The equation of vapor-phase sorption on heterogeneous surfaces with local Guggenheim–Anderson–De Boer model

Colloids and Surfaces A: Physicochem. Eng. Aspects 296 (2007) 182–190

The equation of vapor-phase sorption on heterogeneous surfaces with local Guggenheim–Anderson–De Boer model Artem A. Mishchenko a , Nikolay D. Yakimov b , Konstantin A. Potashev b , Vladimir A. Breus a , Irina P. Breus a,∗ b

a Institute of Chemistry, Kazan State University, Kazan 420008, Russia Institute of Mathematics and Mechanics, Kazan State University, Kazan 420008, Russia

Received 27 January 2006; received in revised form 21 September 2006; accepted 27 September 2006 Available online 1 October 2006

Abstract Recently, we proposed a new four-parametric MGAB equation (modified Guggenheim–Anderson–De Boer equation, GAB) for fitting of experimental isotherms of vapor-phase sorption on geosorbents. In the present study, the theoretical meaning of MGAB equation was elucidated. For this purpose, the properties of the equation of multilayer adsorption on a heterogeneous surface with local characteristics described by GAB model and Zeldovitsh–Roginskiy energy distribution function ρ(ε) were derived and the satisfaction of MGAB equation with these properties was shown. MGAB model has two parameters connected with GAB model: vm is the monolayer sorption capacity and k is the constant of the multilayer sorption, as well as two parameters of sorption energy distribution function ρ(ε): α, characterizing heterogeneity of ρ(ε) and C0 , reflecting energy baseline. The influence of parameters α and C0 on the shape of both ρ(ε) and model sorption isotherm was established. The applicability of MGAB model for the description of surface heterogeneity of a natural sorbent by example of 13 different sorbate–sorbent systems dried and moistened has been discussed. For these systems equilibrium sorption isotherms were measured, corresponding energy distribution functions ρ(ε) were calculated, and obtained α values were compared. On the whole, the differences in α values responded to the real tendencies of heterogeneity changes in these systems. © 2006 Elsevier B.V. All rights reserved. Keywords: Vapor-phase sorption; Heterogeneous surfaces; Sorption energy distribution function; MGAB equation

1. Introduction The surface of natural adsorbents, such as soils, clays and other minerals, is energetically heterogeneous due to complex chemical composition and structural irregularities [1]. Moreover, surface heterogeneity is a fundamental feature of almost all solid surfaces [2]. This fact is now generally recognized by scientists investigating adsorption at gas–solid interface [3–9]. The problem of description of gas-phase sorption in heterogeneous systems has been actively discussed [6,10–14]. For solids having heterogeneous surfaces it is not possible to describe sorption precisely using simple models. Such models derived for homogeneous surfaces can be employed to the experimental data of a heterogeneous sorbent only over a limited surface coverage

∗

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range [15]. For the description of experimental isotherms the mathematical models are usually proposed to understand the nature of adsorption equilibrium between gases (vapors) and solids [16–21]. Some of such models are semi-empirical and others are derived from a fundamental adsorption theory. An advantage of a theoretical equation over a semi-empirical one is the definite physical meaning of its parameters. But sometimes theoretical models lose the simplicity in form and fail to explain highly complicated practical cases [21]. An ideal sorption model should simultaneously satisfy two conditions: it should describe well an isotherm in a whole range of concentrations and have parameters with a physical meaning. However for heterogeneous sorbents, it is difficult to combine both these requirements. Thus, an approximation equation usually satisfies only one of two these conditions. Recently [22], we have proposed the four-parametric empirical MGAB equation (modified Guggenheim–Anderson–De Boer equation, GAB) for the approximation of vapor-phase

