Multilayer Adsorption on Solid Surfaces: Calculation of Layer Thickness on the Basis of the Athermal Parallel Layer Model

Journal of Colloid and Interface Science 243, 37–45 (2001) doi:10.1006/jcis.2001.7741, available online at http://www.idealibrary.com on

Multilayer Adsorption on Solid Surfaces: Calculation of Layer Thickness on the Basis of the Athermal Parallel Layer Model Ferenc Berger1 and Imre D´ek´any Department of Colloid Chemistry and Nanostructured Materials, Research Group of the Hungarian Academy of Sciences, University of Szeged, Szeged, Aradi Vt. Tere 1., H-6720 Hungary Received February, 27, 2001; accepted May 25, 2001; published online September 7, 2001

and lack of data for several decades. The aim of the present work was to study multimolecular adsorption from (nonelectrolytic) binary liquid mixtures on solid surfaces and to make an attempt to calculate the so-called equivalent thickness of the S/L multimolecular adsorption layer with the help of the available thermodynamic functions.

A method for calculating the dependence of equivalent layer thickness on bulk phase composition in athermal binary mixtures has been devised. The prerequisite to the procedure is knowledge of the adsorption excess isotherm and of the cross-sectional area of one component. After integration of the excess isotherm according to the Gibbs equation, the equations of the parallel layer model are used to calculate the composition of the monolayer in contact with the surface, which in turn is used to calculate the thickness of the adsorption layer. The proposed method was tested on a hypothetical experimental system with calculated excess isotherms and a known function of equivalent layer thickness. Reliable results were obtained, especially when adsorption was preferential, the surface was relatively homogeneous, and the ratio of the cross-sectional areas of the individual components was close to the ratio of molar fractions. °C 2001 Academic Press Key Words: adsorption excess; excess isotherm; layer thickness; multilayer adsorption.

MULTIMOLECULAR ADSORPTION ON THE S/L INTERFACE

Knowledge of the composition and thickness of the adsorption layer at the S/L interface is essential for the description of not only adsorption but also other related phenomena such as the stability of disperse systems, the magnitude of forces operating between particles or surfaces, and the structure and rheological behavior of suspensions (4–6). Whether the adsorption layer is mono- or multimolecular and whether its thickness depends on the composition of the bulk phase are questions of basic importance in the interpretation of liquid sorption isotherms. The classical models for adsorption from liquid phase formulated by Kipling, Everett, and later by Schay and Nagy all assume monomolecular adsorption (5–7). In many cases, however, analysis of the isotherms revealed that the monomolecular layer model is inapplicable because it hinders the precise determination of adsorption capacity in many binary mixtures (1–3). Our aim is to extend the possibility of isotherm analysis and to determine the thickness of the multimolecular adsorption layer. For this, a suitable layer model is needed. One of the models most often used to describe the equilibrium between the adsorption layer and the bulk phase is the so-called adsorption space-filling model (1, 2). This model considers the adsorption layer as a homogeneous phase, separated from the bulk phase by a sharp boundary. The volume of the adsorption layer according to this model is

INTRODUCTION

Over the years the problem of multilayer adsorption has been addressed by numerous studies, both experimental and theoretical. Although at least a dozen equations have been formulated to describe multimolecular vapor adsorption, the solutions assigned to multimolecular adsorption on the S/L interface have been at best adaptations from S/V adsorption. Relatively highly developed simulation methods (Monte Carlo, molecular dynamics) have also been used predominantly to describe adsorption on the S/V interface (1–3). Even fewer data are available on the heat effect associated with adsorption from binary mixtures. In the case of adsorption from liquid phase, due to the limited nature of adsorption temperatures only direct calorimetric measurements can be taken into consideration, and without calorimetric data it is impossible to calculate enthropies. All in all, it is little wonder that the field of the thermodynamics of mono- and multilayer adsorption from binary mixtures has been dominated by uncertainty

V s = t s a s = n s1 Vm,1 + n s2 Vm,2 = n s1,0 Vm,l ,

[1]

where t s is the thickness of the adsorption layer, a s is the specific surface area of the adsorbent, n is is the specific adsorbed amount of the ith component in the interfacial layer, Vm,i is the partial molar volume of the ith component in the interfacial layer (the

1 To whom correspondence should be addressed. E-mail: berger@chem. u-szeged.hu.

37

0021-9797/01 $35.00

C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.

