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Ind. Eng. Chem. Res. 2006, 45, 2046-2053
Application of the Fractional Statistical Theory of Adsorption (FSTA) to Adsorption of Linear and Flexible k-mers on Two-Dimensional Surfaces F. Roma´ , J. L. Riccardo, and A. J. Ramirez-Pastor* Departamento de Fısica, UniVersidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina
The adsorption of linear and flexible polyatomics on honeycomb, square, and triangular lattices is studied by combining theoretical modeling, Monte Carlo simulations, and experimental results. In the case of flexible k-mers, simulation data are compared with the corresponding ones from the well-known multisite Langmuir model (MSL) of Nitta et al. [J. Chem. Eng. Jpn. 1984, 17, 45], while in the case of rigid k-mers, computational results are correlated from the model for linear adsorbates (MLA) developed by Ramirez-Pastor et al. [Phys. ReV. B 1999, 59, 11027]. On the other hand, it is shown that it is possible to interpret all adsorption isotherms (for linear and flexible adsorbates) by using the recently reported fractional statistical theory of adsorption of polyatomics (FSTA) of Riccardo et al. [Phys. ReV. Lett. 2004, 93, 186101], with an adequate choice of the fundamental parameter of the model, g, being the measure of the statistical exclusion of adsorption minima. In addition, experimental adsorption isotherms for O2 and C3H8 adsorbed in 5A and 13X zeolites, respectively, were analyzed in terms of FSTA. The more general character of FSTA with respect to the existing models and the satisfactory comparison with simulation and experimental results make the new theoretical model an attractive one for the description of adsorption of polyatomic particles with different sizes and shapes. 1. Introduction A major difficulty in dealing with lattice gases of polyatomic species within the framework of classical statistical mechanics is to properly calculate the configurational entropic contributions to the thermodynamic potentials; this means the degeneracy of the energy spectrum compatible with a given number of particles and adsorption sites. In the Langmuir adsorption model or its generalizations,1-3 this is usually done assuming that the adsorbate occupies only one adsorption site. It is the fact that a polyatomic particle excludes more that one state of the states available to the remaining particles upon adsorption that makes the problem of counting the number of configurations extremely difficult. However, the entropic effects in multisite-occupancy adsorption are by no means negligible. For instance, the coverage dependence of configurational entropy of linear k-mers in one dimension is strongly dependent on k even for noninteracting particles.4 Moreover, the consideration of multiple occupation of sites in multilayer adsorption5 leads to rather significant effects on the adsorption isotherms and, consequently, on the surface and adsorption heat obtained from experiments, compared with the well-known BET model.6 Several theoretical models have been developed in the past in order to treat the k-mers adsorption problem: namely, Flory’s model of adsorption of binary liquids in two dimensions,7,8 Nitta’s model of k-mer adsorption on heterogeneous surfaces,9 the model of polymer mixtures,10 the one-dimensional model and extension to higher dimensions,4,11 the occupation balance approximation,11 virial expansion,11 etc. In general, these studies can be separated in two groups, according to the shape (or flexibility) of the adsorbate molecule considered: (i) those dealing with flexible k-mers4,7-9,11 and (ii) those dealing with linear k-mers.4,11 More recently, a new theory to describe adsorption with multisite occupancy has been introduced,12 * To whom all correspondence should be addressed. Tel. number: 54-2652-436151. Fax number: 54-2652-430224. E-mail: antorami@ unsl.edu.ar.
which incorporates the configuration of the molecule in the adsorbed state as a model parameter. The fractional statistical theory of adsorption of polyatomics (FSTA)12 is based on a generalization of the formalism of quantum fractional statistics (QFS), proposed by Haldane13 as an extended form of Pauli’s exclusion principle. FSTA has been proposed to extend these statistics to describe a broad set of classical systems, such as the adsorption of polyatomics at the gas-solid interface. The aim of this work is to apply the theoretical description put forward in ref 12 to analyze the thermodynamic properties of systems with multisite-occupancy and show that its treatment can be significantly oversimplified from this perspective. This study consists of two main parts: In the first one, linear and flexible k-mers adsorbed on honeycomb, square, and triangular lattices are studied by using Monte Carlo (MC) simulations in a grand canonical ensemble and analyzed in terms of FSTA. The simulation scheme provides a test for the analytical model. Comparisons with previous models of multisite adsorption are carried out as well. In the second part, experimental isotherms for O215-17 adsorbed in a 5A zeolite and C3H818 adsorbed in a 13X zeolite, as well as adsorption heats, are used to test the reliability of the theoretical model. The paper is organized as follows: the adsorption model, theoretical approaches, and the simulation method are presented in section 2. In section 3, adsorption properties predicted by MC simulations (for dimers, trimers, and tetramers) are compared to analytical calculations. In addition, experimental results for oxygen adsorbed in a 5A zeolite and propane adsorbed in a 13X zeolite are analyzed by using the model of ref 12. Finally, general conclusions are given in section 4. 2. Basic Definitions: Model and Theoretical Approach In this section, we describe the lattice-gas model for the adsorption of particles with multisite-occupancy in the monolayer regime. The surface is represented as an array of M ) L × L adsorptive sites in a honeycomb, square, or triangular lattice arrangement, where L denotes the linear size of the array.
