Langmuir 1996, 12, 6119-6126
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Adsorbed Mixtures on a Heterogeneous Surface. The Lattice Gas Model V. A. Bakaev† and W. A. Steele* Department of Chemistry, The Pennsylvania State University, 152 Davey Laboratory, University Park, Pennsylvania 16802 Received April 24, 1996. In Final Form: August 20, 1996X A lattice gas model of mixed adsorption on a heterogeneous surface is considered. The assumption that the differences between the energies of the components on each adsorption site are the same makes this model an ideal mixture. It is shown that mixing at constant spreading pressure in this case is equivalent to that at constant coverage. Interaction between adsorbed molecules induces deviations from ideality, but the excess chemical potentials of the components calculated in the random mixing (mean field) approximation depend linearily upon mole fractions, contrary to the regular bulk solutions which in the same approximation have a quadratic dependence on mole fractions. A new method of describing the experimental data is proposed in which two experimental isotherms are fitted by two model equations with adjustable parameters under the additional constraint that some parameters in the two equations must be equal.
I. Introduction 1 we considered a one-dimensional
In our previous paper, model of a mixture of hard rods of different lengths on a homogeneous and heterogeneous one-dimensional surface (line). The latter was modeled by a random external field, with the strength of the field at the centers of different hard rods being proportional to their lengths. This model admits an exact solution and reproduces qualitatively the peculiarities of mixed adsorption. On a homogeneous surface (in a uniform external field), the mixture of hard rods of different lengths with different adsorption energies is ideal. On a heterogeneous surface (in a random external field), the mixture displays negative deviation from Raoult’s law. As seen from this model study, the deviation of the adsorbed solution from ideality may be caused not only by the attractive interaction between adsorbed molecules which is neglected in the model but also by the heterogeneity of the surface. This actually means that one may not hope to gain information on the nonideality of mixtures adsorbed on heterogeneous surfaces only from the deviation of the corresponding bulk solutions from Raoult’s law which depends upon the interaction between molecules. The important part of this information (pertinent to the heterogeneity of a surface) is contained in the adsorption isotherms of the pure components. In this way we return to a long standing problem of predicting the properties of mixed adsorption from the isotherms of the pure components. This is an important task because it is much harder to measure the adsorption of mixtures than the adsorption of pure components, and consequently, the information on mixed adsorption is much scarcer than that on adsorption of pure substances (cf., e.g., ref 2). There are several theories of adsorption which allow one to evaluate parameters of mixed adsorption from individual isotherms.3 Such theories as the ideal adsorption solution (IAS) or vacancy adsorption solution (VAS) † Permanent address: Institute of Physical Chemistry, Russian Academy of Sciences, Leninsky Prospect 31, Moscow 117915, Russia X Abstract published in Advance ACS Abstracts, November 15, 1996.
(1) Bakaev, V. A.; Steele, W. A. Langmuir, in press. (2) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice-Hall: Englewood Cliffs, NJ, 1989.
S0743-7463(96)00403-9 CCC: $12.00
(for reviews of IAS and VAS see ref 3, and for the history of development of the VAS see ref 4) are based on thermodynamic considerations and do not even make proper distinction between homogeneous and heterogeneous surfaces. On the other hand, the Langmuir theory of mixed adsorption3 is based on statistical thermodynamic considerations and its application to the adsorption of mixtures on heterogeneous surfaces has a long history.4 The general situation of the adsorption of mixtures at present is such that the problem of the evaluation of parameters of mixed adsorption from an ab initio calculation or from individual isotherms is far from a reliable solution. The statistical mechanics (one fluid van der Waals model5 or density functional6) calculations and computer simulations7,8 give reliable results for welldefined models and in many cases5,8 confirm IAS (see also discussion in ref 1 and references therein). These are, however, calculations on model systems with regularly varying external fields. Many important real adsorbents like active carbons, silica gels, and other amorphous surfaces are not expected to have such regular patterns of adsorption potential as assumed in the model calculations of refs 5-8. Qualitatively their adsorption fields are expected to be random as, e.g., that considered in previous work.1 With those irregular adsorbents and even with some regular ones like zeolites, there are many cases when IAS or any other theory of mixed adsorption fails to provide a correct answer3 (see also discussion in ref 1). The worst of it is that one does not know in advance how reliable is the prediction of a particular theory of mixed adsorption. This is especially true for thermodynamics-based (actually, semiempirical) theories like IAS. On the other hand, IAS is very appealing in that it allows one to use information obtained only from individual isotherms to predict the parameters of mixed adsorption. In the present paper we consider a statistical thermo(3) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; J. Wiley & Sons: New York, 1984. (4) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988; Chapter 4. (5) Monson, P. A. Chem. Eng. Sci. 1987, 42, 505. (6) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845. (7) Gilyazov, M. F.; Tolmachev, A. M.; Tovbin, Yu. K. In Fundamentals of Adsorption; Suzuki, M., Ed.; Kodansha: Tokyo, 1993. (8) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Simul. 1994, 13, 161.
