Langmuir 1996, 12, 5433-5443
5433
Isotherm Models for Localized Monolayers with Lateral Interactions. Application to Single-Component and Competitive Adsorption Data Obtained in RP-HPLC Igor Quin˜ones† and Georges Guiochon*,‡ Departamento de Tecnologı´a, Centro de Quı´mica Farmace´ utica (CQF), Calle 200 y 21, Atabey, Playa; P.O. Box 16042, La Habana, Cuba, C.P. 11600, Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Chemical and Analytical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120 Received April 8, 1996. In Final Form: August 7, 1996X Single-component and competitive adsorption data of 2-phenylethanol and 3-phenylpropanol on ODSsilica with methanol-water as the mobile phase, which had been previously reported, were reinterpreted. These data were fitted to several isotherm models derived from statistical thermodynamics which consider lateral adsorbate-adsorbate interactions on homogeneous surfaces: the Fowler, Ruthven, and Moreau et al. models. The single-component Kiselev model for specific lateral interactions was extended to account for the competitive adsorption of binary mixtures and for the finite or infinite dimension of the adsorbed associates. These last models were tested using the same set of experimental data. A comparison was made regarding the ability of these models to predict mixed equilibria using only the identified parameters of the single-component isotherms. In this regard, the best results were obtained with the Kiselev model, which considers the formation of both binary and ternary associates on the surface.
Introduction The use of adsorption-based processes, such as preparative chromatography and simulated moving bed chromatography, for large-scale separations or purifications is becoming common in the pharmaceutical industry.1 The scale-up and optimization of these separation processes requires the ability to characterize accurately the adsorption equilibria.2 In the general case, these equilibria are competitive. The determination of competitive isotherms is long, costly, and time-consuming. However, there are methods available which allow the derivation of reasonable approximations of the competitive isotherms from single-component isotherms which are much easier to measure. These methods allow a considerable reduction in the time and cost required by the data acquisition.3-5 Different isotherm models have been used to correlate single component isotherm data and to predict competitive adsorption isotherms from data acquired in liquid chromatography. The majority of those models were proposed originally for the treatment of gas mixtures; however, their application to liquid mixtures is supported by both theoretical and practical reasons.39 Among them are found (i) simple isotherm models for homogeneous surfaces without lateral interactions, like the Langmuir,6-19 Jovanovic,20-22 and IAST models6,15,16,23 as well as modified * Author to whom correspondence should be sent at the University of Tennessee. † Centro de Quı´mica Farmace ´ utica (CQF). ‡ University of Tennessee and Oak Ridge National Laboratory. X Abstract published in Advance ACS Abstracts, October 15, 1996. (1) Stinson, S. C. Chem. Eng. News 1995, 73 (41), 44. (2) Guiochon, G.; Golshan Shirazi, S.; Katti, A. M. Fundamentals of Nonlinear and Preparative Chromatography; Academic Press: Boston, MA, 1994. (3) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley-Interscience: New York, NY, 1984. (4) Valenzuela, D. P.; Myers, A. L. Adsorption Equilibrium Data Handbook; Prentice Hall: Englewood Cliffs, NJ, 1989. (5) Bellot, J. C.; Condoret, J. S. Proc. Biochem. 1993, 28, 365. (6) Zhu, J.; Katti, A. M.; Guiochon, G. J. Chromatogr. 1991, 552, 71. (7) Katti, A. M.; Ma, Z.; Guiochon, G. AIChE J. 1990, 36, 1722. (8) Jacobson, J. M.; Frenz, J. P.; Horva´th, Cs. Ind. Eng. Chem. Res. 1987, 26, 43. (9) Jacobson, J. M.; Frenz, J. P. J. Chromatogr. 1990, 499, 5. (10) Huang, J. X.; Guiochon, G. J. Colloid Interface Sci. 1989, 128, 577.
