Statistical Thermodynamics of Linear Adsorbates in Low Dimensions

Langmuir 1999, 15, 5707-5712

5707

Statistical Thermodynamics of Linear Adsorbates in Low Dimensions: Application to Adsorption on Heterogeneous Surfaces† A. J. Ramirez-Pastor, V. D. Pereyra, and J. L. Riccardo* Departamento de Fı´sica, Universidad Nacional de San Luis, CONICET Chacabuco 917, C.C. 236, 5700 San Luis, Argentina Received September 29, 1998. In Final Form: January 7, 1999

Exact forms for the thermodynamic functions of linear adsorbates in one dimension are presented in the lattice gas description of localized adsorption with multisite occupancy. A modified form of the adsorption isotherm for linear adsorbates on heterogeneous substrates is proposed, by introducing the rigorous expression for the Helmholtz free energy of the adlayer on a homogeneous one-dimensional lattice into the multisite occupancy model proposed by Nitta et al. (1984). Comparisons between the original and modified isotherms and Monte Carlo simulations for dimers and 4-mers on heterogeneous surfaces with two type of sites are carried out. Fitting of the adsorption isotherm to experimental data for CO, N2, and O2 on zeolites 5A and 10X is performed and discussed as well.

I. Introduction Multisite occupancy adsorption on heterogeneous surfaces is a topic of major interest in surface science since it comprises most of the features present in experimental situations. Most adsorbates, except noble gases, are polyatomic. Furthermore, surfaces generally present inhomogeneities due to irregular arrangement of surface and bulk atoms, the presence of various chemical species, etc., which can significantly affect the entropic contribution to the adsorbate’s free energy. Typical examples are O2, N2, CO, and CO2 absorbed in carbon and zeolite molecular sieves1,2-5 and oligomers in activated carbons.1,6 Despite the obvious significance of polyatomic adsorption, most developments in adsorption theory have mainly dealt with monatomic adsorption.1,7,8 Contributions to this subject concerning adsorption energy distributions, adsorption equilibrium, and kinetics were presented in terms of lattice-gas approximations.9-17 The complexity of these † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, Held in Poland, August 9-16, 1998. * To whom correspondence should be addressed. E-mail: [email protected].

(1) Rudzin´ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (2) Danner, R. P.; Wenzel, L. A. AIChE J. 1969, 15, 515. (3) Miller, G. W.; Knaebel, K. S.; Ikels, K. G. AIChE J. 1987, 54, 2. (4) Schalles, D. G.; Danner, R. P. Adsorpt. Ion Exch. AIChE Symp. Ser. 1987, 84 (264), 83. (5) Goulay, A. M.; Tsakiris, J.; Cohen de Lara, E. Langmuir 1996, 12, 371. (6) Koble, R. A.; Corrigan, T. E. Ind. Eng. Chem. 1952, 44, 383. (7) Steele, W. A. The interaction of gases with solid surfaces; Pergamon Press: New York, 1974. (8) Jaroniec, M. J.; Madey, R. Physical Adsorption on Heterogeneous Surfaces; Elsevier: Amsterdam, 1988. (9) Tovbin Y. K. Russ. J. Phys. Chem. 1974, 48, 1239. (10) Tovbin Y. K. Theor. Exp. Chem. 1981, 17, 28. (11) Nitta, T.; Kuro-oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 45. (12) Nitta, T.; Yamaguchi, A. J. J. Chem. Eng. Jpn. 1992, 25, 420. (13) Nitta, T.; Yamaguchi, A. J. Langmuir 1993, 9, 2618. (14) Nitta, T.; Kiriyama, H.; Shigeta, T. Langmuir 1997, 13, 903. (15) Marczewski, A. W.; Derylo-Marczewska, M.; Jaroniec, M. J. J. Colloid Interface Sci. 1986, 109, 310. (16) Rudzin´ski, W.; Zajac, J.; Hsu, C. C. J. Colloid Interface Sci. 1985, 103, 528.