10.1021/ie0489397 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/14/2006
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2047
Figure 2. (a) Plot of ln θ vs µ/kBT for linear and flexible k-mers adsorbed at low coverage on honeycomb lattices. (b) Plot of the same parameters in part a for square lattices. (c) Plot of the same parameters in part a for triangular lattices. The symbol associations are indicated in the figure.
Figure 1. (a) Examples of linear k-mers adsorbed on square lattices. 1, 2, and 3 correspond to dimer, linear trimer, and linear tetramer, respectively. (b) Available configurations for flexible tetramers adsorbed on square lattices. 4, 5, and 6 correspond to step-shaped, U-shaped, and L-shaped tetramers, respectively. 3 is the same as in part a.
The adsorbate molecules are assumed to be composed by k identical units arranged in two types of configurations: (i) as a linear array of monomers, which we call a “linear k-mer” and (ii) as a chain of adjacent monomers with the following sequence. Once the first monomer is in place, the second monomer occupies one of the c nearest-neighbors of the first monomer. The third and successive monomers occupy one of the c - 1 nearest-neighbors of the preceding monomer. This process continues until k monomers are placed without creating an overlap. We call this feature a “flexible k-mer”. [Only flexible k-mers can be adsorbed on a honeycomb lattice.] As an example, Figure 1 shows linear and flexible k-mers (k ) 3, 4) adsorbed on a square lattice. To describe a system of N k-mers adsorbed on M sites at a given temperature T, let us introduce the occupation variable ci which can take the values ci ) 0 or 1, if the site i is empty or occupied by a k-mer unit, respectively. The k-mer retains its structure upon adsorption and desorption. The Hamiltonian of the system is given by
H ) (o - µ)
∑i ci
(1)
where o is the adsorption energy of a k-mer unit (ko being the total adsorption energy of a k-mer) and µ is the chemical potential. In previous work,4 we presented a multisite adsorption model for linear adsorbates (MLA). The MLA model is based on the exact solution for the thermodynamics functions of linear adsorbates, or linear k-mers, on a one-dimensional discrete space. The thermodynamics was further extended to higher
dimensions based upon the one-dimensional form and a connectivity ansatz. The resulting MLA adsorption isotherm was
[
(k - 1) θ k (1 - θ)k
θ 1km(c, k) exp[β(µ - ko)] )
]
k-1
(2)
where β ) 1/kBT, kB being the Boltzmann constant, and θ ) kN/M and m(c, k) represent the surface coverage and number of available configurations (per lattice site) for a k-mer at zero coverage, respectively. The term m(c, k) is, in general, a function of the connectivity and the size of the adsorbate. It is easy demonstrate that
m(c, k) )
{
c/2 for linear k-mers [c(c - 1)(k-2)]/2 - m′ for flexible k-mers
(3)
the term m′ is subtracted in eq 3 since the first term overestimates m(c, k) by including m′ configurations providing overlaps in the k-mer. Figure 2 shows ln θ vs µ/kBT for linear and flexible k-mers adsorbed at low coverage on honeycomb, square, and triangular lattices. The symbols and lines correspond to the results obtained from MC simulations and eq 2, respectively. The agreement between the theoretical curves and the simulation data supports the results given by eq 3. Due to the fact that eq 2 was obtained from exact calculations in 1-D, it shows a coverage dependence of all thermodynamic functions identical to the exact one-dimensional one [notice that lattice connectivity and configuration of the adsorbate only enter through the constant m(c, k)]. For this reason, it is expected that eq 2 provides a good description for adsorption of linear k-mers. On the other hand, Nitta et al.3,9 modified the Langmuir equation, exp[β(µ - o)] ) θ/(1 - θ),19 including multisiteoccupancy adsorption, that is, assuming that the adsorbate can occupy k sites on the surface. On the basis of Flory’s theory7,8 for polymer solutions, the multisite Langmuir (MSL)
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isotherm9 results in
Ck exp[β(µ - ko)] )
θ (1 - θ)k
(4)
where Ck is a constant. On the basis of the analysis of Figure 2, we set Ck ) km(c, k). The MLA and MSL models were developed for two limiting cases: (i) linear k-mers (eq 2) and (ii) flexible k-mers (eq 4). In the following, we summarize the basis of the FSTA description,12 which allows us to describe the configurational entropy through a single function (parameter), namely, the statistical exclusion, g, accounting for the configuration of the molecules in the adsorbed state. In this approximation, the interaction of one isolated molecule with a solid surface confined in a fixed volume is represented by an adsorption field having a total number G of local minima in the space of coordinates necessary to define the adsorption configuration. Thus, G is the number of equilibrium states of a single molecule at infinitely low density. In general, more than one state out of G are prevented from occupation upon adsorption of a molecule. Furthermore, because of possible concurrent exclusion of states by two or more molecules, the number of states excluded per molecule, g(N), being a measure of the “statistical” interactions, depends in general on the number of molecules N within the volume. From the definition of the number of states available for an Nth molecule after (N - 1) ones are already in the volume N-1 V, dN ) G - ∑N′)1 g(N′) ) G - G0(N), which is a generalization of the relation recently established by Haldane,13 the generalized configurational factor, W(N) ) (dN + N - 1)!/ [N!(dN - 1)!], can be calculated. Consequently, the Helmholtz free energy function can be expressed as βF(N, T, V) ) -ln W(N) + N ln qi + βNo, qi being the partition function from the internal degrees of freedom of a single molecule in the adsorbed state. The general form for the chemical potential of noninteracting adsorbed polyatomics proposed in ref 12 is obtained from βµ ) (∂F/∂N)T,V, as
