Potential theory isotherms

Potential Theory Isotherms for Single and Mixed Gases... 

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). 

The surfaces defined by (j> = constant, where is the scalar field, and resembling contour lines on a topographic map, may be called isotimic surfaces. In potential theory they are referred to as equipotentials in heat conduction they are isothermals, etc. They form a family of non-intersecting surfaces. The gradient of the scalar field measures the rapidity with which the field changes as a function of position. The most rapid change occurs along a family of lines normal to the isotimic surfaces. 

The three isotherms discussed, BET, (H-J based on Gibbs equation) and Polanyi s potential theory involve fundamentally different approaches to the problem. All have been developed for gas-solid systems and none is satisfactory in all cases. Many workers have attempted to improve these and have succeeded for particular systems. Adsorption from gas mixtures may often be represented by a modified form of the single adsorbate equation. The Langmuir equation, for example, has been applied to a mixture of n" components 11). 

The differences observed in the adsorption isotherm are also qualitatively and quantitatively significant for the entropy. It has been recently shown that the isotherm of adsorption of an ideal adsorbate on a heterogeneous surface can be appreciably improved by taking into account the exact form of S from Eq. (7) instead of the approximate one arising from F-H theory [22], Results for the coverage dependence of the chemical potential (adsorption isotherm) and entropy per site are shown in figs. 1-2 for various fc-mer s sizes and interaction energies [attractive (w<0) as well as repulsive (w>0)]. 

Which theory is suitable for a certain application The adsorption theory of Henry is applicable at low pressure. This, however, is natural since it can be viewed as the first term in a series of the adsorption function. A widely used adsorption isotherm equation is the BET equation. It usually fits experimental results for 0.05 < P/P0 < 0.35. For very small pressures the fit is not perfect due to the heterogeneity. For higher pressures the potential theory is more suitable at least for flat, homogeneous adsorbents. It often applies to P/Po values from 0.1 to 0.8. Practically for P/Po > 0.35 adsorption is often dominated by the porosity of the material. A more detailed description of adsorption is obtained by computer simulations [382],... 

Dubinin was the pioneer of the concept of micropore filling. His approach was based on the early potential theory of Polanyi, in which the physisorption isotherm data were expressed in the form of a temperature-invariant characteristic curve . 

Fractal geometry has been used to describe the structure of porous solid and adsorption on heterogeneous solid surface [6-8]. The surface fractal dimension D was calculated from their nitrogen isotherms using both the fractal isotherm equations derived from the FHH theory. The Frenkel-Halsey-Hill (FHH) adsorption isotherm applies the Polanyi adsorption potential theory and is expressed as ... 

Single-component adsorption equilibria on activated carbon of the n-alkanes Q-C4 and of the odorant tert-butyl mercaptan were measured at the operating conditions expected in a large-scale facility for adsorbed natural gas (ANG) storage. The experimental data were correlated successfully with the Adsorption Potential theory and collapsed into a single temperature-independent characteristic curve. The obtained isotherm model should prove to be very useful for predicting the adsorption capacity of an ANG storage tank and to size and optimize the operation of a carbon-based filter for ANG applications. 

The main purpose of the present work is twofold (i) to report an extensive set of singlecomponent adsorption isotherm data of the more common natural gas components on activated carbon, and (ii) to present a means of extrapolating the measured data to higher alkanes in order to be able to span the whole composition of a typical natural gas. There is experimental evidence that for the n-alkanes series this can be done using the Adsorption Potential theory, as demonstrated recently by Holland et al. on Westvaco BAX-1100 carbon, and assumed previously by us. - ... 

Figure 1 displays the experimental characteristic curve obtained and the p values employed to generate it. The existence of very little scatter in the data demonstrates that the isotherms of the various adsorbates were successfully correlated as a single temperature-independent characteristic curve. This fact corroborates the applicability of the Potential theory to the carbon under study. 

