Particle distribution in a microporous material

Equilibria in zeolites and other microporous materials are discussed. We present an independent particles ill a box approach, which leads to a thermodynamic description of internal occupation equilibria of the type ZX(i-1) + ZX(i+1) reversible arrow 2ZX(i), where Z denotes the framework of the material and X the particles that can interchange places. The independent particles in a box an defined by considering a crystal consisting of a finite number of unit cells or boxes each of which can be filled with a specific number of particles. All empty sites in a box have equal probability to be occupied, independent of the number of particles present, as long as sites are available. Each time a particle does fall in a box, the probability for a next one to hit this box is reduced by 1 divided by the number of sites available in an empty box. Hence, as soon as a box is filled, the probability for a particle to hit it becomes zero. The maximum number of particles in the system is equal to the maximum number of sites in a box multiplied by the number of boxes. This allows equilibrium constants and the decrease of entropy as a function of the equivalent fraction of exchanging species to be calculated. We show that the plot of the logarithm of the equilibrium constant versus the equivalent fraction of exchanging species is not linear and that the nonlinearity is caused by the decrease of entropy. On the basis of this observation, we suggest the independent particles in a box to be used as a reference for "ideal behavior" and to serve as a reference for determining activity coefficients. The generalization of the theory leads to the independent particles in boxes with different sites. It is discussed in detail with regard to two nonequivalent sites corresponding to the internal equilibria ZX1(rho 1)X2(rho 2) reversible arrow ZX1(rho+1)X2(rho 2-1) in which XI and X2 are the same species but occupy site 1 and 2, respectively, of a box. We show the solution of this problem and explain the distribution of the particles among the different sites as a function of the average exchange degree.