\sect{Pressure of Lenard Jones gas}
A gas of $N$ particles is confined in a box
of volume $V$ at temprature of $T$.
The two-body interaction between the particles is
given by the Lenard Jones expression:
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\[ u(r)\ \ =\ \ \frac {a} {r^{12}} - \frac{b}{r^6} \]
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Note that this interaction is characterized by
a length scale $r_0$ and an energy scale $\epsilon_0$
that correspond to the position and the depth of the potential.
\begin{itemize}
\item[(a)]
Find an expression for the pressure via the Virial theorem,
assuming that the moments $\langle r^n \rangle_T$ are known.
\item[(b)]
Using the Virial expansion, find an explicit expression
for the pressure assuming low temperatures.
\item[(c)]
Using the Virial expansion, find an explicit expression
for the pressure assuming high temperatures.
\item[(d)]
Comparing your answers to items (a) and (c) deduce
explicit expressions for the $n=-6$ and for the $n=-12$ moments.
Express your result in terms of ${ (V, r_0, \epsilon_0, T) }$.
\end{itemize}