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Langmuir 1997, 13, 1054-1063
Hard Rods on a Line as a Model for Adsorption of Gas Mixtures on Homogeneous and Heterogeneous Surfaces† V. A. Bakaev‡ and W. A. Steele* Department of Chemistry, The Pennsylvania State University, 152 Davey Laboratory, University Park, Pennsylvania 16802 Received October 19, 1995. In Final Form: February 7, 1996X It is shown that a mixture of hard rods on a line in a homogeneous external field is ideal. The composition of such an adsorbed mixture is determined not by the relative values of the energies of the adsorbed molecules but by the relative values of a function similar to the enthalpy of a single molecule. In contrast, the hard rod mixture in an inhomogeneous external field is not ideal but displays minor negative deviations from Raoult’s law. It is suggested that the deviation from ideality depends significantly upon the dimensionality of the space.
I. Introduction The thermodynamics of liquid solutions is conventionally described by describing deviations from ideal behavior in terms of excess functions and activity coefficients.1 Myers and Prausnitz have extended this method to adsorbed solutions.2 They first defined the ideal adsorbed solution as a mixture where the chemical potentials of components are
βµi ) βµi°(π,T) + ln xi
(1)
Here xi is a mole fraction, T is temperature (β ) 1/kT), π is a two-dimensional pressure (spreading pressure), and µi° is the chemical potential of the adsorbed pure ith component at the values of π and T equal to those of the adsorbed solution. This definition of an ideal adsorbed solution differs from that of an ideal three-dimensional mixture3 only in that the pressure in the chemical potentials of pure components is replaced by the spreading pressure. The original intention of the authors of ref 2 was to calculate activity coefficients from the available experimental data on adsorption equilibria and then to interpret the resulting activity coefficients by a suitable theory. Surprisingly, the activity coefficients obtained in this way were found to be equal to unity within the experimental error. Thus, the adsorbed solutions which are formally two-dimensional were found to be much closer to ideality than their three-dimensional counterparts. As Myers and Prausnitz concluded: “On the basis of physicochemical considerations it is very surprising that mixture adsorption equilibria should be so closely approximated by the concept of an ideal solution. The only way to explain this result is to compare the predictions of a realistic statistical mechanical model with those of an ideal solution.2” They also believed that there was a significant problem connected with the fact that the available experimental data for mixed adsorption equilibria referred to heterogeneous surfaces while the statistical models were developed mainly for homogeneous surfaces. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. ‡ Permanent address: Institute of Physical Chemistry, Russian Academy of Sciences, Leninsky Prospect 31, Moscow 117915, Russia. X Abstract published in Advance ACS Abstracts, September 15, 1996.
(1) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969. (2) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (3) Prigogine, I. The Molecular Theory of Solutions; North Holland Publishing Co.: Amsterdam, 1957.
S0743-7463(95)00911-5 CCC: $14.00
In the ensuing papers, the approximation of the ideal adsorbed solution (IAS) has become very popular because it affords an opportunity to evaluate the composition of adsorbed mixture without tedious experimental measurements, using only the data obtained from the adsorption of the pure components which is much simpler to measure than adsorption of mixtures. For the majority of systems, the IAS approximation provides predictions that are in good agreement with experiment but there are also some exceptions (see, e.g., ref 4). On a homogeneous surface in a submonolayer region, a mixture of simple molecules at low temperatures may be considered as a two-dimensional solution. The welldeveloped methods of statistical thermodynamics of bulk solutions can be adapted to this system. In particular, a van der Waals one fluid theory has been successfully employed for considering adsorption on graphitized carbon black.5 Mixed adsorption on graphitized carbon black was also studied by computer simulation.6 Both methods confirmed the validity of IAS for adsorption on homogeneous surfaces when the components do not display a large deviation from the Raoult’s law in the bulk solution. In some cases, the validity of the IAS approximation has been confirmed by computer simulations even for a heterogeneous porous system.7 On the other hand, new experimental evidence and computer simulations have shown that a number of adsorbed solutions are far from ideality. Some of these results are summarized in a recent review article8 (see also ref 9 and references therein and ref 10). The present paper may be considered as a response to the suggestion in the quotation from ref 2. Thus, we develop here an exact statistical mechanical model capable of clarifying the problem of the ideality of a class of adsorbed mixture. This approach is somewhat different from the majority of works in this direction which were dedicated to the search of recipes (mainly semiempirical) that provided quantitative estimates of nonideality of adsorbed solutions (cf., e.g., refs 8 and 10). The model considered here is a one-dimensional model of hard rods on a line. We have already dealt with this (4) Valenzuela, D. P.; Myers, A. L.; Talu, O.; Zwiebel, I. AIChE J. 1988, 34, 397. (5) Monson, P. A. Chem. Eng. Sci. 1987, 42, 505. (6) Finn, J. E.; Monson, P. A. Mol. Phys. 1991, 72, 661. (7) Kaminsky, R. D.; Monson, P. A. Langmuir 1994, 10, 530. (8) Talu, O.; Li, J.; Myers, A. L. Adsorption 1995, 1, 103. (9) Cracknell, R. F.; Nicholson, D. Adsorption 1995, 1, 7. (10) Tien, C. Adsorption Calculations and Modeling; ButterworthHeinemann: Boston, 1994.
