Generalized adsorption isotherms

A similar approach, in this case, based on the DR equation (Equation 4.16), was proposed by Cazorla-Amoros et al. [10] for the determination of the micro-PSD. Similarly, compared to other methods of determining PSDs (i.e., density functional theory [DFT], see below), this method assumes that the experimental isotherms can be described by the general adsorption isotherm (GAI) [58] ... 

Reinmuth [465] arrived at a solution for the Langmuir isotherm in the form of a series, involving the beta function. Levich et al. [362] had an approximate solution for a general adsorption isotherm. 

In general, adsorption isotherms obtained with this and other zeolitic substrates are of Brunauer s Type I, the simple hyperbolic form also known as the Langmuir isotherm. Consequently, the asymptotic limit of adsorption is used instead of the value of Vm normally derived from the BET evaluation of specific surface area. It is, of course, not possible to define exact monolayer or multilayer adsorption in these three-dimensional interconnected pore systems. 

Stoeckli (1981), McEnaney and Mays (1991), Hutson and Yang (1997) and others (see Rudzinski and Everett, 1992) have attempted to provide a theoretical basis for the DR and DA equations in terms of an integral transform or a generalized adsorption isotherm, which may be expressed in the form of Equation (4.52). However, in practice the DR and DA equations are usually applied empirically and consequently the derived quantities (micropore volume, characteristic energy and structural constant) are not always easy to interpret. 

Of a slightly different nature is the temperature congruence test, which virtually is a test for ideality. Very generally, adsorption isotherms can be written as some function of x = x. 6 =... 

Wang, C.-H. Hwang, B.J. A general adsorption isotherm considering multi-layer adsorption and heterogeneity of adsorbent. Chem. Eng. Sci. 2000, 55, 4311-4321. 

Application of the general adsorption isotherm to experimental data requires the determination of four parameters, namely, r, Fajh, ad> d B d- The value of fAm is often apparent from the experimental data. If the size factor r is also known, equation (8.6.18) is rearranged so that A d and B d can be determined from a linear plot. 

The prediction of multicomponent equilibria based on the information derived from the analysis of single component adsorption data is an important issue particularly in the domain of liquid chromatography. To solve the general adsorption isotherm, Equation (27.2), Quinones et al. [156] have proposed an extension of the Jovanovic-Freundlich isotherm for each component of the mixture as local adsorption isotherms. They tested the model with experimental data on the system 2-phenylethanol and 3-phenylpropanol mixtures adsorbed on silica. The experimental data was published elsewhere [157]. The local isotherm employed to solve Equation (27.2) includes lateral interactions, which means a step forward with respect to, that is, Langmuir equation. The results obtained account better for competitive data. One drawback of the model concerns the computational time needed to invert Equation (27.2) nevertheless the authors proposed a method to minimize it. The success of this model compared to other resides in that it takes into account the two main sources of nonideal behavior surface heterogeneity and adsorbate-adsorbate interactions. The authors pointed out that there is some degree of thermodynamic inconsistency in this and other models based on similar -assumptions. These inconsistencies could arise from the simplihcations included in their derivation and the main one is related to the monolayer capacity of each component [156]. 

This integral equation represents a general adsorption isotherm of a multicomponent adsorption system involving heterogeneous surfaces [37]. 

Equations (116) and (118) are special cases for mixed gas adsorption. The more general adsorption isotherm for a patchwise heterogeneous surface may be written as ... 

An alternative approach that may be applied to determine AH (Mupu) is similar to the one presented above for the enthalpy of adsorption of (see equation 9) and it is based on the Gibbs-Helmholtz formula and the general adsorption isotherm presented in equation 15. Thus by experimental determination of pairs of values of E and T at which the coverage is constant, 0 = const, and... 

We now generalize the model treated in section 2.10, but this time the polymers and the ligands are in the liquid phase. Again, the simplest way to obtain the generalized adsorption isotherm as well as other thermodynamic quantities of the system is to view the adsorption process as a chemical reaction, exactly as in the previous section, i.e.,... 

Further application of the proposed approach to predict different sets of adsorption experimental data of methane in BPL carbon, reported in the existing literature, showed that the approach overpredicts some sets of experimental data of methane in BPL carbon such as those obtained by Gusev et al. [22]. From this observation, it was inferred that such overprediction was due to finite pore coimectivity that leads to occurrence of regions in the BPL carbon that are inaccessible to methane, even though the smallest pore size of the BPL carbon is larger than the methane molecule. Accordingly, we proceeded to introduce the fraction of accessible carbon volume into the generalized adsorption isotherm, as earlier proposed by Davies and Seaton [23]. It was found that the current approach predicts correctly the sets of adsorption experimental data of methane in BPL carbon at different temperatures with a constant fraction of accessible volume. 

For slit-shaped pores, the PSD can be estimated by the generalized adsorption isotherm (GAI) ... 

FIG. 13 The generalized adsorption isotherms of CELj on activated carbon. 

Figure 12-32. Generalized adsorption isotherms for heavy, intermediate, and light components showing effect of pressure swing on net adsorbate loading. From Haun et al. (1990)...

The shapes of the charge-potential and the capacity-potential curves follow from the Gibbs adsorption isotherm immediately. The shape of the surface-excess-potential curve is more difficult to derive. This will be the subject to be discussed next and in order to do that we shall consider the general adsorption isotherm... 

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