\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:

\[ u(r)\ \ =\ \ \frac {a} {r^{12}} - \frac{b}{r^6} \]

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}