PROBABILITY-DISTRIBUTION IN THE CELL THEORY OF AN INTERACTING LATTICE-GAS - APPLICATION TO XE-129 NMR OF XENON IN ZEOLITES

A cell theory of an interacting lattice gas is derived from the basic principle of statistical mechanics that the equilibrium distribution of N particles among M cells, when both N and M are very large, is given by the most probable distribution, with the constraints that the total number of particles and the average energy remain constant. The theory provides a direct way to calculate the probability distribution of the gas particles among the cells. A gas-liquid transition is observed at low temperatures where the probability distribution develops a bimodal shape indicating the separation of the cells into two populations, with one population having higher particle densities than the other. The derivatives of the Helmholtz free energy per cell with respect to the average number of particles per cell are examined. When the interaction energy between particles within a cell is proportional to the square of the number of particles in the cell, the smallest possible second derivative of the free energy is 4kT/J2, where J is the maximum number of particles that can be accommodated by the cell and T is the absolute temperature. The probability distribution of the interacting lattice gas is applied to the adsorption of xenon in zeolites. In order to describe Xe-129 nuclear magnetic resonance data reported in the literature for xenon trapped in the alpha-cages of the NaA zeolite, it is necessary that, in addition to the attractive interactions between the particles, the repulsive interactions should also be included when the xenon atoms begin to fill the alpha-cages of the zeolite.