Isotherms Dubinin-Radushkevich

Fe Oj, FejO, nanocomposites, oxidation polymerization, Langmuir adsorption isotherm, Freundlich adsorption isotherm, Dubinin-Radushkevich adsorption isotherm, Tempkin adsorption isotherm, pseudo-first-order kinetic. Pseudo-second-order kinetic, removal efficiency, adsorption capacity... 

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). 

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... 

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. 

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 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). 

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. 

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. 

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. 

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.

Nitrogen isotherms on some activated carbons and the corresponding as-plots and Dubinin-Radushkevich (DR) plots are shown in Figure 9.11. The isotherms on Carbosieve and the carbon cloth JF005 are of well-defined Type I in the IUPAC classification, but the isotherms on the carbon cloth JF517 and the superactive carbon AX21 are evidently more complex. 

Over the past 30 years many organic molecules of different size, shape and polarity have been used as molecular probes. A high proportion of the experimental isotherms on porous carbons have been analysed by application of the Dubinin-Radushkevich (DR) equation or, in a few cases, by the Dubinin-Astakhov (DA) equation. So far, the more sophisticated Dubinin-Stoeckli (DS) treatment (Stoeckli, 1993) has been applied by very few other investigators. 

Various attempts were made by Dubinin and his co-workers to apply the fractional volume filling principle and thereby obtain a characteristic curve for the correlation of a series of physisorption isotherms on a zeolite (Dubinin, 1975). As was noted in Chapter 4, the original Dubinin-Radushkevich (DR) equation (i.e. Equation (4.39)) was found to be inadequate and in its place the more general Dubinin-Astakhov (DA) equation was applied (i.e. Equation (4.45)). 

Various procedures have been used to evaluate the micropore capacity from the experimental isotherm data (e.g. the Dubinin-Radushkevich plot), but in practice these are all empirical methods. It should be kept in mind that no theoretical significance can be deduced from the fact that a particular equation gives a reasonably good fit over a certain range of an isotherm determined at only one temperature. In our view, a safer approach is to plot the amount adsorbed against standard data determined on a non-porous reference material (i.e. to construct a comparison plot or Os-plot)-... 

Two kinetic (CMS-Kl, CMS-K2) and one equilibrium (CMS-R) carbon molecular sieves, used originally for separation of gaseous mixtures, were investigated. The adsorption Nj isotherms at 77 K, in static conditions where obtained. In the case of the two first sieves the adsorption was so low that the calculation of parameters characterizing the texture was impossible. The volume of nitrogen adsorbed on the sieve CMS-R is remarkable From obtained results parameters characterizing micropore structure according to Dubinin -Radushkevich equation and Horvath - Kawazoe method were determined. 

From obtained isotherm were determined parameters characterizing micropore structure according to Dubinin - Radushkevich equation [6] and Horvath - Kawazoe method [7] which are presented below ... 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

Table 2 Parameters of Dubinin-Radushkevich Isotherm for Phe Adsorption on CA-3 Sample at Different Steam Percentages.

For a carbonaceous material, the higher the steam percentage (in volume) in the gas stream, the lower its Phe adsorption capacity. The isotherms shape suggests that the presence of moisture in the gas stream seems to avoid the multilayer adsorption. The best model to fit the Phe adsorption capacities on carbonaceous materials is the Dubinin-Radushkevich model. 

Several methods have been proposed for the characterisation of the Micropore Size Distribution (MPSD) that take into account the energetic heterogeneity of solid surfaces [9,10]. The Dubinin-Radushkevich (DR) and Dubinin-Astakhov (DA) equations have been used to describe the adsorption process on structurally heterogeneous solids [11,12]. From these equations, the adsorption isotherm can be expressed as follows ... 

The applicability of the Dubinin-Radushkevich equation to the very low pressure region of isotherms of various microporous solids. 

Over the years there has been a lot of debate concerning the applicability of the Dubinin-Radushkevich equation on the very low pressure region of isotherms of microporous solids. The experimental downward deviation of the DR-plot for very low pressures is generally attributed to kinetic barriers, especially in the case of nitrogen adsorption at 77K. This low pressure region of isotherms of various adsorbents can be fitted with the Langmuir equation. Hence it is shown that the downward deviation is not due to experimental factors but reflects a different adsorption mechanism. 

All the nitrogen and argon isotherms could be fitted with the Dubinin-Radushkevich equation between, typically, 10 and p/po 10. At higher pressures capillary condensation causes the isotherm to diverge. At lower pressures the typical deviation described in Fig 1 was observed. The carbon dioxide isotherm showed only a minor deviation from the DR-plot (one point) and was hence excluded from this study. 

At first, equilibrium time was increased in order to try to eliminate the low-pressure deviation. This proved to be impossible. The upward shift of the isotherm predicted in Fig 2 was only observed when the initial equilibrium times were very short. Once a reasonable equilibrium time was reached (in the order of several hours per point) the isotherm stayed identical. Even for equilibrium times as high as 12 hours per point the isotherm did not shift towards the Dubinin-Radushkevich line. From this, it was concluded that the deviation for low partial pressures was not an experimental artifact. In fact, this has already been suggested by several authors [6,7], but little attention has been given to these results. Partly because their observations and conclusions were mainly directed towards the fractal characterisation of porous materials. 

The results of the fitting are shown in Figs 3-6. Figure 3 shows a Dubinin-Radushkevich plot for an activated carbon with an additional fitting of the deviation by Eq 2. Figure 4 shows the same three series (experimental isotherm, DR-plot and Eq 2) but on a linear scale. Figure 5 shows only the low pressure part of Fig 4. Finally, Fig 6 is similar to Fig 5 but for an argon isotherm on a zeolite. 

Sorption of nitrogen Nitrogen isotherms were measured using a ASAP 2010 (Micromeritics) at —196 °C. Before the experiment the samples were heated at 120 °C and then outgassed overnight at this temperature under a vacuum of 10 Torr to constant pressure. The isotherms were used to calculate the surface area and pore (DFT [10]) and characteristic enei of adsorption, (Eg) (Dubinin-Radushkevich method [11]). 

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