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Nomenclature BET C, k C0

Brunauer–Emmett–Teller equation sorption parameters of GAB equation (–) parameter of sorption energy distribution function ρ(ε), reflects energy baseline (C value at ε = 0) (–) d, k3 fitting parameters of MGAB equation (–) GAB Guggenheim–Anderson–De Boer equation k1 , k2 fitting parameters of MGAB equation (g/mkl) MGAB modified GAB equation p pressure of sorbate vapors (Pa) v sorption value for a local isotherm in an integral sorption equation (mkl/g) vm monolayer sorption capacity (mkl/g) v(p, ε) or v(x, ε) local sorption isotherm (mkl/g) V sorption value (sorbed amount) (mkl/g) V(p) or V(x) integral sorption isotherm (mkl/g) x ≡ p/p0 relative pressure of sorbate vapors (–) Greek letters α parameter of sorption energy distribution function ρ(ε) characterizing heterogeneity of this distribution (–) ε, ε1 , εp characteristic sorption energy (ε1 for monolayer sorption, εp for multilayer sorption) (kJ/mol) θ surface coverage (–) ρ(ε) function of sorption energy distribution (mol/kJ)

multilayer sorption on heterogeneous sorbents: V =

xd , (k1 + k2 xd )(1 − k3 x)

(1)

here V is a sorption value, x is a relative partial pressure of sorbate vapors (0 ≤ x ≤ 1), and k1 , k2 , k3 and d are empirical parameters (d, k1 , k2 , k3 ≥ 0; d ≤ 1; k3 < 1) [22]. We carried out a complex statistical estimation of the approximation accuracy of 103 vapor-phase sorption isotherms (86 isotherms measured and 17 isotherms reported) of 27 different organic sorbates on soil and mineral sorbents. For comparing approximation adequacy of an isotherm dataset by different equations three criteria were used: s5% is the average probability of the location of experimental sorption values within the range of 5% of a theoretical curve; two interconnected parameters: R2 the square of correlation coefficient and k is the slope tangent for the linear regression (VE , VT ) at various sorbate activities; and D is the relative sample variance as a characteristics of an optimal ratio between numbers of parameters of a fitting equation and its approximation accuracy. For assessing an equation optimal 2 E T ity, the sample variance D = (1/(m − l)) m i=1 [Vi − V (xi )] (here m is a number of experimental dots and l is a number of approximation parameters in certain equation VT (x)) as a major criterion was used. Comparing different fitting equations on the base of D values, the equations with lesser D value should be given preference to. As a result, the proposed MGAB Eq. (1) was demonstrated to provide the best approximation of experimen-

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tal data upon different adsorption systems in the whole isotherm range, in comparison with such well-known models as BET [23], modified BET [24], Aranovich–Donohue [25], GAB [26–28] and Pickett–Dellyes [29,30] equations. It is noteworthy that the Eq. (1) has a power parameter d allowing a good description of nonlinearity of investigated sorption isotherms in an initial isotherm region. It has been frequently noted, in this range power relations often indicate surface heterogeneity of a sorbent [6,10]. The goal of the present study was to elucidate a theoretical meaning of the proposed MGAB equation. For this purpose (i) the properties of the equation of multilayer adsorption on a heterogeneous surface with local characteristics described by GAB model and Zeldovitsh–Roginskiy function of distribution of this local characteristics over the surface were derived, (ii) the satisfaction of MGAB equation with these properties was shown, (iii) the relationships between parameters of MGAB equation, GAB model and Zeldovitsh–Roginskiy distribution function were found, and (iv) the influence of MGAB parameters on the shape of both an energy distribution function and a sorption isotherm was established. 2. Theoretical The problem of adsorption energy heterogeneity of surfaces has arisen more than 70 years ago [7]. The equilibrium adsorption theory for heterogeneous surfaces is being extensively developed, and there is a number of monographs and reviews in this area (e.g. [3,4]). On the whole, the theoretical approach to the quantitative description of energy heterogeneity of a sorbent surface is related to the extensively used integral sorption equation [6,7,10,15,31–33]:  ∞ V (x) = v(x, ε)ρ(ε) dε, (2) 0

where v(x, ε) is a local sorption isotherm deduced theoretically, and ρ(ε) is a function characterizing the distribution of sorption energy, ε. For a theoretical description of experimental isotherms different distribution functions were used. Besides, a number of local adsorption models were offered which can be substituted into Eq. (2) to represent the adsorption process on a homogeneous patch of a surface. Here, the attention is focused at the initial isotherm range (monolayer region), where the influence of heterogeneity on sorption is most striking. As far as we know, for surfaces with random heterogeneity no analytical model has yet been proposed, which would adequately describe sorption behavior in the whole range of sorbate concentrations (including multilayer region). Usually for the sake of convenience simpler models are used to approximate an experimental isotherm of multilayer sorption on heterogeneous surfaces. But such models (BET, Frenkel–Halsey–Hill and others) do not yield good results because they are developed for homogeneous surfaces. Zeldovitsh [34] and Sips [35] were the first authors who noted deviations from linearity in the initial region of an isotherm of gas-phase sorption on a heterogeneous surface. To describe energy heterogeneity of a sorbent surface in the monolayer

