Prausnitz theory

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. 

For clarity sake, we rewrite below the basic equations of the Myers and Prausnitz theory... 

Table 5.3-1. The number of equations in the Myers-Prausnitz theory...

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. 

Olivet, E. D., Diffusional Separation Processes Theory, Design and Evaluation. Wiley, New York, 1966. Reid, R. C., Prausnitz, J. M., and Sherwood, T. K., The Properties of Gases and Liquids. McGraw-Hill, New York, 1977. 

Prausnitz and coworkers [91,92] developed a model which accounts for nonideal entropic effects by deriving a partition function based on a lattice model with three categories of interaction sites hydrogen bond donors, hydrogen bond acceptors, and dispersion force contact sites. A different approach was taken by Marchetti et al. [93,94] and others [95-98], who developed a mean field theory... 

Equation (16-36) with y, = 1 provides the basis for the ideal adsorbed-solution theory [Myers and Prausnitz, AIChE /., 11, 121 (1965)]. The spreading pressure for a pure component is determined by integrating Eq. (16-35) for a pure component to obtain... 

Lue L, Prausnitz JM. Structure and thermodynamics of homogeneous-dendritic-polymer solutions computer simulation, integral-equation, and lattice-cluster theory. Macromolecules 1997 30 6650-6657. 

Theories or computer simulations used to calculate the potential of mean force W(r) are typically based on numerous simplifying assumptions and approximations (de Kruif, 1999 Bratko et al., 2002 Prausnitz, 2003 de Kruif and Tuinier, 2005 Home et al., 2007 Jonsson et al., 2007). Therefore they can provide only a qualitative or, at best, semi-quantitative description of the potential of mean force. Such calculations are nevertheless useful because they can serve as a guide for trends in the factors determining the interactions of both biopolymers and colloidal particles. Thus, an increase in the absolute value of the calculated negative depth of W(r) may be attributed to a predominant type of molecular feature favouring aggregation or self-association. To assist with such a theoretical analysis, expressions for some of the mean force potentials will be presented here in the discussion of specific kinds of interactions occurring between pairs of colloidal particles covered by biopolymers in food colloids. 

Over the years, various other theories and models have been proposed for predicting salt effect in vapor-liquid equilibrium, including ones based on hydration, internal pressure, electrostatic interaction, and van der Waals forces. These have been reviewed in detail by Long and McDevit (25), Prausnitz and Targovnik (31), Furter (7), Johnson and Furter (8), and Furter and Cook (I). Although the electrostatic theory as modified for mixed solvents has had limited success, no single theory has yet been able to account for or to predict salt effect on equilibrium vapor composition from pure-component properties alone. 

Several other studies have also been made in an attempt to account theoretically for the phase transition in terms different from those of the Flory-Huggins theory. Otake et al. [55] thus proposed a theoretical model that takes hydrophobic interaction into account in explaining the thermally induced discontinuous volume collapse of hydrogels. In addition, Prausnitz et al. [56] proposed a lattice model, an improvement of which was made to explain the swelling curves of gels consisting of /V,/V -methylenebis(acrylamide) (MBA)-crosslinked copolymers of AAm with [(methacrylamide)propyl]trimethyl-ammonium chloride (MAPTAC) [57],... 

Parrish and Prausnitz apply vdWP theory to natural gases... 

Donohue M.D., Prausnitz J.M., "Perturbed hard chain theory for fluid mixtures Thermodynamic properties for mixtures in natural gas and petroleum technology", AIChEJ. 1975, 24(5), 850. 

When gas solubility data are lacking or are unavailable at the desired temperature, they can be estimated using available models. The method of Prausnitz and Shair (1961), which is based on regular solution theory and thus has the limitations of that theory. The applicability of regular solution theory is covered in detail by Hildebrand et al. (1970). A more recent model, now widely used, is UNIFAC, which is based on structural contributions of the solute and solvent molecular species. This model is described by Fredenslund et al. (1977) and extensive tabulations of equilibrium data, based on UNIFAC, have been published by Hwang et al. (1992) for aqueous systems where the solute concentrations are low and the solutions depart markedly from thermodynamic equilibrium. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

John Prausnitz I d like to make one quick addendum. I want to defend the use of molecular simulations because we have gained insight from them as well. Let me mention one outstanding example. Until about 20 years ago, we believed that you could only condense a phase with attractive forces, and no one ever questioned that myth. Then computer simulations were done in the 1960s by Bemi Alder and his associates. The results showed that even for hard spheres, without any attractive forces, you can get a phase transition. This was never present in the van der Waals theory. I want to emphasize that simulations also add to our conceptual knowledge. 

