)] Show that the isothermal compressibility
$\kappa_T=-\frac{1}{V}\left(\frac{\partial V}{\partial
P}\right)_T$ is always positive.\\
\end{itemize}
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