systems represents a major difficulty to the development of approximate solutions for the thermodynamic functions. To this respect, simple solvable models of adsorption on homogeneous surfaces are useful for devising alternative approaches for heterogeneous surfaces.7,11-14,17 The description of thermodynamics of polyatomic adsorbates in low dimensions is also gaining much attention since the recent advent of modern techniques for building singlewalled and multiwalled carbon nanotubes,18-21 which has considerably encouraged the investigation of the gassolid interaction (adsorption and transport of simple and polyatomic adsorbates) in such a low dimensional confining adsorption potentials. The design of carbon tubules, as well as of synthetic zeolites and aluminophosphates such as AlPO4-5 22 having narrow channels with diameters of a few angstroms, literally provides a route to the experimental realization of one-dimensional adsorbents. Although experimental studies of adsorption isotherms and thermodynamics of polyatomic adsorbates in nanotubes are still very limited, interesting results have recently been reported for noble gases and molecules with spherical symmetry adsorbed in AlPO4-5 pores.23 A remarkable feature of the gas-solid interaction in single-walled nanotubes (SWN) is an adsorption potential significantly larger than the one on a planar layer of bulk graphite (for instance, for atomic hydrogen isosteric heat of adsorption in a SWN is 19.5 kJ/mol while its value on bulk graphite is 4.9 kJ/mol24). For theoretical purposes adsorption of molecules with a cross sectional typical size comparable to the nanotube diameter can in principle be treated in the one-dimensional lattice gas approach. In this work we present the exact solution for the thermodynamics (17) Ramirez-Pastor, A. J.; Nazzarro, M. S.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1995, 341, 249. (18) Ijima, S. Nature 1991, 345, 56. (19) Ijima, S.; Ichihaschi, T. Nature 1993, 363, 603. (20) Ajayan, P. M.; et al. Nature 1993, 361, 333. (21) Ebbensen, T. W. Science 1994, 265, 1850. (22) Martin, C.; Coulomb, J. P.; Grillet, Y.; Kahn, R. Fundamentals of Adsorption; Le Van, M. D., Ed.; Kluwer Academic: Publishers: Boston, 1996. (23) Martin, C.; Tosi-Pellenq, N.; Patarin, J.; Coulomb, J. P. Langmuir 1998, 14, 1774. (24) Dillon, A. C.; Jones, K. M.; Bekkedahl, T. A.; Kiang, C. H.; Bethune, D. S.; Heben, M. J. Nature 1997, 386, 377.

10.1021/la981346e CCC: $18.00 © 1999 American Chemical Society Published on Web 04/24/1999

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functions of linear adsorbates assumed as linear chains (k-mers) of homonuclear units on a infinite one-dimensional discrete space. The thermodynamics is further extended to higher dimensions based upon their onedimensional form and a connectivity ansatz. Then the rigorous analytical solution for the Helmholtz free energy in one dimension is included into Nitta’s formalism of multisite occupancy adsorption11-14 on heterogeneous surfaces to properly account for the homogeneous contribution to the total Helmholtz free energy. Comparisons with Monte Carlo (MC) simulations show that this leads to a modified adsorption isotherm equation appreciably improved with respect to the standard formulation leading to the Flory-Huggins isotherm in the homogeneous case. The calculation of thermodynamic functions for noninteracting chains adsorbed on a one-dimensional lattice and extension to higher connectivities is carried out in section II. In section III, a modified version of the adsorption isotherm for heterogeneous surfaces is outlined based upon Nitta’s approach. Comparisons between analytical isotherms, model MC simulations, and experimental ones are carried out in section IV. Discussion of results follows, and summary and conclusions are drawn in section V. II. Noninteracting k-mers on a Homogeneous Lattice Let us assume a one-dimensional lattice of M sites with lattice constant a (M f ∞) and periodic boundary conditions. Under this condition all lattice sites are equivalent; hence border effects will not enter our derivation. N linear k-mers are adsorbed on the lattice in such a way that each mer occupies one lattice site and double site occupancy is not allowed as to represent the monolayer regime. Since different k-mers do not interact with each other through their ends, all configurations of N k-mers on M sites are equally probable; henceforth, the canonical partition function Q(M,N,T) results

(

Q(M,N,T) ) Ω(M,N) exp -

)

Nk kBT

(1)

where Ω(M,N) is the total number of configurations and  is the interaction energy between an individual unit of the k-mer and the lattice site. Ω(M,N) can be readily calculated as the total number of permutations of the N indistinguishable k-mers out of ne entities, ne being25

ne ) number of k-mers + number of empty sites ) N + M - kN ) M - (k - 1)N

(2)

N indistinguishable k-mers (N!) and the (M - kN) indistinguisable empty sites ((M - kN)!). By close inspection of eq 3 it is clear that no overcounting of configuration is done according to the way k-mers are adsorbed on the lattice. In the canonical ensemble framework the Helmholtz free energy F(M,N,T) relates to Ω(M,N) through

