| The lattice gas model is used to formulate the grand partition function for various two-dimensional lattices in terms of a transfer matrix. The one-dimensional system is first examined in detail in order to illustrate how the transfer matrix can be used to formulate the grand partition function as a polynomial in the fugacity, z. The method is then extended to two-dimensional lattices. The computer is used to calculate the grand partition function as a polynomial in z, from which the pressure, P, and the number density, (rho), can also be obtained as polynomials in z. An inversion is then obtained via Lagrange's theorem to give an expression for the pressure as a function of the specific volume, which is the virial equation of state. The pressure and density expansions for an mxn lattice are shown to be identical to those for an mx('(INFIN)) lattice for terms up to z('n-1). Thus the virial expansion is accurate for infinite lattices for terms up to (rho)('n-1). The first eight virial coefficients for square, triangular and union-jack lattices, and the first three coefficients for the hexagonal lattice are calculated. |
