Size-Dependent Adsorption Models in Microporous Materials. 1

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I n d . E n g . C h e m . Res. 1996,34,3848-3855

Size-DependentAdsorption Models in Microporous Materials. 1. Thermodynamic Consistency and Theoretical Analysis Massimiliano Gionat and Manuela Giustiniani*” Dipartimento d i Ingegneria Chimica, Universita d i Cagliari, Piazza d’Armi, 09123 Cagliari, Italy, and Dipartimento di Ingegneria Chimica, Universita di Roma “La Sapienza”, V i a Eudossiana 18, 00184 Roma, Italy

We analyze size-scaling effects on the functional form of adsorption isotherms in microporous materials. In particular, Keller adsorption isotherms are carefully discussed, focusing attention on the description of energetidgeometric heterogeneity and on thermodynamic consistency (lowpressure behavior). In order to justify the Freundlich behavior shown by the Keller isotherms, a Monte Carlo model of preferential adsorption is discussed. Despite its simplicity, this model exhibits extremely rich behavior.

1. Introduction The thermodynamics of adsorbed mixtures constitutes a vast and well-established theory. Most of the efforts (ideal adsorbed solution theory (IAST), real adsorbed solution theory (RAST))treat the equilibrium between a fluid and an adsorbed phase as similar to gas-liquid equilibria and, thus, introduce activity coefficients for the adsorbed phase (Myers and Prausnitz, 1965; Myers, 1973, 1983; Talu and Myers, 1988). This approach is intrinsically phenomenological, in the sense that the activity coefficients are mostly obtained by data fitting and it is hard, if not impossible, to relate them to the adsorbate/adsorbent interactions and to energetic/geometric properties of the adsorbents. Another school of thought in the study of adsorption on heterogeneous solids makes extensive use of the concept that a real adsorbent is characterized by a continuous distribution, g(E),of adsorption energies so that the resulting isotherm is the average of a local adsorption energy (for a fured value of E ) over the probability density function (pdf) g ( E ) (Jaroniec and Madey, 1988;Valenzuela et al., 1988; Cerofolini and Re, 1993). This approach is usually adopted for deriving heuristically empirical isotherms by making use of a condensation approximation. The assumption of a broad spectrum of adsorption energies is highly plausible in microporous materials. An important aspect that has recently attracted increasing interest in connection with the development of fractal theories in the physical sciences is the influence of the molecular size of the admolecules on adsorption properties. This topic was first addressed by the group of Avnir, Farin, and Pfeifer, who introduced the m o l e c u l a r t i l i n g m e t h o d in order t o determine the surface fractal dimension of porous adsorbents representing a characteristic parameter related to surface roughness (Avnir et al., 1983, 1984; Pfeifer et al., 1983, 1984). Starting from the possibility of describing rough surfaces by means of a single intensive scaling exponent, many other authors have developed models of adsorption isotherms to include the effects of fractality (Pfeifer and Cole, 1990; Rotschild, 1991). Almost all these models are limited to single-component adsorption and cannot be extended to cover the case isotherms (AIS) of mixtures. ’ Universita di Cagliari. Fax +39-70-280774. E-mail: m d giona2.ing.uniromal.it. Universita di Roma “La Sapienza”. Fax: +39-6-44585339. E-mail: [email protected].

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0888-588519512634-3848$09.00/0

The analysis of the simultaneous influence of energetic disorder and complex geometric roughness has been considered by Keller (1988,19901,who proposes a class of thermodynamically consistent AIS. Keller isotherms satisfy the Maxwell equations and are characterized by a power-law (Freundlich) behavior at low pressures. In particular, the Freundlich exponent depends on the size of the molecules. The properties of the Keller model have been discussed by Staudt (1994). In this article, we critically discuss the functional form of Keller AIS, specially in the low-pressure limit. Our goal is 2-fold: to establish theoretically that deviation from Henry’s law is thermodynamically plausible and that at low pressures Henry’s law should be regarded exclusively as a model assumption regarding the nature of the adsorbed phase and to analyze the dependence of the Freundlich exponent appearing in Keller AIS on size-scaling effects and energetic disorder. This article is organized as follows. First, we briefly review the description of size-scaling effects in adsorption and the concept of thermodynamic consistency. Then we discuss the functional form of Keller AIS and their low-pressure behavior. A statistical mechanical model of dissociation adsorption is presented, whose resulting adsorption isotherms are formally equivalent to Keller AIS. In order to analyze the influence of energetic heterogeneity on low-pressure behavior, a realistic model of preferential adsorption is proposed, and some initial results are presented. An extensive experimental analysis of the validity of Keller AIS is presented elsewhere.

