Langmuir 1996, 12, 5969-5970
5969
Notes A Simple Langmuir Type Model for Mixed Adsorption of Unequal Size Molecules V. A. Bakaev Department of Chemistry, 152 Davey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802 Received April 15, 1996. In Final Form: August 21, 1996
In statistical thermodynamics, the meaning of the Langmuir adsorption equation is that the surface may be separated into independently adsorbing partssadsorption sites or simply sitessthe adsorption capacity of each site being only one molecule. These sites are places on a surface where adsorption activity is concentrated. It was shown that such sites really exist on heterogeneous surfaces and are marked by sharp spikes of adsorption activity (cf., e.g., Figure 2 from ref 1). In the case of adsorption of mixtures, however, this model runs into the notorious difficulty that it requires that the sizes of the components must be the same. This is because the adsorption capacity in the Langmuir model is clearly equal to the number of sites and must be the same for all the components of a mixture, but it is well-known that the adsorption capacities of different size molecules are different in real adsorption experiments. There have been some attempts to obviate this difficulty.2 The more advanced approach leans on the ideas of the lattice model of solutions of flexible polymer molecules: a molecule is subdivided into segments and each segment is assumed to adsorb on a separate site. This approach, however, destroys the main advantages of the Langmuir model. First, adsorption sites are not independent any more. Second, these sites are more closely connected with an adsorbed molecule than with the adsorbing surface and thus partially lose the realistic character they have in the Langmuir model. Indeed, in the lattice models of polymer solutions the vertices of the lattice are imaginary rather than real possible positions of the sections of a molecule. Finally, not all of the molecules of different sizes which are studied in adsorption of mixtures are flexible. The purpose of the new model of mixed adsorption considered in this note is to preserve the main advantages of the Langmuir modelsthe independence and real character of sitessfor mixed adsorption. Its idea is reminiscent of the author’s early work on the statistical thermodynamics of adsorption on zeolites.3 It was assumed there that the cavities of zeolites (micropores in general) may be modeled as independently adsorbing units. On the grounds of that assumption, the calculation of the grand partition function of a zeolite may be reduced to that of the partition function of a single cavity. A simple approximation for the latter has been found by Ruthven, (1) Bakaev, V.; Steele, W. In Adsorption on New and Modified Inorganic Sorbents; Dabrowski, A., Tertykh, V. A., Eds.; Elsevier: Amsterdam, 1996. (2) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992; Chapter 12. (3) Bakaev, V. A. Dokl. Akad. Nauk SSSR 1966, 167, 369.
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and in this way an approximate theory of adsorption on zeolites has been developed.4 Here we try to apply essentially the same idea to the Langmuir model of mixed adsorption. First, consider two hard spheres of different radii adsorbed on a model surface whose atomic structure is a dense random packing of hard spheres. This may be a model of the physical adsorption on an amorphous oxide surface, and we called such a surface the Bernal surface (BS).1 A stable position of a hard sphere modeling an adsorbed molecule on BS is one where it touches three underlying hard spheres representing, e.g., the oxide ions of amorphous oxide surface. Then the center of the molecule is at an adsorption site. Even if two molecules of different sizes lay on the same oxide ions, their centers are in different positions. One can also easily imagine that on such an irregular surface as BS there will be cases when molecules of different radii touch different triangles of oxide ions but their centers are still very close to each other. In other words, the adsorption sites may be different for the molecules of different sizes on the same surface. Now we consider a lattice model of binary mixed adsorption. We designate the larger molecules by A and the smaller ones by B. Consider also two sublattices A and B. Molecules A adsorb on the vertices of the sublattice A and molecules B adsorb on the vertices (sites) of the sublattice B. The two sublattices are not independent: if a molecule B is adsorbed on a B site, then the A site nearest to it must be empty. On the other hand, if a molecule A is adsorbed on an A site, then there are two possibilities: either only one B site nearest to the A site must be empty or two B sites nearest to the occupied A site must be empty. We assume that the complexes of adsorption sites consisting of one A site and two B sites closest to it or one A site and one B site closest to it act independently as adsorption sites in the Langmuir model or micropores in the model of ref 3. Finally, we designate by p1 and p2 (p1 + p2 ) 1) the fractions of complexes of the first and second kind. We will also call them sites although they differ from the Langmuirian sites in the sense that some of them (whose ratio is p1) can adsorb up to two molecules of B or only one molecule of A. The total number of these new sites will be Ns. The statistical thermodynamics of the model is almost as simple as that of the Langmuir model. The grand partition function of the whole surface is the product of those for separate statistically independent sites so that
Ns-1 ln Ξ ) p1 ln{1 + λA exp(-A/kT) + 2λB exp(-B/kT) + [λB exp(-B/kT)]2} + p2 ln{1 + λA exp(-A/kT) + λB exp(-B/kT)} (1) It gives the isotherms of mixed adsorption
NA ) λA
∂ ln Ξ ; ∂λA
∂ ln Ξ NB ) λB ∂λB
(2)
The deviation of the solution from ideality may be characterized by the excess molar Gibbs free energy (4) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; J. Wiley & Sons: New York, 1984; p 78.