βF(M,N,T) ) -ln Q(M,N,T) ) -ln Ω(M,N) + βNk (4) where β ) 1/kBT. The remaining thermodynamic functions can be obtained from the general differential form26

dF ) -S dT - Π dM + µ dN

where S, Π, and µ designate the entropy, spreading pressure, and chemical, potential respectively, which, by definition, are

∂F (∂T )

S)-

M,N

∂F (∂M )

Π)-

µ)

T,N

∂F (∂N )

T,M

βF(M,N,T) ) -{ln[M - (k - 1)N]! - ln N! ln[M - kN]!} + βNk (7) which can be written in terms of the Stirling approximation as follows

βF(M,N,T) ) -[M - (k - 1)N] ln[M - (k - 1)N] + [M - (k - 1)N] + [N ln N - N] + [(M - kN) ln(M - kN) - (M - kN)] + βNk ) -[M - (k - 1)N] ln[M - (k - 1)N] + N ln N + (M - kN) ln(M - kN) + βNk (8) Henceforth, from eq 6

S(M,N) ) [M - (k - 1)N] ln[M - (k - 1)N] kB N ln N - (M - kN) ln(M - kN) (9) βΠ ) ln[M - (k - 1)N] - ln[M - kN]

( )

ne [M - (k - 1)N]! ) N N![M - kN]!

(3)

It should be noted that the numerator in eq 2 amounts to the total number of permutations of N k-mers plus the empty sites regardless that the k-mers, on one hand, and the empty sites, on the other hand, are indistinguisable. Thus, the two factors in the denominator account for the (25) Ramirez-Pastor, A. J.; Eggarter, T. P.; Pereyra, V. D.; Riccardo, J. L. Statistical Thermodynamics and Transport of Linear Adsorbates. Submitted for publication Phys. Rev. B.

(6)

Thus, from eqs 3 and 4

Accordingly,

Ω(M,N) )

(5)

βµ ) ln

(10)

N kN + (k - 1) ln 1 - (k - 1) M M kN k ln 1 + βk (11) M

[

]

[

]

Then, by defining the lattice coverage θ ) kN/M, the free energy per site f ) F/M and the entropy per site s ) S/M, eqs 8-11 can be rewritten in terms of the intensive (26) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley Publishing Company: Reading, MA, 1960.

Thermodynamics of Linear Adsorbates

Langmuir, Vol. 15, No. 18, 1999 5709

variables θ and T.

{[

] [

[

]

(k - 1) (k - 1) θ ln 1 θ k k θ θ ln - (1 - θ) ln(1 - θ) + βθk (12) k k

βf(θ,T) ) - 1 -

}

] [

]

(k - 1) (k - 1) s(θ) θ θ θ ln 1 θ - ln ) 1kB k k k k (1 - θ) ln(1 - θ) (13)

[

]

(k - 1) θ k (1 - θ)

1-

exp(βΠ) )

[

(14)

]

(k - 1) θ θ 1k Ck exp(β(µ - k)) ) (1 - θ)k

θ (1 - θ)k

(15)

(17)

Accordingly, we resolve ln Ω(M,N,c) by setting c′ ) 2 and using ln Ω(M,N,2) from eqs 4 and 8

(18)

It is straightforward from eqs 4, 6-15, and 18 that

θ ln[K(c,k)] k

(19)

sc s θ ) + ln[K(c,k)] kB kB k

(20)

βfc ) βf -

(16)

(where µFl holds for Flory’s approximation) developed by Flory27 for polymer solutions when the solvent is monomeric with unitary molar volume. This is isomorphous with the case analyzed here where the empty sites of the lattice formally correspond to the solvent monomers in Flory’s solution. Concerning thermodynamic functions such as the free energy, entropy, and spreading pressure, their exact forms present appreciable quantitative as well as qualitatively discrepancies with the ones from Flory’s approach. Particularly, the exact molar configurational entropy s(θ) differs already for very small k-mers (dimers, trimers, etc.) at all coverages. The behavior can be summarized as follows: in the limits θ f 0 and θ f 1 the entropy tends to zero. For very low coverages s(θ) is an increasing function of θ, reaches a maximun at θm, then decreases monotonically to zero for θ > θm. The position of θm, which is θm ) 0.5 for k ) 1, shifts to higher coverages as the adsorbate size k increases. This represents a major distinction between the exact solution and Flory’s approach since in the latter, the larger the k the more the maximun in the entropy shifts to lower coverages. Another difference relates to the entropy per site of Flory’s approach in one dimension, which attains negative values for all k > 1. The range of θ where sFl(θ) becomes negative broadens as k increases. Adsorption of linear adsorbates (i.e., dimers, oligomers, etc.), in very narrow carbon nanotubes or cylindrical pores of aluminophosphates, can be thought as a physical system to which the present thermodynamic results can be applied. On the other hand, as we will see in section IV an accurate calculation of the entropy in a homogeneous one-dimensional lattice enable us to improve the description thermodynamics of multisite occupancy adsorption adsorption on heterogeneous lattices. Approximate forms for the Helmholtz free energy of k-mers in higher dimensions can be proposed on the grounds of its rigorous expression in one dimension by approximating the way the total number of configurations factorize upon a change of lattice connectivity.25 Here we briefly reproduce (27) Flory, P. J. Chem. Phys. 1942, 10, 51.

