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.

An approach to adsorption in micropores is developed on the basis of the condensation approximation method. In the case of micropores with half-width less than 0.8 nm, a volume filling of the single micropore and a two-dimensional (2D) condensation on its walls occur at the same critical pressure, pc. From this point of view, the statistical mechanical theories of adsorption on homogeneous surfaces considering lateral interactions may be taken to be a starting point to the description of physical adsorption in micropores. The increased well depth, *, in micropores causes the 2D condensation on their surface to occur at a smaller pc compared with a nonporous surface. As a consequence, the overall adsorption isotherm is determined by the distribution of micropore walls over adsorption energies resulting from a random distribution of micropore widths, d. Proceeding from a model of heterogeneity and the normal distribution of micropore widths, the log-normal distributions of micropore volumes over * and pc are obtained, and an occupied adsorption volume as a function of outer pressures is found. This approach has been successfully applied to seven benzene adsorption isotherms on four species of active carbons. It provides a correct description of equilibrium data in a wide temperature range, leads to reasonable distribution function of * and d, and gives values of an average micropore size that are close to the experimental one.S0743-7463(97)01211-0 CCC: $15.00

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The relationships between the geometric form and saturation pressure are derived on the basis of thermodynamics and molecular models. A comparison of the relationships with the classical Laplace and Kelvin equations shows that there is a profound conceptual diversity between approaches: the classical equations contain the surface tension, γ, as a parameter, whereas developed equations include the difference of the potential energies of a nanoparticle and a semi-infinite planar body with respect to the gas phase, which is equal to the difference of energies of autoadsorption on surfaces of bodies. The latter circumstance affords ground for calculating the change of interest proceeding from well-developed methods of adsorption theory. Molecular models are discussed for simple spherical bodies and applied to the processes of nucleation and growth of drops. In the cases of (i) water and mercury drops or capillaries with diameters of ≈1 µm and (ii) cyclohexane meniscuses on mica cylinders, the classical and new relations predict close results, which are in agreement with experiments; nevertheless, for organic substances, especially at elevated temperatures, noticeable deviations must be expected. The divergence of the approaches makes its appearance in the case of condensation in narrow capillaries with radii ≈1-10 nm when the new equation provides a correct description of experimental facts whereas the Kelvin model predicts unrealistic effects. Considering the growth of surface areas due to the extension or adsorption shows that γ does not take into account all changes associated with alterations of surface areas, and it is this fact that underlies the probable defect in the classical equations.

Changing droplet radii in a liquid-vapor system is due to the phase transition on the droplet surface. As a variation of the internal energy does not depend on the way the change occurs, we can imagine that a gas condenses on a droplet surface in two stages: in the first stage, autoadsorption occurs on the liquid surface, and in the second stage, adsorbed molecules transfer into the volume by diffusion. Assuming that the energetic effects of the diffusion are independent of the surface curvature, one may conclude that if two liquid bodies differ only with respect to their geometry, the difference of enthalpies of condensation on their surfaces, DeltaH(bd), is equal to the variation of energies of autoadsorption. An estimation of DeltaH(bd) for the simple bodies is presented, and the relationship between the saturation pressure and droplet radii is derived. In the range of micrometer dimensions, the new equation and the Kelvin model lead to close results; for nanocapillaries, the Kelvin equation predicts a divergence of hysteresis loops, whereas the new equation adequately describes the observations. The classical model presumes that a surface area, A, affects the free energy, while the new approach is based on the assumption that A is the repository for the internal energy.