βµ ) ln
[
]
n[1 - G ˜ 0(n) + n](G˜ 0′-1) [1 - G ˜ 0(n)]G˜ 0′
- ln K(T)
(5)
where n ) N/G is the density (n becomes finite as N, G f ∞), G ˜ 0(n) ≡ limN,Gf∞ G0(N)/G, G ˜ 0′ ≡ dG ˜ 0/dn, and K(T) ) qi exp(-βo). Furthermore, by taking the simplest approximation within FSTA, namely, g ) constant, a particular isotherm function arises from eq 5
K(T) exp[βµ] )
aθ[1 - aθ(g - 1)]g-1 [1 - aθg]g
(6)
where n ) aθ, θ being either the ratio N/Nm or the ratio V/Vm, where N(V) is the number of adsorbed molecules (adsorbed amount) at given (µ, T) and Nm(Vm) is the one corresponding to monolayer completion. The parameters a and g in the last equation have a precise physical meaning, can be obtained from adsorption experiments, and are related directly to the spatial configuration of a polyatomic molecule in the adsorbed state. Alternatively, eq 6 can be used by assuming some approach to calculate g as a function of the model’s parameters. Thus, given the shape and size of the adsorbate, the adsorption isotherm it is straightforwardly obtained. In the next section, eq 6 will be used in both ways.
For molecules made out of k identical units, each of which can occupy an adsorption site, θ ) kN/M and G ) mM, where m (≡ m(c, k)) is the number of distinguishable configurations of the molecule per lattice site (at zero density) and depends on the lattice/molecule geometry. Then, 1/a ) km [as an example, straight k-mers adsorbed “in registry” on sites of a square lattice would correspond to m ) 2, g ) 2k, and a ) 1/(2k)]. A simple approximation for g can be obtained, assuming independence of the adsorption sites. Under this consideration, if one molecule has m different ways of adsorbing on one site, g ) mk states are excluded when one k-mer is adsorbed occupying k sites on the lattice. For linear k-mers adsorbed on a one-dimensional lattice, it is straightforward that m ) 1, and g ) k, yields the exact adsorption isotherm obtained independently in ref 4. The proposed formalism is simple, easy to apply in practice, and allows us to calculate the entropic contribution to the thermodynamic potential arising from the spatial structure of the molecules, regardless of whether the adsorption is assumed to take place on either a continuous or latticelike volume. In addition, FSTA appears as a general model which allows more complex adsorbates to be dealt with, beyond linear and flexible k-mers, which cannot be studied by using existing approaches such as the MLA and MSL models. [Some examples of such complex adsorbates are the following: (i) g ) 1 and a ) 1/k, representing the case of end-on (normal to the surface) adsorption of k-mers, and (ii) g ) k′ and a ) 1/k, instead representing an adsorbed configuration in which k′ out of k units of the molecule are attached to a one-dimensional (groovelike) surface and (k - k′) units at the end are detached and tilted away from it.] In section 3, analysis of lattice-gas simulations and experimental results has been carried out in order to bear the significance of the parameters a and g in terms of the adsorbate/ surface geometry. 2.1. Monte Carlo Simulation in Grand Canonical Ensemble. The adsorption process is simulated through a grand canonical ensemble MC method.20,21 For a given value of the temperature and chemical potential, an ideal gas phase of polyatomic molecules is put in contact with a lattice of M f ∞ adsorption sites. In the grand canonical ensemble, µ, T, and the volume V (or M) of the physical system are the thermodynamic parameters. In adsorption-desorption equilibrium there are two elementary ways to perform a change of the system state, namely, adsorbing one molecule onto the surface (adding one molecule into the adsorbed phase volume M) and desorbing one molecule from the adsorbed phase (removing one molecule from the volume M). The algorithm to carry out an elementary step in a MC simulation (1 MCS) is the following: Given a square lattice of M adsorption sites with energies already assigned, (i) Set the value of µ and temperature T. (ii) One of over the m possible configurations for the k-mer is randomly selected. (iii) Choose randomly one of the k-uples on the lattice with the form selected in step ii and generate a random number ξ ∈ [0, 1]: (a) If k sites are empty, then adsorb a molecule if ξ e W(Hi f Hf). (b) If k sites are occupied by atoms belonging to the same molecule, then desorb the molecule if ξ e W(Hi f Hf). W(Hi f Hf) is the transition probability given by the Metropolis22 rule:
W(Hi f Hf) ) min{1, exp(-β∆H)}
(7)
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2049 Table 1. Table of Parameters Used in the Fitting of the Simulation Data of Figures 3-7 adsorbate
geometry
m(c, k)
g ) mk
experimental g
g%
dimers flexible trimers flexible tetramers dimers linear trimers linear tetramers flexible trimers flexible tetramers dimers linear trimers linear tetramers flexible trimers flexible tetramers
c)3 c)3 c)3 c)4 c)4 c)4 c)4 c)4 c)6 c)6 c)6 c)6 c)6
1.5 3 6 2 2 2 6 18 3 3 3 15 69a
3 9 24 4 6 8 18 72 6 9 12 45 276
2.917(6) 8.65(2) 22.87(8) 3.739(4) 5.70(1) 7.71(6) 16.87(4) 68.3(4) 5.53(1) 8.54(3) 11.58(7) 41.7(1) 265(2)
2.8 3.9 4.7 6.5 5 3.6 6.3 5.1 7.8 5.1 3.5 7.3 4
a
This value of m(c, k) was obtained from eq 3 with m′ ) 3.
