Modeling the concentration dependence of diffusion in zeolites. I. Analytical theory for benzene in Na-Y

We have developed an analytical expression for the diffusion coefficient of benzene in Na-Y at finite loadings in terms of fundamental rate coefficients. Our theory assumes that benzene molecules jump among SII and W sites, located near Na+ ions in 6-rings and in 12-ring windows, respectively. We assume that instantaneous occupancies in different supercages are identical, a mean field approximation yielding Dθ=16kθaθ2 where aθ≅11 Å is the mean intercage jump length and 1/kθ is the mean supercage residence time. We show that kθ=κ⋅k1⋅P1, where P1 is the probability of occupying a W site, k1 is the total rate of leaving a W site, and κ is the transmission coefficient for cage-to-cage motion. We assume κ=12 for all loadings, and derive analytical formulas for the T and θ dependencies of k1 and P1, assuming that SII and W site occupancies are either 0 or 1 and that benzenes do not otherwise interact. Exact expressions for P1 in the canonical and grand canonical ensembles are related for finite systems with a new correspondence rule. For θ<2/3, the SII→W→SII process contributes no loading dependence to kθ, while the SII→W→W process gives an increasing loading dependence of 1/(2−3θ). For θ>23, kθ initially increases due to enhanced W population, then decreases due to blocking of target W sites. In the article that follows this one we show that our theory agrees quantitatively with simulation, and agrees qualitatively with experiment for low to moderate loadings.