© 1996 American Chemical Society
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dynamic model of adsorption on a heterogeneous surface which produces ideal mixing. This is a lattice gas (LG) model. Its basic assumptions other than those of the Langmuir model include the rather sweeping approximation that the differences of the free energies of interaction of different molecules with an adsorption site are the same for all the sites. This approximation has already been discussed in the literature.4,9 It is known to reduce the problem of mixed adsorption to that of the adsorption of a pure substance4 and to produce ideal mixing with constant selectivity as for the Langmuir model.10 The purpose of the present paper is to reconsider that model from a more general statistical thermodynamics point of view than previously. Since our model produces ideal mixing, it is a particular case of IAS. Thus we can discuss some characteristic features of IAS in statistical mechanical terms, e.g., its basic requirement that an adsorbed mixture should be considered at constant spreading pressure. We also consider the deviations of the model from ideality caused by attractive interaction between molecules adsorbed on a heterogeneous surface. Finally, we try a new approach to description of experimental data differing from the usual approach to binary adsorption, in which one fits two experimental isotherms by two equations like, e.g., UNILAN (UNI for uniform distribution and LAN for the Langmuir local isotherm),2 each having three disposable parameters, and then uses IAS. Thus the total number of parameters determined from experimental data is 6. We try here an approach in which the six parameters of two UNILAN equations are determined under the constraint that two pairs of these parameters are the same. That is, we determine four parameters from two experimental isotherms simultaneousely. The meaning of the constraint is that we demand the number of adsorption sites to be the same for the two components (a requirement of the Langmuir model). The heterogeneity parameters of the UNILAN equation are also to be the same for two components which guarantees the fullfillment of the above mentioned approximation that makes an adsorbed mixture ideal. The paper is organized as follows. In section II we consider in general terms a LG of noninteracting and interacting molecules on a heterogeneous surface. In section 3 we introduce the approximation that makes a noninteracting LG an ideal mixture. It is shown that the requirement of constant spreading pressure (in the IAS) is equivalent in this case to that of constant total coverage (section III.A). Then we show that the interaction between adsorbed molecules (under additional assumptions) in this LG model causes the excess chemical potentials of the components to depend linearily upon the mole fractions in contrast to bulk regular solutions where this dependence is quadratic (section III.B). In subsection III.C we compare the deviation from ideality of the adsorbed mixture at constant spreading pressure (as in IAS) and at constant total coverage. Finally, in section IV we consider a new method of comparing our model with experimental data. II. General A. Noninteracting LG on a Heterogeneous Surface. We consider a set of B generally different adsorption sites numbered by i ) 1, 2, ..., B. Each site can either be empty or occupied by a molecule of one of m different species. We now make a basic assumption of the Langmuir model that the states (empty or full) of sites are inde(9) Jaroniec, M.; Narkiewicz, J.; Rudzinski, W. J. Colloid Interface Sci. 1978, 65, 9. (10) Valenzuela, D. P.; Myers, A. L.; Talu, O.; Zwiebel, I. AIChE J. 1988, 34, 397.
Bakaev and Steele
pendent of each other. In fact, this is the noninteracting LG model in a nonuniform external field. The grand partition function of the ith site is m
χ(i) ) 1 +
∑ λR exp[-R(i)/kT] R)1
(1)
Here λR is the absolute activity of the Rth species: λR ) exp(µR/kT) where µ is the chemical potential; exp[-R(i)/ kT] is a partition function so that R(i) is the free energy of one molecule on an adsorption site. The well-known Langmuir equation for mixtures that follows from eq 1
∂ ln χ(i) λν
∂λν
λν exp[-ν(i)/kT]
) θν(i) )
m
1+
(2)
∑ λR exp[-R(i)/kT] R)1
will be interpreted below as the probability that the ith site is occupied by a molecule of the νth species. Since the states of different sites are independent of each other, the grand partition function of the whole heterogeneous surface is just the product of those functions for separate sites so that B
ln Ξ )
ln χ(i) ∑ i)1
(3)
where χ(i) is determined by eq 1 and the adsorption of the νth component is B
Nν )
θν(i) ∑ i)1
(4)
where θν(i) is determined by eq 2. Thus if one knows the chemical potentials of all components of an adsorbed mixture and the free energies R(i) for i ) 1, ..., B and R ) 1, ..., m, one can calculate the total coverage of the adsorbent m
θ ) N/B;
N)
∑ NR
(5)
R)1
and the LG analog of the spreading pressure
πB ) kT ln Ξ
(6)
B. Interacting LG on a Heterogeneous Surface. If molecules on the adsorption sites interact with each other, the site grand partition functions are not independent and eqs 3 and 4 are no longer valid. However, one can still find an easy solution to the problem provided the surface is strongly heterogeneous. In this case, one can employ the van der Waals (or mean field) approach to adsorption on heterogeneous surfaces11 similar to that widely used in the theory of liquids. The essence of the approximation is that the structure of the adsorbate on a strongly heterogeneous surface is imposed by the surface itself and may be taken as independent of the interaction between adsorbed molecules. In other words, the probability that the ith adsorption site is occupied by a molecule of the νth species and simultaneously the jth adsorption site is occupied by a molecule of the γth species may be taken as a product θν(i)θγ(j), where θν(i) and θγ(j) can be calculated from the Langmuir equation (2). These functions are assumed to be independent of the interaction (11) Bakaev, V. A.; Steele, W. A. J. Chem. Phys. 1993, 98, 9922.