The potential theory has been widely used in the last decades Such a success is mainly due to the predictive character of this method Once v f, f, and (3 are known, it is possible to calculate any isotherm if the reference characteristic curve of the adsorbent is known. A lot of research teams have tried to improve the initial theory by proposing different correlations for the determination of v> [15,16,17] The calculation of Ps and thus f, for supercritical fluids is another problem which has been much debated [18,19,20] Besides, it is possible to relate the reference characteristic curve to the porous structure of the adsorbent so that the theoretical treatments are very ofen used as characterization methods The way the mathematical expression of the reference characteristic curve can be related to the micropore size distribution function of the adsorbent has also been widely discussed in the literature [4,5,6] In this paper we consider two different methods for the calculation of v, and f ... 

The thermodynamic description of the adsorption isotherm of a supercritical gas was shown in the above subsection. The thermodynamic approach cannot explain a more physical meaning of Wl. The molecular potential theory treats the interaction between an admolecule and the pore surface as a function of the distance, as mentioned before. If we use the model of the two parallel semi-infinite slabs of graphite as the micropore walls of activated carbon, the additive form gr(z) of the 9-3 potentials from both graphite slabs is obtained [43] ... 

Chen, S.G. and Yang, R.T. (1994). Theoretical basis for the potential theory adsorption isotherms. The Dubinin-Radushkevich and Dubinin-Astakhov equations. Langmuir, 10, 4244-9. 

The recombination of Martynov s theory, which is based on a Boltzmann distribution and a Lennard-Jones potential, the isotherm for a "smeared adsorption" can simplified to... 

However, commercial adsorbents do not have a smooth surface but rather are highly porous solids with a very irregular and rugged inner surface. This fact is taken into account by the potential theory which forms the basis of the isotherm equation introduced by Dubinin. 

The Frenkel-Halsey-Hill (FHH) isotherm was originally developed to describe the growth of thick films and wetting phenomena on a flat surface and was later extended to studying adsorption on fractal surfaces [3, 55]. In contrast to BET theory, FHH theory applies to long-range adsorbate-absorbent interactions and its approach is closely related to the so-called potential theory of adsorption of Eucken and Polanyi (see Ref. [35]). 

The last three chapters deal with the fundamental and empirical approaches of adsorption isotherm for pure components. They provide the foundation for the investigation of adsorption systems. Most, if not all, adsorption systems usually involve more than one component, and therefore adsorption equilibria involving competition between molecules of different type is needed for the understanding of the system as well as for the design purposes. In this chapter, we will discuss adsorption equilibria for multicomponent system, and we start with the simplest theory for describing multicomponent equilibria, the extended Langmuir isotherm equation. This is then followed by a very popularly used IAS theory. Since this theory is based on the solution thermodynamics, it is independent of the actual model of adsorption. Various versions of the IAS theory are presented, starting with the Myers and Prausnitz theory, followed by the LeVan and Vermeulen approach for binary systems, and then other versions, such as the Fast IAS theory which is developed to speed up the computation. Other multicomponent equilibria theories, such as the Real Adsorption Solution Theory (RAST), the Nitta et al. s theory, the potential theory, etc. are also discussed in this chapter. 

The extension of the potential theory was studied by Bering et al (1963), Doong and Yang (1988) and Mehta and Dannes (1985) to multicomponent systems. We shall present below a brief account of a potential theory put forward by Doong and Yang (1988). The approach is simple in concept, and it results in analytical solution for the multicomponent adsorption isotherm. The basic assumption of their model is that there is no lateral interaction between molecules of different types and pure component isotherm data are described by the DA equation. With this assumption, the parameters of the DA equation (Wq, Eq, n) of each species are not affected by the presence of the other species, but the volume available for each species is reduced. This means that the volume available for the species i is ... 

The analysis of the cases 4, 5 and 7 are utilised in the study of adsorption isotherm of a nonpolar adsorbate in a microporous solid having slit-shaped micropores, such as activated carbon. To complete the potential theory analysis, we now deal with solids having cylindrical pores of molecular dimension. 

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