Dubinin/Radushkevich

Not all of the isotherm models discussed in the following are rigorous in the sense of being thermodynamically consistent. For example, specific deficiencies in the Freundhch, Sips, Dubinin-Radushkevich, Toth, and vacancy solution models have been identified (14). 

The most popular equations used to desci ibe the shape of a charac-tei istic cui ve or a correlation cui ve are the two-parameter Dubinin-Radushkevich (DR) equation... 

Lichen biomass from Parmelina and Cladonia genera have resulted good biosorbents of Pb(II), Cr(III), and Ni(II) ions. The Langmuir, Freundlich, and Dubinin-Radushkevich (D-R) models... 

EOS, LRC (loading ratio correlation), Dubinin-Radushkevich (D-R), and Dubinin-Astakhov (D-A) are more suitable. 

Dual nickel, 9 820—821 Dual-pressure processes, in nitric acid production, 17 175, 177, 179 Dual-solvent fractional extraction, 10 760 Dual Ziegler catalysts, for LLDPE production, 20 191 Dubinin-Radushkevich adsorption isotherm, 1 626, 627 Dubnium (Db), l 492t Ductile (nodular) iron, 14 522 Ductile brittle transition temperature (DBTT), 13 487 Ductile cast iron, 22 518—519 Ductile fracture, as failure mechanism, 26 983 Ductile iron... 

The porosity of the samples is characterised by N2 adsorption at 77 K, and CO2 adsorption at 273 K, using an automated volumetric system (Quantachrome Autosorb-6). From N2 adsorption the DR (Dubinin-Radushkevich) equation is applied for calculating the total micropore volumes, and the BET... 

Porous texture characterization of all the samples was performed by physical adsorption of N2 at 77K. and CO2 at 273K, using an automatic adsorption system (Autosorb-6, Quantachrome). The micropore volume, Vpp (N2), was determined by application of Dubinin-Radushkevich equation to the N2 adsorption isotherm at 77K up to P/Po< 0.1. The volume of narrow micropores, Vnpp (DR,C02>, (mean pore size lower than 0.7 nm) was calculated from CO2 adsorption at 273 K. 

No current theory is capable of providing a general mathematical description of micropore fiUirig and caution should be exercised in the interpretation of values derived from simple equations. Apart from the empirical methods described above for the assessment of the micropore volume, semi-empirical methods exist for the determination of the pore size distributions for micropores. Common approaches are the Dubinin-Radushkevich method, the Dubinin-Astakhov analysis and the Horvath-Kawazoe equation [79]. 

The fresh and spent catalysts were characterized with the physisorption/chemisorption instrument Sorptometer 1900 (Carlo Erba instruments) in order to detect loss of surface area and pore volume. The specific surface area was calculated based on Dubinin-Radushkevich equation. Furthermore thermogravimetric analysis (TGA) of the fresh and used catalysts were performed with a Mettler Toledo TGA/SDTA 851e instrument in synthetic air. The mean particle size and the metal dispersion was measured with a Malvern 2600 particle size analyzer and Autochem 2910 apparatus (by a CO chemisorption technique), respectively. 

This equation is different from the Wheeler equation. The first term on the right-hand side is identical and is the stoichiometric time t, but the second term includes the Langmuir coefficient K explicitly and in R. Thus no link with the Wheeler equation can be found. In addition this equation is valid solely with the Langmuir isotherm. This is a serious limitation because it has been recognized that Dubinin-Radushkevich (DR) approach is very useful. No analytical solution exists for the particular case of DR equation. A solution to this problem is to solve the system of equations by numerical methods. 

Comparison Between the Cohen-Kisarov and Dubinin-Radushkevich Equations. In a plot of log q vs. e2 the experimental points for one adsorption isotherm on zeolite frequently do not give a straight line, which would verify the Dubinin-Radushkevich equation. In this case, two distinct lines of different slopes are found (4). 

Let us assume that an experimental isotherm is perfectly described by the Cohen-Kisarov equation. When plotting the experimental points with the previous coordinates, three different cases may occur (1) if cmlA < 2, a case which was not yet found ((4) and Table I), the curve exhibits a constant convex curvature towards the ordinate axis (2) if cm /A > 2, the curve exhibits two distinct inflection points (Figure 3) where the experimental curve may easily be confused with the tangent to the inflection point, thus explaining the previous observations (3) if cm /A decreases to a value of 2, these inflection points are unified to give a large linear section, and the Dubinin-Radushkevich equation behaves as a limiting case of the Cohen-Kisarov equation. 