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between adsorbed molecules, which is usually considerably weaker than the interaction between those molecules and the surface. In this approximation we can calculate the average energy of interaction between adsorbed molecules as
W)
uRβ(i, j)θR(i)θβ(j) ∑ij ∑ Rβ
(7)
where uRβ(i,j) is the interaction energy between two molecules of the Rth and βth species adsorbed on the ith and jth sites. The free energy in the van der Waals approximation is11
F ) F0 + W
(8)
It is easy to find the change of chemical potential due to interaction of adsorbed atoms in a one-component system (m ) 1)
∆µ )
∂W ∂W/∂λ ) ∂N ∂N/∂λ
If m > 1, differentiate eqs 4 and 7 with respect to the absolute activities to obtain
dN ) J‚dλ;
dW ) ∇W‚dλ
(9)
where J is the Jacobian JRβ ) ∂NR/∂λβ, ∇W is a one row matrix ∇WR ) ∂W/∂λR, and N and λ are one column matrices defined above. Now invert the first of eqs 9 and substitute the result in the second equation to obtain m
where F0 is the free energy of the Langmuir model, which is our reference system. Formally, eqs 7 and 8 are identical with the BraggWilliams (mean field) approximation very well known in the theory of LG and adsorption on homogeneous surfaces.12 The point is, however, that on heterogeneous surfaces this approximation is much more accurate than on homogeneous surfaces. The reason is that the interaction between LG atoms on a homogeneous surface brings about clustering between them which may considerably change the internal structure of the LG and consequently its entropy. If, for example, one site is occupied, the occupation of the neighboring sites becomes more probable when adsorbed atoms attract each other. The ratio of the ocupation probability of a pair of nearest neighbor sites to the pair of distant sites does not depend upon the energy of adsorption on a homogeneous surface where all sites are identical. (It only depends on temperature and the strength of interaction between adsorbed atoms.) This is not the case on a heterogeneous surface. Now the energy of a pair of atoms adsorbed on neighboring as well as distant sites depends mainly on the adsorption energies of the sites, and those energies can be very differentsso much so that the energy of interaction between adsorbed atoms is usually much less than this difference. Thus the probability of occupation of two adsorption sites on a heterogeneous surface does not, in effect, depend upon the distance between those sites but is determined by their adsorption interaction with the sites. This is the reason why the interaction between adsorbed atoms hardly influences the structure and entropy of the adsorbate and why eqs 7 and 8 are more nearly correct for heterogeneous surfaces than for a homogeneous surface.11 Another peculiarity of those equations is that eq 7 corresponds to a grand canonical ensemble since the probabilities θν(i) and θγ(j) are determined by eq 2, which depend on the vector of absolute activities λ ) (λ1, ..., λm). On the other hand, eq 8 corresponds to a canonical ensemble with a fixed adsorption vector N0 ) (N1, ..., Nm). The components of the adsorption vector are determined by eq 4, which means that they correspond to a system of noninteracting particles (reference system) with the free energy F0. Switching on of interaction between adsorbed atoms should not change their numbers which are equal to those for the noninteracting system. This can only be achieved if one changes the chemical potential of the νth component by the value µν ) ∂W/∂Nν, where both W and Nν depend on the vector of absolute activities as described by eqs 2, 4, and 7, corresponding to the reference system of noninteracting atoms.11 (12) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974.
