Dubinin - Radushkevich model

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. 

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. 

Sobolik, J.L. Ludlow, D.K., and Hessevick, W.L.. Parametric sensitivity comparison of the BET and Dubinin-Radushkevich models for determining char surface area bv carbon dioxide adsorption. Fuel, 71(10), 1195-1202(1992). 

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) ] ... 

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). 

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... 

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). 

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. 

Dubinin-Radushkevich. This model (29) is the same as the more general Dubinin-Astakhov equation (30) (see below), with n = 2. Dubinin-Astakhov ... 

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]. 

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.). 

The benzene adsorption/desorption data were used to analyze the porous structure of activated carbons. The BET specific surface area, Sbet, was estimated from the linear BET plot. The adsorption process in microporous materials is well described by the pore filling model. Taking into account the heterogeneity of micropore structure, a special form of Dubinin-Radushkevich equation, the two-term DR isotherm was applied [6,7] allowing for determination of micropore volumes and adsorption energies ... 

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. 

Cerofolini GE (1975) A model which allows for the Freundlich and the Dubinin-Radushkevich adsorption isotherms. Surface Sci 51 333-335... 

Once the isotherm is obtained, a number of calculation models can be applied to different regions of the adsorption isotherm to evaluate the specific surface area (i.e., BET, Dubinin, Langmuir, and the like) or the micro- and mesopore volume and size distributions (i.e., Barett-Joyner-Halenda, Dubinin-Radushkevich, Horvath and Kawazoe, Saito and Foley, and the like). 

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] ... 

Two researches studied the adsorptive properties of montmorillonite clay modified by tetra-butyl ammonium (Akgay, 2004, 2005). The adsorption of p-chlorophenol in this clay was done in batch with 20 mL of pollutant solution to 0.1 g of clay, at 25°C for 16 h. The adsorption isotherms were adjusted according to the models of Freundlich and Dubinin-Radushkevich. The kinetic and thermodynamic parameters pointed to the application of organoclay as adsorbent effective of phenolic compounds in contaminated effluents. 

A simplified model of equilibrium surface suggests that the DR behaviour is observed in low-pressure adsorption on patchwise, weakly heterogeneous surfaces which were grown in equilibrium conditions and hence were quenched at the adsorption temperature. At higher pressures, these surfaces should exhibit the Freundlich behaviour, while in the case of strong heterogeneity adsorption should be described by the Temkin isotherm. The three classic empirical isotherms, Freundlich, Dubinin-Radushkevich, Temkin, seem therefore to be related to adsorption on equilibrium surfaces, and the explanation of these experimental behaviours can be seen as a new chapter of the theory of adsorption the theory of physical adsorption on equilibrium surfaces. 

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),... 

The BET model has been generalized by using of various monolayer isotherms for heterogeneous surfaces Langmuir-Freundlich [94], Toth [95], generalized Freundheh (GF) [94], Dubinin-Radushkevich [67,95,96], and others [5]. These equations have been apphed to the interpretation of experimental data [5,6]. The above-discussed procedure has also extended to the adsorption with lateral interactions on randomly heterogeneous surfaces also [5]. 

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. 

Powder XRD on a heated sample of 25d showed no ehange in the observed pattern. This would infer a robust architecture, but given the proposed strueture, void space would necessarily have to be generated in this desolvated solid, 25e. To further confirm the proposed strueture of 25d and definitively illustrate the porosity of this system, CO2 and N2 sorption isotherms were performed on 25e (Fig. 35). Both yielded type 1 isotherms eharaeteristie of microporous solids. For CO2, surface areas of 326, 373, and 380 m /g for BET, Langmuir, and Dubinin-Radushkevich (DR) models, respeetively, were obtained. 

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. 

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... 

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