´ ANY ´ BERGER AND DEK

38

parameter typically used for the calculations is the molar volume of the pure ith component), and n s1,0 is the adsorption capacity normalized to component (1). When a molecule adsorbed on a surface, (2)s , is exchanged for a molecule (1) of the bulk phase, an equilibrium constant K ∗ may be defined (6–9), r (1) + (2) = r (1) + (2), ¡ s s ¢r x γ x2 γ2 ∗ K = 1 1 r s s, (x1 γ1 ) x2 γ2 s

s

[4]

[5]

[6]

Adsorption on the solid/liquid interface is described by the reduced adsorption excess, according to the Ostwald–Izaguirre equation (7): ¡ ¢ n σ1 (n) = n s1 − n s x1 = n s x1s − x1 ,

V s /Vm,1 + n σ1 (n) (r − 1)

.

[8]

Another often used factor is the so-called separation factor,

A simplified equilibrium constant may be defined by neglecting the activity coefficients: ¡ s ¢r x x2 K = 1 r s. (x1 ) x2

r n σ1 (n) + x1 V s /Vm,1

[3]

This approximation is usually attributed to Everett; it was, however, first proposed by Kiselev and Pavlova, after Zhukovitsky (12), saying that the behavior of the surfacial phase is closer to ideal than that of the bulk phase. In much later publications Everett himself questions its validity (13, 14). In our opinion the following approximation is at least as good as and in certain cases considerably better than the previous one: ¡ s ¢r γ1 γ2 ≈ 1. (γ1 )r γ2s

x1s =

[2]

where xi , xis , ai , and ais are the molar fractions and activity coefficients of the ith component in the bulk phase and in the surface layer (superscript s). It is assumed that the stoichiometry of exchange is determined by the ratio of the molar volumes, i.e., r = Vm,2 /Vm,1 . The use of the equilibrium constant K ∗ is made rather difficult by the surface activity coefficients γis , which, unlike the bulk phase activity coefficients, are not well known. Attempts have been made to experimentally determine the surface activity coefficients (10, 11) but a universally valid solution is yet to be found to this problem. The following approximation is widely used in the special literature: ¡ s ¢r γ1 ≈ 1. γ2s

n σ1 (n) is experimentally determined and the value of the adsorption volume V s is known (from the value of n s 1,0 and from the relationship n s = n s1 + n s2 ), the composition of the surface layer x1s may be calculated in the following way:

[7]

where n s is the material content of the surface phase and n s = n s1 + n s2 . When the specific reduced surface excess amount

S=

x1s x2 . x2s x1

[9]

Since V s = n s1,0 Vm,1 , it follows from Eq. [1] that n s1,0 = n s1 + r n s2 = n s x1s + r n s x2s .

[10]

Starting from Eqs. [7], [9] and [10], the Everett–Schay equation may be obtained: x1 x2 n σ1 (n)

=

1

·

n s1,0

¸ S −r r + x1 . S−1 S−1

[11]

If the value of S is independent of the bulk phase composition x1 , the function x1 x2 /n 1σ (n) vs x1 is linear (7). The values of n s1,0 and S may be calculated from the slope and the intersection. Based on Eqs. [6] and [9] it is easy to show that in case of r = 1, S = K. The space-filling model is suitable for determining adsorption layer thickness or adsorption layer thickness as a function of bulk phase composition only if certain approximations are made regarding the equilibrium constant and the activity coefficients operating in the layers. It was Rusanov (15) who recognized that it is possible to calculate in an exact manner a minimal layer thickness for the basic principles of thermodynamics not to be violated. Rusanov’s inequality is quite simple: µ

∂ xis ∂ xi

¶ > 0.

