Statistical models, adsorption

Thus from an adsorption isotherm and its temperature variation, one can calculate either the differential or the integral entropy of adsorption as a function of surface coverage. The former probably has the greater direct physical meaning, but the latter is the quantity usually first obtained in a statistical thermodynamic adsorption model. 

Application of this, or the equivalent statistical models, to actual polymer adsorption processes is further complicated by very imprecise knowledge of the solid surface area which is actually available for polymer adsorption. Surface roughness etc. can certainly be expected to have much more complex effects than on the adsorption of small molecules due to restrictions on... 

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. 

Equation 1 can be used to determine the pore diameter of an MCM-41 sample which exhibits capillary condensation at a certain relative pressure, or to determine the capillary condensation pressure for an MCM-41 sample of a certain pore diameter. To construct model adsorption isotherms for MCM-41, one also needs a description of the monolayer-multilayer formation on the pore walls. This description can be based on the experimental finding that the statistical film thickness in MCM-41 pores of different sizes (especially above 3 nm) is relatively constant for pressures sufficiently lower from those of the capillaiy condensation and can be adequately approximated by the t-curve for a suitable reference silica [29-31], for instance that reported in Ref. 35. In these studies [29-31], the statistical film thickness in MCM-41 pores, tMcM-4i, was calculated according to the following equation [29] ... 

The integral molar quantities are of importance for modelling adsorption systems or in the statistical mechanical treatment of physisorption. For example, they are required for comparing the properties of the adsorbed phase with those of the bulk... 

The last topic to comment is the difference between the optimum compositions in Ru of the Pt-Ru alloys for the oxidation of CO (50 at.% Ru) and methanol (10-20 at.% Ru). Gasteiger et al. " proposed an explanation based on a statistical modelization of the surface of an alloy containing x at.% Ru. The stmctural atomic model proposed by these authors needs a maximum number of 3-fold platinum sites for the adsorption of the CH3OH molecule, and adjacent to 1 Ru atom to adsorb the OH species. This would correspond to a surface with 8-14 at.% Ru depending on the orientation of the low index single crystal planes, 8% for (111), 10% for (100)... 

This study demonstrated that statistical models may be successfully applied to set up QSPRs available for prediction of adsorption enthalpy of an organic specie on one type of activated carbon. But for generalization, the specific influence of the properties of the GAC have also to be taken into account. 

To determine the surface area of dry powders, it is only necessary to record the first part of the adsorption branch, reducing the experimental time significantly (to less than 0.5 h). When increasing the partial pressure of the adsorbate over the sample, a monolayer of adsorptive builds up, while with increasing relative pressure, multilayer adsorption occurs. Brunauer et al. derived a relation from gas-kinetic and statistical models on how this monolayer coverage can be determined from the mentioned experiment, which nowadays is often called BET-isotherm ... 

The difficulty of gas adsorption methods lies in the fact that purely monomolecular layers are never formed. Already before the adsorbent is completely covered, multiple layers build up locally. Brunauer, Emmett and Teller derived a relation between gas pressure and the amount of gas adsorbed at the surface which is known as the BET isothermal line. They used both a gas kinetic and a statistical model. 

Evaluation of the electrocatalytic isotopic reaction of alkenes was facilitated by using Kemball s statistical model to determine the origin of each species and the probability of each step (380,381). The model assumes olefin adsorption and reaction with deuterium or hydrogen to form ethyl radicals. These can revert to ethylene or they can add H or D to give ethane (i = 0, 1,...,4) ... 

Ustinov, E.A., Vashchenko, L.A., and Polyakov, N.S. (2001). Statistical model of equilibrium adsorption of non-ideal mixtures on zeolites. Russ. Chem. Bull., 50, 220-7. 

Two distinct approaches have been used to model precursor state kinetics. (1) A successive site statistical model, introduced by Kisliuk [426] for adsorption and adapted by King [298] for desorption. (2) The chemical reaction kinetics approach, involving rate coefficients and the stationary state approximation, followed by Becker and Hartman [424], Ehrlich [425] and recently developed by Gorte and Schmidt [297] and Cassuto and King [421], It has recently been shown by Schon-hammer [427] and Cassuto and King [421] that the two approaches produce the same kinetic expressions. Variants of these models have... 

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

The combination of classical electrochemical measurements with ex situ transfer experiments into UHV [242], and in situ structure-sensitive studies such as electroreflectance [25], Raman and infrared (IR)-spectroscopies [29, 243], and more recently STM and SXS [39] provided detailed knowledge on energetic, electronic and structural aspects of (ordered) anion adsorption and phase formation. These experimental studies have been complemented by various theoretical approaches (1) quantum model calculations to explore substrate-adsorbate interactions [244-246] (2) computer simulation techniques to analyze the ion and solvent distribution near the interface [247] (3) statistical models [67] and (4) MC simulations [38] to describe phase transitions in anionic adlayers. 

The energy distribution according to MIAST is not based on a statistical model but on the single component adsorption isotherm. With respect to the corrderrsation of adsorptive molecules, the following approximations are irttroduced (Cerofolirri... 

In the past 30 years, great efforts have been expended to develop techniques for predicting the multicomponent adsorption equilibria based on pure component data. However, until now only limited success has been achieved. Several publications provide good reviews of the work in this area [1,2,5]. Generally speaking, these models can be classified into four groups (1) Vacancy solution theory, (2) statistical models, (3) ideal adsorbed solution theory (lAST), (3) Polanyi theory, and (4) various empirical or semiempirical models,... 

The statistical models are based on thermodynamics arguments. But these models are normally specific to adsorbents with well-defined structures such as zeolites [12] or carbon black [13]. There was also a statistical model reported for the multi-component adsorption equilibria of vapors on activated carbon, but it was validated only for a particular system [10]. 

STATISTICAL MECHANICS DERIVATION OF SELECTED THERMODYNAMIC PROPERTIES OF MODEL ADSORPTION PROCESSES... 

With nonpolar sorbates an increase in heat of adsorption with coverage is commonly observed, as illustrated in Figure 4.5 and this is commonly attributed to the effect of intermolecular attraction forces. The statistical model isotherm, however, suggests an alternative explanation. If the effective molecular volume increases with temperature, as it generally does, the isosteric heats... 

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... 

FIGURE 4.22. X- F diagram for adsorption of isobulane-elhyleiie on I3X sieve at 50 C, 20 psi (absolute), showing comparison of predictions from statistical model ideal adsorbed solution... 

In unpublished work the generalized statistical model [Eq. (4.17)] has been successfully applied to the correlation of liquid phase adsorption equilibrium data for Cg aromatics on faujasite zeolites. For these systems the saturation limit corresponds to approximately three molecules/cage, and at equilibrium with the liquid the adsorbent is essentially saturated so that each cage can be assumed to contain three sorbate molecules. This simplifies the model since only the terms corresponding to / + y = 3 in Eq. (4.17) need be retained, and the expression for the separation factor, assuming an ideal binary fluid phase, becomes... 

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