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Variable thickness of the liquid sorption layers on solid surfaces
Colloids and Surfaces A: Physicochemical and Engineering Aspects 141 (1998) 305–317
Variable thickness of the liquid sorption layers on solid surfaces Ferenc Berger, Imre De´ka´ny * Department of Colloid Chemistry, Attila Jo´zsef University, H-6720 Szeged, Hungary Received 17 January 1997; accepted 7 May 1997
Abstract Calculations of the adsorption capacity and the thickness of the adsorption layer are of primary importance for studies of the adsorption of binary mixtures. Earlier methods were applicable to the determination of real adsorption capacity only in the case of monolayers and/or layers of constant thickness, independent even of mixture composition. In the case of the adsorption isotherms of binary mixtures of non-electrolytes these conditions are often not fulfilled, and neither the Schay–Nagy extrapolation method nor reciprocal isotherm representations can be used for the determination of adsorption capacity on the basis of adsorption isotherms. Here we present a method of calculation suitable for the computation of surface composition and adsorption layer thickness in the whole composition range. Thus, instead of a single value of adsorption capacity, the system is characterized by the layer thickness isotherm given as a function of the composition of the bulk phase. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Adsorption; Multilayers; Binary mixtures; Adsorption layer thickness; Surface layer composition; Adsorption capacity; Free enthalpy of adsorption
1. Introduction Two fundamental questions concerning the interpretation of liquid sorption isotherms of solid–liquid interfaces are whether the adsorption layer is monolayer or multilayer and whether the thickness of the adsorption layer depends upon the composition of the bulk phase. Results published in the literature are not uniform in this respect, since both monolayer [1–3] and multilayer models [4–7] are employed in the calculations. In the present work we would like to point out that it is only in certain extreme cases that a monolayer structure or multilayer with constant thickness is possible and that this structure is strictly associated with the linear region of the excess isotherm [1–4,8–12]. Furthermore, we present those liquid * Corresponding author.
sorption excess isotherms (registered in benzene– n-heptane mixtures) in the case of which the thickness of the adsorption layer demonstrably ranges from a monomolecular to a multilayer coverage on the surface of the solid adsorbent. Our calculations are based on the so-called parallel layer model, where intermolecular interactions are also taken into consideration for calculating layer thickness. When adsorption excess isotherms have no linear regions, it may be postulated, in general, that realistic information about the adsorption equilibrium will be obtained only by applying the model evoking an adsorption layer of variable thickness. Naturally, the appearance of multilayer adsorption needs to be proven by independent methods. The composition dependence of mono- or multilayer adsorption is unambiguously proven by interlamellar adsorption between pillared clay lamellae,
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since values of basal spacing may be determined by X-ray diffraction ( XRD) measurements [10–12].
2. Theoretical background 2.1. The space filling model One of the most often used model for describing of the equilibrium between the adsorbed layer and the bulk phase is the ‘‘space filling’’ model [1–4,13–17]. In this model the volume of the adsorption layer is Vs=tsas=ns V +ns V =ns V (1) 1 m,1 2 m,2 1,0 m,1 where ts is the thickness of the adsorption layer, as is the specific surface area of the adsorbent, ns i is the amount of the ith component in the surface layer, V is the partial molar volume of the ith m,i component in the surface layer (in general the molar volume of the pure ith component is used) and ns is the adsorption capacity related to 1,0 component (1). When a molecule (2)s adsorbed on a solid surface is displaced by a molecule (1), an equilibrium constant K* can be defined [13,18–21]: r(1)+(2)sPr(1)s+(2)
(2)
(xs cs )rx c 1 1 2 2 (3) (x c )rxs cs 1 1 2 2 x , xs , c , cs are the mole fractions and the activity i i i i coefficients of the ith component in the bulk phase, and in the surface layer (index s). It is assumed that the stoichiometry of the displacement is governed by the ratio of the molar volumes, r=V /V . Neglecting the activity coefficients, a m,2 m,1 simplified equilibrium constant can be defined:
where ns is the material content of the surface phase and ns=ns +ns . If the ns(n) excess amount 1 2 1 is determined experimentally and the Vs adsorption volume (from ns or ns=ns +ns ) is known, 1,0 1 2 the xs surface layer composition can be calculated 1 as follows: rns(n) +x Vs/V 1 1 m,1 xs = (6) 1 Vs/V +ns(n) (r−1) m,1 1 There is also often used the so-called separation factor: xs x S= 1 2 (7) xs x 2 1 While Vs=ns V from Eq. (1) follows 1,0 m,1 ns =ns +rns =nsxs +rnsxs (8) 1,0 1 2 1 2 Starting from Eqs. (5), (7) and (8), we can get the Everett–Schay equation [2,3,13,15,16 ]:
A
B
x x r 1 S−r 1 2= (9) + x ns(n) ns S−1 S−1 1 1 1,0 When the value of S is independent of x , the 1 x x /ns(n) vs. x function is linear [2,3]. The values 1 2 1 1 of ns and S can be calculated from the slope and 1,0 intercept. It is easy to show on the basis of Eqs. (4) and (7) that, in the case of r=1, S=K. 2.2. Calculation of the model isotherms
K*=
(xs )rx 1 2 (4) (x )rxs 1 2 The solid/liquid interfacial adsorption is characterized by the reduced adsorption excess amount after the Ostwald–Izaguirre equation [2–4]: K=
ns(n) =ns −nsx =ns(xs −x ) 1 1 1 1 1
(5)
In order to test the proposed method for calculating the adsorption layer thickness, synthetic excess isotherms have been generated by means of the equations of the space filling model. For our model calculations the specific surface area of the adsorbent is considered constant (as=100 m2 g−1). Based on the r value of the mixture and an arbitrarily chosen K value, the xs 1 vs. x function can be calculated from Eq. (4) by 1 iteration. In addition, if the ts layer thickness, e.g. the Vs=tsas layer volume, is also fixed, the adsorption excess can be calculated as follows: Vs ns(n) = (xs −x ) (10) 1 1 1 xs V +(1−xs )V 1 m,1 1 m,2 Of course, ts does not have to be constant, it
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can also be a function of the bulk phase composition. 2.3. Analysis of the space filling model isotherms
(a)
(b)
In the first two examples, model isotherms have been generated at ts=constant, first in the case of r#1 then in that of r≠1. In both of them the monomolecular layer thickness of the component (1) has been chosen as the ts value. As a first example, the toluene(1)–cyclohexane(2) mixture has been chosen at which r=1.0165 and the monomolecular layer thickness calculated from the cross-sectional area value (0.46 nm2/molecule [22]) was 0.384 nm. When model isotherms ns(n) =f (x ) are calculated for this 1 1 binary liquid according to Eq. (10), the isotherm series shown in Fig. 1(a) and the equilibrium composition diagrams xs =f (x ) presented in Fig. 1(b) 1 1 are obtained for the various equilibrium constants (K=103 to K=3). Changes in the equilibrium constant K are dependent on the hydrophobicity of the surface, since we have previously established that, in the case of the adsorption of identical binary liquids, the value of the constant K is a function of surface polarity [8–12]. It follows, therefore, that in the case of the excess isotherms shown in Fig. 1(a), when K=103 the surface is hydrophilic, since toluene is preferentially adsorbed on the surface. The selectivity of toluene adsorption at various values of K is welldemonstrated by Fig. 1(b). The Everett–Schay function is used widely for the reciprocal representation of adsorption excess isotherms [2,8,9,13]. The advantage of representation according to Eq. (9) is its suitability for the determination of K and the monomolecular adsorption capacity ns . The function shown in 1,0 Fig. 1(c) is linear throughout the entire range of compositions. When the size of the components of
Fig. 1. (a) Calculated adsorption excess isotherms with different equilibrium constants at constant ts monolayer value (0.384 nm) in toluene(1)–cyclohexane(2) mixtures on solid. (b) The adsorption equilibrium diagram in toluene(1)–cyclohexane(2) mixtures on solid. (c) The Everett–Schay representation of the calculated excess isotherms in toluene(1)–cyclohexane(2) mixtures.