2. Size-ScalingEffects and Fractality The fractal theory of natural objects and dynamic phenomena in disordered structures can be rightly regarded as a theory of relativity with respect to size. In its geometrical meaning, the fractal description of natural objects is an approximate application of the HausdorfTmeasure theory. In connection with dynamic phenomena in disordered structures, it is a theory of the space-time propagation of fields (concentration fields in diffusion with or without reaction, displacement fields in oscillations, Schrodinger wave fields in the solution of quantum mechanical problems) in structures (fractals) which possess no characteristic length scale. The first application of fractal concepts in adsorption, the molecular tiling method, falls within the geometrical interpretation of fractal scaling: admolecules are regarded as yardsticks of characteristic size ri (or equiva-

0 1995 American Chemical Society

Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3849 N

lently of cross section ai),and the monolayer coverage n, should follow the scaling law

n,

- ri-D

dGa = -Sa dT

(3)

i=l

ai-012

(1)

The application of eq 1 is particularly useful when the probing molecules (labeled with the subscript i) belong to the same homologous sequence (e.g., alkanes CnHzn+2,n = 1, 2, ...1. The choice of the values for the molecular radii gives rise to a subtle question since the radii ri may depend on the energetic interaction in the adsorbed phase. The adsorption cross section may be different from the molecular cross section deriving from force-constant estimation of the Lennard-Jones potential determined from viscosity or second virial coefficients (Hirschfelder et al., 1954). A detailed analysis of this topic is developed by Giustiniani et al. (1995a). Further research on fractal models of adsorption has focused on the interpretation of the exponents appearing in semiempirical expressions of adsorption isotherms in terms of an intrinsic fractal dimension of the adsorbents. In particular, Fripiat et al. (1986) and Pfeifer et al. (1989a,b) extend the BET AI to fractal media, and Jaroniec et al. (1990) and Yin (1991) develop fractal models of multilayer adsorption within the framework of Frenkel-Halsey-Hill (FHH) isotherms, showing that the exponent v appearing in the FHH isotherm, n = n, log[PdPl", is related to the fractal dimension of the adsorbent D as v = 3 - D (see also Avnir and Jaroniec (1989)). Fractal models of multilayer adsorption are also discussed by Cheng et al. (1989). Meakin (1989) and Vlad (1993) apply multifractal analysis to the characterization of adsorption, and Sokolowska (1989) points out the problems associated with the simultaneous presence of energetic heterogeneity and fractality. The main limitation of all these models is that they are restricted to single-component adsorption and cannot be generalized t o multicomponent mixtures. The first consistent thermodynamic model of sizescaling effects in adsorption equilibria which is not limited to monolayer coverage scaling (as in eq 1)and to single-component adsorption was proposed by Keller (see section 4). Before analyzing this class of isotherms, it is advisable to review briefly the problem of thermodynamic consistency.

where n is the spreading pressure, A the extension factor, and ni the concentration in the adsorbed phase [mol/g of adsorbent]. In writing eqs 2 and 3, we use the notation adopted by Keller. Equations 2 and 3 are fully equivalent with the definition of Ga reported by Ruthven (1984), where G" = -An ZZ, &hi and, consequently, dG" = -Sa dT n dA CZ+: dni. By developing dG" and considering the Maxwell equations

+

3. Thermodynamic Consistency Thermodynamic consistency means the set of relationships deriving from thermodynamics that a wellformulated and thermodynamically correct AI model is required to satisfy. Thermodynamic constraints do not derive from model assumptions as t o the nature of the adsorbed phase and are intrinsically related to the foundation of thermodynamic functions (G, H, U,F) as state functions. The fundamental constraints in the developments of AI stem from the fact that the Gibbs free energy (G) is a state function (this constraint can be derived from the GibbsDuhem equation). If Ga is the Gibbs free energy of the adsorbed phase, we have N