© 1997 American Chemical Society
Adsorption of Gas Mixtures on Surfaces
model in a previous paper11 where we considered a onecomponent system in an inhomogeneous external field. The foundation of the statistical mechanics of such a system is the integral Percus equation. The extension of this equation to the case of a mixture of hard rods of different lengths has been given in ref 12. Reference 11 contains a short review of applications of the onedimensional hard rod model and the Percus equation to adsorption problems including the adsorption of mixtures. The one-dimensional hard rod model of adsorbed mixtures has been considered in refs 12-15. Also the exact solution obtained for a mixture of hard rods interacting through a square-well nearest-neighbor potential16 has been used to test an approximate statistical thermodynamics theory of adsorption of mixtures in zeolites.17 However, in all these models, the inhomogeneity of external field was concentrated only at the end points of a line (in fact, it was a one-dimensional model of a slit-shaped micropore). The novelty of the present paper is that we consider here a binary adsorbed mixture of hard rods of different lengths in a random external field. A one-component system of hard rods in a random external field has already been considered in ref 18. The paper consists of two main parts: in section II we consider the binary mixture of hard rods in a constant external field which is a model for adsorption on a onedimensional homogeneous “surface”, and in section III we consider the same mixture in a random external field which is a model for a heterogeneous surface. The subsections of section II include the description of the statistical mechanical model as well as the application of the thermodynamic IAS method of ref 2 to the model and a numerical example which illustrates some general features of mixed adsorption. The subsections of section III include a description of the method of simulation of the random external field, a numerical method for solving the system of coupled integral equations from ref 12, a numerical example showing how the heterogeneity of a surface induces nonideality in an adsorbed mixture, and a discussion of this nonideality in comparison to that obtained in other approaches to this problem. Finally, in the Conclusion, we formulate a hypothesis for the unusually widespread ideality of adsorption solutions that is suggested by the results of the present work. II. Homogeneous External Field A. Rigid Molecules at a Surface Step: Statistical Mechanical Model. Consider first an m-component mixture of hard rods on a line confined between points x ) 0 and x ) L. The configurational integral for this system is known (see, e.g., refs 19 and 12)
ZN ) (L - lN)N for L > lN and ZN ) 0 for L e lN (2) where (11) Bakaev, V. A.; Steele, W. A. J. Chem. Phys. 1995, 103, 751. (12) Vanderlick, T. K.; Davis, H. T.; Percus, J. K. J. Chem. Phys. 1989, 91, 7136. (13) Monson, P. A. Mol. Phys. 1990, 70, 401. (14) Tang, Z.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1992, 97, 5732. (15) Kaminsky, R. D.; Monson, P. A. Langmuir 1993, 9, 561. (16) Monson, P. A. Mol. Phys. 1990, 70, 401. (17) Kaminsky, R. D.; Monson, P. A. AIChE J. 1992, 38, 1979. (18) Bakaev, V. A.; Steele, W. A. In Fundamentals of Adsorption, (Proc. 5’th Internat. Conf., Asilomar, California, May 13-18, 1995); LeVan, M. D., Ed.; Kluwer Academic Publishers: Dordrecht, in press. (19) Burgos, E.; Bonadeo, H. J. Chem. Phys. 1988, 88, 1163.
Langmuir, Vol. 13, No. 5, 1997 1055 m
lN )
Niai ∑ i)1
(3)
is the length of all N hard rods, Ni of which belong to the ith component characterized by the length ai. For completeness, we outline here the derivation of eq 2 since it is omitted in refs 19 and 12. Consider the region of configuration space determined by inequalities
0 < x1 < x2 < ... < xN-1 < xN < L
(4)
Assume that the center of a hard rod of species R(i) is located at each point xi. Thus, a sequence R(1), R(2), ..., R(N) determines the species of the hard rods situated at points of the region (4), many of R(i) being the same since N is much larger than the number of species. The configurational integral for the part of configuration space determined by inequalities (4) is
ZN* )
∫l′L-a
R(N)/2
N
dxN
∫l′x -a N
N-1
N-1,N
dxN-1 ...
∫l′x -a 3
2
2,3
dx2
∫l′x -a 2
1,2
1
dx1 (5)
where
l′1 ) aR(1)/2 and for i > 1, l′i ) l′i-1 + ai-1,i; ai-1,i ) (aR(i-1) + aR(i))/2 A substitution yi ) xi - l′i which is a generalization of that for the one-component case gives (exactly as for one component)20
Z N* )
∫0L-l dyN ... ∫0y dy1 ) (L - lN)N/N! N
2
for
L > lN and ZN* ) 0 for L e lN (6) N where lN ) ∑i)1 aR(i) is the same length of a stick composed of all N hard rods as defined in eq 3 since it does not depend on a permutation of hard rods. For the same reason, all the regions of configuration space determined by N! inequalities (4) differing only by permutations of the xi (which taken together cover all of configuration space) make equal contributions to configurational integral. This proves eqs 2 and 3. This system models adsorption of repulsive molecules on a step of a surface which occurs in reality in a threedimensional space. However, the one-dimensional configurational integral of eq 2 constitutes the main configurational contribution to the partition function of the three-dimensional system of rods at a surface step. This can be well approximated as
QN )
exp(-βNRuR°) qRNR(L - lN)NR 3NR NR!ΛR
∏ R
(7)
Here Λi is the thermal wavelength, ui° is the average energy of a molecule adsorbed on the step, qi is the partition function of a single molecule (that has a dimension of area and makes the whole expression dimensionless), and index i designates the ith species. Both ui° and qi depend on temperature. (20) Lieb, E. H.; Mattis, D. C. Mathematical Physics in One Dimension; Academic Press: New York, 1966.
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The Helmholtz free energy corresponding to eq 7 is
βF )
∑R
NR[βuR° + ln(ΛR3NR/qR) - ln(L - lN) - 1] (8)
The one-dimensional (spreading) pressure π ) -∂F/∂L is
βπ ) N/(L - lN)
(9)
where N ) ∑RNR and the Gibbs free energy G ) F + πL is
βG )
qi ) z02
∑R NR[βuR° + ln(ΛR3βπ/qR) + βπaR] + ∑R NR ln xR
(10)
2π m βui°nm n
m
∑
m
yR ) 1;
R)1
(11)
and the mole fraction of the ith component as xi ) Ni/N. Upon differentiation of eq 10 with respect to Ni, we obtain eq 1. This means that the mixture of hard rods of different lengths on a homogeneous surface is an ideal adsorbed solution. This is an interesting peculiarity of the onedimensional model since in the spaces of higher dimensions, mixtures of hard disks or hard spheres of different sizes are not ideal but display negative deviations from Raoult’s law.3,21 Also it is apposite to emphasize here that the homogeneity of the external field (surface) is a necessary precondition of ideality of the hard rod mixture. In an inhomogeneous random external field (on a heterogeneous surface) the hard rod mixture is nonideal (see below). It is nonideal even on a uniform line of finite length where the heterogeneity is concentrated at the end points of the line.12 The chemical potential of the ith component in the ideal gas phase is
βµi ) ln βpyi + ln Λi3
(12)
where yi is the mole fraction of the ith component in the gas phase. We set the chemical potentials of the gas and adsorbed phases equal to obtain
βµi°(π,T) + ln xi ) ln βpyi + ln Λi3
(13)
Now, define the dimensionless variables
p* ) βpq1a1; π* ) βπa1; qi* ) qi/q1; ai* ) ai/a1 (14) and substitute eqs 11 and 14 into eq 13 to obtain
xi p*qi* ) exp[-(βui° + π*ai*)] yi π*
(15)
In eqs 11 and 14, qi designates the configurational integral of the hard rods of species i with respect to the two coordinates normal to the step. Since we included q1 in the definition of the dimensionless pressure p*, we need not know the absolute values of qi but only the ratios qi* (cf. eq 14). Thus we make only a rough evaluation of qi. It is convenient (but not essential) to assume that qi is the product of configurational integrals corresponding to the harmonic oscillation of a hard rod normal to the line. Then (see ref 22 eqs 3.6.27 and 3.8.4) (21) Lebowitz, J. L.; Rowlinson, J. S. J. Chem. Phys. 1964, 41, 133.