(c)
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(a)
(b)
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the binary liquid is varied and, for example, a binary liquid made up of benzene–n-heptane is considered, where r=1.6488, then the monomolecular layer thickness of benzene calculated from its cross-sectional area value (0.38 nm2/molecule [22]) is 0.388 nm. The shape of the excess isotherms calculated for the conditions described above already differs from those presented in Fig. 1 (see Fig. 2(a)). At identical values of the equilibrium constant K, the adsorption of benzene from nheptane is less preferential than it was from toluene (cf. Fig. 2(b)). On the other hand, the linear representation shown in Fig. 2(c) indicates that if the value of r is other than unity and the value of K is low, then the function x x /ns(n) =f (x ) is no 1 2 1 1 longer linear, i.e. it may not be used for the determination of either ns or K. 1,0 We now return to the examination of the toluene–cyclohexane binary liquid series with the difference that the thickness of the adsorption layer is no longer considered constant; according to the FHH isotherm [23] ts=0.384 nm {1+[A/ln(1/x )]1/B} (11) 1 where the B parameter is related to the rate of fall-off of the adsorption potential with the distance from the surface and A is an intermolecular interaction constant. With different A and B parameters, layer thickness will be a function of the composition x (see Fig. 3(a,b)). When calcula1 tions are done at identical values of K=30, it is apparent from the figures that the shape of the excess isotherms calculated according to Eq. (10) is altered: they exhibit a curvature in the range x =0.21.0. Near linearity is observed in the 1 range x =0.2–0.8 only if the FHH layer thickness 1 is independent of composition, i.e. in Eq. (11) A=0 and B=2.0. If layer thickness (volume) is a function of composition—as can be seen in Fig. 3(b)—then the function x x /ns(n) =f (x ) is 1 2 1 1 not linear but has a backward section (Fig. 3(c)). Changing the value of A in Eq. (11) naturally Fig. 2. (a) Calculated adsorption excess isotherms with different equilibrium constants in benzene(1)–n-heptane(2) mixtures on solid. (b) The adsorption equilibrium diagram in benzene(1)– n-heptane(2) mixtures on solid. (c) The Everett–Schay representation of the excess isotherms.
(c)
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(a)
(b)
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means a change in adsorption capacity ns , since 1,0 the course of the isotherms ns(n) =f (x ) (see 1 1 Fig. 3(a)) and the slope of the straight lines (see Fig. 3(c)) are significantly different. If in Eq. (11) only parameter B, characteristic of the slope of the layer thickness function ts=f(x ), is varied and the value of A is kept 1 constant (A=1), a series of isotherms is obtained which intersect at x =0.37. The isotherm corre1 sponding to B=1000 has a linear region at x =0.2–1.0, i.e. it is justifiable to use the Schay– 1 Nagy extrapolation method for the determination of adsorption capacity [2–4]. According to Fig. 4(b), as parameter B decreases the isotherms incline because the adsorption layer becomes thicker than monomolecular. Model computations reveal a significant increase in the number of layers especially in the range of x 0.6. The isotherms 1 correspondingly exhibit a curvature and the representation x x /ns(n) =f (x ) will also be non-linear 1 2 1 1 (see Fig. 4(c)). The model calculations described above unambiguously prove that it is only in the case of a constant layer thickness or a monomolecular coverage, at r#1 and K=103–102 (i.e. under conditions of preferential adsorption) that both the Schay–Nagy extrapolation method and the Everett–Schay function are suitable for the determination of the adsorption capacity of the pure component ns . 1,0 Experiments, however, often yield data similar to the isotherms presented in Figs. 3(a) and 4(a), but, as we have seen, excess isotherms with bending (non-linear) sections are not suitable for the determination of adsorption capacity. 2.4. The proposed model for calculating the adsorption layer thickness function The strategy of our calculations was based on the following assumptions:
Fig. 3. (a) Calculated adsorption excess isotherms in toluene(1)– cyclohexane(2) mixtures on solid with different A values of the FHH equation, K=30, B=3. (b) The adsorption layer thickness function in toluene(1)–cyclohexane(2) mixtures on solid with different A values of the FHH equation, K=30, B=3. (c) The Everett–Schay representation of the excess isotherms.
(c)
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(1) the adsorption layer is made up of parallel monolayers ( Fig. 5(a)); (2) surface enrichment decreases nearly exponentially ( Fig. 5(b)); (3) surface tension (i.e. the free enthalpy of displacement D G) is restricted mostly to the 21 first layer (Fig. 5(d )): 2 s= ∑ s (12) i i=1 s#s (13) 1 where s is the surface tension and s is the i surface tension in the ith monolayer. (a)
There is the following relation between the free enthalpy of displacement and the surface tension: D G=(s−s* )as (14) 21 (2) where s* is the surface tension of pure compo(2) nent (2). Eqs. (12) and (13) can be written in another form:
(b)
2 D G= ∑ D G (15) 21 21 i i=1 D G #D G (16) 21 1 21 where D G is the free enthalpy of the displace21 i ment in the ith monolayer. In order to characterize the layer thickness in the case of a diffuse adsorption layer, we propose the use of the ‘‘equivalent layer thickness’’ ts . equ This means the thickness of an homogeneous surface layer which has the same composition as the first monolayer (i.e. xs =xs ) and contains an 1 1,1 equivalent excess amount (Fig. 5(b,c)). In the proposed method the calculation consists of the following principal theoretical steps. (1) The function D G vs. x is obtained by 21 1
Fig. 4. (a) Calculated adsorption excess isotherms in toluene(1)– cyclohexane(2) mixtures on solid with different B values of the FHH equation, K=30, A=1. (b) The adsorption layer thickness function in toluene(1)–cyclohexane(2) mixtures on solid at different B values of the FHH equation, K=30, A=1. (c) The Everett–Schay representation of the excess isotherms.
(c)
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Fig. 5. The schematic representation of the solid–liquid interfacial layer structure: (a) solid–liquid interface with multilayers; (b) the surface layer composition xs as function of the distance z from the surface; x is the equilibrium bulk composition, and ts is the 1 1 equ equivalent layer thickness; (c) the excess density function against the distance; the area of A is equivalent to the area of B; the total area under the density function dns(n) /dz is the surface excess amount; (d) the free enthalpy distribution function from the surface. 1
integrating the Gibbs equation [1–7,21]: a1 n s(n) 1 da* (17) D G=−RT 1 21 x a* 2 1 a*1=0 where a is the activity of component (1) in the 1 bulk phase and a =c x . Activity coefficients for 1 1 1 the toluene(1)–cyclohexane(2) and benzene(1)– n-heptane(2) systems were calculated from the Redlich–Kister equation of five parameters from the vapour–liquid equilibrium data [24,25]. (2) The calculation of the composition xs of 1,1 the first monolayer is based on the so-called ‘‘athermal parallel layer model’’. This model [26,27] is basically a monolayer model describing the thermodynamic behaviour of the first layer of molecules adjoining the solid surface. It is in fact a generalization of the well-known lattice model for systems containing molecules of different sizes [28,29]. The parallel layer model makes two essential assumptions: (a) supposing that the molar volume of component (2) is larger than that of component (1), component (2) is treated as an r-mer of component (1); (b) it is assumed that these r-mers are lined up parallel with the surface (‘‘flat’’ r-mers) and a position perpendicular to the surface (‘‘erect’’ r-mers) is excluded.