G" = &'ni

+ +

and by further expressing the equilibrium conditions between the fluid phase and the adphase pf = p:, with pf = pupf RT log(f/Po), we arrive at the set of equations

+

(i ' j = 1)...,N) ( 5 ) which a well-formulated AI model is necessarily required to satisfy. In eq 5, Pi are the partial pressures and fi the fugacities. In the simplified case of the extension factor being independent of the partial pressures and of ideal gas behavior, fi = Pi,eq 5 attains the simplified form

It should be observed that in the definition of thermodynamically consistent models of adsorption, practically all the literature on adsorption add an extra condition deriving from the Henry's law limit at low pressures (see, e.g., Rudisill and LeVan (1988, 1992)) n(T,P) lim-=H P-0

i=l

+ A dn + 5' dni

p

(7)

where H is a finite non-zero constant (Henry's constant). The references usually quoted to validate eq 7 both for homogeneous and heterogeneous adsorbents are an article by Hill (1952) (see also Hill (1960)) and the thermodynamic analysis of Myers and Prausnitz (1965). In their analysis, Myers and Prausnitz consider a liquidlike model equation for the chemical potentials in the adsorbed phase:

,"= p; (79 + RT log(Pe ( n ) y ~ J

(8)

where xi is the molar fraction in the adsorbed phase, yi are the activity coefficients, and P,P(n)is the equilibrium pressure for the pure ith component at spreading pressure n. It is therefore clear from these assumptions that linear Henry behavior at low pressures should follow. The arguments, presented by Hill (1952) in the case of adsorption on heterogeneous solids, read as follows.

3850 Ind. Eng. Chem. Res., Vol. 34,No. 11,1995

Let us regard a heterogeneous surface as composed of patches of energy between E and E dE. The number dn(E) of molecules adsorbed on each patch (which can be regarded as an homogeneous surface) at low pressures is dn(E,P) = g(E)h(E)PdE,where h(E) is Henry's constant for each patch of energy E. By integrating over all the energies, it follows that the overall AI will still display linear Henry behavior n(P) = HP, with H = Gg(E)h(E)dE. Hill's analysis, which is based on the superposition principle, is correct, but does not consider one important aspect: the superposition principle can be violated if the adsorbed molecules are capable of selecting the most favorable adsorption sites, in which case the distribution of the molecules over the patches of different energy depends on the covering itself. A n example of this kind of phenomenon will be discussed in section 5 in connection with a physically (thermodynamically) admissible model of adsorption driven by surface diffusion. The case of adsorption with dissociation is another straightforward example of an adsorption phenomenon in which thermodynamics is not violated but the resulting isotherms exhibit a power-law behavior depending on the dissociation degree. This case will also be considered below in connection with the Keller model of AIS. To sum up, we regard as a proper definition of thermodynamic consistency only those limitations deriving from the fundamental laws of thermodynamics and not from model assumptions on the nature of the adsorbed phase. This implies that all the models satisfying eq 5 or, in the simplified case of ideal gas behavior and A = A(T), eq 6 are to be regarded as thermodynamically consistent.

+

4. Keller Adsorption Isotherms 4.1. Multicomponent Isotherms. Keller proposed a class of AIS based on consistency with the Maxwell equations in which size-scaling effects are encompassed by means of the fractal dimension of the adsorbent. For a generic multicomponent mixture, the Keller AIS read as

extension area are expressed by

where Y is an arbitrary differentiable function of its argument. The function Y obviously has the dimension N/cm and A cmVg of adsorbent. The size-scaling hypothesis enters the model as a closure condition to determine the exponents ai. In the case of monolayer covering of a single adspecies i, limp-, n = n, and n,ai, and therefore by eq 1,