S0743-7463(96)00333-2 CCC: $12.00
Langmuir models derived from rate equations;24,25 (ii) isotherm models for homogeneous surfaces with lateral interactions, like the Fowler6 and the Quadratic6,7,13,15,16,20 models; and (iii) isotherm equations for heterogeneous surfaces, like the bilangmuir16,17,23,26,27 model or the sum of the Langmuir and Quadratic models.28 Other isotherms for homogeneous surfaces with lateral interactions have remained unused so far for the prediction of competitive data. Among these models, the most important are the Ruthven3 and the Moreau et al.29 models. A model for single-component adsorption on homogeneous surfaces with lateral specific interactions was also derived and applied by Kiselev.30,31 This last model also has never been used for the correlation of single-component RPHPLC adsorption data. This work has two goals. First, the single-component Kiselev model will be extended to the case of binary mixtures. Second, this new competitive model will be tested using previously reported adsorption (11) Katti, A. M.; Czok, M.; Guiochon, G. J. Chromatogr. 1991, 556, 105. (12) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1988, 60, 2636. (13) Seidel-Morgenstern, A.; Guiochon, G. J. Chromatogr. 1993, 631, 37. (14) Ma, Z.; Guiochon, G. J. Chromatogr. 1992, 603, 13. (15) Seidel-Morgenstern, A.; Guiochon, G. Chem. Eng. Sci. 1993, 48, 2787. (16) Bellot, J. C.; Condoret, J. S. J. Chromatogr. A 1993, 657, 305. (17) Jacobson, S.; Golshan-Shirazi, S.; Guiochon, G. AIChE J. 1991, 37, 836. (18) Antia, F. D.; Horva´th, Cs. J. Chromatogr. 1989, 484, 1. (19) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1988, 60, 2630. (20) Huang, J. X.; Horva´th, Cs. J. Chromatogr. 1987, 406, 275. (21) Huang, J. X.; Horva´th, Cs. J. Chromatogr. 1987, 406, 285. (22) Quin˜ones, I.; Guiochon, G. J. Chromatogr. A 1996, 734, 83. (23) Charton, F.; Guiochon, G. J. Chromatogr. A 1993, 630, 21. (24) Lin, B. C.; Ma, Z.; Golshan-Shirazi, S.; Guiochon, G. J. Chromatogr. 1989, 475, 1. (25) Gu, T.; Tsai, G.-J.; Tsao, G. T. AIChE J. 1991, 37, 1333. (26) Jacobson, S.; Golshan-Shirazi, S.; Guiochon, G. J. Am. Chem. Soc. 1990, 112, 6492. (27) El Fallah, M. Z.; Guiochon, G. Biotechnol. Bioeng. 1992, 39, 877. (28) Diack, M.; Guiochon, G. Anal. Chem. 1991, 63, 2608. (29) Moreau, M.; Valentin, P.; Vidal-Madjar, C.; Lin, B. C.; Guiochon, G. J. Colloid Interface Sci. 1991, 141, 127. (30) Berezin, G. I.; Kiselev, A. V. J. Colloid Interface Sci. 1972, 38, 227. (31) Berezin, G. I.; Kiselev, A. V.; Sagatelyan, R. T.; Sinitsyn, V. A. J. Colloid Interface Sci. 1972, 38, 335.