P
Several equations theoretically suitable for calculations may be deduced from the parallel layer model. After lengthy comparative tests we came to the conclusion that it is obviously the following equation that yields the best results [15,16 ]: as D G=RT 21 a m,1
C
1
ws r−1 ln 2 + (ws −w ) 2 2 r w r 2
D
(18)
where w is the volume fraction of component 2 (2) in the bulk phase and w =r(1−x )/ 2 1 [ x +r(1−x )], ws is the volume fraction of compo1 1 2 nent (2) in the first monolayer, D G is the free 21 1 enthalpy of displacement in the first monolayer, and a is the molar cross-sectional area of comm,1 ponent (1). When the values of w and D G are 2 21 1 known, ws may be calculated from the equation 2 above. This equation has no analytical solution; however, it can be easily solved numerically, by iteration. It is advisable to write Eq. (18) in the following form: −D G a 1 ws r−1 21 1 m,1 + ln 2 + (ws −w )=0 2 2 RTas r w r 2
(19)
In this case we have a zero-place determination problem which can be solved with one of the
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common methods (e.g. interval-halving method, secant-method, Newton–Raphson method, etc.). (3) After the determination of ws the xs com2 1,1 position in the first monolayer can be calculated as follows: r(1−ws ) 2 (20) xs = 1,1 ws +r(1−ws ) 2 2 According to Eqs. (10) and (20) the equivalent layer thickness is xs +r(1−xs ) ns(n) V 1,1 m,1 1,1 ts = 1 equ as xs −x 1,1 1
(21)
2.5. Testing of the proposed method by model isotherms generated by the space filling model Very few direct experimental data are available about the thickness of the adsorption layer and its changes at various mixture compositions [5– 7,30,28,29,31,32]. Our possibilities for checking the parallel layer model proposed are similarly limited. We do have a possibility, however, to test the method with the help of an isotherm calculated according to Eq. (10) from an arbitrary layer thickness function derived from the space filling model, and to check whether the original layer thickness vs. bulk composition functions are re-obtained. The adsorption space filling method described earlier was chosen as the test method. The differences between the two models are the following. In case of the space filling model: (1) the adsorption layer is homogeneous and unstructured; (2) the characteristics of the layer contacting the surface are not different from those of the other layers; (3) a sharp phase boundary is supposed to divide the adsorption layer and the bulk phase; (4) cross-sectional areas are not taken into consideration, only the molar volumes of the individual components are used in the calculations. In case of the parallel layer model proposed: (1) the adsorption layer is structured both horizontally and vertically;
(2) a special significance as regards solid–liquid interaction is attributed to the layer of molecules contacting the surface; (3) there is a continuous transition between the adsorption layer and the bulk phase; (4) the cross-sectional area of component (1) is taken into consideration. What may be expected from a comparative analysis of this type? Under conditions where the two, very different models are consistent with each other, it is probable that the isotherm determined experimentally by the proposed method will also yield a realistic layer thickness vs. composition function. On the other hand, under conditions where the two models are inconsistent, the layer thickness function derived from the experimental isotherm should be treated with due caution and criticism. In our opinion the model proposed is a better approximation of reality than the space filling model, and the differences are primarily due to the imperfections of the space filling model. An essential precondition of the accuracy of our calculations is the exact determination of the D G function from the isotherm ns(n) =f (x ) which 21 1 1 necessitates the knowledge of bulk activities (a =f x ). Our first example in Fig. 6 represents i i i the layer thickness vs. x functions where r#1 (r= 1
Fig. 6. The equivalent adsorption layer thickness functions calculated from the model isotherms of Fig. 1(a) after Eq. (21) in toluene(1)–cyclohexane(2) mixtures on solid.
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1.0165) and the adsorption layer is supposed to be monomolecular. The values of the D G func21 tion were calculated from the toluene(1)–cyclohexane(2) model isotherm presented in Fig. 1(a) according to Eq. (17). Under these conditions the two models may be expected to be consistent. As demonstrated by Fig. 6, the initial (constant) layer thickness (0.384 nm) was also re-obtained at various values of K (K=3 to 103). The slight difference from theoretical values and the insignificance of standard deviations indicate that the method proposed is not sensitive to lesser integration errors. For the calculations, only a limited number of points (20–30) were taken into consideration and the numerical integration was done using the trapezium formula. In this way, errors occurring at the time of the integration of real, experimentally determined excess isotherms were, so to say, simulated in order to test the sensitivity of the method to integration errors. We chose for our second example the benzene(1)–n-heptane(2) model isotherms shown in Fig. 2(a) where the value of r is significantly different from unity (r=1.6488). Again, the adsorption layer is monomolecular. As demonstrated by Fig. 7, at high values of K (preferential adsorption) the layer thicknesses obtained by the model calculated according to Eq. (21) are in good
agreement with the initial (constant) layer thickness (0.388 nm), while at low values of K this is not true. This result is not surprising, since in the case of non-preferential adsorption (K=3 to 10) a significant part of the value of D G can be attrib21 uted to the entropy term, the value of which is dependent strongly upon the model chosen. It may thus be concluded that at low values of K the accuracy of the result depends on how close the entropy factor of our model approximates the real structure of the actual S/L adsorption layer. So far, the value of K has been considered constant throughout the entire composition range. In reality, however, the value of K may be a function of the equilibrium composition of the bulk phase. Let us, therefore, examine how changes in the value of K influence the functions ts=f(x ). We assume that the value of K changes 1 according to the geometric series K=K1−x1 Kx1 . x1=0 x1=1 As revealed by the layer thickness functions of the toluene(1)–cyclohexane(2) system shown in Fig. 8, non-preferential adsorption (K=5 to 20) does not alter the course of the function significantly. Our next example postulates that layer thickness increases in a linear fashion (ts=0.3836 nm (1+2x ) and r#1. The calculated adsorption 1 excess isotherms are presented in Fig. 9(a). Again
Fig. 7. The equivalent adsorption layer thickness functions calculated from the model isotherms of Fig. 2(a) after Eq. (21) in benzene(1)–n-heptane(2) mixtures on solid.
Fig. 8. The equivalent adsorption layer thickness function after Eq. (21) in toluene(1)–cyclohexane(2) mixtures on solid for variable equilibrium constants.