-

ai ri-D Consequently, the exponents

+

+

can be written as

where a, and rr are respectively the exponent a and the molecular radius of an arbitrarily chosen reference component. In general, a, = a,(T), and consequently, the exponents ai are functions of the temperature. Experimental validation of the proposed model can be obtained by analyzing the temperature behavior of the exponents ai. According t o eq 14, the ratio aila, should be independent of the temperature. The experimental validation of this conjecture is discussed by Giustiniani et al. (1995b). The Keller model exhibits many interesting properties. First of all, the influence of the adsorbent's geometry and roughness, expressed by n, and D, and the influence of energetic heterogeneity, expressed by the nonconstant behavior of the adsorption energies with the partial pressures and ultimately with the coverage, clearly appear in the functional form of the isotherms. The decoupling of these two effects enables us to achieve a clearer picture of the role of energetic/ geometric heterogeneity in the resulting AIS. From eqs 12, the equation of state attains the form

d = n,(T)RT

where @ is a nonnegative, continuously differentiable function, such that the derivative of f *@'(f*) with respect to f * is nondecreasing and otherwise arbitrary (@(x)= (1 x1-l in the case of Langmuir monolayer coverage, @ ( x ) = FJ(1 - x)(l Fx - x ) ] , F =- 0, in the case of BET multilayer adsorption). The parameters Ck are related to the adsorption energies Ei = Ei(T,{Pi}) through the equations

ai

(13)

*)Y.(f*) Y(f *)

@.(f

(15)

Under the assumption that the extension area is independent of the fugacities, Y''(f*) = Y@P(f*),and therefore, Yy*) = Jfo" @(XI dx. By substituting these two expressions in eq 15, it follows that

which represents the equation of state. The enthalpic difference A H = Hf - Ha = -AT(&/ between the fluid and the adphase can be expressed as

anp

AH= and f

* is given by N

(11) k=l

The quantity n,(T) is an intrinsic property of the adsorbent which depends exclusively on temperature. In the Keller model, the spreading pressure and the

This expression allows us to estimate Ei independently of ni once calorimetric measurements of the isosteric heat of adsorption are available.

Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3851 0.6 I

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assume a functional expression of the form a(P)= (1 ~ u P ) / ( l UP)with a, 5 1. Finally, it should be noted that if the adsorption energy does not depend on the pressure, the Keller model reduces to the isotherm proposed by Sips (1948) (see also Rudzinski and Everett (1992)).

+

5. Low-Pressure Behavior and Deviation from Henry's Law

'

l

0.1

180

o 1

220

,

I

260

300

I

340 T [KI

I

380

Figure 1. Dependence of the Freundlich exponent a on the temperature T. Data taken from Trapnell on adsorption of CO on charcoal and reported by Rudzinski and Everett (1992). The solid line represents the linear fit a T proposed by Rudzinski and Everett. The analysis of other experimental data (Giustiniani et al., 1995b) and the results of Monte Carlo simulations (see section 5) do not confirm this linear behavior.

-

The fractal hypothesis equation (1) has practical consequences in the application of the Keller model to experimental data. As extensively discussed by Giustiniani et al. (1995a1, the number of parameters needed t o fit N single-component AIS by making use of the empirical Langmuir-Freundlich isotherms is 3N, while the corresponding Keller isotherms for Ei = constant require only 3 N parameters. This parameter reduction is a consequence of the size-scaling condition and of the requirement of thermodynamic consistency. 4.2. Single-Component Isotherms. Keller AIS exhibit another interesting phenomenon related to sizescaling properties. For the sake of simplicity, let us consider the case of single-component adsorption with monolayer covering and Ei = constant:

+

The size of the admolecules controls not only the monolayer coverage a t high pressures

n-n,a

for P - m

(19)

but also the low-pressure behavior

n(P)-Pa

for P - 0

(20)

in the form of a power-law Freundlich exponent. The thermodynamic consistency of the latter expression will be discussed in detail in the next section. Nevertheless, it is important to point out in this context that it would be completely erroneous to attribute the origin of a Freundlich exponent a less than unity to purely geometric effects. Figure 1 shows the behavior of a vs T obtained by Trapnell for adsorption of CO on charcoal and reported by Rudzinski and Everett (1992). As can be seen from this figure, the exponent a increases up to unity with an increase in temperature. This result has been confirmed by the analysis of other literature data (Giustiniani et al., 1995b) and supports the hypothesis that the interplay between energetic heterogeneity and size effects is the origin of a Freundlich behavior observed a t low pressures. Staudt (1994) and Staudt et al. (1995) have recently extended the Keller model to include the dependence of the exponent a on the pressure. Specifically, they