(16)
Here z0 determines the distance of a hard rod from the surface, which we assume to be equal for all the species, and m and n are exponents of the distance dependence of the intermolecular potential (cf. eq 3.8.3 of ref 22; m and n in eq 16 should not to be confused with m and n in other parts of this paper). Thus qi* ) u1°/ui°. Consider now a m-component mixture and suppose that the composition y1, ..., ym, the total pressure p*, and the inverse temperature (β) of the gas phase are given. The parameters of the adsorption system ui°, qi*, and ai* are also supposed to be known for all i ) 1, ..., m. Thus one has m equations (15) which contain m + 1 unknowns ({xi} and π) plus two equations
Finally, the chemical potential of the adsorbed pure ith component is designated as
βµi° ) βui° + ln(Λi3βπ/qi) + βπai
2/(m-n)
( )
∑ xR ) 1
(17)
R)1
To solve the problem, add up all the equations (15) and use the right eq 17 to obtain m
∑ yRqR* exp[-(βuR° + π*aR*)] R)1
π* ) p*
(18)
This is an equation for π*. When its solution is substituted in eq 15, one readily obtains the composition of the adsorbed phase xi for i ) 1, ..., m. Finally, one obtains from eqs 9 and 3
F* )
π* ; 1 + π*a j*
π* )
F* 1 - F*a j*
(19)
m j * ) ∑i)1 ai*xi. This is the Tonks where F* ) a1N/L and a equation of state for a one-component hard rod fluid, the length of a hard rod being linearly dependent on the composition of the mixture. In other words, eqs 19 are, in fact, a one-fluid model23 of the hard rod mixture which is exact for our one-dimensional system. B. IAS Solution. The above solution is based on the explicit form for the chemical potentials of the components of a mixture of hard rods which follows from the explicit expression for the configurational integral. However, since we have established that the mixture of hard rods is ideal regardless of the values of the interaction energies ui°, the problem may be also solved by a thermodynamic method provided that the adsorption isotherms of the pure components are known.2 The dependence of the chemical potential of the pure ith component upon pressure π (spreading pressure) may be found by integrating the equation dπ ) Fdµi which follows from the definition of the chemical potential. Thus, at a given π and T, the values of µi°(π,T) (as well as y1, ..., ym) in m equations (13) are known so that these equations contain m + 1 unknowns: x1, ..., xm and p. Together with the second eq 17 they give m + 1 equations for m + 1 unknowns. If, for example, the gas phase is ideal, eq 1 takes the form2
pyi ) pi°(π)xi
(20)
which immediately provides a solution to the problem of the mixture adsorption (22) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (23) McDonald, R. I. In Statistical mechanics; Specialist Periodical Reports; Singer, K., Ed.; Chemical Society, Burlington House: London, 1973; Vol. 1.
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Langmuir, Vol. 13, No. 5, 1997 1057
m
[yi/pi°(π)] ∑ i)1
p ) 1/
xi ) pyi/pi°(π) m
1/A )
xi/Ai°(π) ∑ i)1
(21)
Here A is the total adsorption and the equations correspond to a constant composition of the gas phase and some fixed value of π which determines the values of equilibrium pressure pi° and adsorption Ai° of the pure components. The last equation expresses the fact that the molar volume (area) does not change upon ideal mixing at constant spreading pressure.2,1 Thus the isotherm equation A ) f(p) for the mixture is obtained in this method as a parametric equation: A ) A(π); p ) p(π). C. A Numerical Example. Consider now as an example a binary mixture of hard rods, the hard rods of the second component being 1.5 times longer then those of the first (a2/a1 ) 1.5). Assume also that the adsorption energy of a hard rod molecule is proportional to its length so that u2°/u1° ) 1.5. As to the absolute values of the parameters, we chose a1 as a unit of length and assume that βui° ) -10. (This is a typical value for physical adsorption at the boiling temperature of the bulk adsorptive since the energy of the physical adsorption is close to the heat of vaporization. In this case, the above value corresponds to Trouton’s rule.) Calculated isotherms of adsorption are presented in Figure 1. At low values of the reduced total pressure, the isotherm for this mixture is slightly shifted compared to the isotherm of the pure second component. The shift occurs because the mixture isotherm corresponds to a constant composition (y1 ) 0.5) of the gas phase so that the partial pressure of the second component in the gas phase is p*y2 while for the isotherm of the pure second component it is p*. Thus the isotherm of the mixture is shifted with respect to that of the pure second component by -ln y2()ln 2 for the composition of Figure 1). Apart from this shift, the two isotherms are almost identical in the low-pressure region. This is because the adsorption energy of the second component is 1.5 times larger than that of the first component so that the second component adsorbs preferentially at low pressure (and coverage). Indeed, one may see from Figure 2 that the adsorbate hardly contains molecules of the first component in this region. Their mole fraction may be evaluated by substituting eq 18 into eq 15 and neglecting πai terms in the arguments of exponents since here πai , ui°. With increase of the total pressure of the gas phase at constant composition, the composition of the adsorbate changes to give a higher mole fraction of the first component. Gradually, the molecules that are larger and more strongly attracted to the surface are substituted by the less attractive but smaller ones. This behavior is typical for the adsorption of different-sized molecules of a homologous series of compounds in that it is observed that the larger molecules preferentially adsorb at low coverage and the smaller molecules preferentially adsorb at high coverage.8 Figure 1 clearly demonstrates this phenomenon and our model affords an explanation. Consider the variable ui° + πai that is reminiscent of the enthalpy of adsorbed molecule of the ith species. One may consider πai as an analog of the PV term in the enthalpy and ui° in both cases is an average energy (potential energy in our model). Thus we will conditionally call this variable the “enthalpy” of a hard rod of the ith
Figure 1. One-dimensional isotherms on a homogeneous step: dashed lines, pure components; solid line, mixture at y1 ) 0.5; symbols, IAS calculation. The reduced pressure p* is defined in eq 14; coverage is fraction of the line occupied by adsorbed rods (Na1/L). In this and in all the figures except Figures 4, 5, and 8, temperature and gas phase composition are held constant.