[12]

p,T

Making use of the above criterion and its consequences, T´oth (16), Brown et al. (17), and O´scik et al. (18) showed that, in the cases they studied, the adsorption layer is certainly thicker than monomolecular. The theory and practice of multimolecular liquid sorption on heterogeneous surfaces have been addressed by Jaroniec et al. (19). Findenegg approached the problem of determining layer thickness by combining gas and liquid sorption data (20). Theoretical and experimental research in this field has been reviewed by Dabrowski et al. (21). As shown above, the space-filling model makes the assumption that the interfacial phase is homogeneous and is separated from the bulk phase by a sharp interface. In the reality, however, it is quite certain that the transition of the adsorption layer into the bulk phase does not follow a stepwise course but proceeds

CALCULATION OF S/L ADSORPTION LAYER THICKNESS

in a diffuse way, resulting in a gradually changing (decreasing) concentration profile. The most common treatment of the description of the diffuse interfacial layer is to break it down into monomolecular sublayers. This procedure is especially problematic when the molecules are extremely anisometric or when the molecules of the different components are of highly dissimilar sizes. To understand the structure and properties of interfacial layers, it is practical to start with spherical, unstructured molecules of uniform size. This approach is used with the so-called lattice model, the most significant model describing multimolecular layers (22). A version of the lattice model, the so-called parallel layer model, allows molecules of various sizes (23). The “athermal” variety of the latter model is suitable for describing adsorption from binary mixtures with low heats of mixing.

39

ing a specific surface area of a s = 100 m2 /g for the adsorbent. Knowing the value r of the given binary mixture and selecting an arbitrary value for K , the function x1s vs x1 may be calculated from the equation defining the (simplified) equilibrium constant K by iteration. If, subsequently, the layer thickness t s , i.e., the specific layer volume V s = t s a s , is also fixed, the specific adsorption excess may be calculated in the following way: n σ1 (n) =

¡ s ¢ Vs ¡ ¢ x1 − x1 . s s x1 Vm,1 + 1 − x1 Vm,2

[13]

Naturally, t s is not necessarily constant but may also be a function of bulk phase composition. 2. Analysis of Model Isotherms

CALCULATION OF LAYER THICKNESS WITHIN THE FRAMEWORK OF THE ATHERMAL PARALLEL LAYER MODEL

It is essential to determine adsorption capacity and the thickness of the adsorption layer for the investigation of adsorption from binary mixtures. Earlier methods for determination of true adsorption capacity may only be used in the case of monolayers and/or adsorption layers of constant thickness, the thickness of which is independent of mixture composition. In the case of adsorption isotherms determined in binary mixtures of nonelectrolytes, these conditions are often not met, and neither the extrapolation method of Schay and Nagy nor the reciprocal representation of the isotherms is utilizable for the determination of adsorption capacity from the isotherms, except for the case of S-shaped (type IV) excess isotherms. In this section a method for calculating surface composition and adsorption layer thickness in the entire composition range is described. In this way the system is characterized by a layer thickness isotherm as a function of bulk phase composition, rather than by a single value of adsorption capacity. Our calculations are based on the parallel layer model in which, in addition to the molar volumes of the individual components, the cross-sectional area of component (1) is taken into account for the calculation of layer thickness. In case the adsorption excess isotherms do not have a linear section and their Everett–Schay representation is not linear, it may be assumed in general that only a model postulating an adsorption layer of variable thickness may provide a realistic description of the adsorption equilibrium. 1. Calculation of Model Isotherms Due to the theoretical and experimental difficulties involved, data on the thickness of multimolecular adsorption layers on the S/L interface of binary mixtures are rather scarce. In addition to experimental tests, it is therefore advisable to compare any new method for the calculation of layer thickness with other independent models. Synthetic model isotherms were generated with the help of the above-described space-filling model, assum-