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region, Zeldovitsh [34] used the Langmuir equation as a local isotherm and the exponential distribution of sorption energy on a surface. He demonstrated that in the range of small x values, sorption isotherm is significantly nonlinear and can be described by a power equation. To derive the ρ(ε) function, he proposed to approximate the kernel of Eq. (2) as follows:

equation as a local sorption isotherm was chosen:

 ⎧  ε −ε ⎪ ⎪ p exp , 0 < p < exp ⎨ RT RT

 v(p, ε) ≈ . −ε ⎪ ⎪ ⎩ 1, p > exp RT

(i) the monolayer sorption capacity, vm ; (ii) the constant of monolayer sorption, C: ε 1 C = g exp , RT

(3)

As a result, the ρ(ε) function took the form of: α ρ(ε) = · exp RT



−ε · α RT

 (4)

where α (0 < α < 1) is the parameter characterizing heterogeneity of the distribution of sorption energy, ε [6,34]. In Eq. (4), the term α/RT was found ∞from the condition for the probability density distribution 0 ρ(ε) dε = 1. As a result, Eq. (2) turned into the Freundlich equation V = Kf ·pα and described well the initial isotherm region [34]. Eq. (4) is still frequently used for the analysis of adsorption on heterogeneous surfaces. In [6] it was referred to as “Zeldovitsh–Roginskiy distribution”, while [36] named it “exponential distribution”. Recently, Brouers et al. [6] presented the more detailed analysis of Zeldovitsh’s approach. In particular, the physical meaning of parameter α was investigated in detail.

v=

x ((1/vm C) + ((C − 1)/vm C)x)(1 − kx)

(5)

The GAB equation has three parameters:

(6)

where ε1 is the characteristic energy of monolayer sorption, g a constant, and R and T represent universal gas constant and absolute temperature, correspondingly; and (iii) the constant of multilayer sorption, k: ε p , (7) k = g exp RT where εp is the characteristic sorption energy of multilayer sorption, and g is a constant. Now let us demonstrate that the choice of the GAB Eq. (5) as a local sorption model allows elucidating the theoretical meaning of the proposed MGAB Eq. (1). The GAB Eq. (5) can be rewritten as follows: θ≡

v x = , vm ((1/C) + ((C − 1)/C)x)(1 − kx)

(8)

3. Results and discussion

where θ is the sorbent surface coverage degree. For the monolayer sorption θ values are less or equal to 1, and for the multilayer sorption they are higher than 1. The maximal θ value, θ max (at x = 1) is defined as:

3.1. The theoretical validation of MGAB equation

θmax =

In accordance with the aforementioned principles of a heterogeneity description connected with the sorption integral Eq. (2), we attempted to get a theoretical description of the surface energy heterogeneity for multilayer adsorption. For this purpose the proper local isotherm and the adsorption energy distribution function were required. In contrast to Zeldovitsh [34] and other authors [10,36,37] used the Langmuir equation as a local isotherm, for a description of multilayer sorption on a heterogeneous sorbent surface another function v(x, ε) as a kernel of the integral in Eq. (2) is necessary. The first multilayer sorption model was represented by the BET equation [23]. However, being simplified this equation has a number of significant disadvantages. It describes well only a restricted isotherm range (x ∈ [0.05; 0.35]) and besides does not have the entire thermodynamical correctness, i.e. after BET substitution into the integral of Gibbs’s equation this integral does not have a finite value [38]. Recently, the GAB equation representing the simplest modification of BET model has been offered for describing equilibrium multilayer sorption [39,40]. Similarly to BET, the GAB equation has been derived for homogeneous surfaces [26–28], but in contrast to BET it is thermodynamically correct. Considering all above mentioned the GAB