Figure 2. Dependence of the composition of adsorbate (mole fraction of the first component) shown as a function of reduced pressure for the solid curve of Figure 1.
species. One may see from eqs 15 and 18 that it is not the energy of the adsorbed molecule ui° that determines the adsorption behavior but its “enthalpy” which is an argument of the exponents in those equations. Figure 3 shows how the “enthalpies” of the adsorbed molecules change vs pressure of the gas phase. At low pressure the “enthalpy” of the larger molecule is lower because its energy is lower and the spreading pressure is too low to make the πa2 term comparable in magnitude to the energy. However, with increase of pressure, the spreading pressure π rises until finally the pressure term in the “enthalpy” overwhelms the energy term. At this stage, the “enthalpy” of the smaller molecule is smaller than that of the larger molecule and correspondingly the concentration of the smaller molecules in the adsorbate begins to increase as Figures 1 and 2 show. (The reason for this change of the composition of the adsorbate is of course the minimization of the Gibbs free energy as seen from eq 10.) We observed a similar influence of the spreading pressure on the behavior of one-component adsorbed hard rods in our previous work when some hard rods were squeezed out of a favorable adsorption site by their neighbors.11 Finally, the symbols in Figure 1 were calculated from the IAS theory using the isotherms of adsorption of the pure components. Naturally these exactly coincide with
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Figure 3. “Enthalpy” of a particle on a homogeneous step plotted versus the logarithm of the reduced pressure for the solid curve of Figure 1. Dashed line is for the smaller component and solid line, for the larger.
the solid curve because the mixture of hard rods in a homogeneous external field is an ideal mixture. III. Inhomogeneous External Field Now we consider the adsorption of hard rods in an inhomogeneous external field. (As for the system of hard rods in a homogeneous external field, hard rods in an inhomogeneous external field may be considered as a model of adsorption on a surface step. However, now the step should not be considered as straight but as meandering. Such systems do exist in reality as mentioned in our previous paper,18 but as in that paper, we are interested here not in the peculiarities of adsorption on a step but in the general properties of the adsorption of mixtures.) A. Random External Field. The atomic arrangements near strongly heterogeneous surfaces are irregular (e.g., amorphous) so that the external field (adsorption potential) for an adsorbed molecule can be taken to be primarily a random function. To simulate such a field, either one may first simulate an irregular profile and then calculate the random adsorption potential by evaluating the interaction of an adsorbed particle with solid in the half-space limited by this profile24 or one can directly simulate the random field as follows:24,18 Generate a stationary Markovian random sequence with a normal distribution using the recurrence formula25
ui+1 ) Fui + σγx1 - F2
(22)
Here γ is a Gaussian random variable with zero average value and unit standard deviation that may be generated by a standard subroutine, F is the correlation coefficient between the nearest neighbors of the sequence (not to be confused with densities in other parts of the paper), and σ is the standard deviation in the random sequence {ui}, with u0 ) 0. The values of ui may be taken as ordinates of the field at discrete points xi chosen arbitrarily. (Values of the field at other points will be obtained by interpolation.) In the current work, we have chosen F ) 0.2, σ ) 1.0, and the distance between xi+1 and xi equal to 1, the unit of length being the length of a shorter hard rod (a1). One of the realizations of the random field (rescaled to umax ) 0) is shown in Figure 4. This picture resembles the variation of the adsorption potential of an atom on an (24) Bakaev, V. A. Dokl. Akad. Nauk SSSR 1984, 279, 115; Dokl. Phys. Chem. (Engl. Transl.) 1985, 279, 983. (25) Feller, W. An Introduction to Probability Theory and Its Applications, 2nd ed.; J. Wiley & Sons: New York, 1971; Vol. 2, p 97.
Figure 4. A one-dimensional random external field plotted as a function of position along the line, see text for details of the calculation.
amorphous, i.e., atomically irregular surface (cf. Figure 1 from ref 26). The profile of the external field in Figure 4 is of course a random function which is to say that its appearance changes drastically from one realization to another, but since it is stationary, its average characteristics are stable. One characteristic that is very important for adsorption is the energy distribution of the minima of u(X). For example, in Figure 4, there are 30 minima which have different values of βumin. Not all the minima in Figure 4 can be considered as adsorption sites. For instance, the very shallow minimum at x ≈ 12 clearly cannot be. We choose only those minima whose depth (the distance from the lowest neighboring maximum) is larger than 1: ∆u > kT. Thus we generated realizations of the random function u(X) many times until the total number of minima reached ca. 3000. The distribution of these minima in energy normalized to the average number of minima on a one-dimensional surface is presented in Figure 5. In the case of adsorption on a real two-dimensional surface, the minima of the adsorption potential U(X,Y,Z) are usually considered to be adsorption sites,27 i.e., points at which the adsorption is concentrated. It is interesting to note that despite the fact that the function U(X,Y,Z) depends on three parameters while u(X) depends only upon one, the distribution of minima for u(X) as presented in Figure 5 has a shape typical of the energy distributions found for adsorption site distributions of real heterogeneous surfaces:27 it is a bell-shaped curve skewed to higher absolute values of energy. B. Numerical Solution of the Integral Equations for a Heterogeneous Binary Mixture. The statistical thermodynamics of a mixture of hard rods in an arbitrary external field was developed in ref 12. At a given value of chemical potential, one must solve a system of coupled integral equations for the density profiles of all components of the mixture. When the density profiles are known, one can calculate the composition of the adsorbate and all its thermodynamic functions. It has also been shown that, in fact, one needs to solve only two decoupled integral equations for a mixture with any number of components to find all the density profiles.14 In general, this should drastically simplify the problem. However, our attempt to solve equations of ref 14 for the random external field described above failed. (The reason for that failure is not yet clear to us.) Thus, we consider (26) Bakaev, V. A. Surf. Sci. 1988, 198, 571. (27) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic: London, 1992.