Figure 3. (a) Adsorption isotherms of dimers on a honeycomb lattice (c ) 3). (b) Same isotherms as in part a for trimers. (c) Same isotherms as in part a for tetramers. The symbols represent MC results, and the lines correspond to different approaches (see inset of part a).
where ∆H ) Hf - Hi is the difference between the Hamiltonians of the final and initial states. (iv) Repeat this process from step iii M times. The equilibrium state can be well reproduced after discarding the first r′ ) 105-106 MCSs. Then, averages are taken over r ) 105-106 successive configurations. The adsorption isotherm, or mean coverage as a function of the chemical potential [θ(µ)], is obtained as a simple average:
θ(µ) )
1
M
∑〈ci〉 ) M i
k〈N〉 M
(8)
where 〈N〉 is the mean number of adsorbed particles, and 〈...〉 indicates the time average over the r Monte Carlo simulation runs. 3. Results and Discussion 3.1. Comparison between Theory and Monte Carlo Simulation in Grand Canonical Ensemble. Computational simulations have been developed for square, honeycomb, and triangular L × L lattices, with L ) 150 and periodic boundary conditions. With this lattice size, we verified that finite size effects are negligible. Figure 3 presents a comparison between the simulation adsorption isotherms and the corresponding ones obtained from the analytical approaches of eqs 2, 4, and 6 for dimers, trimers, and tetramers adsorbed on honeycomb lattices. The symbols represent the simulation data, and the lines correspond to theoretical isotherms. In all cases, simulation curves present an intermediate behavior between the predictions of eqs 2 and 4. A more accurate correlation with the simulation data is obtained by using eq 6 with 1/a ) km. For this purpose, m is obtained from eq 3 and only the parameter g is adjusted. The values obtained for g (noted as gexp) are given in Table 1 and are in good agreement with those predicted from the approximation of independent sites (gideal ) mk). The differences between the experimental and theoretical results can be easily rationalized
Figure 4. (a) Adsorption isotherms of dimers on a square lattice (c ) 4). (b) Same isotherms as in part a for linear trimers. (c) Same isotherms as in part a for linear tetramers. The symbols represent MC results, and the lines correspond to different approaches (see inset of part a).
with the help of the percentage error, g%, which is defined as
g% ) 100
|
|
gideal - gexp gideal
(9)
The values of g% increase as the size k increases, ranging between 3 and 5%. In addition, gexp < gideal, given that gideal ) mk overestimates the number of excluded states because of simultaneous exclusion of neighboring particles. Figure 4 is similar to Figure 3 for the case of dimers (4a), linear trimers (4b), and linear tetramers (4c) adsorbed on squares lattices. The model of eq 2 provides a good approximation with small differences between the simulation and theoretical results. With an adequate choice of g (see Table 1), FSTA reproduces very accurately the curves from the MLA model. As in Figure 3, the discrepancies between gideal and gexp are small. On the other hand, the MSL theory predicts a smaller θ than the simulation data over the range 0-1. This separation increases when the size of the adsorbate is increased, and the disagreement turns out to be significantly large for k > 3. The case of flexible trimers and tetramers adsorbed on squares lattices is shown in Figure 5a and b, respectively. Clearly, the standard models of eqs 2 and 4 do not reproduce the simulation
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Figure 5. (a) Adsorption isotherms of flexible trimers on a square lattice (c ) 4). (b) Same isotherms as in part a for flexible tetramers. The symbols represent MC results, and the lines correspond to different approaches (see inset of part a).