Our theoretical model is based on the following schematic concept of the adsorption process on an energetically heterogeneous surface. At x → 0, sorbate molecules in the first place occupy most high-energy sites and then, with the increasing of sorbate pressure more and more low-energy sites of a sorption system are occupied. If the energy of interaction of a sorbate molecule with high-energy site lays over the sorption energy of non-occupied low-energy site, the multilayer formation takes place. This process spreads until all surface sites will be occupied, that is until a sorbent surface may be considered as homogeneous. The further increase of sorbate pressure is followed by a layer-by-layer completing and by gradual aligning (decreasing differences) of sorption values over all sites (at the P → P0 , θ → θ max ). According to our model these conditions are issued in the following. As a consequence of (6) the variability of ε parameter corresponds to differences of parameter C of sorption Eq. (8) on different surface sites. Therefore, it is possible that at the same x value on some surface sites a sorbate did not yet occupy a monolayer (θ < 1) while on the other sites multilayer formation takes place (θ > 1). So, in our sorption model we admit that multilayer formation can commence prior to complete mono-

1 1−k

(9)

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layer surface coverage. The possibility of multilayer formation before monolayer completing was reported by Toth [38,41]. The εp parameter in Eq. (7) and therefore the k constant in GAB equation (Eq. (8)) define the properties of an isotherm near the multilayer completing, at x → 1 (particularly, at θ(1) = 1/(1 − k)). It is natural to suppose that sorption volume and affinity will not be influenced by a surface heterogeneity after the significant number of layers will be completed. This idea corresponds to the conclusion of paper [42]. In accordance with this, θ max , k and εp values can be considered equal for all sorption sites. On the contrary, ε1 parameter in Eq. (6) (further designated as ε) defines predominantly an adsorption energy profile of the first layer and partly (with the decreasing influence) of the second and next initial layers. Here, the adsorption energy changes from site to site due to a surface heterogeneity. It is clear that the sorption energy of the second and next layers may differ with consequent decreasing of heterogeneity influence. This fact is accounted in our model by considering differences of ε values in Eq. (6) for different surface sites which are described by an appropriate distribution function. In this case, g parameter of Eq. (6) will reflect the energy baseline (at ε = 0) and will be designated further as C0 . Hence, Eq. (6) can be rewritten in the form of:  ε C(ε) = C0 exp . (10) RT Further, the distribution function of Zeldovitsh–Roginskiy (4) [6,34] as a function of the adsorption energy distribution, ρ(ε) was chosen. Hence, taking into account Eqs. (4) and (8), Eq. (2) can be rewritten as the integral sorption equation  ∞ x V (x)

 = Θ(x) ≡ vm (1/C(ε)) + ((C(ε)−1)/C(ε))x (1−kx) 0  α α · exp −ε · dε. (11) RT RT After factoring out the integrand term 1/(1 − kx) and after taking into account Θ(x) ≡ V (x)/vm , Eq. (11) is transformed into  ∞  C(ε) · x vm α α V (x) = · exp −ε · dε. 1−kx 0 1 + (C(ε)−1) · x RT RT (12) Therefore, taking into account the Eq. (10) and using the substitution t = ((1 − x)/(C0 · x))α · exp(−(εα/RT )), i.e.

  εα C0 · x α exp − =t· , RT 1−x

  α dt · dε = − , RT t

C = t −1/α ·



1−x x

 ,

we obtain vm V (x) = ·Φ 1−k·x



C0 · x 1−x

α  ,

(13)

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where the Φ function can be expressed as an integral:  1/z dt , 0 < z < ∞, Φ(z) = z · 1 + t 1/α 0

(14)

Generally, the integral in Eq. (14) is expressed by special (e.g. hypergeometric) functions, but the main properties of the Φ(z) function can be easily determined. Specifically, Φ(z) is an increasing function, whose values lie within the interval 0 < Φ(z) < 1, and whose limits are equal to lim Φ(z) = 0 and z→0

lim Φ(z) = 1. The integral in Eq. (14) increases with 1/z and z→∞ ∞ tends to reach a finite limit Iα = 0 dt/(1 + t 1/α ) at z → 0. Therefore, at x close to zero, function Φ((C0 · x/(1 − x))α ) in Eq. (13) behaves as (C0 · x)α · Iα , while at x close to unity, lim Φ((C0 · x/(1 − x))α ) = 1. These main properties of Φ(z) x→1

function (14) stem from the use of GAB Eq. (5) and distribution function (4) during our construction of the integral sorption Eq. (11). The obtained Eqs. (13) and (14) are quite complex for a practical application. However, it is evident that real sorption values always deviate from a theoretical model to some extent. Thus, to describe sorption one can use a simpler function, as long as it possesses the aforementioned main properties when applied to heterogeneous surfaces. One of such simple functions can be presented by