Ω(M,N,c) ) [K(c,k)]N Ω(M,N,c′)

ln Ω(M,N,c) ) Ω(M,N,2) + N ln[K(c,k)]

k-1

where Ck ) k. The eq 15 represents the exact, so-called, isotherm equation for k-mers in one dimension, which can be compared to the well-known equation

Ck exp(β(µFl - k)) )

the arguments leading to the calculation of approximated thermodynamical functions of linear chains adsorbed on lattices with connectivity c higher than 2 (i.e., higher than one dimension). In general, a number of states Ω for fixed M and N will be also a function of the lattice connectivity, henceforth Ω ≡ Ω(M,N,c). To derive an explicit form for the Ω(M,N,c) that bears the advantages of the exact solution in one dimension, we assume the following connectivity ansatz

[

(k - 1) θ k (1 - θ)

1-

exp(βΠc) )

Ck,c exp(β(µc - k)) )

[

]

(21)

]

(k - 1) θ k (1 - θ)k

θ 1-

k-1

(22)

where the subindex c stands for the thermodynamic quantities in regular lattices with connectivity c and the constant Ck of eq 15 has now the general expression Ck,c ) k K(c,k) (thus, Ck ) Ck,2 for the sake of consistency). Equations 17-22 are the basic thermodynamic functions for noninteracting linear adsorbates in lattices with general connectivity c. K(c,k) can be explicitly approximate as in the Flory-Huggins’s approach. We can think of the exact number of configuration for given M,N,k and c, Ω(M,N,c) as being

Ω(M,N,c) ) ΩFl(M,N,c) ∆Ω(M,N,c)

(23)

where ΩFl(M,N,c) stands for Flory’s counting strategy for the same set of parameters and ∆Ω represents the difference (which is actually unknown at this point). Thus the ratio

ΩFl(M,N,c) ∆Ω(M,N,c) Ω(M,N,c) ) Ω(M,N,c′) ΩFl(M,N,c′) ∆Ω(M,N,c′)

(24)

we now assume that ∆Ω does not depend on c, so the eq 24 becomes

ΩFl(M,N,c) Ω(M,N,c) c-1 ) ) c′ - 1 Ω(M,N,c′) ΩFl(M,N,c′)

[

N(k-1)

]

(25)

It is worth mentioning that the isotherm equation (22), valid for lattices of arbitrary connectivity, shows coverage dependence identical to the exact one-dimensional one (eq 15). The lattice connectivity only enters in the constant Ck,c (actually eq 15 is the only particular case in which eq 22 becomes exact).

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III. Noninteracting k-mers on Heterogeneous Substrates We now turn to the description of multisite occupancy adsorption on heterogeneous surfaces in the lattice gas approximation. Hereafter, we present the main characteristics of Nitta’s approach, provided that we will further introduce the rigorous partition function for the homogeneous system into this approach to more accurately account for the entropic effects in the general case. We assume a lattice with M1, M2, ..., Mw sites of kind 1, 2, ..., w, respectively.1,9-1 The total of adsorption sites amounts

w

Mi

θij ) θtj; ∑ i)1 M

Mi ∑ i)1

(26)

The polyatomic adsorbate can be assumed as consisting of s type of units (groups) that adsorb individually on the lattice sites. Provided the total number of units in a molecule is k, then

j ) 1, ..., s

θij

∑jθij)

)

θ1j expβ(ij - 1j)

i ) 1, ..., w;

(1 -

∑jθ1j)

j ) 1, ..., s

Yj

s

βµ ) ln θ - k ln(1 - θ) +

kj ln ∑ Y* i)1

(35)

j

kj ∑ j)1

(27)

kj being the number of units of kind j in the molecule. The interaction energy of a group of jth type on an adsorption site of the ith kind is denoted ij. The partition function in the canonical ensemble is given by