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In the next example, a parabola-shaped layer thickness function with one maximum is used in the calculations. The functions ts=f(x ) may be 1 calculated from ts=0.3836 nm (1+8x −8x2 ). The 1 1 calculated excess isotherms are shown in Fig. 10(a). The result obtained ( Fig. 10(b)) and its explanation are similar to those described for the case of linear layer thickness vs. composition functions. It is noteworthy that the character of the curve is not dependent upon the value of K
(a)
(a)
(b) Fig. 9. (a) The calculated adsorption excess isotherms in toluene(1)–cyclohexane(2) mixtures with linear layer thickness function, ts=0.3836(1+2x ) nm. (b) The equivalent adsorp1 tion layer thickness functions after Eq. (21) in toluene(1)– cyclohexane(2) mixtures on solid for different equilibrium constants, ts=0.3836(1+2x ) nm. 1
the two models are consistent in the case of preferential adsorption (see Fig. 9(b)). The reason for the difference at low values of K is that the two models differ regarding the distribution of the energy of solid–liquid interaction [18,20,31,32] within the adsorption layer: the space filling model assumes a uniform distribution, whereas according to our model a significant part of D G is concen21 trated in the layer of molecules in contact with the surface.
(b) Fig. 10. (a) The calculated adsorption excess isotherms in toluene(1)–cyclohexane(2) mixtures with ts=0.3836(1+8x 1 −8x2 ) nm. (b) The equivalent adsorption layer thickness func1 tions after Eq. (21) in toluene(1)–cyclohexane(2) mixtures on solid for different equilibrium constants.
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and the position of the maximum is not displaced either. Further calculations also were made in order to check the sensitivity of the proposed method to deviations originating from experimental error in determinations of the points of the excess isotherm and to uncertainties in values of specific surface area and cross-sectional area. The details of these calculations are not expounded here. In neither case was extreme sensitivity observed, i.e. the method is quite stable numerically. We present two examples to illustrate the correctness of the determination of the proposed function ts=f(x ). 1 The isotherm determined on pillared clay is a good example for constant values of ts: in this case the constancy of adsorption interlayer volume may be proven by an independent XRD measurement, ˚ since the value of basal spacing is d =17.8 A L throughout the entire composition range. On the other hand, the hydrophilic aerosil/benzene–n-heptane system serves as an appropriate example for variable layer thickness. Let us next examine how layer thickness in the aerosil R972/benzene(1)–n-heptane(2) system may be calculated, with the help of the equations obtained as described in Section 2.4, in the case of an excess isotherm with a non-linear section. The isotherm is shown in Fig. 11. Activities for the benzene–n-heptane liquid pair used in the Gibbs equation are based on the liquid–vapour equilibrium data [24,25]. When layer thickness is calculated by numerical iteration of Eqs. (18) and (19)
on a computer, the layer thickness vs. composition function shown in Fig. 11 is obtained.
Fig. 12 shows the excess isotherm determined in the benzene(1)–n-heptane(2) binary mixture on Al–PILC and the equivalent layer thickness function calculated therefrom. Since basal spacing ˚ and the thickness determined by XRD is 17.8 A 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 [8–12], the cross-sectional area of one benzene molecule is 0.38 nm2 [22], then this value means 1.05 molecular layers, i.e. an exactly monomolecular adsorption layer. This is in excellent agreement with the value predicted by the model.
Fig. 11. Adsorption excess isotherm on R972 aerosil in benzene(1)–n-heptane(2) mixtures and equivalent layer thickness function calculated by Eq. (21).
Fig. 12. Adsorption excess isotherm on Al–PILC in benzene(1)–n-heptane(2) mixtures and equivalent layer thickness function calculated by Eq. (21).
3. Materials and methods Benzene and n-heptane of analytical purity, on a Linde 4A molecular sieve were used for the experiments. Three different adsorbents were used. The first one was the hydrophobized silicate designated R972 (Degussa AG, Germany), as =115 BET m2 g−1. The second adsorbent was pillared Al–PILC prepared from Na-montmorillonite, as =132.8 m2 g−1. BET 4. Results
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Fig. 11 presents the excess isotherm determined on adsorbent R972 and the changes in the specific layer thickness as a function of changing mixture composition. On the basis of the so-called Rusanov criterion [2–4], a minimal layer thickness and a minimal layer number necessary for a thermodynamically consistent description of adsorption may be calculated for each excess isotherm. In the present case the value of this minimal layer number at low values of x is about unity, or even lower, 1 whereas at high values of x it is about three. The 1 function describing equivalent layer thickness calculated on the basis of the parallel layer model satisfies this criterion.
5. Conclusion Adsorption of toluene–cyclohexane and benzene–n-heptane binary mixtures on solid adsorbents was studied. Model isotherms were calculated by changing the equilibrium constant of adsorption with the help of known bulk activities. The thickness of the adsorption layer was at first assumed monomolecular and independent of composition. It was established that it is only in this case that the model isotherms have a linear section and that the Everett–Schay function x x /ns(n) =f (x ) may be applied for the determi1 2 1 1 nation of adsorption capacity. The higher the value of K, the longer the linear section of the isotherm is. If adsorption is not preferential and the value of K decreases, the range suitable for linear extrapolation is decreased. If model isotherms are not calculated according to the space filling model assuming a constant layer thickness but, conversely, layer thickness is a function of composition, i.e. ts=f(x ), excess 1 isotherms exhibit a strong inclination. The extent of inclination indicates how much the thickness of the adsorption layer depends upon the composition of the bulk phase. The aerosil/benzene–n-heptane system is a good example of this phenomenon: adsorption layer thickness was calculated by numerical iteration of Eqs. (18) and (19) we proposed. The calculation is based on the free enthalpy function determined according to the Gibbs equation using the excess isotherm and bulk activities;
the knowledge of the function allows the calculation of both the composition and the thickness of the adsorption layer. The correctness of the method was checked on an adsorbent (pillared clay) where the existence of an S/L monomolecular layer had been proven by XRD.
Acknowledgment The authors are very grateful to the Hungarian Research Scientific Fund, OTKA I/5 T007531, OTKA I/6 T014159, and to the Hungarian Ministry of Education MKM Pr. No. 219/1995.
References [1] G. Schay, L.G. Nagy, Period. Polytech. Chem. Eng. 4 (1960) 45. [2] G. Schay, in: E. Matijevic ( Ed.), Surface and Colloid Science, vol. 2, Wiley, London, 1969, p. 155. [3] G. Schay, in: D.H. Everett (Ed.), Surface Area Determination, Proc. Int. Symp. 1969, Butterworths, London, 1970, p. 273. [4] J.J. Kipling, Adsorption from Solutions of NonElectrolytes, Academic Press, London, 1965. [5] J. To´th, Acta Chem. Acad. Sci. Hung. 63 (1970) 67. [6 ] J. To´th, Acta Chem. Acad. Sci. Hung. 63 (1970) 179. [7] J. To´th, J. Colloid Interface Sci. 46 (1974) 38. [8] I. De´ka´ny, F. Sza´nto´, L.G. Nagy, G. Schay, J. Colloid Interface Sci. 93 (1983) 151. [9] I. De´ka´ny, L.G. Nagy, G. Schay, J. Colloid Interface Sci. 66 (1978) 197–200. [10] I. De´ka´ny, F. Sza´nto´, A. Weiss, L.G. Nagy, Ber. Bunsenges. Phys. Chem. 89 (1985) 62–67. [11] I. De´ka´ny, F. Sza´nto´, A. Weiss, G. Lagaly, Ber. Bunsenges. Phys. Chem. 90 (1986) 427–431. [12] I. De´ka´ny, F. Sza´nto´, A. Weiss, G. Lagaly, Ber. Bunsenges. Phys. Chem. 90 (1986) 422–427. [13] D.H. Everett, Trans. Faraday Soc., 60 (1964) 1803; 61. (1965) 2478. [14] D.H. Billett, D.H. Everett, E.E.H. Wright, Proc. Chem. Soc. (1964) 216. [15] D.H. Everett, Pure Appl. Chem. 53 (1981) 2181. [16 ] D.H. Everett, in: Colloid Science, vol. 3. Chemical Society, London, 1979, p. 66. [17] D.H. Everett, Prog. Colloid Polym. Sci. 65 (1978) 103. [18] A. Dabrowski, M. Jaroniec, J. Colloid Interface Sci. 77 (1980) 571. [19] A. Dabrowski, Monatsh. Chem. 114 (1983) 875. [20] A. Dabrowski, Monatsh. Chem. 117 (1986) 139. [21] G. Schay, Pure Appl. Chem. 48 (1976) 393–400.
