Dubinin-Radushkevich method

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]. 

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].

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]). 

Since the molecular size of SO2 is around 0.43 nm (LJ parameter o i = 0.429 nm) the presence of pmes smaller than 0.8 nm in the structure of carbon should be crucial for physical adsfflption of this molecule. Indeed the importance of such pores in the proces of SO2 removal was pointed out in the literature. Raymundo-Pinero and coworkers studied the dependence of the amount adsorbed on various carbons on the porosity measured using Dubinin-Radushkevich method, CO2 adsorption and the total pore volume calculated from nitrogen adsorption [26]. The results obtained show the relatively px)d correlation for the volume of micnopotes calculated form carbon dioxide adsorption. It has to be pointed out here that it is believed that CO2 at experimental conditions chosen in that research adsorbs only in pores smaller than 0.7 nm. Such correlation is found only when oxygen is present in the system. Lack of oxygen decreases the amount adsorbed by a factor of two to six depending on the type of carbon. 

On dry gels, standard characterization techniques for porous media are used, several of which have been described in Volume 2 of this series helium pycnometry for pore volume determination (Section 6.3.1.2) as well as nitrogen adsorption at 77 K for surface area (Section 6.3.2.2, BET method), for microporosity (Section 6.3.3.2, Dubinin-Radushkevich method), for pore size distribution (Section 6.3.3.3, BJFl method), and for total pore volume (Section 6.3.3.4). When characterizing gels by nitrogen adsorption, other methods are also used for data interpretation, for example, the t-plot method for microporosity (Lippens and de Boer, 1965) and the Dollimore-Heal method (Dollimore and Heal, 1964) or Broekhoff-de Boer theory for mesoporosity (Lecloux, 1981). 

Conunonly, textural characterization of porous solids is carried out by physical adsorption of gases, which can be analyzed using several theories, to provide detailed information about the carbon micropore structure. A number of attempts have been made to establish standard procedures for the interpretation of the adsorption data in the characterization of porous solids [11-13]. However, there is still a lack of agreement on the assessment and interpretation of the adsorption data [14], and the results found in the literature depend upon the theory used to interpret the isotherms [15-18]. Usually, the PSD of porous solids is evaluated firom N2 adsorption at 77 K, and the structural heterogeneity of the microporosity is determined from the Dubinin-Radushkevich method (DR) and its modifications (Dubinin-Asthakov, DA,... 

The adsorption isotherms of N2 at 77 K (Fig. 1) and CO2 at 194.5 K (Fig. 2) performed on the powder samples are of type I according to lUPAC classification. All the adsorption isotherms exhibit a sharp increase in the adsorbed volume, at low relative pressures, while at higher pressures (0.2

relative pressures elose to 1.0 (p/po>0.95) can be attributed to interparticle condensation. The adsorption data were interpreted on the basis of BET [13], Langmuir [14] and Dubinin -Radushkevich methods [15] and the speeific surface area of the samples is calculated (Table 1), using the value 0.162 nmVmolecule forN2-... 

In Eq. 1.4, Na is Avogadro s number. The specific surface area that can be determined by gas sorption ranges from 0.01 to over 2000 mVg. Determination of pore size and pore size distribution of porous materials can be made from the adsorption/desorption isotherm using an assessment model, such as the t-plot, the MP method, the Dubinin-Radushkevich method and the BJH model, etc. [42], suitable for the shape and structure of the pores. The range of pore sizes that can be measured using gas sorption is from a few Angstroms up to about half a micron. 

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. 

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... 

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)-... 

The characterization of porous materials exhibiting a composite pore structure encompassing micro-meso-and perhaps macro-pore sizes, is of particular significance for the development of separation and reaction processes. Among the characterization methods for materials exhibiting ultramicropore structures, DpDubinin-Radushkevich (DR)[2], Dubinin-Astakov (DA) [3], Dubinin-Stoeckli (DS) [4], as well as the Horvath-Kawazoe (HK), [5] methods are routinely used for the evaluation of the micropore capacity and the pore size distriburion (PSD). 

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 ... 

Specific surface areas of various carbonized materials were measured by nitrogen gas adsorption with BET methods using an automated surface area analyzer (micro-track type 4200, Nikkiso, Japan). For mesopores whose diameter were less than SO nm, the surface areas and pore volumes were measured by carbon dioxide adsorption. The carbon dioxide adsorption at 298 K was measured with Bellsorp 28 (BEL Japan). The pore volume was determined using Dubinin-Radushkevich equation [4], and the surface area was determined by Medek s method [S]. 