© 1996 American Chemical Society
5970 Langmuir, Vol. 12, No. 24, 1996
( )
( )
λA λB gE + xB ln 0 ) xA ln 0 RT λA xA λB xB
Notes
(3)
In eqs 1, 2, and 3 λA ) exp(µA/kT) and λB are the absolute activites of the corresponding components of the adsorbed solution; λA0 and λB0 are the absolute activities of pure components. In other words, index 0 in, e.g., λA0 designates that in eq 1 one must take λA ) λA0 and λB ) 0. Then one may find λA0 from the equation ln Ξ(λA0, 0) ) ln Ξ(λA, λB). In the same way one determines λB0 and finally obtains gE from eq 3 on condition of mixing at constant spreading pressure. Similarily one may find λA0 and λB0 from eq 2 and determine gE on condition of mixing at constant total coverage. In this case, however, a solution to the equation NA(λA, λB) + NB(λA, λB) ) NA(λA0, 0) may not exist because the maximal value of NB is Ns(2p2 + p1) and the maximal value of NA is Ns. In eq 3, xA ) NA/(NA + NB) and xB ) 1 - xA are mole fractions. Finally, A and B in eq 2 are free energies of single molecules adsorbed on corresponding sites (e.g., A ) -kT ln qA, where qA is the partition function of a single A-molecule on a site). The following numerical example shows that the model might be useful for the theory of mixed adsorption. Let A/kT ) -15 and B/kT ) -10 with p1 ) p2 ) 0.5. In Figure 1 the isotherm of adsorption from a gas mixture is compared with isotherms of pure components as obtained from eqs 1 and 2. The isotherms of pure components are obtained when either λA or λB in eq 1 is equal to zero. They obey the Langmuir equation. The isotherm corresponding to the larger molecule is shifted to the left because it has larger absolute value of adsorption energy. The isotherm of the mixture in Figure 1 corresponds to equal chemical potentials of components: µA ) kT ln λA ) µB ) kT ln λB. It is qualitatively similar to that obtained from an exact one-dimensional model of mixed adsorption of hard rods on homogeneous and heterogeneous onedimensional surfaces in ref 5. At the low values of chemical potential the isotherm of a mixture almost coincides with that of a larger and more strongly adsorbed component. In Figures 1 and 6 of ref 5 the former is slightly shifted to the right with respect to the latter because there we considered an equimolar ideal gas mixture where the chemical potentials of components are less than those of pure components at the same pressure due to entropy of mixing. In the present case, we have imposed another condition which makes our gas mixture almost but not exactly equimolar. The increase of the chemical potentials of components in Figure 1 corresponds to increase of total pressure in the gas phase. Since chemical potentials of components are chosen to be equal, the composition of the gas phase changes only a little. However, as seen from Figure 1 (cf. also Figure 2 of ref 5), the composition of adsorbed phase changes drastically. With the increase of chemical potential, the molecules that are larger and more strongly attracted to the surface (sites) are substituted by the less attractive but smaller ones. The deviation of the adsorbed solution from ideality is shown in Figure 2. The curves were calculated from eq 3 with λA ) exp(-µ0/kT)yA and λB ) exp(-µ0/kT)(1 - yA), where yAsthe mole fraction of larger molecules in the gas phaseschanged from zero to unity. Then the total adsorption, the mole fractions of components and the spreading pressure (Ns-1 ln Ξ) can be calculated from eqs 1 and 2. Finally λA0 and λB0 were determined as a solution of eq 1 with fixed left hand side and correspondingly λB (5) Bakaev, V. A.; Steele, W. A. Langmuir, in print.
Figure 1. Isotherms of adsorption: solid line, gas mixture, chemical potentials of components are equal; broken lines, pure components.
Figure 2. Excess molar Gibbs free energy at constant spreading pressure. µ0/kT ) (1) -9.0; (2) -7.0; (3) -5.0; (4) -3.0.
) 0 or λA ) 0 (in which cases it reduces to the Langmuir equation). Apart from the first curve which at yA ≈ 0.5 corresponds to the region of the left inflection point of the mixed isotherm in Figure 1, the curves in Figure 2 display mainly negative deviation from Raoult’s law. Similar behavior of adsorbed mixture was observed for the onedimensional model of the heterogeneous surface discussed in ref 5. It was emphasized there (cf. discussion section in ref 5) that this is typical of adsorption on active carbons and other adsorbents with heterogeneous surfaces. In conclusion, the model of mixed adsorption considered in this note makes it possible to obtain isotherms and other thermodynamic properties of mixed adsorption from individual isotherms of pure components. However, those individual isotherms obey the Langmuir equation which does not always describe experimental data sufficiently well. Thus the model should be modified to allow for nonLangmuirian individual isotherms in order to be of practical utility in predicting binary data from pure component isotherms. This can be done (similarly as in the one-component case (ref 2)) by introducing the distribution of adsorption sites (of both types) in energy. Work is underway in this direction. Acknowledgment. I thank W. A. Steele for discussion of this paper. Support for this research was provided by Grant DMR 902 2681 of the Division of Material Research of the NSF. LA9603582
