Adsorption Dubinin-Polanyi theory

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. 

Dubinin and coworkers, during the course of their extensive studies on activated carbons, have developed the so-called theory of volume filling of micropores. Based on numerous experimental data, Dubinin and collaborators have added a second postulate to the Polanyi theory, which complements it. For an identical degree of filling of the volume of adsorption space, the ratio of adsorption potentials for any two vapors is constant ... 

The Dubinin adsorption isotherm equation is a good tool for the measurement of the micropore volume. This isotherm can be deduced with the help of Dubinin s theory of volume filling, and Polanyi s adsorption potential [11,26], The Dubinin adsorption isotherm equation has the following form [11]... 

The theory has been developed mainly by Dubinin and coworkers over a period of time to describe adsorption of gases and vapors by microporus solids in general and active carbons in particular. Dubinin defined the molar work of adsorption as the change in the Gibbs free energy rather than as adsorption potential (Polanyi). Accordingly, the molar work of adsorption A is given by... 

Correlation of the experimental isotherms by a generalized one is encouraged by the possibility that if the generalized isotherm for a given adsorbate is known, the adsorption values at any temperature and pressure can be predicted. Such a representative generalized isotherm is the Dubinin-Polanyi characteristic curve. According to the potential theory, the plot of adsorbed volume versus adsorption potential is temperature independent and, hence, characteristic. The adsorbed volume is determined by [55]... 

An analytical method for applying Polanyi s theory at temperatures near the critical temperature of the adsorbate is described. The procedure involves the Cohen-Kisarov equation for the characteristic curve as well as extrapolated values from the physical properties of the liquid. This method was adequate for adsorption on various molecular sieves. The range of temperature, where this method is valid, is discussed. The Dubinin-Rad/ush-kevich equation was a limiting case of the Cohen-Kisarov s equation. From the value of the integral molar entropy of adsorption, the adsorbed phase appears to have less freedom than the compressed phase of same density. 

Peculiarities of Adsorption in Microporous Carbons (the Polanyi Potential Theory Dubinin and Related)... 

The temperature invariance of the adsorption potential (fundamental postulate of Polanyi s theory) has been widely proved, especially, by the extensive work led by Dubinin [31-34],... 

The Dubinin-Radushkevieh (DR) equation is usually applied to describe the physical adsorption of organic vapors on microporous adsorbents. It is based on the micropore volume-filling theory and the Polanyi concept of adsorption potential. The DR equation can be expressed as... 

However, Dubinin and co-workers do not accept the concept of monolayer formation in micropores and propose determining the microporous volume, Fq, on the basis of the thermodynamic theory of Polanyi adsorption. However, one can observe that the monolayer volume, Vm, when expressed in liquid nitrogen volume per unit mass, is very close to the Dubinin volume, Vo. The proportionality of the BET monolayer volume, Vm, and the so-called micropore volume, Va, (Vo 11 Vm) has been observed for many materials, as shown in different studies [2, 3]. This means that both variables are correlated, so determining one is equivalent to the determining the other. The discussion on the physicochemical meaning of these parameters may be interesting from a theoretical point of view but as far as practical characterization of porous materials is concerned, both methods can often be considered as equivalent. 

In the Dubiiiin-Radushkevitch (DR) equation [115], an adsorption model derived from a concept of Dubinin [20] based on Polanyi potential theory, the fluid volume V adsorbed in micropores at pressure P is represented empirically as... 

The Dubinin-Radushkevich (DR) equation was originally devised as an empirical expression of the Polanyi adsorption potential theory, and due to its simplicity it has been widely used to correlate adsorption data in many microporous sohds despite its failure in giving the correct Henry constant at extremely low pressures. This equation is based on the premise that adsorption in micropores follows a mechanism of pore filhng rather than the molecular layering and capillary condensation as proposed for mesoporous sofids. It has the form ... 

Now let us overview the theoretical adsorption models for characterization of the pore structures according to the pore size range. For physical adsorption of the gas molecules on such microporous sohds as activated carbons and zeolites, Dubinin and Radushkevich developed an empirical equation, which describes the volume filling process in the micropoies. Their theory incorporates earlier work by Polanyi in regard to the adsorption potential ad defined as... 

In Chapter 2, we discussed the fundamentals of adsorption equilibria for pure component, and in Chapter 3 we presented various empirical equations, practical for the calculation of adsorption kinetics and adsorber design, the BET theory and its varieties for the description of multilayer adsorption used as the yardstick for the surface area determination, and the capillary condensation for the pore size distribution determination. Here, we present another important adsorption mechanism applicable for microporous solids only, called micropore filling. In this class of solids, micropore walls are in proximity to each other, providing an enhanced adsorption potential within the micropores. This strong potential is due to the dispersive forces. Theories based on this force include that of Polanyi and particularly that of Dubinin, who coined the term micropore filling. This Dubinin theory forms the basis for many equations which are currently used for the description of equilibria in microporous solids. 

