Micropore volume filling. A condensation approximation approach as a foundation to the Dubinin-Astakhov equation

A volume filling of a single micropore and a two-dimensional (2D) condensation on its walls occur at the same critical pressure, and the local adsorption behavior may be modeled by the condensation approximation. This phenomenon underlies the approach to the micropore volume filling based on the condensation approximation (VFCA) and the Dubinin-Astakhov (DA) equation. The DA equation follows from the VFCA approach, being an approximate form to the general relationship. The physical meanings of the exponent, n, and the characteristic energy, E, with respect to the surface heterogeneity, are given. As a whole, n is determined only by the standard deviation of the micropore widths: the less the heterogeneity, the larger n, approaching to infinity for the homogeneous carbon. The characteristic energy depends on the average micropore sizes and its standard deviation. In the case of the DA equation with n = 2 or n = 3, standard deviations are equal to 0.4915 or 0.3493, respectively, and E depends only upon an average micropore size or upon a related adsorption potential. For the individual micropore, E determines the critical pressure of a 2D condensation. In the case of water adsorption on active carbons, the base heterogeneity due to variation in the pore widths, as perceived by water molecules, is negligible. Hence, water adsorption may be considered to be an extension of a 2D-condensation into practically homogeneous micropore volumes. It is shown that empirical relationships for calculating of average micropore sizes and an applicability of the DA equation to the adsorption on nonporous surfaces may be also explained in the framework of the VFCA approach.