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Langmuir, Vol. 13, No. 5, 1997 1059
- n1)/2, n j ) (n1 + n2)/2, and n1 and n2 are even. Now shift the index i in the second eq 25 by m points backward to obtain
β[µ2q - u2(i - m)] ) ln n1w2(i - m) + i
∑ c2(j - i + n2)w1(j) + j)i-n 2
i-m
∑
c2(j - i + m + n2)w2(j) (26)
j)i-m-n2
Figure 5. Distribution of the number of external field minima with a given value of energy for the field shown in Figure 4.
the original integral equations of ref 12 for a binary mixture and use a method for their solution which is a generalization of that described in ref 11 for the onecomponent hard rod system. Equations (23) presented below differ from the coupled integral equations of ref 12 only by a factor a1 needed to make the arguments of the ln functions dimensionless. Correspondingly, the chemical potentials µ1q and µ2q are shifted with respect to ref 12.
β[µ1q - u1(X)] ) ln a1w1(X) +
X-∆ X w1(Y) dY + ∫X-aj ∫X-a
21
1
F1(X) ) w1(X)[1 -
X+a j F2(Y) dY] ∫XX+a F1(Y) dY - ∫X+∆
F2(X) ) w2(X)[1 -
X+a X+a j F1(Y) dY - ∫X F2(Y) dY] ∫X+∆
12
w1(Y) dY +
X w2(Y) dY ∫X-a 2
(23)
Here a j ) 1/2(a1 + a2) and ∆21 ) 1/2(a2 - a1); w1(X) and w2(X) are auxiliary functions so that after solving these equations one has to solve eqs 27 to find the densities of components. The relation of the µiq to the absolute values of the chemical potentials µi is readily obtained on the condition that at constant values of ui(X) ) ui° and correspondingly constant values of wi(X) ) wi° and Fi(X) ) Fi in eqs 23 and 27, these equations should give eq 1. Taking into account eq 19, one obtains from eqs 27: wi° ) xiβπ (F ) F1 + F2; xi ) Fi/F). Substitute this value of wi° in eqs 23 for wi(X) and wi(Y) and compare the resulting equation with eqs 11, 13, and 14 to obtain
βµiq ) βui° + ln a1xiβπ + aiβπ ) ln p*qi*yi (24)
1
12
2
21
(27)
These equations can be solved by the same method as eqs 23 but much more simply because eqs 27 are linear. Discretize them as eqs 25 and 26 to obtain
F1(i)/w1(i) ) i+n j
i+n1
1-
β[µ2q - u2(X)] ) ln a1w2(X) + X-∆ ∫X-a j
w2(Y) dY
Assume that values of w1(j) are known for j < i and values of w2(j) are known for j < i - m, then the first eq 25 and eq 26 are two nonlinear equations which may be solved for w1(i) and w2(i - m) by the Newton-Raphson method. After that w1(i + 1) and w2(i - m + 1) may be found by the same method, etc. Now, when w1(X) and w2(X) are known, we must solve two linear integral equations to find F1(X) and F2(X)12
∑ j)i
c1(j - i)F1(j) -
∑ c2(j - i + m)F2(j)
j)i-m
F2(i - m)/w2(i - m) ) i+n j
i+n1
1-
∑ j)i
c1(j - i)F1(j) -
∑ c2(j - i + m)F2(j)
(28)
j)i-m
where c1(i) and c2(i) are the same weights of the quadrature formula as in eqs 25. The right-hand sides of eqs 28 are equal, so that
F2(i - m) ) F1(i)w2(i - m)/w1(i)
(29)
and, from the first of eqs 28 j)i+n1
F1(i) ) w1(i)[1 -
∑
c1(j - i)F1(j) -
j)i+1 i+n j
Let the step of integration in eqs 23 be the unit of length. Evaluate integrals by quadrature formulae to obtain
∑ j)i-n
i-m
c1(j - i + n1)w1(j) +
1
∑j c1(j - i + nj )w2(j) j)i-n
(25)
β[µ2q - u2(i)] ) ln n1w2(i) + i
i+m
∑
c2(j - i + n j )w1(j) +
j)i-n j
c2(j - i + m)F2(j)]/[1 + c1(0)w1(i) + c2(0)w2(i - m)] (30)
β[µ1q - u1(i)] ) ln n1w1(i) + i
∑
j)i-m+1
∑
c2(j - i + n2)w2(j)
j)i-n2
where c1(i) and c2(i) are the weights of a quadrature formula (e.g., for the trapezoid rule c1(0) ) c2(0) ) c1(n1) ) c2(n2) ) 1/2 and all the other weights are 1); m ) (n2
Here it is assumed that the values of F1(j) for j > i and of F2(j) for j > i - m are known. Thus eq 30 may be solved for F1(i); after that we find F2(i - m) from eq 29 then F1(i - 1), etc. That is, eqs 25 and 26 are solved by the stepby-step method like eq 25, only in the opposite direction. The effectiveness of the Newton-Raphson method used for the solution of the two nonlinear equations for w1(j) and w2(j) strongly depends upon the initial approximation to the solution. We have here employed the local density approximation which means that each value of Fi(j0) was found by the method described in subsection IIA, i.e., by assuming that the external field has the same value ui(j0)
1060 Langmuir, Vol. 13, No. 5, 1997
Bakaev and Steele
Figure 6. Isotherms on the one-dimensional heterogeneous surface illustrated in Figures 4 and 5: dashed lines, pure components; solid line, mixture at y1 ) 0.5; symbols, IAS calculation.