dW ) ∇W‚J-1‚dN )
∑ ∆µR dNR R)1
(10)
This equation gives the perturbations of the chemical potentials of the νth species as the νth component of the one row matrix ∇W‚J-1. Thus the solution to the problem of the m component mixture of interacting molecules adsorbed on B different sites is as follows: First, assume that the molecules do not interact with each other and find the adsorption vector N ) (N1, ..., Nm) vs the vector of activities λ ) (λ1, ..., λm). This is a well-known procedure of application of the Langmuir equation to a mixture on a heterogeneous surface which gives, in fact, the composition and the total coverage of adsorbed phase vs composition as well as the total pressure of the gas.4 Then invert the solution, that is, find λ vs N and use eqs 7 and 10 to make a correction to the chemical potentials of the components of the mixture due to interactions between the adsorbed molecules. Thus, in the final form the solution to the problem is given as the composition and the total pressure of the equilibrium gas phase vs composition and the total coverage of the adsorbed phase. III. One-Component Model of Adsorbed Mixtures A. Noninteracting LG. To solve the problem of a mixture of noninteracting molecules on a set of different adsorption sites, one has to know all the free energies 1(i), 2(i), ..., m(i) for any adsorption site with i ) 1, ..., B. For that, one has to have an idea of what actually an adsorption site is. In physical adsorption, the ith adsorption site may be modeled as a dimple on a surface, e.g., a hemispherical cup of radius Ri. If all the molecules can also be modeled as spheres and if their radii are smaller than Ri, then the energy of, say, the first species is determined by Ri. (It also depends on atomic parameters, together with combination rules for them, etc. which are supposed to be known.) Thus if one knows 1(i), one can in principle calculate Ri, and subsequently, all the energies 2(i), ..., m(i). In other words, we take the first component as a reference and consider the energy of a molecule on an arbitrary adsorption site to be a function of the energy of the first species. This approach is well known in the theory of the adsorption of mixtures on heterogeneous surfaces.4 Now define an effective absolute activity λ as follows m
λ ) λ1 +
∑ λR exp(-R1/kT) R)2
ν1 ) ν - 1
(11)
(In fact, ν, 1, and correspondingly λ in this equation
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depend on an adsorption site so that λ may be designated as λ(i).) The basic eqs 1 and 3 take the form
ln Ξ )
∫ ln[1 + λ exp(-1/kT)]f(1) d1
(12)
where f(1) d1 is the probability that the free energy of a molecule of the first (reference) component on a given adsorption site is close to 1. The probability density f(1) is normalized to the total number of adsorption sites B. This equation is very similar to that for a one-component system. The essential difference is that λ as determined by eq 11 depends upon 1 and is different for different adsorption sites. Thus it cannot play the role of activity. Now we make a sweeping approximation by assuming that all the ν1 are independent of 1. This means that all the distributions f(ν) have the same shape and differ from each other only by a shift along the energy axis.4 It is known that this approximation formally reduces the problem of multicomponent adsorption to that of adsorption of one component.4 The number of molecules of the νth component in the system is
Nν ) λν
∂ ln Ξ ∂ ln Ξ ) λν exp(-ν1/kT) ∂λν ∂λ
(13)
We sum up all the eqs 13 and use eq 11 to obtain m
N)
∂ ln Ξ
∑ NR ) λ R)1
∂λ
(14)
This equation shows that λ really plays the role of the absolute activity in that it determines the total number of particles in the system as if the multicomponent system is effectively one component. Now divide eq 13 by eq 14 to obtain
xν )
λν exp(-ν1/kT) λ
(15)
This equation determines the composition (as the mole fractions xν ) Nν/N) of the system. It is a somewhat more general form of eq 4.35 in ref 4, and was obtained first in ref 9. It follows from eqs 12 and 14 that λ is a function of the total coverage θ ) N/B: λ ) λ+(θ). Then from eq 15 we obtain
µν ) µν+(θ) + kT ln xν
(16)
µν+(θ) ) ν1 + kT ln λ+(θ)
(17)
where
is the chemical potential of the pure νth component (xν ) 1), adsorbed at the same coverage as in the mixture. One also may see from eqs 6 and 12 that λ may be written as a function of the spreading pressure: λ ) λ0(π). In this case eq 15 may be expressed as
µν ) µν0(π) + kT ln xν
(18)
where
µν0(π) ) ν1 + kT ln λ0(π) is the chemical potential of the pure νth component at the same spreading pressure as that of the mixture. From eqs 16 and 18 it follows that
µν+(θ) ) µν0(π)
(19)
which means that the spreading pressure π is a universal function of the total coverage: π ) π(θ). This function should be the same for all the pure components and all mixture compositions. It is apposite to recall here that for the Langmuir model of mixed adsorption on a homogeneous surface, the spreading pressure is really a universal function
π ) -kT ln(1 - θ)
(20)
that does not depend either on the microscopic constants ν (cf. eq 1) describing interactions of different molecules with an adsorption site or on the mixture composition. Equation 18 shows that the mixture of molecules modeled as a noninteracting LG on the model heterogeneous surface described above is an ideal mixture as defined in ref 13. Thus one can calculate the composition and coverage of adsorbate from the data obtained from the isotherms of adsorption of the pure components by the IAS method. This method considers the adsorption of pure components at the spreading pressure π of the mixture.13 In other words, the isotherm equations in IAS are considered as parametric equations, with the parameter being the spreading pressure.1 However, eq 16 shows that, at least on the model heterogeneous surface considered above, IAS is not the only choice. One can use a similar method based on consideration of the adsorption of pure components at the same total coverage. To put it differently, one may chose the total coverage as a parameter instead of spreading pressure. B. Interacting LG. Now we consider eq 7 in the approximation of the one-component model. To evaluate the energy of interaction of molecules adsorbed on sites of a heterogeneous surface, one has to know not only the distribution of those sites in energy but also the distances between sites which determine uRβ in eq 7. The latter information is related to the topography of a heterogeneous surface.14 A realistic model of adsorption on heterogeneous surfaces should take account of the fact that apart from distribution in energy, the locations of adsorption sites are scattered over the surface. Such a model of adsorption on a disordered surface has been considered (see, e.g, section 13.3 of ref 14) with the help of Monte Carlo simulation. In our analytic approach we follow the conventional route, i.e., we assume that adsorption sites, though randomly distributed in energy, are located at the vertices of a regular net. This is a rather sweeping approximation which, however, is almost always implicitly made in the theory of adsorption on heterogeneous surfaces.14 Besides, we will consider only the interaction of the nearest neighbors. The purpose of these additional assumptions is to make the interaction energy uRβ(i,j) in eq 7 to be constant with respect to the variation of i and j. With all these approximations eq 7 takes the form
W)
uRβ ∫ d1 ∫ d1′f(1)g(1′|1)θR(1)θβ(1′) ∑ Rβ
(21)
Here g(1′|1) is the probability that the adsorption energy for the reference component on a site which is the nearest neighbor of another (central) site is 1′ on the condition that the adsorption energy of the central site is 1. (13) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (14) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992.