Figure 6. (a) Adsorption isotherms of dimers on a triangular lattice (c ) 6). (b) Same isotherms as in part a for linear trimers. (c) Same isotherms as in part a for linear tetramers. The symbols represent MC results, and the lines correspond to different approaches (see inset of part a).
data well. An appreciably more accurate result is obtained by the use of the eq 6 and the values of g reported in Table 1. It is worth mentioning that this is the lowest approximation in the framework of FSTA, and it already does clearly better than the existing models for adsorption of polyatomics. Figures 6 and 7 collect the results for linear and flexible dimers, trimers, and tetramers adsorbed on triangular lattices. In this case, the high connectivity of the lattice produces an increase in the number of available states to a single molecule and makes the problem more difficult. Consequently, the theoretical approximations are less accurate in the triangular lattices than in the honeycomb or square geometries. In Figures 3-7, m has been obtained from eq 3 according to the type of adsorbate and the connectivity of the lattice, while
Figure 7. (a) Adsorption isotherms of flexible trimers on a triangular lattice (c ) 6). (b) Same isotherms as part a for flexible tetramers. The symbols represent MC results, and the lines correspond to different approaches (see inset of part a).
Figure 8. (a) Adsorption isotherms from eq 6 for fixed values of m, flexible tetramers, c ) 4, and different values of g, as indicated in the figure. (b) Comparison between the FSTA model (eq 6) and simulation data for adsorbed particles on a square lattice. The adsorption isotherms correspond to k ) 4 and two different shapes, as indicated in the figure. The symbols represent MC results, and the lines correspond to the theoretical approach.
g has been obtained by fitting the simulation data. When this procedure is used, g is a measure of the capacity of the molecule to be adsorbed. This situation can be understood from Figure 8a, where adsorption isotherms from eq 6 are plotted for fixed values of m, flexible tetramers, c ) 4, and different values of g. The curves from top to bottom correspond to increasing values of g. Two main conclusions can be extracted from the figure: (i) The model of independent sites provides the maximum value for g () mk). In this case, the number of excluded states by an adsorbed molecule is the maximum. Consequently, the capacity of adsorption is minimum and the corresponding adsorption
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applicability of the FSTA description to interpret experimental data of adsorption with multisite-occupancy. For this purpose, experimental adsorption isotherms of oxygen16,17 and propane18 in 5A and 13X zeolites, respectively, were modeled in terms of the FSTA model. In our analysis, eq 6 was used assuming that (i) the adsorption sites are independents (g ) mk), the values of g% in Table 1 justify this approximation, and (ii) the interaction between two sorbed molecules is introduced as a mean-field contribution. In addition, given that the experimental isotherms were reported in the amount adsorbed as a function of the pressure, we rewrite eq 6 in a more convenient form:
Figure 9. Configurational entropy versus coverage for dimers on square lattices. The symbols represent simulation data, and the lines provide theoretical results for different values of g, as indicated in the text.
isotherm constitutes the lower limit of the theoretical isotherms. (ii) As g is diminished, the capacity of adsorption is increased. Figure 8b shows that the predictions of the FSTA model are observed in simulation data. For this purpose, two adsorption isotherms have been simulated on a square lattice. One of them corresponding to U-shaped tetramers, the other one, to stepshaped tetramers (see the legend in the figure). The term m ) 4 in both cases; however, step-shaped tetramers are less “compact” and exclude more states than U-shaped tetramers. Consequently, the slope of the U-shaped isotherm is more pronounced than slope of the step-shaped isotherm. From the point of view of the FSTA model, the values of g obtained by adjusting the simulation data were 14.53 ( 0.06 and 15.58 ( 0.09 for U- and step-shaped tetramers, respectively. As was expected from the above arguments, gU < gstep. Finally, to test the sensitivity of the parameter g, Figure 9 shows a comparison between simulated values of the configurational entropy per site, s, versus coverage and the corresponding ones obtained from FSTA (being S ) (∂F/∂T)N,M and s ≡ S/M) for k ) 2, c ) 4 (see Figure 4a), and different values of the exclusion parameter g. As it was obtained for the isotherm, the best fit corresponds to g ) 3.74 (solid line). The three curves in dashed (dotted) lines correspond to increments (decrements) of 2%, 4%, and 6% with respect to the fitted g ) 3.74. As can be clearly visualized, small variations of g provide notable differences in the entropy curves. Thus, the statistical exclusion parameter results in being highly sensitive to the spatial configuration of the adparticles. This represents evidence of the physical and experimental significance of g. The more compact the configuration of the segments attached to the surface sites, the smaller g value. For instance, g may vary from g ) 4 for dimers on a square lattice to g ) 6 and g ) 8 for straight trimers and tetramers, respectively (see Table 1). Additionally, g increases up to g ) 18 and g ) 72 for flexible trimers and tetramers, respectively. However, g would decrease if the number of segments detached from the surface increases. The exhaustive analysis of Figures 2-9 reveals two fundamental properties of the FSTA model: that is, (1) the possibility of studying adsorption of molecules with different shapes (including no flat adsorption), beyond those of linear and flexible k-mers, and (2) the possibility of obtaining valuable information about the adsorbate from the combination of a ) (km)-1 and g. In the next section, we will apply these ideas in the discussion of experimental results. 3.2. Comparison between the FSTA Model and Experimental Results. The goal of the present section is to test the