© 1996 American Chemical Society
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data of phenyl alcohols on ODS-silica6 and compared to the other models listed above in terms of the ability to predict correctly competitive data using only the parameters derived from the single-component adsorption data. Theory The adsorption isotherm equations relate the equilibrium concentrations of the studied components in the bulk liquid phase, Ci, and in the adsorbed phase, qi. The equilibrium concentration in the adsorbed phase is usually expressed through the surface coverage θ ) qi/qs,i, where qs,i is the monolayer capacity for the corresponding component.3 In this study, we compare the results obtained using different competitive adsorption isotherm models, the competitive extensions of the Fowler model32 and the Kiselev models,30,31 as well as the competitive models described by Moreau et al.29 and by Ruthven.3 1. Fowler Model. One version of the Fowler32 model was derived in terms of the so-called Bragg-Williams approximation. It is the simplest model assuming localized monolayer adsorption with lateral attractive interactions. The single component model is defined as32
C)
θ e-zθ/RT K(1 - θ)
(1)
where K is the low-concentration equilibrium constant (LCEC), is the interaction energy between two molecules adsorbed on two nearest-neighbor sites, z is the number of nearest-neighbor sites, R is the universal gas constant, and T is the absolute temperature. The assumptions underlying the Fowler model are that adsorbed molecules form a localized monolayer on an energetically homogenous surface and that every adsorbed molecule interacts with a limited number of other adsorbed molecules distributed randomly on the nearest-neighbor sites. The term -zθ/RT is a measure of the average force field acting on an adsorbed molecule and created by the surrounding molecules on the nearest sites. It is usually represented as -χθ. Note that the classical Henry constant is equal to the product Kqs,i for the isotherm which tends toward the Henry law at low concentrations, e.g., the Langmuir and the Fowler models.4 Two extensions of the single-component isotherm for multicomponent adsorption have been reported in the recent chromatographic literature: the Jaroniec et al.33 and Zhu et al.6 equations. The extension suggested by Zhu et al. is the simplest one. It is defined as6
θ1
1 ) Ke-χ1(θ1+θ2) C1 1 - (θ1 + θ2)
(2a)
θ2 1 ) Ke-χ2(θ2+θ1) C2 1 - (θ1 + θ2)
(2b)
the first one and conversely. Equations 2a and 2b are equivalent to eqs 3a and 3b with K ) K1 ) K2, χ12 ) χ1, and χ21 ) χ2. 2. Model of Moreau et al. Moreau et al. have described several models which are extensions of the two-dimensional lattice models with adsorbate-adsorbate interactions to the treatment of multicomponent adsorption. These models were derived using the theory of lattice statistics.29 The assumptions made are those usually applied in lattice models: The surface is considered to be homogeneous, and the coverage is limited to a monolayer. If the adsorbate-adsorbate interactions are limited to those between pairs of molecules, the model gives for the singlecomponent adsorption isotherm
θ)
KC + b(KC)2 1 + 2KC + b(KC)2
(4a)
where b is the adsorbate-adsorbate interaction parameter. In the case of binary, competitive adsorption, the model gives
θ1 ) [K1C1 + b11(K1C1)2 + b12K1K2C1C2]/[1 + 2(K1C1 + K2C2) + b11(K1C1)2 + b22(K2C2)2 + 2b12K1K2C1C2] (4b) θ2 ) [K2C2 + b22(K2C2)2 + b12K1K2C1C2]/[1 + 2(K1C1 + K2C2) + b11(K1C1)2 + b22(K2C2)2 + 2b12K1K2C1C2] (4c) The parameters bii of this model are obtained from the correlation of single-component data (eq 4a). The cross-coefficient b12 accounts for the interaction between both components in the mixed adsorbed phase and must be derived from measurements of binary adsorption. If we assume triple-cell nearest-neighbor interactions, the single-component isotherm becomes
θ)
KC + 2b(KC)2 + (bKC)3 1 + 3KC + 3b(KC)2 + (bKC)3
(5a)
In the case of binary, competitive adsorption, the same model gives
θ1 )
Θ1 Θ3
(5b)
θ2 )
Θ2 Θ3
(5c)
with
where the subscripts 1 and 2 correspond to the first and second component, respectively. Note that equations 2a and 2b are empirical and purely predictive, since they consider only the single-component parameters which can be identified from singlecomponent adsorption data. Within this model, the LCEC constants should be equal for both compounds, which is somewhat unrealistic.6 The Jaroniec extension is defined as33
θ1 1 ) K1e-(χ1θ1+χ12θ2) C1 1 - (θ1 + θ2)
(3a)
θ2 1 ) K2e-(χ2θ2+χ21θ1) C2 1 - (θ1 + θ2)
(3b)
where the coefficients χ12 and χ21 account for the different influence of the molecules of the second component on those of (32) Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, U.K. 1960. (33) Jaroniec, M.; Borowko, M.; Patrykiejew, A. Surf. Sci. 1978, 78, L501.