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Equation 21 is already tractable, but one still has to know the conditional probability g(1′|1). At this stage we just make the simplest assumption that the probabilities for 1′ and 1 are statistically independent
g(1′|1) ) (z/B)f(1′)
(22)
This means that f(1′) and g(1′|1) differ only in normalization constants: the former is normalized to the total number of sites on the surface B and the latter is normalized to the number of the nearest neighbors of a site z. Now the double integral in eq 21 factorizes into two identical integrals and since
∫ d1f(1)θR(1) ) NR and ∑RNR ) Bθ we obtain
∑R ∑β uRβxRxβ
W ) zBθ2
(23)
This is the energy of adsorbate/adsorbate interaction for a mixture with a composition determined by the mole fractions. For adsorption of the pure νth component at the same total coverage θ as the mixture, one should put xν ) 1 in eq 23 and all other mole fractions equal to zero. Then one obtains the adsorbate/adsorbate energy of mixing
WM ) W -
∑R xRWR
(24)
where WR is the adsorbate/adsorbate energy of the pure Rth component at the same total coverage as that for the mixture. Since in eq 8 F0 is the free energy of an ideal mixture which has zero energy of mixing, eq 24, in fact, determines the excess free energy of the adsorbed solution E
M
F )W
(25)
Consider now the special case of a binary mixture. Equations 23-25 give the following result
FE ) 2zBθ2wxAxB
FE ) 2(z/B)wNANB the excess chemical potentials are
µBE ) 2zwθxA
µA ) µA+(θ) + kT ln xA + 2zwθxB µB ) µB+(θ) + kT ln xB + 2zwθxA
(27)
Thus, in our case, the dependence of the excess chemical potential on the mole fraction of the other component is linear, while for a regular solution it is quadratic.15 We determined the excess chemical potentials in eq 27 directly as derivatives with respect to the number of particles, e.g., µAE ) ∂FE/∂NA at constant NB. This became possible because we made the assumption of eq 22 and factorized the double integral in eq 21. This assumption (15) Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, 1952. (16) Prigogine, I. The Molecular Theory of Solutions; North-Holland Publishing Co.: Amsterdam, 1957; Chapter 3.
(28)
Let the numbers of molecules of the first and second components NA and NB (and correspondingly the mole fractions xA and xB) be given. If the total number of adsorption sites B is known, one determines the total coverage θ. Then one determines (e.g., from an experiment) the chemical potentials of the pure components µA+(θ) and µB+(θ) at the same total coverage and calculates the chemical potentials in the mixture by eq 28, taking zw, e.g., as an adjustable parameter. This is essentially the same procedure as in IAS in the sense that it does not require analytical expression for the adsorption isotherms of the pure components. The difference, however, is that in IAS the parameter which is kept constant is the spreading pressure while here it is the total coverage. C. Constant Coverage vs Constant Spreading Pressure. Now we consider how eq 28 changes if one holds the spreading pressure constant rather than the total coverage. Given the total coverage and the composition of a binary mixture, one may calculate spreading pressure from the following:
π)
[
∫0θ xA
]
∂µA ∂µB θ dθ + xB ∂θ ∂θ
(29)
This equation follows from eq 6 if one integrates d ln Ξ in eq 13 (or the Gibbs-Duhem equation) along the line of constant composition. Equations 28 and 17 are substituted into eq 29 to obtain
(26)
where w ) uAB - (uAA + uBB)/2 is the interchange energy.15,16 This equation differs only by the factor 2Bθ2 from the expression for the excess molar free energy (FE/N) of a strictly regular solution.15,16 However, this makes a qualitative difference in the composition dependence of the excess chemical potentials. Since eq 26 can be written as
µAE ) 2zwθxB;
made W in eq 23 be a function of the total coverage θ and mole fractions xν. Generally W is a function of absolute activities and determination of the excess chemical potentials would require the more complex procedure described in section IIB. Adding the ideal part of the chemical potentials to these excess chemical potentials, we obtain equations that solve (for our model) the problem of binary adsorbed mixtures on heterogeneous surfaces:
π ) πid(θ) + 2zwθ2xAxB
(30)
Here πid(θ) is the ideal solution component of the spreading pressure which follows from integration of eq 17 in eq 29. This part does not depend on the composition of the adsorbed mixture, in line with section IIA. Another part of the spreading pressure results from the interaction between adsorbed molecules. This part depends upon the composition of the adsorbed mixture. Thus, if one keeps the spreading pressure constant and changes the composition of the adsorbate (e.g., xA), one will have the total coverage θ changing and the excess thermodynamic functions, e.g., FE from eq 26, will differ when they are taken at constant total coverage or at constant spreading pressure. This raises the question at which condition they are smaller, that is, when the mixture is closer to ideal. The problem is a quantitative one which depends on the relative values of two terms on the right of eq 30. To gain some insight we have to consider a numerical example. To this end, we consider a special case of the uniform distribution of adsorption sites in energy for the first component
B if min e 1 e max max - min ) 0 otherwise
f(1) )
(31)
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Figure 1. Spreading pressure: solid line, total, on a heterogeneous surface; dashed line, Langmuir spreading pressure on a homogeneous surface; dash-dotted line, the interaction component.
