Adsorption isotherms Dubinin-Astakhov equation

Stoeckli (1993) has pointed out that the Dubinin-Astakhov equation (Equation (4.45)) can be derived from Equation (4.52), but McEnaney (1988) and others (e.g. Jaroniec et al. 1997) have drawn attention to the difficulty in arriving at an unambiguous interpretation of the energy distribution function. Indeed, Stoeckli et al. (1998) have now pointed out that Equation (4.45) can be usefully applied to a number of adsorption isotherms on non-porous solids. A comprehensive review of the significance and application of Equation (4.52) is given by Rudzinski and Everett (1992). 

The pore volumes of the obtained hard carbons were measured using the molecular probe method [3]. Adsorption isotherms of the probe molecules were measured at 298 K using an adsorption apparatus (Bel Japan, Belsorp 28). The employed probe molecules were CO2, C2H6> n-C4H o and (-C4H10 (minimum molecular dimensions 0.33, 0.40, 0.43 and 0.50 nm, respectively). By applying the Dubinin-Astakhov equation (n=2) [1] to the measured isotherms, the limiting micropore volumes corresponding to the minimum size of the adsorbed molecules were determined. 

Chen, S.G. and Yang, R.T. (1994). Theoretical basis for the potential theory adsorption isotherms. The Dubinin-Radushkevich and Dubinin-Astakhov equations. Langmuir, 10, 4244-9. 

Adsorption equilibrium data of SO2, NH3 and CO2 were analyzed by the linearized forms of the Langmuir and Freundlich equations (see Eqns. 2 and 4). The water vapour isotherm was analyzed by the linearized forms of the Dubinin-Astakhov equation (6), because of the above mentioned condensation phenomena. 

Langmuir model is an excellent fit for the adsorption of SO2 and NH3 on Pentalofos tuff and a fairly good representation in the case of CO2. In the latter case, the Freundlich model appears, however, to be more suitable. On the contrary, as expected, in the fitting of the water vapour adsorption isotherm, best results are obtained using the Dubinin-Astakhov equation. 

Siemieniewska, T., et ah. Application of the Dubinin-Astakhov equation to evaluation of benzene and cyclohexane adsorption isotherms on steam gasified humic acid chars from brown coal. Energy Fuels, 4(1), 61-69(1990). 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

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]. 

Textural characterisation of the samples was carried out by measuring apparent density (mercury at 0.1 MPa), mercury porosimetry and N2 and CO2 adsorption isotherms, at -196 and 0 °C, respectively. The apparent surface areas of the samples were obtained by using the BET equation [5]. The micropore size analysis was performed by means of the t-plot and the Dubinin-Astakhov methods [6]. 

The Dubinin-Astakhov (D-A) [6] equation was applied to the N2 adsorption isotherms. The accessible pore width, L, was calculated fi om the expression proposed by Stoeckli and Ballerini [9]. 

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 study of a particular adsorption process requires the knowledge of equilibrium data and adsorption kinetics [4]. Equilibrium data are obtained firom adsorption isotherms and are used to evaluate the capacity of activated carbons to adsorb a particular molecule. They constitute the first experimental information that is generally used as a tool to discriminate among different activated carbons and thereby choose the most appropriate one for a particular application. Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum [3]. Therefore, all equations derived firom monolayer gas adsorption remain vafid. Some of these equations, such as the Langmuir and Dubinin—Astakhov, are widely used to determine the adsorption capacity of activated carbons. Batch equilibrium tests are often complemented by kinetics studies, to determine the external mass transfer resistance and the effective diffusion coefficient, and by dynamic column studies. These column studies are used to determine system size requirements, contact time, and carbon usage rates. These parameters can be obtained from the breakthrough curves. In this chapter, I shall deal mainly with equilibrium data in the adsorption of organic solutes. 

The isotherms developed by Dubinin and coworkers employ a power to which the adsorption potential is raised that indicates the prevalent type of pores. The Dubinin-Radnshkevich eqnation [1 in Table 14.3] was intended for microporons adsorbents, since the exponent is 2. The Dnbi-nin-Astakhov equation (m) allows the exponent B to vary, bnt a reasonable lower limit is unity (for macroporous adsorbents). The Dnbinin-Stoeckli equation (n) allows a distribution of pore sizes, which is a feature of many adsorbents. 

The potential theory of adsorption was introduced by Polanyi in 1914. Dubinin [48,49] and Stoeckli et al. [50] improved the theory and termed it the theory of volume filling of micropores (TVFM). This theory has been widely used in correlating the effect of temperature on the adsorption isotherms of pure gases. The modern formulationof TVFM is the Dubinin-Astakhov (DA) equation, which is expressed as... 

A promising equation for the adsorption isotherm has been proposed by Dubinin and Astakhov [80,81] ... 

The Dubinin-Astakhov isotherm (18) has also been extended to adsorption from solution [200,206] moreover, the equations obtained for the values7=1 and j = 2 have been applied to... 

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