Dubinin-Radushkevich potential

Pore diameter Average pore diameter Dubinin-Astakhov Dubinin- Radushkevich Potential... 

Semiernpirical Isotherm Models. Some of these models have been shown to have some thermodynamic inconsistencies and should be used with due care. They include models based on the Polanyi adsorption potential (Dubinin-Radushkevich, Dubinin-Astakhov, Radke-Prausnitz, Toth, UNI LAN. and BET). 

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 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 ... 

Since N2 adsorption is done at 77 K and CO2 at 273 or 298 K, the experiments cannot be directly compared, which introduces strong concerns about the similarities and differences among both adsorptives. Thus, a better way to compare the two experiments is to plot the characteristic curves [33—35, 37], These characteristic curves, obtained applying the Dubinin-Radushkevich (DR) equation [47] to the adsorption isotherms, are the plot of the logarithm of the volume of liquid adsorbed versus the square of the adsorption potential corrected for the affinity coefficient ((3) of the adsorptive ((/l//3) = (RTln(/o//)/[3), T being the temperature, / the fugacity, and/ the saturation fugacity). 

On the basis of volume-filling mechanism and thermodynamic considerations, Dubinin and Radushkevich [19] found empirically that the characteristic curves obtained using the Potential Theory for adsorption on many microporous carbons could be linearized using the Dubinin-Radushkevich (DR) equation. 

Dubinin-Radushkevich (D-R) isotherm model is more generally applicable than the Freimdlich isotherm since it is not limited by the homogeneous surface and constant adsorption potential assumption. The D-R equation has the general expression as Equation (11.6) [6] ... 

The best utility of the Dubinin-Radushkevich equation lies in the fact that the temperature dependence of such equation is manifested in the adsorption potential A, defined as in eq. (3.2-30), that is if one plots adsorption data of different temperatures as the logarithm of the amount adsorbed versus the square of adsorption potential, all the data should lie on the same curve, which is known as the characteristic curve. The slope of such curve is the inverse of the square of the characteristic energy E = PEq. 

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 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 . 

The value of jS = 2 corresponds to the standard Dubinin-Radushkevich (DR) potential. Apart from adsorption in micropores, this potential may also be used to describe adsorption at low coverage of a surface [8]. 

One of the best known potential expressions is the semi-empirical Dubinin/Radushkevich-Astakhov (DA) potential (Shapiro and Stenby, 1998) ... 

If n = 2, the DA potential results to the weU-known Dubinin-Radushkevich (DR) potential ... 

Various methods for estimating micropore sizes in activated carbons from a single adsorption isotherm are reviewed. The methods include (i) single parameter estimates of micropore size based upon Dubinin s theory of the volunae filling of micropores (ii) estimates of micropore size distributions based upon a generalised Dubinin-Radushkevich equation, and (iii) the use of intermolecular potentials in model nrucropares. 

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) ] ... 

The similarity in the shapes of the characteristic curves, shown in Fig. 9.2, and the positive side of a Gaussian curve, led Dubinin and Radushkevich to postulate that the fraction of the adsorption volume V occupied by liquid adsorbate at various values of adsorption potentials E can be expressed as a Gaussian function. Thus,... 

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... 

Thus, the Dubinin and Radushkevich equation states the distribution of the adsorption space W according to the differential molar work of adsorption. A t)q)ical plot of the adsorption potential versus the reduced pressure is shown in Figure 4.2-1 for T = 77, 273 and 473 K. For low reduced pressure, the adsorption potential is high, while it is low for high reduced pressure. The latter means that less molar work is required for adsorption via micropore filling when the gas is approaching the vapour pressure. 

Dubinin and Radushkevich suggested that for microporous active carbons (or porous materials), volume of adsorption space can be expressed as a Gaussian function of the corresponding adsorption potential W=W (s), so that the characteristic curve for microporous active carbons can be written as... 

The DR isotherm, initially proposed by Dubinin and Radushkevich for the description of adsorption on porous adsorbents, was found by Hobson to describe adsorption in submonolayer range on non-porous surfaces too. This empirical discovery was a challenge to theoreticians since the DR isotherm was expressed in terms of the Polanyi potential, which is expected not to apply to adsorption in the submonolayer range. 

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... 

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