One may use eqs 12 and 14 to obtain the total coverage θ vs effective activity λ (in the noninteracting lattice gas approximation)
1 + λ exp(-min/kT) kT θ) ln max - min 1 + λ exp(-max/kT)
Figure 2. Excess Helmholtz free energy: solid line, at constant spreading pressure; dashed line, at constant total coverage.
adsorption. This is known to describe experimental data with corresponding (“Langmuirian”) shape of the isotherm at least qualitatively correctly. The chemical potential can be expressed via the fugacity fν as17
µν ) µν0 + kT ln fν
(33)
(32)
One may also numerically integrate eq 12 to obtain the ideal solution component of spreading pressure as a function of λ. This is actually an equation that was called UNILAN in ref 2 and successfully used there for the description of pure gas isotherms. Eliminating λ from both equations, one obtains πid(θ) in eq 30. The dependence of spreading pressure upon coverage is presented in Figure 1. (In the LG model π/kT is a dimensionless variable as shown in eq 6.) The solid curve was calculated from eq 30. The parameters of eq 32 were emin/kT ) -11 and emax/kT ) -9 and the interchange energy parameter multiplied by the coordination number is 2zw/kT ) 5 (cf. eqs 26-28 and 30). (These are values which are often encountered in physical adsorption.) The dashed curve represents the spreading pressure corresponding to the Langmuir equation (20). Finally, the dotdashed curve corresponds to the second term on the right of eq 30 calculated at an equimolar composition. If one takes the coordination number z ) 5, then the value of the interchange energy w will be one order of magnitude smaller than the usual value of the adsorbate/adsorbent energy in physical adsorption. Although the value of w normally should be even smaller than that, we choose this enlarged value of w to make the figures more legible. Now one can numerically solve the equation f(θ,xA) ) π at constant xA (where f(θ,xA) is the right-hand side of eq 30) to obtain θ ) θ(π,xA) and to calculate the excess thermodynamic functions, e.g., FE from eq 26 vs xA at constant π. The result is presented in Figure 2. The solid curve is calculated on the condition π/kT ) 4 (cf. Figure 1). One may see that the excess thermodynamic function is less at constant spreading pressure than at constant total coverage. This might have been an explanation of the success of IAS were it not that the difference between both values so small. Given the fact that we deliberately enlarged the value of w (which is the reason why the excess free energy is so large in Figure 2), the difference is almost negligible. IV. Comparison with Experiment and Discussion. Consider first eq 2, the Langmuir equation for mixed
where µν0 is the chemical potential -kT ln p+ for the νth species in the reference state which we choose to be the ideal gas state of the pure νth component at unit pressure (unit fugacity). (Here µν0 is expressed not per mole as in ref 17 but per molecule.) We also introduce the fugacity coefficient φν17 as fν ) Pyνφν and evaluate it for the gas phase mixture by the virial expansion to the second virial coefficient. For a binary gas mixture (the only one considered here), this approximation gives17
2 ln φ1 ) (y1B11 + y2B12) - ln zmix v 2 ln φ2 ) (y1B12 + y2B22) - ln zmix v
(34)
where the Bij values are second virial coefficients, v is the molar volume of the gas mixture, and zmix is its compressibility coefficient that also is evaluated in the second virial coefficient approximation.17 Now the absolute activities in eq 2 take the form λν ) Pyνφν exp[µν0/kT] and the Langmuir equation for mixed adsorption is
θν )
Pyνφν exp[-∆ν/kT] m
1+P
(35)
∑ yRφR exp[-∆R/kT]
R)1
where ∆ν ) ν - µν0 is the difference between the free energy of one molecule on an adsorption site (all sites are assumed identical here so that the index marking a site is unnecessary) and the chemical potential per molecule in the ideal gas reference state described above. For a binary mixture it is convenient to define a separation factor3 (selectivity2): (17) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969.