Keqp exp(-βwq/qmax) )
(q/qmax)[g - (g - 1)q/qmax]g-1 [g - g(q/qmax)]g
(10)
where exp(-βwq/qmax) is the mean-field term taking into account the lateral interactions between adsorbed molecules with w being the total ad-ad interaction energy per particle at full coverage (negative for attraction and positive for repulsion); p is the pressure, p ) po exp(βµ); (qmax) q is the (saturation) amount adsorbed; and Keq is the equilibrium constant, Keq ) K∞ exp(-Hst/RT), Hst being the isosteric heat of adsorption (kcal/ mol) and R being the ideal law gas constant. Under the restrictions given in eq 10, the FSTA model has the same number of fitting parameters as the MLA and MSL models. The main difference between the models is that, in the case of the MLA and MSL models, the adsorbate is taken into account by means of a parameter, k, which is the number of active sites occupied by the molecules, whereas, in the case of the FSTA model, the corresponding parameter, g, is a mixing between the size of the adsorbate and the surface geometry. In other words, for a given k, eqs 2 and 4 show identical coverage dependence, regardless of the configuration of the adsorbed state, which only enters in the constant Ck. On the other hand, beyond of the size k of the adsorbate, the lattice/molecule geometry is explicitly taken into account in the FSTA model, being responsible for the shape of the adsorption isotherm. This property provides the basis to investigate the changes of the configuration of adsorbates upon changes in density (configurational spectroscopy) from thermodynamic data (work in this sense is in progress). Physically, the advantages of FSTA are a consequence of properly considering the configurational entropy of the adsorbate. This treatment, in which the entropic effects of the adsorbate size are accounted for, represents a qualitative advance with respect to the existing models of multisiteoccupancy adsorption. Adsorption isotherms of O2 adsorbed in 5A zeolite are shown in Figure 10. The symbols are experimental data, and the lines represent theoretical results from eq 10. Experimental data were taken from two different sources in the literature; namely, empty and full symbols correspond to the data of Miller et al.16 and Danner et al.,17 respectively. In the first set of data,16 the amount adsorbed was measured in units of the number of adsorbate molecules per cavity. In the other case,17 the amount adsorbed was measured in units of ccSTP per gram of adsorbent. To homogenize the plots, we have represented the amount adsorbed by using the adimensional surface coverage θ ) q/qmax. The fitting was done in two stages: (1) On the basis of previous studies of Monte Carlo simulations,24 we fix g ) 4 (k ) 2 and m ) 2). Then, we determine, by using multiple adjustments, the set of parameters (Keq, qmax, w) which gives the best fit to the experimental data of Miller et al.16 in the
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Figure 10. Comparison between the FSTA model (eq 6) and experimental adsorption isotherms16,17 for O2 adsorbed in 5A zeolite. The lines correspond to the FSTA model, and empty (full) symbols correspond to experimental data from ref 16 (17). The parameters used in the theoretical model are listed in Table 2. (inset) Temperature dependence of Henry’s constants predicted by the FSTA model for the fitting isotherms. The isosteric heat of adsorption reported in Table 2 was obtained from the slope of the curve. Table 2. Table of Parameters Used in the Fitting of the Experimental Data of Figures 9 and 10a system
k
m
g
qmax
HQFS st
Hexp st
wQFS
wexp
O2/5A C3H8/13X
2 3
2 1
4 3
12b-130.9c 5.75
3.10 6.94
3.37d 6.81g
0.72e 1.27e
0.54f 0.5h
FSTA, and wexp are expressed in kcal/mol. b Value of HFSTA , Hexp st st , w qmax (molecules/cavity) obtained by adjusting experimental data from ref 16. c Value of qmax (ccSTP/g of adsorbent) obtained by adjusting experimental data from ref 17. d Experimental data taken from ref 16. e Values of w calculated at full coverage. f Simulation data taken from ref 24. g Experimental data taken from ref 18. h Value of the molecular interaction C3H8C3H8 in the liquid phase.25 a
entire range of pressure and temperature values. The results are shown in Table 2. The values obtained for Hst (from the plot of ln Keq vs 1/T) and w are in excellent agreement with the experimental value of Hst reported in ref 15 and the simulational calculation of w in ref 24. With respect to qmax, it was not possible from the work of Razmus et al.15 to calculate the saturation amount adsorbed in order to compare with the theoretical result. However, we used an independent way of validating the value of qmax (second stage). (2) We fix the values of g, Keq, and w, according to stage 1. Then, we set only the value of qmax, which adjusts the experimental isotherm measured by Danner et al.17 The result obtained for the saturation amount adsorbed coincides with the monolayer volume reported in ref 17. The quantity and variety of fitting data demonstrate the consistency and robustness of the analysis shown in Figure 10. Figure 11 shows adsorption isotherms of C3H8 in 13X zeolite. The lines correspond to the FSTA model, and the symbols represent experimental data from ref 18. As in previous work, we consider a “bead segment” description of the alkane chain, in which each methyl group is represented as a single site. In this frame, we fix k ) 3 for the propane molecule. In addition, the length of the molecule of propane (6.7 Å) is large with respect to the diameter of the cavity (11.6 Å). This fact suggests that the molecules in the adsorbed phase should be aligned along a preferential direction. Otherwise, it is impossible to adsorb 5-6 molecules per cavity. Then, we propose to fix m ) 1 as in the one-dimensional case and, consequently, g ) km ) 2. Under these considerations, the theoretical curves in Figure 11 were obtained by adjusting the set of parameters Keq, qmax, and w as in Figure 10.