There are many experimental results indicating a Freundlich behavior at low pressures. For example, Baker and Fox have analyzed krypton and xenon adsorption on Pyrex glass and nickel at ultravacuum pressures (up to 5 x Torr) and found values of a close to V 3 . A review of other literature data can be found in Rudzinski and Everett (1992). Amongst the theoretical contributions in the statistical mechanical explanation of the Freundlich exponent, attention should be drawn t o the work of Nitta et al. (1984a,b). The authors consider multisite occupancy on heterogeneous surfaces characterized by random or patchwise energetic topography and derive a linear dependence of the Freundlich exponent on temperature and on the number of carbon atoms in the hydrocarbon chain. In this section, we analyze the origin of anomalous nonlinear behavior at low pressures by considering two simple models. The first is the case of adsorption with dissociation, which is well-known in the literature. We perform a statistical mechanical analysis of this model in order to recover Keller AIS, at least formally. A model of preferential adsorption driven by nearest-neighbor hopping is then proposed, and some initial results are presented. The analysis of this model is particularly interesting in order to discuss the nature of the anomalies which give rise t o the Freundlich behavior in connection with energetic heterogeneity and phase transitions. 5.1. Statistical Mechanical Analysis of Dissociative Adsorption and Derivation of Keller AIS. It is known from kinetic arguments that dissociative adsorption gives rise to power-law adsorption isotherms at low pressure. In this section, we consider the statistical mechanical analysis of adsorption with dissociation since it will highlight some formal properties of the Keller model. Let us consider the single-component adsorption (the gas phase consists of a single species) of a species Ai on a lattice composed of M sites, and let pi be the dissociation degree (the number of subparts into which an admolecule splits at the surface). The partition function Q" of this sistem takes the form

From eq 21, the chemical potential L,( sorbed phase can be readily obtained as

of the ad-

where k g is the Boltzmann constant and 8 = NpJM the coverage. By equating this expression with the expression for the chemical potential in the gas phase (under ideal gas conditions), we obtain the adsorption isotherm

3852 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995

(23) where

a,= llpi

(24)

c ( T ) = q(T) exp(a&T)lkBT) = P I , exp(a,E,lkBT)

(25) As can be seen from these expressions, the AI obtained coincides with the Keller AIS in the case of adsorption energies independent of pressure. This equivalence is, however, purely formal since Keller AIS and dissociative adsorption do have different physical origins. However, the above calculations highlight two points: (1)By means of plausible physical assumptions on the nature of the adphase, it is possible to derive from rigorous arguments a power-law AI at low pressures, confirming once and for all that the constraint of linear Henry behavior a t low pressures is not an intrinsic thermodynamic result but rather a model assumption. (2) In eq 25, as in the case of Keller AIS, the exponential Arrhenius term in c ( T ) depends on the group a&/kBT, Le., explicitly on the product of aztimes the adsorption energy E,. Bearing in mind the physical differences between Keller AIS and the hypothesis underlying the derivation of eq 23, it is possible to use eq 23 as a starting point to derive the more general equation (91, in the case of E, = Ei(Z',{P,}). For this purpose, let us put the AI in the form

n,(T,{P,>>= n,Wf*)[qc,Plai+ P,Mll

heterogeneous energetic distribution of the adsorbent material. In this section, we present a model of preferential adsorption showing Freundlich behavior at low pressure and capable of explaining many phenomena of adsorption on heterogeneous solids. This article is confined to model formulation and to discussion of the relevant numerical results in connection with the thermodynamic theory of adsorption. We define as preferential adsorption models (following Pfeifer (1988))those models in which particles adsorb preferentially on the most energetic sites. The ideal case is that in which all the admolecules adsorb at the most energetical sites. In this case, there exists a closed-form relation between the total coverage 8t and the lower value of the adsorption energy E* up to which particles are adsorbed. In other words, we can define a coverage distribution function 8(E)such that 8(E)dE represents the fraction of adsorption sites, having energy between E and E dE,filled with admolecules. In the case of ideal preferential adsorption, 8(E) is a step function

+

(29) where E* is defined in terms of the energy distribution function F(E) = SEg(E)dE by means of the relation F(E*) = 1 - e,

(30)

In particular, if the pdf is exponential, g ( E ) = /3/kB Te-DElkBT,then

(26)