Adsorbed Mixtures on a Heterogeneous Surface
S ) x2y1/x1y2
Langmuir, Vol. 12, No. 25, 1996 6125
(36)
where xν and yν are the mole fractions of the νth component (ν ) 1, 2) in the adsorbate and in the corresponding gas phase, and it is assumed that the second component is more strongly adsorbed than the first (S > 1). Since xν ) θν/θ (θ is the total coverage), one obtains from eqs 35 and 36 that
S)
φ2 exp[-(∆2 - ∆1)/kT] φ1
(37)
If the gas phase is assumed to be ideal, then φ1 ) 1 and φ2 ) 1 so that the separation coefficient for the Langmuir representation of the adsorption of a binary mixture is a constant equal to the ratio of the Henry constants of the pure components.3 This is true only for a homogeneous surface. A similar argument gives eq 15 for a heterogeneous surface with an arbitrary distribution in free energy of adsorption sites for a single molecule on condition that the differences between free energies of two different molecules on the same site are the same for all adsorption sites. In this case
Figure 3. Experimental isotherms of pure methane and ethane on active carbon:18 solid line, UNILAN; dashed line, the Langmuir equation; crosses, methane; diamonds, ethane. Table 1. Selectivity of Methane-Ethane Mixtures on Active Carbon y1
Sexp
(38)
0.981 0.945 0.827
15.1 16.4 17.1
Now we compare eq 38 with experimental data. In the first place, we write eq 32 for two (ν ) 1, 2) pure components
0.980 0.916 0.699
16.4 13.5 13.5
φ2 exp[-(21 - µ20 + µ10)/kT] S) φ1
Ns cν + Pφν(P) exp(s) ln 2s cν + Pφν(P) exp(-s)
S
SIAS
11.1 11.1 11.1
21.1 18.9 17.1
P ) 5 bar
P ) 20 bar 11.1 11.1 11.1
We could use eq 39 to describe the experimental isotherms of methane and ethane on an active carbon as presented, e.g., as in ref 18. However in this case we obtain different values for the parameters s and Ns for these gases. This will contradict our model: the adsorption capacity Ns is equal to the number of adsorption sites and must be the same (e.g., B in eq 32) for both gases. Also s measures the width of the distribution of the adsorption site energy and must be the same in our model because, as assumed in section IIIA, the distributions in 1 and 2 (methane and ethane) have the same shape and differ only by a shift along the energy axis. Thus to apply our model consistently, we have to determine four parameters c1, c2, s, and Ns simultaneously from two adsorption isotherms under the constraint that s ) s(1) ) s(2) and Ns ) Ns(1) ) Ns(2). If we can successfully describe experimental isotherms for pure gases in this way, we may hope that
the mixed adsorption will be really ideal and the calculation of selectivity will be reliable. The results of this method of describing experimental data are presented in Figure 3. Equation 39 depends nonlinearily upon the parameters. To find them, we used the Marquardt method19 which, however, needs a reasonable initial approximation for success. As an initial approximation, we used s ) 0 in eq 39, which reduces it to the Langmuir equation2 that can be easily linearized. Then we found the parameters of the two coupled Langmuir equations which have the same adsorption capacity but different constants c1 and c2 in the limit of eq 39 at s f 0. We then fitted the two isotherms of CH4 and C2H6 from Table 1 of ref 18. These are presented in Figure 3. Then we increased the value of the parameter s by, e.g., steps of 0.2 and at each step used the Marquardt method to find the three other parameters. The merit function was χ2, which is an output of the standard subroutine from ref 19. The result corresponding to the lowest value of χ2 is presented in Figure 3. Neither the Langmuir equation nor eq 39 (UNILAN of ref 2) are able to describe reasonably the experimental isotherms of pure methane and ethane from ref 18 under the above mentioned constraints. Thus one cannot expect a very good description of the mixed adsorption. The separation factor calculated from eq 40 is given in Table 1 as S. The values of S obtained from the experimental data in Table 3 of ref 18 are designated by Sexp. In the last column of Table 1, the separation factor calculated by the IAS method is presented. (To calculate this, we described the isotherms of methane and ethane from ref 18 by the Dubinin-Radushkevich (DR) equation.) The extrapolated vapor pressures of those substances at 303.15 K were taken from ref 18 although there is no explanation there how they were obtained. The parameters of the DR
(18) Richter, E.; Schu¨tz, W.; Myers, A. L. Chem. Eng. Sci. 1989, 44, 1609.
(19) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes; Cambridge University Press: Cambridge, 1992.