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[28] D.H. Everett, G.H. Findenegg, Nature 223 (1969) 52. [29] G.H. Findenegg, J. Chem. Soc. Faraday Trans. 1: 69 (1973) 1069. [30] G.H. Findenegg, J. Colloid Interface Sci. 35 (1971) 249. [31] W. Rudzinski, S. Partyka, J. Chem. Soc. Faraday Trans. 1: 77 (1981) 2577. [32] S.W. Rudzinski, J. Zajac, I. De´ka´ny, F. Sza´nto´, J. Colloid Interface Sci. 112 (1986) 473.
Variable thickness of the liquid sorption layers on solid surfaces Ferenc Berger, Imre De´ka´ny * Department of Colloid Chemistry, Attila Jo´zsef University, H-6720 Szeged, Hungary Received 17 January 1997; accepted 7 May 1997
Abstract Calculations of the adsorption capacity and the thickness of the adsorption layer are of primary importance for studies of the adsorption of binary mixtures. Earlier methods were applicable to the determination of real adsorption capacity only in the case of monolayers and/or layers of constant thickness, independent even of mixture composition. In the case of the adsorption isotherms of binary mixtures of non-electrolytes these conditions are often not fulfilled, and neither the Schay–Nagy extrapolation method nor reciprocal isotherm representations can be used for the determination of adsorption capacity on the basis of adsorption isotherms. Here we present a method of calculation suitable for the computation of surface composition and adsorption layer thickness in the whole composition range. Thus, instead of a single value of adsorption capacity, the system is characterized by the layer thickness isotherm given as a function of the composition of the bulk phase. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Adsorption; Multilayers; Binary mixtures; Adsorption layer thickness; Surface layer composition; Adsorption capacity; Free enthalpy of adsorption
1. Introduction Two fundamental questions concerning the interpretation of liquid sorption isotherms of solid–liquid interfaces are whether the adsorption layer is monolayer or multilayer and whether the thickness of the adsorption layer depends upon the composition of the bulk phase. Results published in the literature are not uniform in this respect, since both monolayer [1–3] and multilayer models [4–7] are employed in the calculations. In the present work we would like to point out that it is only in certain extreme cases that a monolayer structure or multilayer with constant thickness is possible and that this structure is strictly associated with the linear region of the excess isotherm [1–4,8–12]. Furthermore, we present those liquid * Corresponding author.
sorption excess isotherms (registered in benzene– n-heptane mixtures) in the case of which the thickness of the adsorption layer demonstrably ranges from a monomolecular to a multilayer coverage on the surface of the solid adsorbent. Our calculations are based on the so-called parallel layer model, where intermolecular interactions are also taken into consideration for calculating layer thickness. When adsorption excess isotherms have no linear regions, it may be postulated, in general, that realistic information about the adsorption equilibrium will be obtained only by applying the model evoking an adsorption layer of variable thickness. Naturally, the appearance of multilayer adsorption needs to be proven by independent methods. The composition dependence of mono- or multilayer adsorption is unambiguously proven by interlamellar adsorption between pillared clay lamellae,
0927-7757/98/See front matter. Elsevier Science B.V. All rights reserved. PII S0 9 2 7 -7 7 5 7 ( 9 7 ) 0 0 17 1 - 4
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since values of basal spacing may be determined by X-ray diffraction ( XRD) measurements [10–12].
2. Theoretical background 2.1. The space filling model One of the most often used model for describing of the equilibrium between the adsorbed layer and the bulk phase is the ‘‘space filling’’ model [1–4,13–17]. In this model the volume of the adsorption layer is Vs=tsas=ns V +ns V =ns V (1) 1 m,1 2 m,2 1,0 m,1 where ts is the thickness of the adsorption layer, as is the specific surface area of the adsorbent, ns i is the amount of the ith component in the surface layer, V is the partial molar volume of the ith m,i component in the surface layer (in general the molar volume of the pure ith component is used) and ns is the adsorption capacity related to 1,0 component (1). When a molecule (2)s adsorbed on a solid surface is displaced by a molecule (1), an equilibrium constant K* can be defined [13,18–21]: r(1)+(2)sPr(1)s+(2)
(2)
(xs cs )rx c 1 1 2 2 (3) (x c )rxs cs 1 1 2 2 x , xs , c , cs are the mole fractions and the activity i i i i coefficients of the ith component in the bulk phase, and in the surface layer (index s). It is assumed that the stoichiometry of the displacement is governed by the ratio of the molar volumes, r=V /V . Neglecting the activity coefficients, a m,2 m,1 simplified equilibrium constant can be defined:
where ns is the material content of the surface phase and ns=ns +ns . If the ns(n) excess amount 1 2 1 is determined experimentally and the Vs adsorption volume (from ns or ns=ns +ns ) is known, 1,0 1 2 the xs surface layer composition can be calculated 1 as follows: rns(n) +x Vs/V 1 1 m,1 xs = (6) 1 Vs/V +ns(n) (r−1) m,1 1 There is also often used the so-called separation factor: xs x S= 1 2 (7) xs x 2 1 While Vs=ns V from Eq. (1) follows 1,0 m,1 ns =ns +rns =nsxs +rnsxs (8) 1,0 1 2 1 2 Starting from Eqs. (5), (7) and (8), we can get the Everett–Schay equation [2,3,13,15,16 ]:
A
B
x x r 1 S−r 1 2= (9) + x ns(n) ns S−1 S−1 1 1 1,0 When the value of S is independent of x , the 1 x x /ns(n) vs. x function is linear [2,3]. The values 1 2 1 1 of ns and S can be calculated from the slope and 1,0 intercept. It is easy to show on the basis of Eqs. (4) and (7) that, in the case of r=1, S=K. 2.2. Calculation of the model isotherms
K*=
(xs )rx 1 2 (4) (x )rxs 1 2 The solid/liquid interfacial adsorption is characterized by the reduced adsorption excess amount after the Ostwald–Izaguirre equation [2–4]: K=
ns(n) =ns −nsx =ns(xs −x ) 1 1 1 1 1
(5)
In order to test the proposed method for calculating the adsorption layer thickness, synthetic excess isotherms have been generated by means of the equations of the space filling model. For our model calculations the specific surface area of the adsorbent is considered constant (as=100 m2 g−1). Based on the r value of the mixture and an arbitrarily chosen K value, the xs 1 vs. x function can be calculated from Eq. (4) by 1 iteration. In addition, if the ts layer thickness, e.g. the Vs=tsas layer volume, is also fixed, the adsorption excess can be calculated as follows: Vs ns(n) = (xs −x ) (10) 1 1 1 xs V +(1−xs )V 1 m,1 1 m,2 Of course, ts does not have to be constant, it
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can also be a function of the bulk phase composition. 2.3. Analysis of the space filling model isotherms
(a)
(b)
