Statistical Model Isotherm

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. 

The statistical data shown in Table 5.2 were obtained for an isothermal continuously stirred reactor (CSR) with a spatial time of 1.5 h. With these experimental data, we can formulate a relationship between the reactant conversion (y) and the input concentration (x). For the establishment of a statistical model based on a... 

Hill (1949) extended the statistical thermodynamic treatment of the Langmuir isotherm to model localized unimolecular adsorption on a random heterogeneous surface, including the case of an adsorbate such... 

De Pablo, Laso, and Suter studied the behavior of polyethylene above and below the melting point." A variation of the MC technique was used to study polyethylene and to sample its configuration space. The technique is suitable for the study of long chains at high densities. The simulations were carried out in an isobaric—isothermal statistical mechanical ensemble, which allows the calculation of density at a given pressure and temperature. A series of simulations at different temperatures indicated a phase transition. The polyethylene model employed in the simulations crystallizes spontaneously at low temperatures. At temperature higher than the melting point, the simulated melt is described accurately by the model. 

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 ... 

For faujasite-type zeolites and zeolite-A, where several molecules can occupy one cage, a special type of isotherm has been derived, the statistical model isotherm [49]. This isotherm treats each cage in the zeolite as a subsystem. Each cage can contain a fixed maximum number of molecules, and the molecules can interchange between the cages. Within a cage, interaction between the molecules is taken into account, which is not accounted for in the Langmuir description. 

However, a solution in the Laplace domain has been derived by Kucera [30] and Kubin [31]. The solution cannot be transformed back into the time domain, but from that solution, these authors have derived the expressions for the first five statistical moments (see Section 6.4.1). For a linear isotherm, this model has been studied extensively in the literature. The solution of an extension of this model, using a macro-micropore diffusion model with external film mass transfer resistance, has also been discussed [32]. All these studies use the Laplace domain solution and moment analysis. 

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. 

In this paper, we present an exact calculation of the statistical mechanics of a lattice model of hydrocarbon adsorption in the quasi one-dimensional pores of zeolites, based on a matrix method that utilises the Constant Pressure partition. The model is tested on benzene adsorption, where it reproduces experimentally observed steps in isotherms. The model has been extended also to linear alkanes where it reproduces very accurately experimental adsorption isotherms as well as Monte-Carlo simulation results of ethane. 

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. 

Statistical models of the sorption behavior of foods have been applied for predicting the sorption behavior of foods. One of the most well-known and used model is the equation proposed by Brunauer, Emmett, and Teller known as the BET sorption isotherm model. This model is used extensively in food research. Another model extensively used for foods proposed by Guggenheim, Anderson, and de Boer is known as the GAB model or equation. The predicted values by either... 

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... 

Any isotherm that has explicit expression for the spreading pressure can be used in the FastlAS formalism, which we will describe in details below. Langmuir, O Brien and Myers, Sips, and dual Langmuir equations fall to this class (Table 5.3-3). Another isotherm, called the statistical model obtained by Ruthven (Ruthven and Goddard, 1986) also belongs to this class ... 

The equations of motion for a fluid whose internal energy is a function only of density, entropy and their spatial derivatives are able to explain motions through non-isothermal liquid-vapour interfaces. Molecular and statistical models which take the local state of molecules into consideration lead to such an internal energy of the fluid... 

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... 

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... 

The same is true of the classical Myers-Prausnitz theory with activity coefficients introduced in order to account for nonideality of the adsorbed phase and of the general statistical model [Eq. (4.17)] with the cross coefficients retained as parameters. Since the cross coefficients cannot, as yet, be predicted theoretically from the single-component isotherms, this reduces somewhat the predictive value of these models. However, it has been shown that, for the system N2-O2-CO-IOX, the vacancy solution theory with the cross coefficients evaluated from limited binary data provides a good prediction of the ternary equilibrium data. The same approach may be extended to multicomponent systems provided data for all constituent binaries are available. The vacancy solution theory thus provides a practically useful means of data correlation and makes possible the prediction of multicomponent equilibrium behavior from binary data. The potential for the application of classical solution theory or of the statistical models in a similar way has not yet been investigated to the same extent. 

The four possible stereomers of a chiral surfactant with two asymmetric centers within the polar head group have been synthesized and their absolute configuration determined by X-ray diffraction. One of the diastereomers exhibits a chiral discrimination when spread on water interface the monolayer racemic film undergoes a phase transition from a liquid-expanded towards a liquid-condensed phase upon compression, while the pure enantiomers only have a liquid-expanded phase, as revealed by the measured pressure-area isotherms. The transition pressure-composition diagram indicates that heterochiral interactions are favored. Our results are compared to predictions of Andelman and de Gennes based upon a statistical model. 

Figure 3.14 Comparison of generalized thermodynamic. statistical model with experimental results for n-heptane-cyclohexane mixtures adsorbed on a 13X zeolite. The points represent experimental data while the continuous and discontinuous lines are predictions based on single component isothermal data (source Goddard and Ruthven 1984).

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. 

Thermodynamically Consistent Isotherm Models. These models include both the statistical thermodynamic models and the models that can be derived from an assumed equation of state for the adsorbed phase plus the thermodynamics of the adsorbed phase, ie, the Gibbs adsorption isotherm,... 

Statistical Thermodynamic Isotherm Models. These approaches were pioneered by Fowler and Guggenheim (21) and Hill (22). Examples of the appHcation of this approach to modeling of adsorption in microporous adsorbents are given in references 3, 23—27. Excellent reviews have been written (4,28). 

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). 

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