The first two model isotherms were calculated at t s = constant, first for r ≈ 1 and next for r 6= 1. In both cases the monomolecular layer thickness of component (1) was chosen as the value of t s . Our first example is the toluene (1)/cyclohexane (2) mixture for which r = 1.02 and the monomolecular layer thickness calculated from the cross-sectional area (0.46 nm2 /molecule (24)) is 0.384 nm. The isotherms n 1σ (n) vs x1 may be calculated on the basis of Eq. [13] (Fig. 1a) for various equilibrium constants (K = 1000 . . . K = 3). The different equilibrium constants correspond to surfaces with different polarities and surface energies (25). The above choice is also rendered practical by the fact that toluene is preferentially adsorbed on the majority of ordinary surfaces. The Everett–Schay function [11] is widely used for the socalled reciproque representation of adsorption excess isotherms (6, 7, 25). In the case where layer thickness is constant and selectivity is not dependent on composition, the function x1 x2 /n 1σ (n) vs x1 is linear, making it possible to calculate the equilibrium constant K and the (monomolecular) adsorption capacity n s1,0 from the slope and the intersection. As shown in Fig. 1b, this representation indeed yields a straight line in the entire concentration range, irrespective of the value of K . Let us next consider the benzene (1)/n-heptane (2) mixture in the case where the molar volumes of the two components differ (r = 1.65) and the monomolecular layer thickness of benzene, calculated from its cross-sectional area (0.38 nm2 /molecule (24)), is 0.388 nm. The shape of the excess isotherms obtained (Fig. 2a) is different from that of the isotherms shown in Fig. 1a. The reciproque representation (Fig. 2b) indicates that if r 6= 1 and the value of K is small, the function x1 x2 /n 1σ (n) vs x1 is no longer linear; i.e., it may no longer be used for the determination of either n s1,0 or K . Let us now return to the case of r ≈ 1, i.e., to the toluene (1)/cyclohexane (2) mixture, and let us examine what happens if layer thickness is variable rather than constant. Considering that no single explicit multimolecular isotherm equation is known for miscible binary mixtures, still less one describing the layer

´ ANY ´ BERGER AND DEK

40

where parameter B is the power of the exponential equation describing the decay of adsorption potential with increasing distance from the surface and we let the value of B be 3, as expected on theoretical grounds. Let the equilibrium constant be K = 30, and let us vary the value of parameter A. The various arbitrary layer thickness functions obtained and the corresponding isotherms are shown in Figs. 3a and 3b, respectively. In the case of A 6= 0 the isotherms obtained no longer have a linear section and in this case the Schay–Nagy representation no longer gives a straight line (Fig. 3c). The reason for this is that changes in layer thickness result in the alteration of adsorption capacity.

FIG. 1. (a) Excess isotherms and (b) their reciprocal representations calculated at constant layer thicknesses in toluene/cyclohexane mixtures.

thickness function, one of the equations formulated for S/V interfaces must be adopted. One of these is the semiempirical Frenkel–Halsey–Hill (FHH) equation (26), the original version of which describes the thickness of the multimolecular layer building in the course of vapor adsorption. In spite of the fact that conditions in binary mixtures are quite different, let us employ this equation for the generation of arbitrary layer thickness functions. Naturally, it must be taken into consideration that the adsorption layer on the S/L interface is at least monomolecular, whereas this is not so on the S/V interface. The FHH equation modified with this in mind, applied to our chosen mixture, is t s = 0.384 nm [1 + (A/ ln(1/x1 ))1/B ],

[14]

FIG. 2. (a) Excess isotherms and (b) their reciprocal representations calculated at constant (monomolecular for benzene) layer thicknesses in benzene/nheptane mixtures.

41

CALCULATION OF S/L ADSORPTION LAYER THICKNESS

FIG. 3. (b) Excess isotherms and (c) their reciprocal representations calculated from (a) layer thickness functions based on the FHH equation in toluene/cyclohexane mixtures.

These model calculations demonstrate that the Schay–Nagy extrapolation method and the Everett–Schay function may be used to determine the adsorption capacity relative to pure component (1), n s1,0 , only when layer thickness is constant or in the case of monomolecular coverage (even then only when r ≈ 1 if the value of K is not large enough; i.e. adsorption is not sufficiently preferential).

3. A Model for Calculating the Adsorption Layer Thickness Function The strategy of the calculation is based on the following approximations:

(i) The entire adsorption layer consists of a series of monolayers parallel with each other and the surface (Fig. 4a). (ii) The concentration profile characterizing surface enrichment approaches bulk phase composition according to a monotonous function (Fig. 4b). (iii) Surface tension (i.e., the free enthalpy of adsorption, 121 G) may be attributed predominantly to the first sublayer (Fig. 4d). This also means that surface tension may in some way be distributed among the sublayers; i.e., it is partitionable: σ =

∞ X i=1

σi ,

[15a]

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42

FIG. 4. Structure of the S/L interfacial layer. (a) The interfacial layer as a series of monolayers. (b) The x1s vs z function. (c) The interfacial excess-density function. (The areas A and B are equal.) (d) The distribution of the free enthalpy of adsorption.