Q)

(34)

Using the Flory-Huggins’s approximation for the first term in the right-hand side of eq 31, one obtains1,11

s

k)

(33)

and

(1 -

w

M)

and Y1j ) Y1. θij is the coverage of site of type i by units of type j, θtj the coverage of units type j on the whole surface, and Yj* ) (Mj/M)/(1 - kN/M). The θij and θtj come from constrain equations

g(N,M,Nij) exp∑∑ βNijij ∑ N i j

(28)

However from the exact combinatorial factor of k-mers on a homogeneous surface lattice, it yields25

[

βµ ) ln θ + (k - 1) ln 1 -

(k - 1) k

]

θ -

k ln(1 - θ) +

s

Yj

kj ln ∑ Y* i)1

(36)

j

ij

where N is, as before, the total number of k-mers and Nij the number of pairs group j-site i (for simplicity, we have also assumed the internal and vibrational contributions to the partition factor to be a unitary factor in eq 28). The term g can be approximated by following the configurationcounting procedure of the quasi-chemical approximation. Thus,

Nij!

ln g(N,M,Nij) ) ln go(N,M) -

IV. Results and Discussion

∑i ∑j ln N *!

(29)

)]

(30)

ij

and

βF(N,M,T) ) -ln Q )

[

- ln go +

∑i ∑j

(

ln

Nij! Nij*!

+ βNijij

where go holds for k-mers on a homogeneous surface and the N* does for the random distribution of N molecules on M sites. Accordingly the adsorption isotherm can be obtained by equalizing the chemical potentials of the adlayer, µ, and the gas phase, µg

βµg ) βµ )

ln Q (∂ ∂N )

M,T

( )

)-

∂ ln go ∂N

s

+

M,T

∑ j)1

kj ln

[] Yj

Yj*

(31)

where

Yij )

θij /θtj 1-

∑j)1θij

where the term corresponding to the contribution of heterogeneity has been left identical in both equations. As it will shown in section IV, by comparison of eqs 35 and 36 with MC simulations for dimers and 4-mers, the modified isotherm (eq 36) performs significantly better than eq 35 in fitting simulation results in one and two dimensions using the same set of parameters in all cases.

(32)

The results can be sorted in two sets. In Figures 1-4 comparisons between MC simulation for dimers (Figures1 and 3) and 4-mers (Figures2 and 4) in one and two dimensions and analytical isotherms from eqs 35 and 36 are depicted. In all cases simulation data are represented by solid circles, the results from the original (eq 35) and modified isotherm (eq 36) are plotted in dashed lines and solid lines, respectively. MC simulation was done by following the standard Metropolis algorithm in the Grand Canonical Ensemble.17,28 Three sets of curves are compared in each of these figures; one corresponds to the homogeneous surface, β∆ ) 0, while the two remaining do for the simplest heterogeneous surfaces consisting of two type of sites whose relative adsorption energy difference is β∆ ) 2 or 4 (since we are here dealing with homonuclear k-mers and only two type of sites β∆ ) β(11 - 12)). This enables us to observe the general effects of heterogeneity on the adsorption isotherm (which have been extensively discussed in refs 11-14) as well as the accuracy of the model isotherms for fitting the simulation data. For dimers (Figures 1 and 3) both isotherms reproduce qualitatively well the trend of the data, even for fairly strong heterogeneous surfaces (β∆ ) 4). The modified (28) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H. J. Chem. Phys. 1953, 21, 1087.

Thermodynamics of Linear Adsorbates

Figure 1. Adsorption isotherm for multisite occupancy adsorption on homogeneous and heterogeneous surfaces. The surface is assumed to consist of two types of sites, with equal concentration distributed at random. The surface heterogeneity is related to the site energy difference β∆. In this case MC simulations of dimer adsorption on a one-dimensional lattice are compared with analytical isotherms from eqs 35 and 36 for various values of surface heterogeneity; β∆ ) 2, 4. The case of a homogeneous surface, β∆ ) 0, is also shown. Solid circles correspond to MC simulation in the Grand Canonical Ensemble; dashed lines are isotherms from Nitta’s eq 35; solid lines are results from modified eq 36.

Figure 2. Same as Figure 1 for 4-mers in one dimension.

