Thermodynamics, adsorption models

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. 

At the level of approximation invoked by the simple geometric model, the mathematical problem becomes one of inverting Eq. (14), a linear Fredholm integral equation of the first kind, to obtain the PSD. The kernel r(P, e) represents the thermodynamic adsorption model, r(P) is the experimental function, and the pore size distribution /(//) is the unknown function. The usual method of determining /(H) is to solve Eq. (14) numerically via discretization into a system of linear equations. 

In the special case of Langmuir isotherm we have P = 0, and then =1.) The Bntler eqnation is nsed by many authors as a starting point for development of thermodynamic adsorption models. It shonld be kept in mind that the specific form of the expressions for n, and which are to be snbstituted in Equation 5.16, is not arbitrary, but must correspond to the same thermodynamic model (to the same expression for F,— in our case Equation 5.11). At last, snbstitnting Equation 5.16 into Equation 5.9 we derive the Frumkin adsorption isotherm in Table 5.2, where K is defined by Equation 5.3. 

While a thermodynamic treatment can be developed entirely in terms of f(P,T), to apply adsorption models, it is highly desirable to know on a per square centimeter basis rather than a per gram basis or, alternatively, to know B, the fraction of surface covered. In both the physical chemistry and the applied chemistry of the solid-gas interface, the specific surface area is thus of extreme importance. 

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). 

An analysis of the thermodynamic stability models of various nickel minerals and solution species indicates that nickel ferrite is the solid species that will most likely precipitate in soils (Sadiq and Enfield 1984a). Experiments on 21 mineral soils supported its formation in soil suspensions following nickel adsorption (Sadiq and Enfield 1984b). The formation of nickel aluminate, phosphate, or silicate was not significant. Ni and Ni(OHX are major components of the soil solution in alkaline soils. In acid soils, the predominant solution species will probably be NE, NiS04°, and NiHP04° (Sadiq and Enfield 1984a). 

The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

Multilayer adsorption models have been used by Asada [147,148] to account for the zero-order desorption kinetics. The two layers are equilibrated. Desorption goes from the rarefied phase only. This model has been generalized [148] for an arbitrary number of layers. The filling of the upper layer was studied with allowance for the three neighboring molecules being located in the lower one. The desorption frequency factor (CM) was regarded as being independent of the layer number. The theory has been correlated with experiment for the Xe/CO/W system [149]. Analysis of the two-layer model has been continued in Ref. [150], to see how the ratios of the adspecies flows from the rarefied phases of the first and the second layers vary if the frequency factors for the adspecies of the individual layers differ from one another. In the thermodynamic equilibrium conditions these flows were found to be the same at different ratios of the above factors. 

Adsorption model of HPLC retention mechanism allows clear definition of the column void volume as the total volume of the liquid phase in the column, but this model requires the use of the surface-specific retention and the correlation of the HPLC retention with the thermodynamic (and thus energetic) parameters, which is not well-developed. This model requires the selection of the standard state of given chromatographic system and relation of all parameters to that state. 

Model unit operation to optimize operating -ariables (c.g. loading, bed length, flow rate and required plate count) ba.sed on thermodynamics (adsorption isotherm, separation factor, the column saturation capacity). 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

First theoretical interpretations of Me UPD by Rogers [3.7, 3.12], Nicholson [3.209, 3.210], and Schmidt [3.45] were based on an idealized adsorption model already developed by Herzfeld [3.211]. Later, Schmidt [3.54] used Guggenheim s interphase concept" [3.212, 3.213] to describe the thermodynamics of Me UPD processes. Schmidt, Lorenz, Staikov et al. [3.48, 3.57, 3.89-3.94, 3.100, 3.214, 3.215] and Schultze et al. [3.116-3.120, 3.216] used classical concepts to explain the kinetics of Me UPD and UPD-OPD transition processes including charge transfer, Meloiy bulk diffusion, and nucleation and growth phenomena. First and higher order phase transitions, which can participate in 2D Meads phase formation processes, were discussed controversially by various authors [3.36, 3.83, 3.84, 3.92-3.94, 3.98, 3.101, 3.110-3.114, 3.117-3.120, 3.217-3.225]. 

The two-dimensional gas model assumes no mutual interaction of the adsorbed molecules. It is believed that the adsorbent creates a constant (across the surface) adsorption potential. Thus, in the framework of statistical thermodynamics, the model describes adsorption as the transition of a gas with three translational degrees of freedom into an adsorbed state with one vibrational and two translational degrees. Assuming ideal behavior and using molar quantities, one obtains the standard entropy in the adsorbed phase as the sum of the translational and vibrational entropies from Eqs. 5.28 and 5.29 ... 

A certain distribution of the adsorbate between the gas and solid phases exists in a gas-solid system. The distribution depends on temperature and pressure and is customarily expressed as the moles or weight adsorbed per unit weight of solid either as a function of pressure at a constant temperature (isotherm) or as a function of temperature at a constant pressure (isobar). Thermodynamic treatments can be used to develop adsorption models to describe the distribution of the species between the gas and solid phases. Some of the more commonly used adsorption models will be discussed in the following sections. 

To check the performance characteristics of the program, we have performed a series of calculations for the thermodynamic adsorption properties of n-alkanes on a model microporous graphite surface. 

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