σ ≈ σ1 .

[15b]

Here σ is surface tension and σi is its fraction associated with the ith sublayer. There is the following relationship between the (specific) free enthalpy of adsorption and surface tension: ¡ ∗ ¢ 121 G = σ − σ(2) a s ,

[16]

∗

Here σ (2) is the surface tension of pure component (2) (on the S/L interface in question). Equations [14] and [15] may be converted as 121 G =

∞ X

121 G i

[17]

i=1

121 G 1 ≈ 121 G,

[18]

where 121 G i is the fraction of the free enthalpy of adsorption associated with the ith monolayer. To be able to characterize the thickness of the diffuse adsorption layer, we propose the introduction of “equivalent layer thickness,” t sequ . This stands for the thickness of a hypothetical, homogeneous layer with a graded concentration profile, the composition of which is identical with that of the first monolayer (i.e., x s1 = x s1,1 ) and contains an equivalent excess amount (Figs. 4b and 4c). The proposed calculation consists of the following main steps: (I) The function 121 G vs x1 is obtained by integration according to the Gibbs equation Za1 121 G = −RT ∗ a1=0

n σ1 (n) ∗ da , x2 a1∗ 1

[19]

where a1 is the activity of component (1) in the bulk phase and a1 = γ1 x1 . Activity coefficients for the toluene (1)/cyclohexane (2) and benzene (1)/n-heptane (2) mixtures were calculated from activity functions published in the literature (27). s is calculated (II) The composition of the first monolayer, x1,1 with the help of the so-called athermal parallel layer model. This is a monolayer model (23) describing the thermodynamical behavior of the molecules in the first sublayer adjoining the (not necessarily solid) surface. It is essentially an extended version of the lattice model: the latter operates with molecules of uniform size, whereas the extended version is capable of describing mixtures of molecules with different sizes (volumes) (20). The parallel layer model involves two important approximations: (i) it is assumed that the molar volume of component (2) is larger than that of component (1), and component (2) is treated as an r -mer of component (1); (ii) it is assumed that these r mers are arranged parallel with the surface (this is where the name of the model originates; i.e., originally it had nothing to do with the series of sublayers parallel with the surface). Conformations perpendicular to the surface, extruding from the plane of the monolayer, are excluded from the statistical mechanical derivation of the enthropy term. Several theoretically useful equations may be obtained from the basic equations of the parallel layer model (23). Comparative calculations led us to the conclusion that best results are obtained with the equation 121 G 1 = RT

¸ · ¢ a s 1 φ2s r − 1¡ s + ln φ 2 − φ2 , am,1 r φ2 r

[20]

where φ2 is the volume fraction of component (2) in the bulk phase and φ2 = r (1 − x1 )/(x1 + r (1 − x1 )); φ2s is the volume fraction of component (2) in the first sublayer, 121 G 1 is the free enthalpy of adsorption in the first sublayer, and am,1 is the molar cross-sectional area of component (1). If the values of φ2 and 121 G 1 are known, φ2s may be calculated from Eq. [20]. The

CALCULATION OF S/L ADSORPTION LAYER THICKNESS

43

equation does not have an analytic solution but may be easily solved numerically, by means of iteration. For this purpose it is conventient to convert Eq. [20] to the form ¢ 1 φ2s r − 1¡ s −121 G 1 am,1 ln φ + + − φ = 0. 2 2 RT a s r φ2 r

[21]

This is essentially a problem of finding the intercept of a function which may be solved with any of the usual procedures (interval bisection, rule of false position (regula falsi), Newton–Raphson method). s in (III) After the determination of φ2s , the composition x1,1 the first sublayer may be calculated in the following way: ¡ ¢ r 1 − φ2s s ¡ ¢. x1,1 = s [22] φ2 + r 1 − φ2r

Based on Eqs. [13] and [22], the equivalent layer thickness is: ¡ ¢ s s + r 1 − x1,1 n σ1 (n) Vm,1 x1,1 s tequ = . [23] s as x1,1 − x1 4. Checking the Proposed Method with the Help of Model Isotherms Generated by the Space-Filling Model It is possible to test the validity of the method with isotherms calculated according to Eqs. [6] and [13], i.e., the space-filling model, using an arbitrary but known layer thickness function (t s vs x1 ) and to check whether the original layer thickness function is recovered. The space-filling model, selected as a control method, is significantly different from the model to be tested. The main differences in the case of the space-filling model are:

FIG. 5. Equivalent layer thickness functions calculated from the isotherms of Fig. 1a (r = 1.02).

