Dubinin-Polanyi theory

Dubinin, Polanyi, and Radushkevich proposed about 1947 a simple but very useful empirical theory allowing one to calculate the amount of gas adsorbed in a microporous sorbent. The theory was based on a pore filling model. Today it is used for both characterization of porous solids and also for engineering purposes. It has been extended by several authors among them predominantly Astakhov (1970). The theory is still the subject of further investigations, mainly by statistical mechanics and computational methods (DFT) [7.1-7.3, 7.48-7.55], 

The adsorbate is considered as a fluid phase in the sense of thermodynamics, cp. Sect 7.4, which is exerted to the external forces of the atoms and molecules located on the surface of a sorbent material. The general condition for thermod)mamic equilibrium of such a phase (a) against a fluid or gaseous sorptive phase (f) is [7.56, 7.57]  

Here i. are the chemical potentials of the adsorbate and the sorptive phase and 0 (for attractive forces) is the mechanical potential per unit of mass of the forces acting near the surface of the sorbent on the admolecules. Ideal gas and liquid approximations [7.17] for p lead via (7.76) to the approximate expression 

This is a somewhat generalized version of the so-called Dubinin-Radushkevich (RD) AI [7.48]. The mass of gas adsorbed in the pores then is 

Here po is the density of the sorptive medium in a reference liquid state which may be chosen as the density of the saturated boiling liquid at the chosen temperature, i. e. po = Ps (T), [7.5, 7.3]. The parameter a in the characteristic curve of the sorbent material (7.78) is the reciprocal of a specific energy the exponent N normally is limited to 2 N 6 and for zeolites and activated carbons often has numerical values about N = 3. Both parameters are characteristic for a sorbent material and the micropore spectrum included in it. Details of practical applications of (7.79) and a variety of generalizations to real gas adsorptives and multicomponent systems can be found in the (still growing) literature in this field [7.53 7.55, 7.58]. 

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Fig. 5 Characteristic curve for benzene on activated carbon at five different temperatures (1 = 20 °C, 2 = 50 °C, 3 = 80 °C, 4 = 110 °C, 5 = 140 °C) showing conformity with the Dubinin-Polanyi theory. From Kiselev [21] with permission...

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. 

The Dubinin-Polanyi theory was extended to binary mixtures by Bering, Serpinsky, and Surinova. It was suggested that if the characteristic curves for the single components are given by Eq. (3.107) then the curve for the mixture should be given by... 

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]... 

At present the Polanyi theory has rather historial meaning. However, the theory of volume filling micropores (TVFM) also called the Dubinin-Radushkevich theory [136] which is generally accepted, though always improved, originates from the Polan3d theory. TVFM has significant importance for the characteristic of most industrial adsorbents which have a well developed porous structure. 

Nonpolar He, Ar, Nj 02,CH4 C02 Langmuir virial Expansion Dubinin-Polanyi BET Integral Equation Ideal Adsorbed Solution Theory... 

The relationship to the Freundlich isotherms is important for two reasons. First the question as to whether the % theory can predict isotherms such as the Freundlich (of which = 1 is a special case), Dubinin-Astakov, Dubinin-Radushkevich and Toth isotherms All but the Toth isotherm will be referred to as the Dubinin-Polanyi (DP) isotherm. Second, the reason for the observation that in most cases P appears to approach 0 as approaches 0. Even though there are cases where P approaches a finite value, thus disproving the universal application of Henry s law , this is not convincing without an explanation as to why it is observed in many cases. 

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. 

The method proposed here for applying Polanyi s theory analytically agrees well with experiments at temperatures not too far above the critical temperature of the adsorbate. In this domain, the Dubinin-Radushkevich... 

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],... 

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 . 

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. 

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