at all points. The values of w1(j) and w2(j) were then approximated as (cf. eqs 27): wi(j) ) Fi(j)/[1 - a1F1(j) a2F2(j)]. After finding the density profiles F1(X) and F2(X) by solving eqs 23 and 27, we integrated them to obtain the amount of adsorption of both components A1 and A2. This was done as a function of the pressure of the gas phase p* (cf. eq 14), its composition, and the temperature being kept constant. Finally, we noticed that the numerical procedure described here can yield small negative (unphysical) values of F1 and F2 when the sums on the right-hand sides of eqs 28 exceed unity. This is probably a result of the roundoff errors. However, these seemingly small errors may be the source of serious discrepancies in the final results (see the end of the next subsection). C. Excess Gibbs Free Energy for the Adsorbed Mixture. Some calculated isotherms of adsorption are presented in Figure 6 for a binary mixture of hard rods of the same lengths and with the same gas phase composition as in Figure 1. The external field in Figure 6 corresponds to that shown in Figure 4. Both the individual isotherms and the mixed isotherm are qualitatively the same as for the homogeneous case shown in Figure 1. However, now the symbols which represent the IAS calculation do not coincide exactly with the solid curve as in the case of the homogeneous external field. It means that while a given mixture of hard rods on a line is an ideal mixture in a homogeneous external field, it is not ideal in an inhomogeneous field. The deviation of an m-component mixture from ideality is conventionally measured by the excess molar Gibbs free energy:1 m
βgE )
xi ln γi ∑ i)1
(31)
Here γi is the activity coefficient of the ith component which is defined by the modified eq 1
βµi ) βµi°(π,T) + ln xiγi(π,T,x1, ..., xm)
(32)
To calculate γ1 and γ2 from this equation for the particular case of a binary mixture (m ) 2 in eq 31), we calculated 19 isotherms of mixed adsorption similar to that presented in Figure 6 for constant compositions of the gas phase corresponding to y1 from 0.05 to 0.95 in steps of 0.05. Along each of those isotherms as well as along the individual isotherms presented in Figure 6
Figure 7. Spreading pressures for isotherms of Figure 6: solid lines, y1 ) 0 and y1 ) 1 (see text); dashed line, y1 ) 0.5.
(corresponding to y1 ) 0 and y1 ) 1), the spreading pressure π was calculated as a function of the total pressure by integrating the Gibbs-Duhem equation:
dπ ) n1 dµ1 + n2 dµ2
(33)
where dµi depends on p* according to eq 12 (y1 is constant along an isotherm) and ni (i ) 1,2) is the adsorption per unit length. The spreading pressures for the isotherms of Figure 6 are presented in Figure 7. One may see that at relatively low pressures p* the spreading pressure curve of the adsorbed mixture almost coincides with that of the pure second component. This is because at those pressures the concentration of the first component in the adsorbate is really small. The situation is similar to that depicted in Figure 2 for a homogeneous adsorbed mixture and can be also deduced from Figure 6. At the pressures above the point of intersection of all the curves in Figure 7, the curve for the mixture is close to that for the first component which again corresponds to Figures 2 and 6. Now, from eqs 32 we obtain γi(π) along the isotherms when xi are also determined by π (and correspondingly by p* as, e.g., in Figure 2). A substitution of these γi(π) and xi(π) into eq 31 gives βgE(π) and x1(π) along isotherms. These may be considered as a parametric form of the equation for βgE(x1) which is the goal of the calculation. To calculate this, we interpolated βgE(π) and x1(π) along the isotherms and found 19 values of βgE(x1) at several values of π. The results are presented in Figure 8. Values of γi(x1) may be found from the curve of βgE(x1) by the tangent intercepts method (cf., e.g., ref 1, p 219). We used this method to check the thermodynamic consistency of the final results. The average deviations of the activity coefficients calculated by the tangent intercepts method from their original values in Figure 8 were 5-9% for all the curves of Figure 8. These are probably results of the roundoff, interpolation, etc., errors in the long computation process from the numerical solution of the integral equations 23 and 27 to the final results since we have found that thermodynamic consistency is very sensitive to small computational errors. The excess thermodynamics functions of Figure 8 were obtained for one realization of a random external field similar to that in Figure 4. Other realizations give curves qualitatively the same as those presented in Figure 8 but quantitatively different. (However, in all our trial computations, the maximal value of -βgE never exceeded twice that of the minimum in Figure 8). Thus, to obtain the value of βgE for the infinite one-dimensional heterogeneous surface corresponding to the given characteristics F and
Adsorption of Gas Mixtures on Surfaces
Figure 8. Excess molar Gibbs free energy for a binary mixture of hard rods on the one-dimensional heterogeneous surface illustrated in Figures 4 and 5: solid line, π* ) 5.54; dashed line, π* ) 6.11; dot-dashed line, π* ) 6.69.
σ of the external field determined by eq 22, one has to obtain many curves similar to those in Figure 8 and to average them. This, however, has not been done here because some of those curves (with exceptionally deep minima of external field, cf. Figure 4) did not pass the thermodynamic consistency test. This might be connected with the yet unresolved problems with numerical solution of eqs 23 and 27 mentioned above. Thus, in our opinion, the result presented in Figure 8 is qualitatively reliable but needs further refinement. In general, the variation of βgE with the composition of the adsorbate in Figure 8 is of an order of magnitude typical of simple liquid-liquid mixtures.23 The main difference is that in the liquid mixtures under relatively small pressures around 1 bar the activity coefficients are almost independent of pressure1 while for the activity coefficients and excess function in Figure 8, the dependence upon π is considerable. This peculiarity of adsorbed solutions has been mentioned in ref 8. However, we should emphasize here that the spreading pressure to which the curves in Figure 8 correspond is not small. Comparison with Figures 7, 6, and 3 shows that these spreading pressures are in the region where aiπ are close to the energies of adsorption ui°. As we have already mentioned, energies of adsorptions are generally close to heats of condensation. Thus the spreading pressures for which Figure 8 was obtained correspond to the value of a pressure equal to the heat of condensation divided by the volume of a molecule. For argon at its boiling temperature, e.g., that pressure is ca. 1 GPa. At smaller values of the spreading pressure and y1 > 0.05 (the lower limit of y1 in these calculations), the adsorbate consists mainly of the second component and it is difficult to determine the dependence of βgE on composition. D. Discussion. We have established by statistical thermodynamics arguments in section II that a onedimensional mixture of hard rods of different sizes and different adsorption energies (that are proportional to size) is an ideal mixture. This system models adsorption on a homogeneous surface. From the point of view of phenomenological thermodynamics, this means the following: If one takes, say, two equal portions of different onedimensional hard rod fluid confined by four pistons maintaining equal one-dimensional (spreading) pressures on each portion and brings them in contact (removes two pistons), then neither the total volume nor the total energy of this system changes. Consider now the same conceptual experiment with the similar system but in a random external field. This is the