  C0 · x α Iα · (C0 · x)α Φ = , (15) 1 + (Iα · C0 α − 1) · xα 1−x which gives the following equation after the substitution into the integral Eq. (12): V (x) =

vm Iα · (C0 · x)α · . 1 − k · x 1 + (Iα · C0 α − 1) · xα

(16)

It is evident that the obtained Eq. (16) is identical to the MGAB Eq. (1), if MGAB parameters d, k1 , k2 , k3 are expressed through parameters of Eq. (16) as following relationships: d = α; 1 ; vm · K(C0 , α) k2 = k1 · (K(C0 , α) − 1), k3 = k k1 =

,

(17)

where K(C0 , α) = Iα · C0α

Hence, we have demonstrated that the obtained Eq. (16) completely coincides with the empirical MGAB Eq. (1). The latter was proposed by us earlier to describe the vapor-phase multilayer sorption on heterogeneous sorbents. Therefore, MGAB equation is not only optimal for fitting experimental sorption data, but also has theoretical interpretation as the equation of integral sorption on a heterogeneous surface with local properties described by the GAB model. Based on such theoretical consideration, the meaning of empirical parameters d, k1 , k2 , k3 in MGAB Eq. (1) becomes clear. In particular, MGAB approximation of sorption isotherms allows estimating not only the values of vm and k parameters, generally used as sorption parameters. Additionally, MGAB equation allows determining the values of two parameters which characterize an adsorption energy distribution for

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Fig. 1. The influence of parameter α on the shape of energy distribution function, ρ(C/C0 )RT.

a heterogeneous sorbent. These parameters are: α, characterizing the degree of sorbent surface energy heterogeneity, and C0 , characterizing the energy baseline, i.e. the sorption constant for the weakest sorbent surface site. All these parameters are determined from the following relationships: α = d; 1 ; k1 + k 2 k = k3 ; 

1 + k2 /k1 1/d , C0 = Id vm =

(18)  where Id =

∞

0

dt 1 + t 1/d Fig. 2. The influence of parameter α on the shape of a model adsorption isotherm at different C0 values. The main picture represents the initial isotherm range, and the small picture—the overall isotherm. (a) C0 = 1, (b) C0 = 10.

3.2. The influence of MGAB equation parameters on the shape of an energy distribution function and sorption isotherm

Because the C0 parameter reflects the energy baseline only, it is reasonable to use the normalized C/C0 values instead of C values. As a result, after such a normalization of the distribution function ρ(C)RT the C0 parameter can be eliminated and the influence of the heterogeneity parameter α alone on the shape of an energy distribution function can be determined (Fig. 1). It is evident that with the decrease of α value, the moiety of high energy sites increases, i.e. the sorbent surface becomes more heterogeneous. These relations clearly demonstrate the physical meaning of α as a parameter of heterogeneity of an adsorption system. Then let us discuss the influence of each parameter of energy distribution function (α and C0 ) on the shape of a model adsorption isotherm (sorption values versus relative sorbate pressures). In Figs. 2 and 3 these effects are shown both for an initial

For examination of the influence of C0 and α parameters on the shape of both adsorption energy distribution function ρ (4) and sorption isotherm, the parametric analysis of the theoretical MGAB Eq. (16) was carried out. Such an analysis did not need using any numerical or statistical methods. It was sufficient just to know a functional form of the appropriate relations and here both needed relations (the distribution function and MGAB equation) are explicit. As a result, varying one or another parameter one can visually estimate its influence. Taking into account Eq. (10), ρ function (4) can be rewritten as:

−α α C ρ(C) = . (19) RT C0 Table 1 Physical and chemical characteristics of soil sorbents Soil sorbent [22,44]