The potential theory of adsorption was introduced by Polanyi in 1914. Dubinin [48,49] and Stoeckli et al. [50] improved the theory and termed it the theory of volume filling of micropores (TVFM). This theory has been widely used in correlating the effect of temperature on the adsorption isotherms of pure gases. The modern formulationof TVFM is the Dubinin-Astakhov (DA) equation, which is expressed as... 

Based on the Polanyi potential theory, different approaches to describe the adsorption behavior of a purely microporous material (isotherm type I, Figure 21.25) have been undertaken by Dubinin and Stockli in collaboration with different other scientists. The simplest relationship that can be considered the base for all other variants is the Dubinin-Radushkevich equation [58] ... 

The potential theory of adsorption first introduced in 1914 by Polanyi" " and later modified by Dubinin for adsorption on microporous adsorbents is still regarded as fundamentally sound and accepted as correct and better than all the other theories. This longevity of the theory is due to its essentially thermodynamic character and lack of insistence on a detailed physical picture. 

The Polanyi potential theory successfully represents the temperature dependence of adsorption. It is also the only theory that gives quantitative description of physical adsorption on strongly heterogeneous surfaces, such as those of active carbons and oxide gels. However, the significance of the theory had been limited for a long time because it did not provide an analytical expression for the adsorption isotherm. This problem was solved by Dubinin and coworkers. ... 

Adsorption theory of the volume filing of micropores (TVFM theory) has been proposed by Dubinin and Radushkievich [136], but this approach has originated from the potential theory of adsorption introduced by Eucken [112] and Polanyi [113,114]. 

The DR isotherm remained known to a relatively small group of researchers until Hobson called attention to its applicability to non porous surfaces. In a letter to the Journal of Chemical Physics, Hobson, not without wonder, observed that the appearance of the Dubinin-Radushkevich equation in the present context (submonolayer adsorption of nitrogen on Pyrex) is surprising for two reasons. First, it is a particular equation within the Polanyi potential theory, which is a theory of condensation and might not be expected to apply to physical adsorption at very low coverage. Second, most of the adsorbents to which the Dubinin-Radushkevich equation have been applied have been porous, whereas our conclusion suggests that Pyrex is non-porous for nitrogen. Thus, unless and until a basic derivation for this equation is provided, it can only be considered as a useful empirical relation . 

An alternative approach to lAST/RAST which combines a thermodynamic treatment with a specific consideration of fluid-solid interactions is the potential theory of adsorption (Polanyi (1914)-Dubinin (1950)), especially in the form designed for multi-component mixtures pioneered by Shapiro and Stenby (1998). 

This model for mixed adsorption (Grant and Manes 1966) is based upon the idea of equipotential energies among the components of the adsorbed mixture and is thus related to the Polanyi potential theory discussed in Section 3.3.5. As previously recorded, Dubinin and Radushkevich (1947) postulated a direct relation between the affinity coefficient Pi of a component i and the molar volume Vmt of the saturated pure liquid. The equipotential energy concept for two components is thus (eiiPi) = (ey/ft). Hence, by use of equation (3.18) for each component... 

The adsorption and desorption isotherms of water vapor are drawn at 25°C for dealuminated HY zeolites upon framework Si/Al ratio. The isotherms are compared to that of the parent NaY zeolite. The isotherm changes in shape from the type I to the type IV with an hysteresis loop changing from the type H4 to the type H2, as increases the Si/Al ratio. The POLANYI-DUBININ theory is used to determine the micropore volume accessible to water. It decreases with increasing Si/Al ratios, down to zero at a Si/Al ratio of 35. Such a result is accounted by the adsorption on the hydrophilic centers which are the cations (H ) associated with the structural aluminium ions, each cation being coordinated by 8H2O. 

Adsorption of hydrocarbons or nitrogen reflects the same trend as did water adsorption, with respect to the biporous character of dealuminated samples (structural micropores and secondary mesopores) (refs. 2, 4-6). The non polar character of n-hexane leads to use such a molecule for determining the micropore volume by the POLANYI-DUBININ theory. The study of the pore size distributions from adsorption isotherms and mercury porosimetric measurements characterizes two pore diameters of the secondary pore network, the one between 3 nm and 4 nm and the second to about 20 nm (refs. 7, 8). 

The POLANYI-DUBININ adsorption potential theory is used to characterize the micropore network of zeolites (ref. 10). An isotherm at a given temperature T (expressed in volume adsorbed per activated zeolite mass unit, W, as a function of the relative pressure p/Pq) is treated in the DUBININ-RADUSHKEVICH model (ref. 11) (denoted D-R) in the linear form log W = f[(Tlog Po/p) ] ... 

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