The calculation methods for pore distribution in the microporous domain are still the subject of numerous disputes with various opposing schools of thought , particularly with regard to the nature of the adsorbed phase in micropores. In fact, the adsorbate-adsorbent interactions in these types of solids are such that the adsorbate no longer has the properties of the liquid phase, particularly in terms of density, rendering the capillary condensation theory and Kelvin s equation inadequate. The micropore domain (0.1 to several nm) corresponds to molecular sizes and is thus especially important for current preoccupations (zeolites, new specialised aluminas). Unfortunately, current routine techniques are insufficient to cover this domain both in terms of the accuracy of measurement (very low pressure and temperature gas-solid isotherms) and their geometrical interpretation (insufficiency of semi-empirical models such as BET, BJH, Horvath-Kawazoe, Dubinin Radushkevich. etc.). 

We turn now to the analysis of pore structure. For this purpose, various optional computational procedures are incorporated in the software, which is now provided with most commercial adsorption equipment. For example, for micropore size analysis the isotherm can be converted into a t-plot and also displayed in either the Dubinin-Radushkevich (DR) or the Dubinin-Astakov (DA) coordinates. With some packages it is also possible to apply the MP method of Brunauer, the Horvath-Kawazoe (HK) method and/or density functional theory... 

The specific surface area (Sbf.t) has been evaluated by full 3-parameters BET equation and 2-parameters linear BET plot in the range p/p° 0.01-0.2. The total pore volume (Vt) has been evaluated by Gurvitsch rule and by density functional theory (DFT) method. The micropore volume (Vm) has been determined by DFT, Dubinin-Radushkevich (D/R) and Horvath-Kavazoe (H/K) (Saito-Foley) equations at p/p° < 0.168. 

In order to evaluate correctly the textural properties a carefully selection of calculation method is necessary. Evaluation of micropore volume in ERS-8 and SA calculated with Dubinin-Radushkevich and DFT are consistent, instead an overestimate value is observed with Horvath-Kavazoe method. The pore size distribution of MSA, MCM-41, HMS and commercial silica-alumina materials have been evaluated by BJH and DFT method. Only DFT model is effective, in particular for evaluation in the border line range between micro and mesopores. 

In order to estimate the pore size distributions in microporous materials several methods have been developed, which are all controversial. Brunauer has developed the MP method [52] using the de Boer t-curve. This pore shape modelless method gives a pore hydraulic radius r, which represents the ratio porous volume/surface (it should be realised that the BET specific surface area used in this method has no meaning for the case of micropores ). Other methods like the Dubinin-Radushkevich or Dubinin-Astakov equations (involving slitshaped pores) continue to attract extensive attention and discussion concerning their validity. This method is essentially empirical in nature and supposes a Gaussian pore size distribution. 

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 following table 5.3-3 shows the various formula for the spreading pressure and the pure component hypothetical pressure for various commonly used isotherms. Some isotherms such as Langmuir, Freundlich, LRC have analytical expressions for the spreading pressure as well as the pure component hypothetical pressure. Other isotherms, such as O Brien Myers, Ruthven, Toth and Nitta have analytical expression for the spreading pressure, but the pure component hypothetical pressure expressed in terms of the reduced pressure must be determined from a numerical method. For other general isotherms, such as Unilan, Aranovich, Dubinin-Radushkevich, Dubinin-Astakhov, Dubinin-Stoeckli, Dubinin-Jaroniec, one must resort to a numerical method to obtain the spreading pressure as well as the pure component hypothetical pressure. 

Common evaluation tools that provide quantitative information are the Density Functional Theory (DFT) for micropores and mesopores, Hie Dubinin-Radushkevich equation for the extraction of characteristic parameters on micropores, the t-plot and the oLg-plot for the separation of surface area located in micro- and nonmicropores, the method to calculate the so-called BET surface area and the BJH relationship that provides access to the mesopore size distribution. 

The Dubinin-Radushkevich equation with its numerous modifications is very important for the adsorption methods of characteristics of most industrial adsorbents. These adsorbents have a complex and well developed porous structure including pores of different shapes and widths but micropores play the... 

Nitrogen adsorption/desorption isotherms were measured at 77 K and evaluated using a Quantachrome Autosorb-1 computer-controlled apparatus. (Quantachrome, Boynton Beach, FL, USA) The apparent surface area was derived using the Brunauer-Emmett-Teller (BET) model, Sa.BEx- The total pore volume, Vp at, was calculated from the amount of nitrogen vapor adsorbed, at a relative pressure close to unity, on the assumption that the pores are then filled with liquid nitrogen. The average pore radius, rp, was derived from the total pore volume and the BET surface area on the basis of uniform cylindrical pores. The micropore volumes, and Fo dr, were computed by the Dubinin-Radushkevich (DR) and t methods (Halsey), respectively. The characteristic energy, Eo, was derived from the DR plot as well with P =0.34. The slit size, Lq, was derived from the relation = 10-8/(-Eo-H-4),... 

---