Simple Statistical Model Isotherm

In the original version of this model an attractive term based on Lennard-Jones potential constants was included in order to account for intermolecular attraction between adsorbed molecules. However, it was found that the fit of the experimental isotherm was not significantly improved by the inclusion of these factors so they were omitted in later studies. This observation suggests that, at least for nonpolar sorbates, the adsorption equilibrium behavior is governed by the attractive potential of the framework, which determines the Henry constant, and the repulsive interaction between molecules due to their finite size. Attractive forces between adsorbed molecules appear to be of only secondary importance. 

FIGURE 4.5. Variation of isosteric heat of sorption with coverage, calculated from Eq. (3.101), for light paraffins in 5A zeolite. Points are experimental data of Schirmcr el for butanc-5A. (From ref. 10.) 

The basic assumptions of this model may be regarded as a reasonable approximation for nonpolar sorbates at modest concentration levels. The model cannot be expected to apply in the high-concentration region approaching the saturation limit or for highly polar sorbates such as NHj or H2O which are almost certainly localized. The model has, however, been found to provide an adequate representation of the behavior of quite strongly quadrupolar sorbates such as CO2 on several zeolites. At lower temperatures adsorption may be expected to become more localized leading to deviations from the model. 

FIGURE 4.6. Experimental isotherms for benzene in 13X zeolite crystals at 458 and 513 K. Points arc experimental, curves are calculated from Eq. (3.104) with o//3 5.0, K 8.8 moie-cules/cage Torr at 458 K, and K = 1.25 molecule/cage Torr at 513 K.,[Reproduced from ref. 16 by permission of the publishers, Butierworth Co (Publishers) Ltd. .] 

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The extension of the simple statistical model to adsorption of a binary mixture is given by Eq. (3.102) and further extension to multicomponent systems follows naturally. " The parameters of the model (the Henry con-stant and effective molecular volume for each component) are derived from the single-component isotherms so that an a priori prediction of the mixture... 

Comparisons between the binary isotherm predictions derived from the varous theoretical approaches have been presented by Danner and Choi for C2H6-C2H4-I3X sieve, by KauF for mixtures of Oj, CO, CH4, CjH, etc., on activated carbon, and by Sorial, Granville, and Daly for O2-N2-5A sieve (see Section 11.3). When the molecular volumes of both components are similar, there is little difference between the predictions of the ideal adsorbed solution theory and the simple statistical model as is to be expected from Eqs. (4.17)-(4.26). Both approaches generally give good predictions for sorption of mixtures of saturated hydrocarbons and other nonpolar species. However, the... 

Adsorption isotherms obtained from the model have been shown to agree very closely with the predictions of recently published statistical theories (9,13). While there can be no doubt that the more sophisticated, statistical models provide more information on the nature of the adsorption process and the structure of the adsorbed film, because of its simple form, the macroscopic model can offer a powerful tool for the analysis, interpretation and utilization of adsorption data. 

Consider a non-reactive system consisting of a binary liquid alloy A-B and an oxide substrate such as AI1O3 at constant temperature. A simple statistical thermodynamic model has been developed (Li et al. 1989) to predict the contact angle and the work of adhesion isotherms, 0(XB) and Wa(XB), from the known values of contact angles... 

Under electrochemical conditions and T, P = constant, adsorption isotherms can be derived using standard statistical considerations to calculate the Gibbs energy of the adsorbate in the interphase and the equilibrium condition for the electrochemical potentials of the adsorbed species i in the electrolyte and in the adsorbed state (eq. (8.15) in Section 8.2). A model for the statistical considerations consists of a 2D lattice of arbitrary geometry with Ns adsorption sites per unit area. In the case of a 1/1 adsorption, each adsorbed particle can occupy only one adsorption site so that the maximal number of adsorbed particles per unit area in the compact monolayer is determined by A ax = Ng. Then, this model corresponds to the simple Ising model. The number of adsorbed particles, A ads< and the number of unoccupied adsorption sites, No, per unit area are given by... 

Simple statistical thermodynamic models of adsorption [14] suggest that the equilibrium isotherm should be written as the ratio of two polynomials of the... 

The DFT method [15, 77-79] is derived from statistical thermodynamics and offers an efficient means of computing model isotherms for simple pore geometries. The accuracy of the DFT model isotherms rivals that of the isotherms obtained from molecular simulation, but the computational time required by DFT is typically about 1% of the CPU time needed to complete GCMC or GEMC isotherm calculations for a comparable system. The DFT method retains its computational advantage over molecular simulation only for pore shapes of low dimensionality, such as slits, spheres, or... 

There are several other derivations of the Langmuir adsorption isotherm from statistical mechanics and thermodynamics. Although the model is physically unrealistic for describing the adsorption of gases on real surfaces, its successes, just like the success of other adsorption isotherms also based on different simple adsorption models, is due to the relative insensitivity of macroscopic adsorption measurements to the atomic details of the adsorption process. Thus the adsorption isotherm... 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

An approximate analysis of polymer adsorption as a set of sequential reactions leads to a simple equation for the adsorption isotherm expressed in terms of three parameters. Comparison of the model with recently published statistical theories reveals remarkable agreement in both the general shape of the isotherms and the predicted effects of molecular weight. The problems of applying such models to experimental data are discussed. 

The presence of adsorption hysteresis is the special feature of all adsorbents with a mesopore structure. The adsorption and desorption isotherms differ appreciably from one another and form a closed hysteresis loop. According to the lUPAC classification four main types of hysteresis loops can be distinguished HI, H2, H3 and H4 (ref. l). Experimental adsorption and desorption isotherms in the hysteresis region provide information for calculating the structural characteristics of porous materials-porosity, surface area and pore size distribution. Traditional methods for such calculations are based on the assumption of an unrelated system of pores of simple form, as a rule, cylindrical capillaries. The calculations are based on either the adsorption or the desorption isotherm, ignoring the existence of hysteresis in the adsorption process. This leads to two different pore size distributions. The question of which of these is to be preferred has been the subject of unending discussion. In this report a statistical theory of capillary hysteresis phenomena in porous media has been developed. The analysis is based on percolation theory and pore space networks models, which are widely used for the modeling of such processes by many authors (refs. 2-10). The new percolation methods for porous structure parameters computation are also proposed. 

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