121 G vs x1 were calculated from the model isotherms for the toluene (1)/cyclohexan (2) mixture presented in Fig. 1a. Under these conditions the behavior of the two models is expected to be consistent. Indeed the original (constant) layer thickness (0.384 nm) is recovered even with extremely different values of K . In our next example layer thickness is assumed to change according to a linear function (t s = 0.3836 nm [1 + 2x1 ]) and r ≈ 1. The results, shown in Fig. 6, are calculated from excess isotherms shown in Fig. 2a. Again the two models prove to be consistent in the case of preferential adsorption. The difference observed at low values of K is predominantly due to the basic

(i) the adsorption layer is homogeneous and unstructured; (ii) the molecules adjoining the surface are in no way different from other molecules within the layer; (iii) the adsorption layer is separated from the bulk phase by a sharp boundary; (iv) molar cross-sectional area is not used in the calculations, only the molar volumes of the individual components are taken into consideration. In the case of the proposed parallel layer model: (i) the adsorption layer is structured both vertically and horizontally; (ii) the molecules adjoining the surface are of special importance from the point of view of thermodynamic properties; (iii) the transition between the asorption layer and the bulk phase is continuous; (iv) the molar cross-sectional area of component (1) is taken into consideration. In our first example illustrated in Fig. 5, the thickness of the adsorption layer is constant, independent of mixture composition; it is monomolecular; and r ≈ 1 (r = 1.02). The functions

FIG. 6. Equivalent layer thickness functions calculated from the isotherms of Fig. 2a (r = 1.65).

44

´ ANY ´ BERGER AND DEK

FIG. 7. Adsorption excess isotherm on A1-PILC in benzene/n-heptane mixtures and the equivalent layer thickness function calculated with the proposed model.

difference between the two models regarding the distribution of the free enthalpy of adsorption within the layer. The space-filling model assumes a homogeneous distribution, whereas according to the model proposed by us, the greater part of 121 G is concentrated within the contact sublayer. 5. Experimental Control for the Proposed Method Two examples will be presented to illustrate the usefulness of the proposed calculation method. Adsorption on solid surfaces from benzene (1)/n-heptane (2) mixtures was studied in both cases. This binary mixture is nearly athermal; the mixing heat

is low. The adsorption excess quantities were measured with a Rayleigh interferometer. The first adsorbent studied was Al2 O3 -pillared montmorillonite, aBET s = 132.8 m2 /g. Figure 7 shows the excess isotherm determined in the benzene (1)/n-heptane (2) binary mixture on Al-PILC and the specific layer thickness function calculated thereof. Since basal spacing determined by X-ray diffraction is 1.78 nm and the thickness of the silicate layer is 0.96 nm, a value of 0.82 nm is obtained for the thickness of the interlamellar space, i.e., 0.41 nm per surface. If, based on the data published in the literature (24), the cross-sectional area of the benzene molecule is 0.38 nm2 , then this value means 1.05 molecular layers, that is, an exactly monomolecular adsorption layer. This is in excellent agreement with the value predicted by the model. The second adsorbent studied was a silica (silicon dioxide consisting of nearly spherical particles) designated R972, hydrophobized by dimethyldichlorosilane treatment (Degussa AG), aBET s = 115 m2 /g. The excess isotherm and the equivalent layer thickness function calculated thereof are shown in Fig. 8. The minimal equivalent layer thickness or the minimal equivalent monolayer number may be estimated on the basis of Rusanov’s criterion. In the present case at low values of x1 the minimal equivalent monolayer number is about 1, whereas at relatively high values of x1 it is about 3. This requirement is met by the equivalent layer thickness function calculated on the basis of the athermal parallel layer model. CONCLUSION

A method for calculating the equivalent layer thickness as a function of the bulk phase composition has been developed for athermal binary mixtures. It requires knowledge of the adsorption excess isotherm and the cross-sectional area of one of the two components. After integration of the Gibbs adsorption isotherm equation, the composition of the monolayer, in direct contact with the solid surface, and the equivalent layer thickness can be calculated. The procedure was tested by using both generated model excess isotherms, where the equivalent layer thickness functions were known, and selected experimental systems. These tests confirmed the applicability of the method. The results are especially reliable if the adsorption is preferential, the surface is relatively homogeneous, and the ratio of the cross-sectional areas of the two components is not significantly different from that of the molar volumes. ACKNOWLEDGMENT The authors are very grateful for the financial support of Hungarian Research Scientific Fund, OTKA T 025392.