Place of sampling

Leached chernozem Dark-gray forest soil

Tatarstan, Russia

Soil layer (cm)

Clay (%)

Silt (%)

Sand (%)

Texture

Mineralogy

Organic carbon (%)

CEC (mequiv/ 100 of g)

0–20 10–20 75–85 140–150

44 36 35 35

51 52 27 31

5 12 38 34

Silty clay Silty clay loam Clay loam Clay loam

–

4.0 4.4 0.4 0.0

– 60.5 21.3 19.9

Illite–smectite

pH 5.6 5.5 – –

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of a sorbed amount for higher C0 values is being observed. At α = 1 (Fig. 3b) C0 influence primarily results in increasing initial isotherm slopes and in the consequent rising the curves. The similar but some lesser effect one can also see at small α values (Fig. 3a).

Fig. 3. The influence of parameter C0 on the shape of a model adsorption isotherm at different α values. The main picture represents the initial isotherm range, and the small picture—the overall isotherm. (a) α = 0.3, (b) α = 1.0.

isotherm range where they are most demonstrative (the main pictures) and for an overall isotherm (additional small pictures). In both figures the effects of one parameter (α or C0 ) at small (Figs. 2a and 3a) and high (Figs. 2b and 3b) values of another parameter (C0 or α) are shown. Regarding sorption values (a sorbed amount for a hypothetical adsorbate in a model adsorbate–adsorbent system), in Fig. 2a and 2b one can see increasing their growth rate in the initial isotherm range with decreasing α values (the isotherm runs very close to the y-axis). This visual effect has a physical meaning as it expresses the increase of a moiety of highenergy interactions on the surface of a more heterogeneous sorbent. On the other hand, at high relative sorbate pressures all isotherms are very close to each other independently from α values, and the rate of their closing increases with the increase of C0 values. The increasing C0 values (Fig. 2b) leads to the rise of curves which is especially noticeable at high α values in the initial and intermediate isotherm ranges. This turns out lesser differences between isotherms with different α at high C0 values. Similarly, Fig. 3a and 3b illustrate the influence of C0 parameter on the shape of a model adsorption isotherm at different (small and high) α values. Here in the initial isotherm range, in contrast with the effect of α parameter, much sharper increase

Fig. 4. Adsorption of methanol on different solid sorbents. The data for soil samples and kieselgel were taken from our database [22], the data for activated carbon were reported [45]. (A) The differences in α values (error bars show the parameter estimation errors). (B) The energy distribution functions, ρ(C/C0 )RT vs. normalized sorption constant, C/C0 (calculated using the theoretical MGAB equation). (C) Adsorption isotherms, V(x) vs. x. In the legends the corresponding α values are shown.

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3.3. Application of MGAB equation to the experimental data As discussed above, the early proposed MGAB equation is applicable to various sorbate–sorbent systems and fits almost all isotherm types (I–V) except stepwise VI (like a helium adsorption on a solid krypton [32]) and non-monotonic ones [43]. In regard to the area of application of the theoretical MGAB model

(i.e. use of its physically significant parameters), an estimation of an adsorbent surface heterogeneity with the help of α parameter is reasonable for isotherms types II and IV only which are typical for nonporous and mesoporous sorbents. For such isotherm types the surface adsorption is assumed to be predominant. As to isotherm type III (convex to the x-axis), its shape itself indicates a weak interaction between a sorbate molecule and adsorbent surface. Therefore, the differences between the sites on such

Fig. 5. Adsorption of p-xylene on soils with different moisture. The data were taken from our database [22]. (a) The difference in α values (error bars show the parameter estimation errors). (b) The energy distribution functions, ρ(C/C0 )RT vs. normalized sorption constant, C/C0 (calculated using the theoretical MGAB equation). (c) Adsorption isotherms, V(x) vs. x. In the legend the corresponding α values are shown. Soil sorbents: (I) Dark-gray forest soil, layer 145–150 cm. (II) Dark-gray forest soil, layer 75–85 cm. (III) Dark-gray forest soil, layer 10–20 cm. (IV) Leached chernozem, layer 0–20 cm. Soil characteristics are given in the Table 1. In the legend the corresponding α values are shown.

