
\sect{Pressure of hard spheres}

%Baruch's  A22

Consider a one-dimensional classical gas of ${N}$ particles in a
length ${L}$ at temperature ${T}$. The particles have mass $m$ and
interact via a 2-body "hard sphere" interaction (${x_{i}}$ is the
position of the $i$-th particle):
\begin{eqnarray}
V(x_{i}- x_{j}) &=& \infty  \qquad \qquad    |x_{i}-x_{j}|<a \nonumber\\
&=& 0 \qquad \qquad \,\,\, |x_{i}- x_{j}|>a \nonumber
\end{eqnarray}

\begin{itemize}
\item [(a)]
Evaluate the exact free energy F(T,L,N).
\item [(b)]
Find the equation of state and identify the first virial
coefficient; compare with its direct definition.
\item [(c)]
Show that the energy is ${E=Nk_{B}T/2}$. Why is there no effect of
the interactions on ${E}$ ?\\
\end{itemize}

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