Dimitrelis, D and Prausnitz, J.M. Comparison of two Hard-Sphere Reference Systems for Perturbation Theories for Mixtures, Fluid Phase Equilibria. Vol. 31. 1986, pp. 1-21. 

Abbott, M. M. Prausnitz, J. M., "Generalized van derWaals Theory A Classical Perspective," Fluid Phase Equilibria, 37, 29 (1987). 

The ideal adsorbed solution theory of Myers and Prausnitz [13] has been used for developing a multicomponent isotherm model based on the single component isotherm at any pore size. The following result is obtained for the multicomponent isotherm for each component studied in a pore of physical width H (=/f -0.3354)... 

This study firstly aims at understanding adsorption properties of two HSZ towards three VOC (methyl ethyl ketone, toluene, and 1,4-dioxane), through single and binary adsorption equilibrium experiments. Secondly, the Ideal Adsorbed Solution Theory (IAST) established by Myers and Prausnitz [10], is applied to predict adsorption behaviour of binary systems on quasi homogeneous adsorbents, regarding the pure component isotherms fitting models [S]. Finally, extension of adsorbed phase to real behaviour is investigated [4]. 

Like the theory of corresponding states on which it is based, Kay s rule provides only approximate values of the quantities it is used to calculate. It works best when used for mixtures of nonpolar compounds whose critical temperatures and pressures are within a factor of two of one another. Reid, Prausnitz, and Poling (see footnote 1) provide more complex but more accurate mixing rules for systems that do not fall into this category. 

For multicomponent adsorption the most commonly used isotherm is the extended Langmuir isotherm (Eq. 18). Another, frequently used approach is the Ideal Adsorption Solution theory (IAS theory), which was developed by Prausnitz [53] and applied to mixtures of gases by, for example, Kaul [54] and Rees [52,55]. 

The M-S approach allows the prediction of the mixture diffusion based on the pure component M-S diffusivities, along with the mixrnre isotherms (i.e., estimated using the ideal adsorbed solution theory [LAST] of Myers and Prausnitz) [68]. Experimentally, the NMR pulse field gradient technique has been used to calculate the mixture diffusion coefficients. 

Kobayashi and co-workers began to use the van der Waals and Platteeuw theory in the 1960s to predict hydrate formation in ternary systems. Parrish and Prausnitz extended the method to prediction of hydrate incipient formation in natural gas systems causing the widespread industrial adoption of the van der Waals and the Platteeuw statistical method. Many academic (e.g.. Holder and co-workers ) and commercial programs (e.g., D.B. Robinson, and Associates ) enabled the gas and oil industry to predict thermodynamic conditions at the incipient formation point, and thereby to prevent hydrate formation in industrial processes. All of the errors in the van der Waals and Platteeuw theory were placed in the solid phase. It may be argued that the theory s unusual success in prediction inhibited motivations for advances in hydrate phase measurements. 

For the prediction of solubility of polar solutes in polar solvents, the regular solution theory, Eq. (17), has been modified to take into account the additional interactions between the solvent and the solute. Some of these modified methods are discussed by Prausnitz et al. (16). 

General Reeerences Abbott, M. M., and H. C. Van Ness, Schaum s Outline of Theory and Problems of Thermodynamics, 2d ed., McGraw-Hill, New York, 1989. Poling, B. E., J. M. Prausnitz, and J. P. O Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2001. Prausnitz, J. M., R. N. Lichtenthaler, and E. G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 3d ed., Prentice-Hall PTR, Upper Saddle River, N.J., 1999. Sandler, S. I., Chemical and Engineering Thermodynamics, 3d ed.. 

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