In the first two examples, model isotherms have been generated at ts=constant, first in the case of r#1 then in that of r≠1. In both of them the monomolecular layer thickness of the component (1) has been chosen as the ts value. As a first example, the toluene(1)–cyclohexane(2) mixture has been chosen at which r=1.0165 and the monomolecular layer thickness calculated from the cross-sectional area value (0.46 nm2/molecule [22]) was 0.384 nm. When model isotherms ns(n) =f (x ) are calculated for this 1 1 binary liquid according to Eq. (10), the isotherm series shown in Fig. 1(a) and the equilibrium composition diagrams xs =f (x ) presented in Fig. 1(b) 1 1 are obtained for the various equilibrium constants (K=103 to K=3). Changes in the equilibrium constant K are dependent on the hydrophobicity of the surface, since we have previously established that, in the case of the adsorption of identical binary liquids, the value of the constant K is a function of surface polarity [8–12]. It follows, therefore, that in the case of the excess isotherms shown in Fig. 1(a), when K=103 the surface is hydrophilic, since toluene is preferentially adsorbed on the surface. The selectivity of toluene adsorption at various values of K is welldemonstrated by Fig. 1(b). The Everett–Schay function is used widely for the reciprocal representation of adsorption excess isotherms [2,8,9,13]. The advantage of representation according to Eq. (9) is its suitability for the determination of K and the monomolecular adsorption capacity ns . The function shown in 1,0 Fig. 1(c) is linear throughout the entire range of compositions. When the size of the components of
Fig. 1. (a) Calculated adsorption excess isotherms with different equilibrium constants at constant ts monolayer value (0.384 nm) in toluene(1)–cyclohexane(2) mixtures on solid. (b) The adsorption equilibrium diagram in toluene(1)–cyclohexane(2) mixtures on solid. (c) The Everett–Schay representation of the calculated excess isotherms in toluene(1)–cyclohexane(2) mixtures.
(c)
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(a)
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the binary liquid is varied and, for example, a binary liquid made up of benzene–n-heptane is considered, where r=1.6488, then the monomolecular layer thickness of benzene calculated from its cross-sectional area value (0.38 nm2/molecule [22]) is 0.388 nm. The shape of the excess isotherms calculated for the conditions described above already differs from those presented in Fig. 1 (see Fig. 2(a)). At identical values of the equilibrium constant K, the adsorption of benzene from nheptane is less preferential than it was from toluene (cf. Fig. 2(b)). On the other hand, the linear representation shown in Fig. 2(c) indicates that if the value of r is other than unity and the value of K is low, then the function x x /ns(n) =f (x ) is no 1 2 1 1 longer linear, i.e. it may not be used for the determination of either ns or K. 1,0 We now return to the examination of the toluene–cyclohexane binary liquid series with the difference that the thickness of the adsorption layer is no longer considered constant; according to the FHH isotherm [23] ts=0.384 nm {1+[A/ln(1/x )]1/B} (11) 1 where the B parameter is related to the rate of fall-off of the adsorption potential with the distance from the surface and A is an intermolecular interaction constant. With different A and B parameters, layer thickness will be a function of the composition x (see Fig. 3(a,b)). When calcula1 tions are done at identical values of K=30, it is apparent from the figures that the shape of the excess isotherms calculated according to Eq. (10) is altered: they exhibit a curvature in the range x =0.21.0. Near linearity is observed in the 1 range x =0.2–0.8 only if the FHH layer thickness 1 is independent of composition, i.e. in Eq. (11) A=0 and B=2.0. If layer thickness (volume) is a function of composition—as can be seen in Fig. 3(b)—then the function x x /ns(n) =f (x ) is 1 2 1 1 not linear but has a backward section (Fig. 3(c)). Changing the value of A in Eq. (11) naturally Fig. 2. (a) Calculated adsorption excess isotherms with different equilibrium constants in benzene(1)–n-heptane(2) mixtures on solid. (b) The adsorption equilibrium diagram in benzene(1)– n-heptane(2) mixtures on solid. (c) The Everett–Schay representation of the excess isotherms.
(c)
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(a)
(b)
309
means a change in adsorption capacity ns , since 1,0 the course of the isotherms ns(n) =f (x ) (see 1 1 Fig. 3(a)) and the slope of the straight lines (see Fig. 3(c)) are significantly different. If in Eq. (11) only parameter B, characteristic of the slope of the layer thickness function ts=f(x ), is varied and the value of A is kept 1 constant (A=1), a series of isotherms is obtained which intersect at x =0.37. The isotherm corre1 sponding to B=1000 has a linear region at x =0.2–1.0, i.e. it is justifiable to use the Schay– 1 Nagy extrapolation method for the determination of adsorption capacity [2–4]. According to Fig. 4(b), as parameter B decreases the isotherms incline because the adsorption layer becomes thicker than monomolecular. Model computations reveal a significant increase in the number of layers especially in the range of x 0.6. The isotherms 1 correspondingly exhibit a curvature and the representation x x /ns(n) =f (x ) will also be non-linear 1 2 1 1 (see Fig. 4(c)). The model calculations described above unambiguously prove that it is only in the case of a constant layer thickness or a monomolecular coverage, at r#1 and K=103–102 (i.e. under conditions of preferential adsorption) that both the Schay–Nagy extrapolation method and the Everett–Schay function are suitable for the determination of the adsorption capacity of the pure component ns . 1,0 Experiments, however, often yield data similar to the isotherms presented in Figs. 3(a) and 4(a), but, as we have seen, excess isotherms with bending (non-linear) sections are not suitable for the determination of adsorption capacity. 2.4. The proposed model for calculating the adsorption layer thickness function The strategy of our calculations was based on the following assumptions:
Fig. 3. (a) Calculated adsorption excess isotherms in toluene(1)– cyclohexane(2) mixtures on solid with different A values of the FHH equation, K=30, B=3. (b) The adsorption layer thickness function in toluene(1)–cyclohexane(2) mixtures on solid with different A values of the FHH equation, K=30, B=3. (c) The Everett–Schay representation of the excess isotherms.
(c)
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(1) the adsorption layer is made up of parallel monolayers ( Fig. 5(a)); (2) surface enrichment decreases nearly exponentially ( Fig. 5(b)); (3) surface tension (i.e. the free enthalpy of displacement D G) is restricted mostly to the 21 first layer (Fig. 5(d )): 2 s= ∑ s (12) i i=1 s#s (13) 1 where s is the surface tension and s is the i surface tension in the ith monolayer. (a)
There is the following relation between the free enthalpy of displacement and the surface tension: D G=(s−s* )as (14) 21 (2) where s* is the surface tension of pure compo(2) nent (2). Eqs. (12) and (13) can be written in another form:
(b)
2 D G= ∑ D G (15) 21 21 i i=1 D G #D G (16) 21 1 21 where D G is the free enthalpy of the displace21 i ment in the ith monolayer. In order to characterize the layer thickness in the case of a diffuse adsorption layer, we propose the use of the ‘‘equivalent layer thickness’’ ts . equ This means the thickness of an homogeneous surface layer which has the same composition as the first monolayer (i.e. xs =xs ) and contains an 1 1,1 equivalent excess amount (Fig. 5(b,c)). In the proposed method the calculation consists of the following principal theoretical steps. (1) The function D G vs. x is obtained by 21 1
Fig. 4. (a) Calculated adsorption excess isotherms in toluene(1)– cyclohexane(2) mixtures on solid with different B values of the FHH equation, K=30, A=1. (b) The adsorption layer thickness function in toluene(1)–cyclohexane(2) mixtures on solid at different B values of the FHH equation, K=30, A=1. (c) The Everett–Schay representation of the excess isotherms.