Figure 11. Comparison between the FSTA model (eq 6) and experimental adsorption isotherms18 for C3H8 adsorbed in 13X zeolite. The symbols and lines correspond to experimental data and the FSTA model, respectively. The parameters used in the theoretical model are listed in Table 2. (inset) Temperature dependence of Henry’s constants predicted by the FSTA model for the fitting isotherms. The isosteric heat of adsorption reported in Table 2 was obtained from the slope of the curve.
As in Figure 10, Hst was obtained from the slope of ln Keq vs 1/T, in very good agreement with experimental data from ref 18. As in the experiment, the fitting value of qmax is less than 6 molecules per cavity, and the fractional value of qmax () 5.75) is indicative that some molecules may stand across the windows, belonging to two cavities. With respect to the lateral interaction at full coverage, the relation between the fitted value of w and the molecular interaction C3H8-C3H8 in the liquid phase, , given in ref 18 is w/ ≈ 2.5. This value indicates that each propane molecule in the adsorbed phase at qmax should interact with 2.5 neighbors and reinforces the idea that the system can be treated as a quasi-one-dimensional system. It is important to emphasize the following: (1) Experimental data in Figure 10 have been previously used to test Nitta’s wellknown formalism of multisite-occupancy adsorption.9 A good quantitative agreement can also be obtained in ref 9, although nonphysical values of k were necessary for a proper fitting. (2) A highly artificial model with eight adjust parameters was necessary to interpret the experimental data in ref 18. On the contrary, the proposed FSTA model has a strong theoretical foundation, based on arguments of statistical mechanics, and is simple enough to justify its applicability in the description of experimental data in the presence of multisite-occupancy. 4. Conclusions The FSTA model provides a new theoretical framework for understanding multisite-occupancy adsorption. It is therefore interesting to compare results from the FSTA model with existing theoretical predictions, with computer simulations, and with experimental results. Following this line, the paper was organized into two parts: In the first, the adsorption of k-mers of different size and shape on honeycomb, square, and triangular surfaces was studied via MC simulations and the results were compared with predictions from three theoretical models (MLA, MSL, and FSTA). In the second part, FSTA's adsorption isotherms were compared with experimental ones coming from different authors.16-18 The simulation test of FSTA's isotherm was carried out for the particular case of dimers, trimers, and tetramers having two typical structures: so-called linear and flexible molecules. Several conclusions can be drawn from the comparison: (i) The one-dimensional exact result is recovered by using the ap-
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2053
proximation of independent sites for g. In this case, no adjustable parameters exist in the FSTA model, since m and g are analytically obtained. (ii) The FSTA model appears as a general model allowing adsorbates with different sizes and shapes to be treated, beyond those of linear and flexible k-mers, which cannot be studied by using existing approaches such as the MLA model, the MSL model, etc. (iii) In two dimensions, good fittings were obtained with only adjusting the parameter g. The values obtained for g were in good agreement with those predicted from the approximation of independent sites (the differences ranged between 2.8% and 7.8%). (iv) The term g is a measure of the structure of the adsorbed molecule. Small values of g are associated with “compact” molecules, while large values of g are associated with “extended molecules”. From another perspective, for a given chemical potential, the equilibrium coverage increases (decreases) as g is decreased (increased). On the other hand, experimental data of O2/5A16,17 and C3H8/ 13X18 were well fitted by using the FSTA model. It is worth noticing that although there are several parameters in this problem (i.e., m, g, k, Hst, qmax, and w) they can be separated in two categories: those which were either taken from or justify by independent sources (such as, k, Hst, qmax, and w) and those coming from the FSTA model (m and g), which allow information about the mode of adsorption of the k-mers to be obtained. The exhaustive analysis presented here has shown that the FSTA model is a good one considering the complexity of the physical situation which is intented to be described and could be very useful in interpreting experimental data. However, further comprehensive analysis of experimental isotherms involving polyatomics through the proposed formalism appears necessary to discern its applicability, reliability, and accuracy. Two main courses of action are being pursued: (1) to develop an analytical model for g capable of improving the approximation of independent sites and (2) to extend this analysis to the recently reported case of molecules whose segments attach or detach from the surface.26,27 These studies are in progress. Acknowledgment This work was supported in part by CONICET (Argentina) under project PIP 02425 and the Universidad Nacional de San Luis (Argentina) under the projects 328501 and 322000. One of the authors (A.J.R.P.) is grateful to the Departamento de Quımica, Universidad Auto´noma Metropolitana-Iztapalapa (Me´xico, D.F.) for its hospitality during the time that this manuscript was prepared. Literature Cited (1) Hill, T. L. An Introduction to Statistical Thermodynamics; AddisonWesley Publishing Company: Reading, MA, 1960. (2) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (3) Rudzin´ski, W.; Everett, D. H. Adsorption of gases on Heterogeneous Surfaces; Academic Press: London, 1992. (4) Ramirez-Pastor, A. J.; Eggarter, T. P.; Pereyra, V.; Riccardo, J. L. Statistical thermodynamics and transport of linear adsorbates. Phys. ReV. B 1999, 59, 11027.