(31)

where the M,(i = 1, ..., h9 depend on the adsorption energies and on their derivatives with respect to the partial pressures and are unknown functions to be determined by imposing Maxwell equations. By substituting eq 26 in eq 6, one obtains the following conditions:

Up to this point, the analogy between the preferential adsorption model and condensation approximation is evident. Starting from the condition of local equilibrium between adsorption and desorption,

kd(E)e(E) = ka(E)P(l- 8(E)) where fl,,j. =

ai(Zj- M j ) + a,(M, - Zi) + z p i - Z,Mj

and ai = qciPia,, Zi = C~l(aCJ(aPi)P~a~. Since the function CD is arbitrary, eqs 28 generically admit a solution if simultaneously f1,u = 0 and f2,y = 0. The condition fi,,j. = 0 implies Mi = Zi (i = 1,...,N ) , and this solution identically satisfies the second set of conditions f i ,= ~ 0. Rearranging the terms, eq 26 with M i = Zi is exactly the Keller isotherm. 5.2. Preferential Adsorption:Monte Carlo Simulations, Energetic Heterogeneity,and Phase Transitions. Apart from the very peculiar case of dissociation, it can be proved from statistical mechanical arguments that, independently of geometric heterogeneity and fractality, an energetic homogeneous adsorbent always exhibits linear Henry behavior a t low pressures (Cole et al., 1986; Pfeifer, 1988). The physical origin of nonunitary Frendlich exponents should be sought in an

(32)

and integrating with respect to the distribution function g(E),we obtain in the ideal case

where Ka = k,(E)g(E)dE. Equation 33 enables us t o obtain the AIS once k,(E) and k@) are specified. Let us consider the case of an exponential distribution of desorption rates

k&E) = KdeVYEikBT

(34)

and of uniform adsorption rate k, = Ka (the slope of k a ( E ) is irrelevant in the low-pressure limit). For the definition of 8(E),the overall (experimentally measurable) coverage et is given by

8, = hm8(E)gCE)dE By inserting eq 31 into eq 33, we obtain

(35)

Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3853 1.0

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a

1

and therefore at low pressures

e,

-

pa

(37)

with

P a=y+p

(38)

What is the physical reason for this phenomenon? It is clear that the global coherent selection of the most favorable sites should be justified in terms of a microscopic model of local interactions between admolecules and surface structure. The local mechanism of site selection is represented by adparticle hopping between nearest-neighboring adsorbing sites. Surface diffusion phenomena have been discussed by many authors, and a review on surface diffusion has been presented by Kapoor et al. (1989). To simplify the picture, let us consider the case of a one-dimensional model of the adsorbent structure, labeling the spatial distribution of the energies on the surface sites with a subscript i, i.e., Ei. Following the classical way of describing surface diffusion (Yang et al., 19731, an adparticle adsorbed on a site i can move randomly over the surface at the hopping rates hi-j, j = i f 1, (39) where

The parameter controlling diffusion and adsorption rate is the Thiele adsorption modulus, defined as the ratio between the characteristic time for diffusion Zd and the characteristic time for adsorption Za, 4, = Zd/Za. From the above analysis, it can be easily seen that the case of ideal preferential adsorption corresponds to the situation 4, 0, T 0, where the temperature zero limit means the case Elk&” >> 0, in which the random motion induced by the temperature is negligible with respect to the energetic field on the adsorbent surface. Therefore, the ideal case 4, 0, T 0 corresponds t o the deterministic diffusion toward the most energetic sites. Monte Carlo simulation of the adsorption model with hopping rates can thus be carried out followingthe same procedure outlined by Seri-Levy and Avnir (1993) in their study of hysteresis in adsorptioddesorption equilibria on fractal surfaces. In particular, we consider 1 - D lattices of lo5 lattice sites, assuming an exponential distribution of adsorption energies defined in the interval (0, E,,,) and a corresponding distribution of desorption rates, eq 34. Figure 2 shows the behavior of the exponent a as a function of the temperature T (or more precisely of the dimensionless group Emax/kBT).At high temperature, a 1,and with the decreasing of temperature, a l/2. This temperature behavior has been observed experimentally (see Figure 1)and is confirmed by the analysis of the experimental data reported by Giustiniani et al. (199513).