Langmuir, Vol. 13, No. 5, 1997 1061
system considered in this section which models adsorption on a heterogeneous surface. In each portion of the heterogeneous surface, particles tend to occupy the most favorable adsorption sites (minima of external field) but the number of the sites of each energy is limited and determined by the distribution similar to that in Figure 5. Thus if one allows the two portions of the adsorbed fluids to mix, then the molecules having larger attraction to the surface (lengthier hard rods in our case) will be substituted by the less attractive molecules on the stronger adsorption sites of the total surface and the total adsorption energy of the system will decrease. Thus one should expect the negative energy of mixing which actually is demonstrated in Figure 8 for a one-dimensional hard rod mixture on a specific model heterogeneous surface. This argument is not rigorous because it ignores the change of the spreading pressure upon mixing but its validity is confirmed by our computations on a rigorous onedimensional model. But there is nothing in this argument that is specific for our one-dimensional model, so that one may expect that in general the heterogeneity of a surface should induce a negative deviation from the Raoult’s law qualitatively similar to that shown in Figure 8. Adsorption of mixtures on heterogeneous surfaces has been considered in refs 4, 10, and 28 which also contain references and short reviews of earlier papers. In all these papers adsorption of a mixture on a heterogeneous surface was considered as a superposition of adsorption on separate adsorption sites characterized by different adsorption energies. The total (global) isotherm depends in this model on the distribution of adsorption sites in energy and the isotherm on a single site (local isotherm) as a function of that energy. Two approaches have been used for this general model in the case of mixed adsorption:28 one involves calculating only the adsorption of a single component on each site; another is to assume that each site can adsorb many different molecules. In the former case one can consider an adsorption site as a minimum of adsorption potential (e.g., minima in Figure 4) which can adsorb only one molecule. This strictly gives a global adsorption isotherm as a superposition of Langmuir’s isotherms for each site.27 This is a reasonable approach but by no means exact. It works well at small coverages when there are many vacant adsorption sites. In this case, the minima of the adsorption potential are the most probable sites of location of adsorbed molecules. However, when coverage increases, other parts of adsorption space play an increasingly important role. Our model clearly displays this phenomenon. For example, from Figure 6 one may see how at high coverage smaller molecules with weaker attraction to the surface squeeze out larger and more attractive ones. This effect lies outside the realm of the model with one molecule per adsorption site. The approach which assumes that a mixture of molecules can adsorb on a single adsorption site in fact takes this site as a patch of a homogeneous surface. This is the approach of refs 4 and 28. It is logically consistent if one assumes those patches to be sufficiently large so that the areas of border regions where molecules on one patch interact with those on another patch are negligibly small in comparison with the areas of patches. In this case the mutual influence of patches is negligible and the global isotherm is really a superposition of local isotherms on those patches. The problem with this model is that there are very few (if any) real adsorbents whose surfaces are really composed of such patches.26 Anyway, active carbon or zeolite, or silica (these are the adsorbents used for (28) Kapoor, A.; Ritter, J. A.; Yang, R. T. Langmuir 1990, 6, 660.
1062 Langmuir, Vol. 13, No. 5, 1997
comparison of theory with experiment in refs 4 and 28), are not patchwise heterogeneous according to the extant information on their atomic surface structures. Furthermore, the distribution of patches in the adsorption energy which in a logically consistent model should follow from independent information on the atomic structure of an adsorbent is chosen arbitrarily to be a uniform continuous distribution in ref 28 or a discrete binomial distribution in ref 4. Finally, the local isotherm which in both cases was chosen as Langmuir’s isotherm (explicitly in ref 28 and implicitly in ref 4) is thermodynamically inconsistent (as mentioned in ref 28) when the monolayer capacities of pure components are different. This is obvious, since, as mentioned above, the Langmuir model assumes that each site can adsorb only one molecule and the monolayer capacity in this model is just the number of sites. Also there is no evidence that a mixture adsorbed on a homogeneous surface is an ideal mixture as assumed in ref 4. On the contrary, it is known that even a onedimensional hard rod mixture with attraction of nearest neighbors is nonideal3 and the three-dimensional mixture of hard spheres of different radii is also nonideal.21 Thus the approach of refs 4 and 28 to mixed adsorption on a heterogeneous surface is in fact a phenomenological one. The actual purpose of using the patchwise heterogeneous model in this case (as in the majority of adsorption studies) is just to introduce additional disposable parameters in the description of adsorption isotherm in order to better correlate experimental data. This is a reasonable goal and such an approach is effective especially in description of isotherms of pure components. However, one should not expect a consistent explanation of the reasons for the ideality or nonideality of adsorbed mixture for such an approach to mixed adsorption on a heterogeneous surface. Since the patchwise heterogeneous model has been successfully applied to description of experimental data on active coals and zeolites where certainly there are no patches of homogeneous surface, a question arises if our one-dimensional model could also be compared with real experimental data on those adsorbents. To formally do that one has first of all to transfer our dimensionless variables to experimental dimensionalities. For example, our dimensionless spreading pressure π* from eq 14 corresponds to variable ψ ) ΠA/RT from ref 8 where A is a specific surface so that ψ has a dimensionality of mol/ kg. In this paper A designates adsorbed amount and Ai°(∞) is adsorption of ith component at infinite spreading pressure (as, e.g., in eq 21) which is in fact the saturation capacity of an adsorbent with respect to that component. In our one-dimensional model Ai°(∞) ) L/ai and in twodimensional models Ai°(∞) ≈ A/ai2. Niether specific surface A nor L have real physical meaning for such microporous adsorbents as active carbons or zeolites but the saturation capacities Ai°(∞) have. Thus what we have to do is to exclude L from our equations and substitute it by adsorption capacities. For example, the above mentioned spreading pressure ψ is connected to π* as: ψ ) A1°(∞)π* where A1°(∞) is the saturation capacity of the smaller molecules. Now let us compare results presented in Figure 8 with some realistic data. In the review paper,8 there is an approximate expression for the excess Gibbs free energy of the adsorbed binary mixture taken from ref 29 (29) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989; p 216.
