
\sect{Lattice gas}

%Baruch's  C08

Lattice gas model:
Consider $N$ classical particles of mass $m$ where each
particle is located on a unit cell of a simple cubic lattice with a lattice constant $a$. Each unit cell can contain either 0 or 1 particles, providing an "excluded volume" type interaction. The number of unit cells is M, i.e. the volume is $V=Ma^3$. Therefore $0<N<M$ and the density is $0<n<1/a^3$. There is no constraint on the momentum of each particle.
\begin{itemize} \item[(a)] Evaluate the grand partition function
and the density $n(\mu, T)$ where $\mu$ is the chemical potential and $T$ is the temperature.
\item[(b)] Find the pressure $P$ in terms of $T$ and $n$.
Identify the limit $n\rightarrow 0$ and explain what happens in the limit $n\rightarrow 1/a^3$.
\item[(c)] This model does not show a first order transition
as in a full lattice gas model. What ingredient is missing here?
\end{itemize}

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