- -

- -

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0.2

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0.0 0

1

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Figure 2. Freundlich exponent a vs E,,IkBT on a regular spatial distribution of adsorption energies with j3 = 1, y = 1. The solid line is the theoretical prediction, eq 38.

The origin of the dependence of the exponent a on temperature derives from the order-disorder transition in the adphase in the presence of a distribution of adsorption energies. This phenomenon is discussed in detail by Cerny (1983); see also Woodruff et al. (1983) and Nicholson and Parsonage (1982). At very low temperatures, admolecules are adsorbed on the more energetic sites: the adphase behaves like an ordered solidlike phase. As the temperature increases, there appears a mobile, disordered gaslike phase, since at high temperatures the hopping rates in eq 39 are practically insensitive to the energy differences Eikl - Ei. This analysis leads to the conclusion that the low-pressure Freundlich exponent a can be regarded as the macroscopic manifestation of a phase transition in the adphase. The hypothesis of an order-disorder transition in the adphase explains the limitations of eq 8 for the chemical potentials, which is valid only for a gaslike adsorbed phase. 6. Concluding Remarks

In this article, we have analyzed the size-scaling properties of adsorption isotherms in detail, with particular emphasis on the Keller model. We have presented some examples showing that adherence t o Henry’s law at low pressures should not be regarded as a thermodynamic constraint but is a model assumption on the nature of the adphase. To summarize, the main conclusions of this article are as follows: (1) The power-law behavior of ni with pressure Pi shown by Keller AIS a t low pressures should not be considered as a weakness of the Keller model. The Keller model is rigorously thermodynamically consistent since it satisfies the Maxwell equations. (2) The deviation from Henry’s law at low pressures cannot be attributed to geometric effects (fractality) but to energetic disorder and more specifically to orderdisorder transitions in the adphase. (3)The model of preferential adsorption with hopping between nearest-neighboring sites is a simple physical model showing Freundlich behavior at low pressures and all the fundamental features (e.g., temperature behavior) of the order-disorder transition associated with a continuous distribution of adsorption energies. The temperature behavior of the Freundlich exponent predicted by this model finds confirmation in experimental results (Rudzinski and Everett, 1992; Giustiniani et al., 199513). This model provides a clearer

3854 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995

understanding of the properties of this transition in connection with experimentally measurable thermodynamic quantities. The really challenging problem that still remains is the relationship between the Freundlich exponent, the size of the admolecules, and the fractal dimension of the adsorbent surface. Finally, it should be noted that Keller AIS are valid for generic fractal adsorbents and therefore, at fortiori, for regular Euclidean surfaces (D= 2) and for regular three-dimensional adsorbents (D= 3).

Acknowledgment We thank D. K. Ludlow, W. A. Schwalm, M. K. Schwalm, J. U. Keller, R. Staudt, and A. Viola for useful discussions.

Nomenclature n, = monolayer coverage [mol/g of adsorbent] rI = molecular size of the ith species [AI D = fractal dimension A = extension area [cm2/gof adsorbent] n, = concentration in the adsorbed phase of the ith species [mol/g of adsorbent] P, = partial pressure in the gas phase of the ith species [atml T = temperature [Kl fi = fugacities n, = surface adsorption capacity [moVgI c1= parameters defined in eq 10 PI- = parameters entering into eq 10 R = universal gas constant E = adsorption energy f * = quantity defined in eq 11 E* = threshold energy g = pdf of adsorption energies k = rate distribution kg = Boltzmann constant

Greek Letters x = spreading pressure [N/cml p I = chemical potential of the ith species a = Keller exponent defined in eq 14 CD = function entering into eq 9 Y = arbitrary differentiable function, see eq 12 [N/cml NE) = coverage distribution function et = total coverage /3 = exponent in the exponential pdf g(E) y = exponent in eq 34 = Thiele adsorption modulus Superscripts a = adsorbed phase f = fluid phase Subscripts r = reference component

a = adsorption d = desorption

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Received for review December 2, 1994 Revised manuscript received J u n e 13, 1995 Accepted J u n e 27, 1995@ IE940714G

Abstract published i n Advance ACS Abstracts, September 15, 1995. @