Nν )
(39)
where ν ν /kT ) 0ν /kT - s; max /kT ) 0ν /kT + s; cν ) min
exp[(0ν - µ0ν )/kT] Apart from the fugacity coefficients which account for the nonideality of the gas, eq 39 is the UNILAN equation widely used in ref 2 for description of the experimental isotherms of pure gases. One substitutes the coefficients cν from eq 39 into eq 38 (21 ) 20 - 10) to obtain
S)
φ2c1 φ1c2
(40)
6126 Langmuir, Vol. 12, No. 25, 1996
equations given in Table 2 of ref 18 are in error (probably misprint) and were redetermined. The DR equations describe experimental isotherms of pure methane and ethane from Table 1 of ref 18 as well as UNILAN without above mentioned constraints. It is claimed in ref 18 that IAS based on the DR equation describes the mixed adsorption at lower pressure (5 bar) reasonably well. Deviations at a higher pressure (20 bar) were ascribed to nonidealities in the adsorbed phase. Table 1 shows that this might be not exactly the case. Although the IAS values are definitely closer to experiment than our model at lower values of y1, they show a dependence upon the composition of the gas phase opposite to that of the experimental data. Our data presented in the third column of Table 1 do not display any dependence on the composition of the gas phase (y1) or its total pressure p. In fact, they do depend on y1 and p because φ1 and φ2 in eq 40 depend on them. However, this dependence is very weak compared to the dependence of the experimental data on those factors. Thus the separation factor provided by our model is almost constant and lower than its experimental values in Table 1. However, we believe that there is one advantage of our approach over such widely used approximations as IAS. Already from application of our model to the individual isotherms in Figure 3, we see that the model should not be expected to provide a very good description of mixed adsorption. There is no such warning with the majority of attempts to obtain the properties of mixed adsorption from isotherms of pure components. That is because the methods applied are either semiempirical as in IAS or they are based on a definite model like the Langmuir model but are inconsistently applied to the description of experimental data, e.g., without taking care of the intrinsic constraints of the Langmuir model. As one sees from Table 1, the model under discussion can only give rough evaluations of the essential properties of mixed adsorption. It does not reflect the realistic dependence of the separation factor S upon the composition of adsorbed mixture as shown by, e.g., the experimental data of ref 18. This dependence was a subject of a theoretical study6 and computer simulations.7,8 In our opinion, the main defect of the model considered is that of the Langmuir model as a whole: it requires that the adsorption capacity of all the components of an adsorption mixture be the same. There have been some attempts to remove this restriction (see ref 14, Chapter 12) which draw on the ideas of the LG theory of polymer solutions.15,16 However they substitute the very clear and appealing Langmuir picture that every site adsorbs only one molecule by far less clear assumptions. V. Conclusion We have considered a Langmuir type model of mixed adsorption that guarantees the adsorbed mixture to be
Bakaev and Steele
ideal. It means that we can determine the thermodynamic properties of an adsorbed mixture from the isotherms of the pure components. This is similar to the thermodynamics theories like IAS, but the latter cannot tell from the isotherms of pure components how reliable its prediction of mixed isotherms will be. In contrast, the statistical model discussed here, even when based on rather sweeping approximations, can in principle do that. However, to that end one should not consider isotherms of adsorption of pure components independently. For example, if one chooses as a model the Langmuir equation for the mixed binary adsorption, one should not apply the Langmuir equation to both components independently and then determine the separation factor of the mixture as the ratio of their constants. Instead one should apply two Langmuir equations simultaneously to the experimental isotherms for each component under the constraint that their adsorption capacities be the same. If one is successful in matching experimental isotherms of the pure components under this constraint (which will seldom be the case), one may hope that the prediction of the properties of mixed adsorption (e.g., the separation factor) will be also successful. If not, one, at least, will be warned that this particular model cannot provide reliable predictions of the mixed adsorption. This is what actually happened with the model considered in this paper. From Figure 3 one can see that not only the Langmuir equation but even eq 39, which should describe adsorption on heterogeneous surfaces (and usually does this very well), cannot match the experimental isotherms of the pure components under the constraints that guarantee the ideality of the mixed adsorption of those components. Thus we still can evaluate the separation factor but are warned not too expect too much of it. Nevertheless we deem the model considered in this paper not useless. It naturally raises the question whether the constancy of the spreading pressure which is an essential feature of IAS is really so important. For our model, it is not: the condition of constant spreading pressure is equvivalent to that of constant coverage. But this model is restricted by the basic constraint of the Langmuir model that the adsorption capacities of all the components be equal. Without this restriction the condition of constant spreading pressure will not be equivalent to that of constant coverage even for an ideal adsorbed solution. Our model also shows that the LG model of adsorbed solutions which corresponds to the strictly regular15,16 model of bulk solutions may have different dependences of excess thermodynamic functions on composition as against the bulk regular solutions. Acknowledgment. Support for this research was provided by Grant DMR 902 2681 of the Division of Material Research of the N.S.F. LA9604036