(c)
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Fig. 5. The schematic representation of the solid–liquid interfacial layer structure: (a) solid–liquid interface with multilayers; (b) the surface layer composition xs as function of the distance z from the surface; x is the equilibrium bulk composition, and ts is the 1 1 equ equivalent layer thickness; (c) the excess density function against the distance; the area of A is equivalent to the area of B; the total area under the density function dns(n) /dz is the surface excess amount; (d) the free enthalpy distribution function from the surface. 1
integrating the Gibbs equation [1–7,21]: a1 n s(n) 1 da* (17) D G=−RT 1 21 x a* 2 1 a*1=0 where a is the activity of component (1) in the 1 bulk phase and a =c x . Activity coefficients for 1 1 1 the toluene(1)–cyclohexane(2) and benzene(1)– n-heptane(2) systems were calculated from the Redlich–Kister equation of five parameters from the vapour–liquid equilibrium data [24,25]. (2) The calculation of the composition xs of 1,1 the first monolayer is based on the so-called ‘‘athermal parallel layer model’’. This model [26,27] is basically a monolayer model describing the thermodynamic behaviour of the first layer of molecules adjoining the solid surface. It is in fact a generalization of the well-known lattice model for systems containing molecules of different sizes [28,29]. The parallel layer model makes two essential assumptions: (a) supposing that the molar volume of component (2) is larger than that of component (1), component (2) is treated as an r-mer of component (1); (b) it is assumed that these r-mers are lined up parallel with the surface (‘‘flat’’ r-mers) and a position perpendicular to the surface (‘‘erect’’ r-mers) is excluded.
P
Several equations theoretically suitable for calculations may be deduced from the parallel layer model. After lengthy comparative tests we came to the conclusion that it is obviously the following equation that yields the best results [15,16 ]: as D G=RT 21 a m,1
C
1
ws r−1 ln 2 + (ws −w ) 2 2 r w r 2
D
(18)
where w is the volume fraction of component 2 (2) in the bulk phase and w =r(1−x )/ 2 1 [ x +r(1−x )], ws is the volume fraction of compo1 1 2 nent (2) in the first monolayer, D G is the free 21 1 enthalpy of displacement in the first monolayer, and a is the molar cross-sectional area of comm,1 ponent (1). When the values of w and D G are 2 21 1 known, ws may be calculated from the equation 2 above. This equation has no analytical solution; however, it can be easily solved numerically, by iteration. It is advisable to write Eq. (18) in the following form: −D G a 1 ws r−1 21 1 m,1 + ln 2 + (ws −w )=0 2 2 RTas r w r 2
(19)
In this case we have a zero-place determination problem which can be solved with one of the
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common methods (e.g. interval-halving method, secant-method, Newton–Raphson method, etc.). (3) After the determination of ws the xs com2 1,1 position in the first monolayer can be calculated as follows: r(1−ws ) 2 (20) xs = 1,1 ws +r(1−ws ) 2 2 According to Eqs. (10) and (20) the equivalent layer thickness is xs +r(1−xs ) ns(n) V 1,1 m,1 1,1 ts = 1 equ as xs −x 1,1 1
(21)
2.5. Testing of the proposed method by model isotherms generated by the space filling model Very few direct experimental data are available about the thickness of the adsorption layer and its changes at various mixture compositions [5– 7,30,28,29,31,32]. Our possibilities for checking the parallel layer model proposed are similarly limited. We do have a possibility, however, to test the method with the help of an isotherm calculated according to Eq. (10) from an arbitrary layer thickness function derived from the space filling model, and to check whether the original layer thickness vs. bulk composition functions are re-obtained. The adsorption space filling method described earlier was chosen as the test method. The differences between the two models are the following. In case of the space filling model: (1) the adsorption layer is homogeneous and unstructured; (2) the characteristics of the layer contacting the surface are not different from those of the other layers; (3) a sharp phase boundary is supposed to divide the adsorption layer and the bulk phase; (4) cross-sectional areas are not taken into consideration, only the molar volumes of the individual components are used in the calculations. In case of the parallel layer model proposed: (1) the adsorption layer is structured both horizontally and vertically;
(2) a special significance as regards solid–liquid interaction is attributed to the layer of molecules contacting the surface; (3) there is a continuous transition between the adsorption layer and the bulk phase; (4) the cross-sectional area of component (1) is taken into consideration. What may be expected from a comparative analysis of this type? Under conditions where the two, very different models are consistent with each other, it is probable that the isotherm determined experimentally by the proposed method will also yield a realistic layer thickness vs. composition function. On the other hand, under conditions where the two models are inconsistent, the layer thickness function derived from the experimental isotherm should be treated with due caution and criticism. In our opinion the model proposed is a better approximation of reality than the space filling model, and the differences are primarily due to the imperfections of the space filling model. An essential precondition of the accuracy of our calculations is the exact determination of the D G function from the isotherm ns(n) =f (x ) which 21 1 1 necessitates the knowledge of bulk activities (a =f x ). Our first example in Fig. 6 represents i i i the layer thickness vs. x functions where r#1 (r= 1
Fig. 6. The equivalent adsorption layer thickness functions calculated from the model isotherms of Fig. 1(a) after Eq. (21) in toluene(1)–cyclohexane(2) mixtures on solid.
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1.0165) and the adsorption layer is supposed to be monomolecular. The values of the D G func21 tion were calculated from the toluene(1)–cyclohexane(2) model isotherm presented in Fig. 1(a) according to Eq. (17). Under these conditions the two models may be expected to be consistent. As demonstrated by Fig. 6, the initial (constant) layer thickness (0.384 nm) was also re-obtained at various values of K (K=3 to 103). The slight difference from theoretical values and the insignificance of standard deviations indicate that the method proposed is not sensitive to lesser integration errors. For the calculations, only a limited number of points (20–30) were taken into consideration and the numerical integration was done using the trapezium formula. In this way, errors occurring at the time of the integration of real, experimentally determined excess isotherms were, so to say, simulated in order to test the sensitivity of the method to integration errors. We chose for our second example the benzene(1)–n-heptane(2) model isotherms shown in Fig. 2(a) where the value of r is significantly different from unity (r=1.6488). Again, the adsorption layer is monomolecular. As demonstrated by Fig. 7, at high values of K (preferential adsorption) the layer thicknesses obtained by the model calculated according to Eq. (21) are in good
agreement with the initial (constant) layer thickness (0.388 nm), while at low values of K this is not true. This result is not surprising, since in the case of non-preferential adsorption (K=3 to 10) a significant part of the value of D G can be attrib21 uted to the entropy term, the value of which is dependent strongly upon the model chosen. It may thus be concluded that at low values of K the accuracy of the result depends on how close the entropy factor of our model approximates the real structure of the actual S/L adsorption layer. So far, the value of K has been considered constant throughout the entire composition range. In reality, however, the value of K may be a function of the equilibrium composition of the bulk phase. Let us, therefore, examine how changes in the value of K influence the functions ts=f(x ). We assume that the value of K changes 1 according to the geometric series K=K1−x1 Kx1 . x1=0 x1=1 As revealed by the layer thickness functions of the toluene(1)–cyclohexane(2) system shown in Fig. 8, non-preferential adsorption (K=5 to 20) does not alter the course of the function significantly. Our next example postulates that layer thickness increases in a linear fashion (ts=0.3836 nm (1+2x ) and r#1. The calculated adsorption 1 excess isotherms are presented in Fig. 9(a). Again
Fig. 7. The equivalent adsorption layer thickness functions calculated from the model isotherms of Fig. 2(a) after Eq. (21) in benzene(1)–n-heptane(2) mixtures on solid.