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... 

The fractional pore filling of the micropores of radius r at a given pressure P is given by the Dubinin-Radushkevich (DR) isotherm... 

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). 

Here, AG is Gibbs free energy. For carbon materials being comprised of slit-shaped pores, the Dubinin-Radushkevich (D-R) equation is given as... 

Other -more complicated- models to evaluate the microporous volume exist. The Dubinin-Radushkevich model46,47,48,49 is based on thermodynamical considerations concerning the process of micropore filling. Full discussion of this model is beyond the scope of this book. The reader is referred to the standard work of Gregg and Sing50 on adsorption for a detailed treatment. 

Useful information about micropore structures can be derived from nitrogen or argon isotherm data in terms of the C-constant (BET), t or as-plots and the Dubinin-Radushkevich models. 

By combining Equations 4.12 and 4.15, and for plotting purposes, the well-known Dubinin-Radushkevich equation (DR) is obtained ... 

Stoeckli HF. Generalization of Dubinin-Radushkevich equation for filling of heterogeneous micropore systems./. Colloid Interface Sci., 1977 59(1) 184-185. 

At this point, it is feasible to correlate the liquid-phase adsorption equilibrium single component data, with the help of isotherm equations developed for gas-phase adsorption, since, in principle, it is feasible to extend these isotherms to liquid-phase adsorption by the simple replacement of adsorbate pressure by concentration [92], These equations are the Langmuir, Freundlich, Sips, Toth, and Dubinin-Radushkevich equations [91-93], Nevertheless, the Langmuir and Freudlich equations are the most extensively applied to correlate liquid-phase adsorption data. [2,87],... 

The micropore volume is defined as the pore volume of the pores < 2 nm. Microporous volumes calculated from the application of the Dubinin-Radushkevich equation to the N2 adsorption isotherms at 77 K. The mean pore size of each sample obtained from N2 adsorption was determined by applying Dubinin-Radushkevich equation. The hydrogen sorption isotherms were measured with the High Speed Gas Sorption Analyser NOVA 1200 at 77 K in the pressure range 0-0.1 MPa. 

Figure 4. Volume capacity of hydrogen storage for carbon sorbents vs. micropore volume (determined on Dubinin-Radushkevich method) at pressure 0.1 MPa and 77 K -experimental data (Table 1), a continuous line - the linear approximation obtained by authors - experimental data (Table 2), a dashed line - the linear approximation given in [10].

Physisorption on microporous carbons can be described with the Dubinin-Radushkevich equation [13]... 

As a results of the experiments, we obtained hydrogen sorption isotherms for different carbon materials and empirical coefficients for the Dubinin-Radushkevich equation (5), presented in Table 3. 

TABLE 3. The empirical coefficients of Dubinin-Radushkevich equation for hydrogen sorption on the carbon materials... 

The influence of temperature can be seen on Figs. 8-9. The storage capability is increasing for lower temperatures. Figure 9 compares the behaviour of the adsorption isotherms at different temperature levels for two of the more promising samples steam activated Busofit-M8 and wood-based carbon WAC 3-00 . The shape of the isotherms in the two cases is dissimilar. The isotherms for the 77 and 153 K exhibit a classical type 1 isotherm shape indicating a microporous material. The isotherms at room temperature exhibit a much less pronounced curvature (more like type II isotherm). As is seen from plots (Fig. 9) experimental data fit the calculated adsorption values (Dubinin-Radushkevich equation) with an error sufficient for practical purposes. 

Figure 9. Hydrogen adsorption isotherms for active carbon fiber Busofit-M8 (a), wood-based cardon WAC 3-00 (b) and different temperatures (1 - 77, 2 -153, 3 - 193, 4 - 293 K) experimental data - points, calculated data (Dubinin-Radushkevich equation) - lines.

Rearrangement of Equation (4.41) gives the Dubinin-Radushkevich, DR, equation in its usual form... 

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