Chapter 8.2, Problem 23P

The Ising model can be used to simulate other systems besides ferromagnets; examples include antiferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of $-u_0$ to the energy for each pair of neighboring sites that arc both occupied.

(a) Write down a formula for the grand partition function for this system, as a function of $u_0$, T, and $\mu$.

(b) Rearrange your formula to show that it is identical, up to a multiplicative factor that does not depend on the state of the system, to the ordinary partition function for an Ising ferromagnet in the presence of an external magnetic field B, provided that you make the replacements $u_0\to 4\epsilon$ and $\mu \to 2{\mu }_{\text{B}}B-8\epsilon$. (Note that $\mu$ is the chemical potential of the gas while ${\mu }_{\text{B}}$ is the magnetic moment of a dipole in the magnet.)

(c) Discuss the implications. Which states of the magnet correspond to low-density states of the lattice gas? Which states of the magnet correspond to high-density states in which the gas has condensed into a liquid? What shape does this model predict for the liquid-gas phase boundary in the $P-T$ plane?