Fig. 8. The equivalent adsorption layer thickness function after Eq. (21) in toluene(1)–cyclohexane(2) mixtures on solid for variable equilibrium constants.
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In the next example, a parabola-shaped layer thickness function with one maximum is used in the calculations. The functions ts=f(x ) may be 1 calculated from ts=0.3836 nm (1+8x −8x2 ). The 1 1 calculated excess isotherms are shown in Fig. 10(a). The result obtained ( Fig. 10(b)) and its explanation are similar to those described for the case of linear layer thickness vs. composition functions. It is noteworthy that the character of the curve is not dependent upon the value of K
(a)
(a)
(b) Fig. 9. (a) The calculated adsorption excess isotherms in toluene(1)–cyclohexane(2) mixtures with linear layer thickness function, ts=0.3836(1+2x ) nm. (b) The equivalent adsorp1 tion layer thickness functions after Eq. (21) in toluene(1)– cyclohexane(2) mixtures on solid for different equilibrium constants, ts=0.3836(1+2x ) nm. 1
the two models are consistent in the case of preferential adsorption (see Fig. 9(b)). The reason for the difference at low values of K is that the two models differ regarding the distribution of the energy of solid–liquid interaction [18,20,31,32] within the adsorption layer: the space filling model assumes a uniform distribution, whereas according to our model a significant part of D G is concen21 trated in the layer of molecules in contact with the surface.
(b) Fig. 10. (a) The calculated adsorption excess isotherms in toluene(1)–cyclohexane(2) mixtures with ts=0.3836(1+8x 1 −8x2 ) nm. (b) The equivalent adsorption layer thickness func1 tions after Eq. (21) in toluene(1)–cyclohexane(2) mixtures on solid for different equilibrium constants.
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and the position of the maximum is not displaced either. Further calculations also were made in order to check the sensitivity of the proposed method to deviations originating from experimental error in determinations of the points of the excess isotherm and to uncertainties in values of specific surface area and cross-sectional area. The details of these calculations are not expounded here. In neither case was extreme sensitivity observed, i.e. the method is quite stable numerically. We present two examples to illustrate the correctness of the determination of the proposed function ts=f(x ). 1 The isotherm determined on pillared clay is a good example for constant values of ts: in this case the constancy of adsorption interlayer volume may be proven by an independent XRD measurement, ˚ since the value of basal spacing is d =17.8 A L throughout the entire composition range. On the other hand, the hydrophilic aerosil/benzene–n-heptane system serves as an appropriate example for variable layer thickness. Let us next examine how layer thickness in the aerosil R972/benzene(1)–n-heptane(2) system may be calculated, with the help of the equations obtained as described in Section 2.4, in the case of an excess isotherm with a non-linear section. The isotherm is shown in Fig. 11. Activities for the benzene–n-heptane liquid pair used in the Gibbs equation are based on the liquid–vapour equilibrium data [24,25]. When layer thickness is calculated by numerical iteration of Eqs. (18) and (19)
on a computer, the layer thickness vs. composition function shown in Fig. 11 is obtained.
Fig. 12 shows the excess isotherm determined in the benzene(1)–n-heptane(2) binary mixture on Al–PILC and the equivalent layer thickness function calculated therefrom. Since basal spacing ˚ and the thickness determined by XRD is 17.8 A 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 [8–12], the cross-sectional area of one benzene molecule is 0.38 nm2 [22], then this value means 1.05 molecular layers, i.e. an exactly monomolecular adsorption layer. This is in excellent agreement with the value predicted by the model.
Fig. 11. Adsorption excess isotherm on R972 aerosil in benzene(1)–n-heptane(2) mixtures and equivalent layer thickness function calculated by Eq. (21).
Fig. 12. Adsorption excess isotherm on Al–PILC in benzene(1)–n-heptane(2) mixtures and equivalent layer thickness function calculated by Eq. (21).
3. Materials and methods Benzene and n-heptane of analytical purity, on a Linde 4A molecular sieve were used for the experiments. Three different adsorbents were used. The first one was the hydrophobized silicate designated R972 (Degussa AG, Germany), as =115 BET m2 g−1. The second adsorbent was pillared Al–PILC prepared from Na-montmorillonite, as =132.8 m2 g−1. BET 4. Results
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Fig. 11 presents the excess isotherm determined on adsorbent R972 and the changes in the specific layer thickness as a function of changing mixture composition. On the basis of the so-called Rusanov criterion [2–4], a minimal layer thickness and a minimal layer number necessary for a thermodynamically consistent description of adsorption may be calculated for each excess isotherm. In the present case the value of this minimal layer number at low values of x is about unity, or even lower, 1 whereas at high values of x it is about three. The 1 function describing equivalent layer thickness calculated on the basis of the parallel layer model satisfies this criterion.
5. Conclusion Adsorption of toluene–cyclohexane and benzene–n-heptane binary mixtures on solid adsorbents was studied. Model isotherms were calculated by changing the equilibrium constant of adsorption with the help of known bulk activities. The thickness of the adsorption layer was at first assumed monomolecular and independent of composition. It was established that it is only in this case that the model isotherms have a linear section and that the Everett–Schay function x x /ns(n) =f (x ) may be applied for the determi1 2 1 1 nation of adsorption capacity. The higher the value of K, the longer the linear section of the isotherm is. If adsorption is not preferential and the value of K decreases, the range suitable for linear extrapolation is decreased. If model isotherms are not calculated according to the space filling model assuming a constant layer thickness but, conversely, layer thickness is a function of composition, i.e. ts=f(x ), excess 1 isotherms exhibit a strong inclination. The extent of inclination indicates how much the thickness of the adsorption layer depends upon the composition of the bulk phase. The aerosil/benzene–n-heptane system is a good example of this phenomenon: adsorption layer thickness was calculated by numerical iteration of Eqs. (18) and (19) we proposed. The calculation is based on the free enthalpy function determined according to the Gibbs equation using the excess isotherm and bulk activities;
the knowledge of the function allows the calculation of both the composition and the thickness of the adsorption layer. The correctness of the method was checked on an adsorbent (pillared clay) where the existence of an S/L monomolecular layer had been proven by XRD.
Acknowledgment The authors are very grateful to the Hungarian Research Scientific Fund, OTKA I/5 T007531, OTKA I/6 T014159, and to the Hungarian Ministry of Education MKM Pr. No. 219/1995.
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