REFERENCES FIG. 8. Adsorption excess isotherm on Aerosil R972 in benzene/n-heptane mixtures and the equivalent layer thickness function calculated with the proposed model.

1. Machula, G., and D´ek´any, I., Colloids Surf. A 61, 331 (1991). 2. Machula, G., D´ek´any, I., and Nagy, L. Gy., Colloids Surf. A 71, 219 (1991).

CALCULATION OF S/L ADSORPTION LAYER THICKNESS 3. Kir´aly, Z., T´uri, L., D´ek´any, I., Bean, K., and Vincent B., Colloid Polym Sci. 274, 779 (1996). 4. Vold, M. J., J. Colloid Sci. 16, 1 (1961). 5. Kipling, J. J., “Adsorption from Solutions of Non-electrolytes.” Academic Press, London, 1965. 6. Everett, D. H., Trans. Faraday Soc. 61, 2478 (1965). 7. Schay, G., “Surface and Colloid Science” (E. Matijevi, Ed.), Vol. 2, pp. 155–221. Wiley, London, 1969. 8. Vincent, B. J., J. Colloid Interface Sci. 112, 270 (1973). 9. Da¿browski, A., Monatsh. Chem. 117, 139 (1986). 10. Schay, G., Nagy, L. Gy., and Szekr´enyesy, T., Periodica Polytechnica 6, 91 (1962). 11. Larionov, O. G., and Myers, A. L., Chem. Eng. Sci. 26, 1025 (1971). 12. Kiselev, A. V., and Pavlova, L. F., Izvestia Akad. Nauk. S.S.S.R. Ser. Khim. 18, 265 (1965). 13. Everett, D. H., in “Adsorption from solution and gas adsorption,” (J. Rouquerol and K. S. W. Sing, Eds.), pp. 1–20, Elsevier, Amsterdam, 1982. 14. Everett, D. H., Pure Appl. Chem. 53, 2181 (1981).

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15. Rusanov, A. I., in “Progress in Surface and Membrane Science,” Vol. 4, pp. 57–114. Academic Press, London, 1971. 16. T´oth, J., J. Colloid Interface Sci. 46, 38 (1973). 17. Brown, Ch. E., Everett, D. H., and Morgan, Ch. J., J. Chem. Soc. Faraday Trans. I 71, 883 (1975). 18. O´scik, J., Goworek, K., and Kusak, R., J. Colloid Interface Sci. 84, 308 (1981). 19. Jaroniec, M., Da¿browski, A., and T´oth, J., Chem. Eng. Sci. 39, 65 (1984). 20. Findenegg, G. H., J. Chem. Soc. Faraday Trans. I 69, 1069 (1973). 21. Da¿browski, A., Jaroniec, M., and O´scik, J., “Surface and Colloid Science,” (E. Matijevic, Ed.) Vol. 14, p. 83. Plenum, New York, 1987. 22. Guggenheim, E. A., “Mixtures.” Oxford Univ. Press, Oxford, 1952. 23. Defay, R., Prigogine, I., Bellemans, A., and Everett, D. H., “Surface Tension and Adsorption, p. 172. Longmans, London, 1966. 24. Mc. Clellan, A. L., and Harnsberger, H. F., J. Colloid Interface Sci. 23, 577 (1967). 25. D´ek´any, I., Sz´ant´o, F., Nagy, L. Gy., and Schay, G., J. Colloid Interface Sci. 93, 151 (1983). 26. Hill, T. L., J. Chem. Phys. 17, 590 (1949). 27. Scatchard, G., and Ticknor, L. B., J. Am. Chem. Soc. 74, 3724 (1952).