Figure 3. Same as Figure 1 for dimers in two dimensions.

isotherm (eq 36) performs quantitatively better in all the cases studied. It is worth mentioning that (as a general characteristic) the accuracy is better for two dimensions compared to one dimension as the surface becomes more heterogeneous (see the case β∆ ) 4 in Figures 1 and 3).

Langmuir, Vol. 15, No. 18, 1999 5711

Figure 4. Same as Figure 1 for 4-mers in two dimensions.

Figure 5. Comparison between experimental adsorption isotherms from ref 2 for CO, N2, and O2 on zeolite 5A and the theoretical isotherm fron eq 36. Solid squares represent experimental data and solid lines correspond to the theoretical isotherm. The concentrations of strongly and weakly adsorptive sites are f1 ) 0.33 and f2 ) 0.67 taken from ref 1. The site energy differences are ∆/kB ) 0.00, 505.05, and 757.57 K for O2, N2, and CO, respectively. To compare with the experimental values, an ideal gas phase must be assumed, thus βµ ) ln(p/po). In all the cases the adsorbed molecules are assumed to be dimers; thus the corresponding adsorbate size is k ) 2.

This is also valid for 4-mers (case β∆ ) 4 in Figures 2 and 4), and it is expected to be the general behavior for larger molecules. It is also interesting to notice that for a given heterogeneity (a given value of β∆), the larger the adsorbate molecule the better eq 36 reproduces the simulation. This is also observed for eq 35 although in a qualitative sense. This behavior can be traced to the fact that as the size of the adsorbate increases, the adsorption energy distribution smoothes out owing to the appearance of a larger number of adsorption energy levels which decreases the effective heterogeneity seen by the adsorbate (a quantitative measure of this effective heterogeneity could be the ratio jk/∆, where jk is the mean adsortion energy of a k-mer and ∆ the site energy difference for individual units of the k-mer). In Figures 5 and 6 experimental adsorption isotherms of CO, N2, and O2 on zeolites 5A and 10X from Danner et al.2 are fitted by the function eq 36. As discussed in refs 11-14 and 17, this system can be characterized by an adsorption lattice with two types of sites. The agreement is good in all cases although no adsorbate-adsorbate interactions have been accounted for.

5712 Langmuir, Vol. 15, No. 18, 1999

Figure 6. Same as Figure 5 for CO, N2, and O2 on zeolite 10X. The concentrations of strongly and weakly adsorptive sites are f1 ) 0.22 and f2 ) 0.78 taken from ref 1. The site energy differences are ∆/kB ) 0.00, 505.05, and 757.57 K for O2, N2, and CO, respectively. In all the cases the adsorbed molecules are assumed to be dimers thus the corresponding adsorbate size is k ) 2.

It should be mentioning that good quantitative agreement can also be obtained by fitting the data with eq 35, although values of k * 2 are sometime necessary for a proper fitting.1 V. Summary and Conclusions An exact calculation of thermodynamic properties of noninteracting linear adsorbates in one dimension has been carried out. The resulting adsorption isotherm shows significant qualitative differences with respect to the standard Flory’s approach. Further extension to the

Ramirez-Pastor et al.

thermodynamics of linear adsorbates in lattices with connectivity higher than 2 has been discussed as well. Since no adsorbate-adsorbate interactions are included, all the differences may be attributed to entropic contributions which have appreciable effects in the adsorption isotherm and remaining thermodynamic functions. Furthermore, the modification of Nitta’s original isotherm for multisite occupancy adsorption on heterogeneous surfaces, by incorporating the exact form for the homogeneous part of the Helmholtz free energy (instead of the approximate form of it resulting from the Flory-Huggins’s approach), leads to an improved modified adsorption isotherm equation enable to accurately reproduce MC simulations of dimers and 4-mers adsorption on one- and two-dimensional heterogeneous surfaces. Experimental isotherms of dimers adsorbed in zeolites cages with two types of sites can be reproduced by the modified isotherm with a set of parameters that are thermodynamically reasonable and in agreement with other studies.11-14,17 The knowledge of the exact coverage and temperature dependence of the free energy of linear adsorbates in a homogeneous one-dimensional space allows the development of a more accurate description of the adsorption isotherm in higher dimensions even for heterogeneous surfaces. In this case, the observed differences with respect to former approaches can be attributed to the configurational entropy that is more properly taken into account in the present case. Acknowledgment. This work is partially supported by the CONICET (Argentina) and Fundacio´n Antorchas (Argentina). The European Economic Community, Project ITDC-240, is greatly acknowledged for the provision of valuable equipment. LA981346E