Dubinin/Radushkevich isotherm equation

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)). 

Micropores in the lignocellulosic wastes and resulting chars were analysed from CO2 adsorption isotherms by applying the Dubinin-Radushkevich (DR) equation (14). DR plots... 

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). 

In a study achieved by Memon et al. [16] the sorption of carbofuran and methyl parathion on treated and untreated chestnut shells has been studied using high performance liquid chromatography. In this study, the maximum sorption of methyl parathion and carbofuran onto chestnut shells was achieved at a concentration of 0.38.10 and 0.45.10" mol.dm respectively. Adsorption isotherms depicted a better fitting with the Langmuir isotherm. The results of sorption energy obtained from the Dubinin-Radushkevich isotherm pointed out that adsorption was driven by physical interactions. The kinetics of sorption follows a first-order rate equation. The thermodynamic parameters AS and AG indicate that the sorption process is thermodynamically favourable, and spontaneous, whereas the value of AH shows the exothermic nature of sorption process for methyl parathion and endothermic nature of carbofuran. The developed sorption method has been employed in methyl parathion and carbofuran in real surface and ground water samples. The sorbed amount of methyl parathion and carbofuran may be removed by methanol to the extent of 97-99% from the surface of chestnut shells. 

The adsorption isotherms of O2 on ACF s at 77 K are close to those of N2, as shown in Fig. 7 [61]. Although the adsorption isotherm was measured at 77K, which is below the boiling temperature (90.18K), no condensation occurred during the measurement. The O2 adsorption isotherm was described by the Dubinin-Radushkevich (DR) equation [62]... 

Porosity and pore-size distributions were determined by gas adsorption and immersion calorimetry, with the measurement of helium and bulk densities. Volumes of micropores were calculated using the Dubinin-Radushkevich (DR) equation (Section 4.2.3) to interpret the adsorption isotherms of N2 (77 K), CO2 (273 K) and n-C4H o (273 K). Volumes of mesopores were evaluated by subtracting the total volume of micropores from the amount of nitrogen adsorbed at p/p° = 0.95. The two density values for each carbon were used to calculate the volume of the carbon skeleton and the total volume of pores (including the inter-particle space in monolithic disks). Immersion calorimetry of the carbon into liquids with different molecular dimensions (dichloromethane 0.33 run benzene 0.37 nm and 2,2-dimethylbutane 0.56 nm) permits the calculation of the surface area accessible to such liquids and subsequent micropore size distributions. The adsorption of methane has been carried out at 298 K in a VTI high-pressure volumetric adsorption system. Additional techniques such as mercury porosimetry and scanning electron microscopy (SEM) have also been used for the characterization of the carbons. 

The micropore volumes deduced from the Dubinin-Radushkevich (DR) equation are compiled in Table 1. From both the data in this Table 1 and the isotherm shape evolution (Figs. 1-4) it can be deduced that the reaction with air produces an important change... 

The N2 adsorption isotherms for carbons of series P given in Fig. 2 show the development of micro- and mesoporosity from P-7 to P-58 and a decrea.se in adsorptive capacity thereafter. Table 1 shows that there is an important increase in ash content with increasing burn-off so that only a 31% of sample P-70 is carbon. The application of the Dubinin-Radushkevich (DR) equation to the adsorption data of the Nj (77K), CO2 (273K) and n-C H,o (273K) for all carbons leads to the micropore volume (VJ values listed in Table 1. There is an increase in V ... 

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. 

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.

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]. 

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. 

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