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a low-energy surface are negligible, hereupon α parameter is approximately constant (equal to 1). So it is unreasonable to carry out the analysis of surface heterogeneity from an isotherm type III. In Figs. 4 and 5 the area of application of MGAB model to various sorption systems dried and moistened is illustrated, in order to show how our model acts as related to natural sorption systems. For this purpose, 13 different sorbate–sorbent systems from our sorption database were chosen [22]. For those systems equilibrium isotherms were measured at 25 ◦ C using static headspace gas chromatographic analysis method, the experimental procedure was described elsewhere [22,44]. As sorbents silica gel (Kieselgel, Fluka AG) and soil samples having different texture and organic content (characteristics are given in Table 1) were studied. Besides, the reported experimental isotherms of methanol sorption on the activated carbon (macropore porosity 0.31; micropore porosity 0.40; BET surface area 1200 m2 /g; mesopore surface area 82 m2 /g) [45] were used. For each sorption isotherm shown in Figs. 4c and 5c the values of α parameter (Figs. 4a and 5a) using MGAB equation were calculated, and then corresponding energy distribution functions ρ(C/C0 )RT versus normalized sorption constant, C/C0 (calculated using the theoretical MGAB Eq. (16)) were constructed, Figs. 4b and 5b. In Fig. 4 adsorption on four different solids is described and methanol as a molecular probe for evaluation of surface energy heterogeneity is used. It is known that purified activated carbons as well as synthetic silica gels possess a fairly homogeneous surface energy distribution. On the contrary, soils represent extremely heterogeneous sorbents and the extent of its heterogeneity increases with the complexity of soil texture and composition level. The comparison of α values calculated for four sorbents (in Fig. 4a, 0.25 ± 0.11 for leached chernozem, 0.52 ± 0.09 for dark-gray forest soil, 0.83 ± 0.10 for silica gel, and 1.03 ± 0.07 for activated carbon), as well as the shape of obtained energy distribution functions (Fig. 4b) demonstrate the expected differences in heterogeneity of these sorbents as related to methanol adsorption. It has to be noted that in contrast to Fig. 2, no close relation between α values (Fig. 4a) and an isotherm shape (Fig. 4c) should be expected. It is because in the first case the influence of α alone on an isotherm shape is accounted and in the last one the isotherm shape is simultaneously influenced by all four MGAB parameters. As an essentially more illustrative examples, the results upon the influence of moisture on soil surface heterogeneity for pxylene sorption on four different soil samples in Fig. 5 are shown (different layers of a dark-gray forest soil were moistened). It is evident that with increasing moisture soil surface becomes more low-energetic due to water molecules occupy high-energy sites. This effect leads to decreasing surface heterogeneity (e.g. see [46]). As it is shown in Fig. 5a, in the case of both soil samples having small organic carbon content (Ia, IIa) α values for sorption on dried and moistened samples significantly differ. Such differences in α values are in congruence with the real differences in heterogeneity for these systems. For soils having higher organic carbon content (IIIa, IVa) one can see the similar differences which are still displaying here as tendencies only.

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The corresponding isotherms and energy distribution functions are represented in Fig. 5c and 5b, respectively. The results of comparisons show that differences in α values for various sorbate–sorbent systems respond on the whole to the real tendencies of heterogeneity of these systems. Occurring large errors of determination of some α values are apparently connected both with an insufficient quantity of experimental points on an isotherm and with their scatter. Undoubtedly, for a high-precision determination of α values, precise and detailed measurements of experimental sorption values are required. 4. Conclusions The theoretical meaning of the MGAB equation proposed early for fitting experimental sorption isotherms on heterogeneous sorbents was elucidated. On the basis of relationships (17) the MGAB fitting Eq. (1) can be rewritten as vm Iα · (C0 · x)α · 1 − k · x 1 + (Iα · C0 α − 1) · xα ∞ where Iα = 0 dt/(1 + t)1/α is an auxiliary constant; vm the monolayer sorption capacity; k the constant of the multilayer sorption (7); α the parameter of sorption energy distribution function ρ(ε) characterizing heterogeneity of this distribution (4); and C0 is the parameter of sorption energy distribution function ρ(ε) reflecting energy baseline (the sorption constant for the weakest sorption site, C value at ε = 0).

V (x) =

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