(5) Riccardo J. L.; Ramirez-Pastor A. J.; Roma´, F. Multilayer adsorption with multisite occupancy: an improved isotherm for surface characterization. Langmuir 2002, 18, 2130. (6) Brunauer, S.; Emmet, P. H.; Teller, E. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 1938, 60, 309. (7) Flory, P. J. Thermodynamics of high-polymer solutions. J. Chem. Phys. 1942, 10, 51. (8) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (9) Nitta, T.; Kuro-oka, M.; Katayama, T. An adsorption isotherm of multisite occupancy model for homogeneous surface. J. Chem. Eng. Jpn. 1984, 17, 45. (10) Ryu, J.-H.; Gujrati, P. D. Lattice theory of a multicomponent mixture of monodisperse polymers of fixed architectures. J. Chem. Phys. 1997, 107, 3954. Gujrati, P. D. Polydisperse solution of randomly branched homopolymers, inversion symmetry and critical and theta states. J. Chem. Phys. 1998, 108, 5089 and references therein. (11) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Multisite occupancy adsorption: comparative study of new different analytical approaches. Langmuir 2003, 19, 6770. (12) Riccardo, J. L.; Roma´ F.; Ramirez-Pastor, A. J. Fractional Statistical Theory of Adsorption of Polyatomics Phys. ReV. Lett. 2004, 93, 186101. (13) Haldane, F. D. M. “Fractional statistics” in arbitrary dimensions: A generalization of the Pauli principle. Phys. ReV. Lett. 1991, 67, 937. (14) Wu, Y. S. Statistical distribution for generalized ideal gas of fractional-statistics particles. Phys. ReV. Lett. 1994, 73, 922. (15) Razmus, D. M.; Hall, C. K. Prediction of gas adsorption in 5A zeolites using Monte Carlo simulation. AIChE J. 1991, 37, 769. (16) Miller, G. W.; Knaebel, K. S.; Ikels, K. G. Equilibria of nitrogen, oxygen, argon, and air in molecular sieve 5A. AIChE J. 1987, 33, 2. (17) Danner, R. P.; Wenzel, L. A. Adsorption of carbon monoxidenitrogen, carbon monoxide-oxygen, and oxygen-nitrogen mixtures on synthetic zeolites. AIChE J. 1969, 15, 515. (18) Tarek, M.; Kahn, R.; Cohen de Lara, E. Modelization of experimental isotherms of n-alkanes in NaX zeolite. Zeolites 1995, 15, 67. (19) Langmuir, I. The adsorption of gases on plane surfaces of glass, mica, and platinum in 5A zeolite pellets. J. Am. Chem. Soc. 1918, 40, 163. (20) Binder, K. Applications of the Monte Carlo method in statistical physics; Topics in current Physics; Springer: Berlin, 1984; Vol. 36. (21) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (22) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. W.; Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys. 1953, 21, 1087. (23) Silva, J. A. C.; Rodrigues, A. E. Multisite Langmuir model applied to the interpretation of sorption of n-paraffins in 5A zeolite. Ind. Eng. Chem. Res. 1999, 38, 2434 and references therein. (24) Ramirez-Pastor, A. J.; Nazzarro, M. S.; Riccardo, J. L.; Zgrablich, G. Dimer physisorption on heterogeneous substrates. Surf. Sci. 1995, 341, 249. (25) Hirshfelder, J. O.; Curtis, C. F.; Bard, R. B. Molecular Theory of Gases and Liquids; J. Wiley and Sons: New York, 1964. (26) Paserba, K. R.; Gellman, A. J. Kinetics and energetics of oligomer desorption from surfaces. Phys. ReV. Lett. 2001, 86, 4338; Effects of conformational isomerism on the desorption kinetics of n-alkanes from graphite. J. Chem. Phys. 2001, 115, 6737. (27) Gellman, A. J.; Paserba, K. R. Kinetics and mechanism of oligomer desorption from surfaces: n-alkanes on graphite. J. Phys. Chem. B 2002, 106, 13231.
ReceiVed for reView November 2, 2004 ReVised manuscript receiVed December 28, 2005 Accepted January 24, 2006 IE0489397