Bakaev and Steele
gex ) x1x2C(1 - e-βψ) RT
(34)
There is a specific example there which considers the case with C ) -0.5, β ) 0.38 kg/mol, and m ) m1A + m1B ) 2.4 mol/kg. (Here we temporarily preserve the notation of ref 8 where m designates the saturation capacity of the adsorbent with respect to the smaller molecules - our {A1°(∞)}.) At π* ) 6.11 and x1 ) 0.5 in Figure 8, βgE ) -0.093 and ∂βgE/∂π* ) -0.011. Substitution of the values of C, β from ref 8, and ψ ) 2.4 × 6.11 ) 14.7 mol/kg in eq 34 yields gex/RT ) -0.11 and ∂[gex/RT]/∂π* ) -4 × 10-4. Thus the excess molar Gibbs free energy in our model displays a moderate negative deviation from ideal mixing typical for adsorption in micropores. This also was obtained in a computer simulation with the molecules having the size ratio close to our model (see below). However its derivative with respect to π* is considerably larger in our model than in the realistic example from ref 8. The example considered brings us to the discussion of yet another approach to the explanation of the deviation of adsorbed mixtures in micropores from ideality which was termed “partial-exclusion theory” (PET).30 This is by no means an exact statistical thermodynamics theory. The idea of the approach is to divide the volume of a micropore into two parts: the peripheral part near the wall of a micropore which is accessible to the centers of only the smaller molecules in a binary mixture and the central part accessible to all the molecules. Then the adsorbent is considered as a mixture of two independent (this is the most sweeping approximation) fictitious adsorbentssone having the total volume of micropores equal to the above mentioned peripheral part of a micropore multiplied by the number of micropores and the other with that volume proportional to the central part of the micropores. Adsorption in those fictitious adsorbents is described by some empirical equation and the adsorbed mixture in the central part of a micrpore is assumed to be ideal. The theory predicts a negative deviation from Raoult’s law which is really observed in the majority of microporous adsorbents.8 The reason for that again may be illustrated with the help of the conceptual experiment employed above. Thus let two porous adsorbents, one filled by smaller molecules and another filled with larger molecules under the same spreading pressure be brought into contact. The larger molecules are assumed to mix ideally (without change of total volume or energy) in adsorption space accessible to them. But the smaller molecules will have an opportunity to occupy adsorption space in the adsorbent initially filled with larger molecules and unaccessible to them. This is the region near the walls of micropores which is the strongest adsorption site with respect to the smaller molecules. Thus one should expect a negative energy of mixing. There is some similarity in this explanation of the negative deviation from ideality in PET and the explanation model given above for our model. Furthermore, the value of the excess molar Gibbs free energy for an equimolar mixture obtained by computer simulation in ref 30 and explained by PET is -0.16, which is reasonably close to what is presented in Figure 8. Besides, the purpose of the computer simulation30 was to investigate the influence of difference of molecular sizes on deviation of adsorbed mixture from ideality, which in fact our model also establishes, and even the ratio of molecular sizes in ref 30 (1.34) is close to ours. However, in many respects PET and our approach are very different. Ours (30) Dunne, J.; Myers, A. L. Chem. Eng. Sci. 1994, 49, 2941.
Adsorption of Gas Mixtures on Surfaces
is an (almost) exact statistical mechanical theory which is, however, strictly applicable only to a very rare experimental situation (adsorption on a step of a crystal face) while PET, being a sweeping approximation from the very beginning, is intended to explain experiments on the most widely used microporous adsorbents. IV. Conclusion It seems to be a general belief that a mixture is ideal either when it is very rarefied (i.e., in the ideal gas state) or when the molecules of its components are very similar in size and other characteristics (like a mixture of isotopes). However, as shown in section II, the validity of this statement depends upon the dimensionality of space. It is certainly true for conventional three-dimensional mixtures but false for the homogeneous one-dimensional mixture of hard rods. The one-dimensional mixture of hard rods is an ideal mixture at any density and any difference in the particle sizes. This mixture is ideal even when energies of interaction of different hard rods with an external field are different. Thus for a mixture of non-hard-sphere molecules adsorbed on a homogeneous step of a surface, the IAS should be a good approximation. This is because it is the repulsive interaction between adsorbed molecules (which determine the adsorption capacity) and their interaction with the adsorbent (external field) which play the major part in physical adsorption. The van der Waals attraction between adsorbed molecules also manifests itself in physical adsorption. It causes the deviation of a onedimensional mixture from ideality (as the model of onedimensional hard rods with nearest neighbor attraction shows3). However, the overall energy of adsorbateadsorbate attraction is usually considerably smaller than the energy of adsorbate-adsorbent interaction (e.g., less than 15% as in ref 30). This is the reason why some basic models of physical adsorption like, e.g., the Langmuir model, can simply ignore the adsorbate-adsorbate attraction. Thus one may expect that the mixture adsorbed on a homogeneous two-dimensional surface takes an intermediate position between the nonideal three-dimensional solutions and the ideal one-dimensional mixture of hard rods. This is a conclusion that may be drawn from section
Langmuir, Vol. 13, No. 5, 1997 1063
II. It might help to understand the success of the IAS but relates only to homogeneous surfaces where (in stark contrast with the situation in one-component adsorption studies) very little experimental work or even computer simulations have been done. There are many computer simulations of mixtures on homogeneous surfaces but they do not discuss the problem of excess thermodynamic functions of the two-dimensional mixture in comparison to those of the bulk solutions. The general conclusion that may be drawn from section III is that the inhomogeneity of a surface induces the same negative deviation from Raoult’s law that is usually observed in adsorption experiments. However, those experiments are as a rule carried out on industrially important microporous adsorbents like, e.g., active carbons whose atomic structure and real geometry of porous space are still unknown. Thus one is free to speculate on the reasons for the nonideality of adsorbed solutions in those adsorbents and to explain them by different models (like patchwise model, PET, etc.). What is needed here is experimental and computer simulation work on the systems with well-defined atomic structure (not necessarily regular). It is not that such work is totally absent (see ref 30). The problem is with the general understanding of the heterogeneity of an adsorbent. It seems that the prevailing point of view is that the heterogeneity of adsorbents lies exclusively with the models like the adsorption site model. However, any adsorption space where adsorbed molecules of the same species have different energies (like, e.g., in the model spherical micropore with smooth walls considered in ref 30) and where molecules of different species may compete for energetically favorable sites should be considered as heterogeneous. Finally, we hope that our model may be helpful in elucidating the influence of dimensionality of space and heterogeneity of surfaces on excess thermodynamic functions of adsorption equilibrium. Acknowledgment. This work was supported by Grant DMR 9022681 from the Division of Materials Research of the NSF. LA9509110