Θ1 ) K1C1 + 2b11(K1C1)2 + b311(K1C1)2 + b12K2C2(2 + 2b11K1C1 + b22K2C2) (5d) Θ2 ) K2C2 + 2b22(K2C2)2 + b322(K2C2)2 + b12K1C1(2 + b11K1C1 + 2b22K2C2) (5e) Θ3 ) 1 + 3(K1C1 + K2C2) + b11(K1C1)2(3 + b211K1C1) + b22(K2C21)2(3 + b222K2C2) + 3b12K1K2C1C2(2 + b11K1C1 + b22K2C2) (5f) Although more complex than any of the previous ones, this model has only one cross-coefficient which must be determined from competitive equilibrium adsorption data. 3. Ruthven Model. The Ruthven model was derived using a statistical thermodynamic approach, applied to multicomponent adsorption equilibria.3 Within this model, the molecules are localized at different surface sites or cages, with little interchange between sites. Each cage can be considered as a separate subsystem and, in a first approximation, interactions between molecules belonging to neighbor cages may be neglected. This model was applied successfully to account for adsorption in zeolites, where the structure consists of cages interconnected through small windows. A similar model had been suggested to
Localized Monolayers with Lateral Interactions
Langmuir, Vol. 12, No. 22, 1996 5435
account for single-component adsorption in zeolitic systems.41 In this systems, each cage can contain a finite number of adsorbate molecules. For a cage which may accommodate up to two molecules of the adsorbate, the model gives for single-component adsorption
θ)
KC + R(KC)2
(6a)
1 + KC + (R/2)(KC)2
where R is the parameter accounting for adsorbate-adsorbate interactions in a cage and representing the corresponding configuration integral. For binary, competitive adsorption, the model gives
obtained by combining the equilibrium constants for all the reactions in an overall constant that describes the annexation of an adsorbed molecule to the given associated group. In principle, the value of this constant is independent of the dimension of the molecular cluster. For single-component adsorption of component A, the set of reactions will be written
for binary associates A + A h A2
(8a)
A2 + A h A3
(8b)
for ternary associates
2
θ1 ) [K1C1 + R20(K1C1) + R11(K1C1)(K2C2)]/[1 + K1C1 + K2C2 + (R20/2)(K1C1)2 + (R02/2)(K2C2)2 + R11(K1C1)(K2C2)] (6b) θ2 ) [K2C2 + R02(K2C2)2 + R11(K1C1)(K2C2)]/[1 + K1C1 + K2C2 + (R20/2)(K1C1)2 + (R02/2)(K2C2)2 + R11(K1C1)(K2C2)] (6c) where the parameters R20 and R02 are obtained from the correlation of single-component data and the parameter R11 accounts for the interaction between both components in the mixed adsorbed phase. For a cage which may accommodate up to three molecules of the adsorbate, the Ruthven model gives for single-component adsorption
θ)
for binary associates K2a )
K3a )
where R2 and R3 are the parameters corresponding to binary and ternary adsorbate-adsorbate interactions in a cage, respectively. For binary, competitive adsorption, the same model gives
θ A2
(9a)
θAθA
Φ1 P
(7b)
Φ2 θ2 ) P
(7c)
θA3
(9b)
θA2θA
for n-dimensional associates Kna )
θAn
(9n)
θAn-1θA
In principle, it has been shown that we can consider that the values of all the equilibrium constants are equal to an overall association constant, Ka,30 so
K2a ) K3a ) ... ) Kna ) Ka
Φ1 ) K1C1 + R20(K1C1)2 + R11(K1C1)(K2C2) + 2
2
(R30/2)(K1C1) + R21(K1C1) (K2C2) + (R12/2)(K1C1)(K2C2) (7d)
(10)
Taking into account eqs 9a, 9b, 9n, and 10, the surface coverages of the molecular clusters become
for binary associates
Φ2 ) K2C2 + R02(K2C2)2 + R11(K1C1)(K2C2) + (R03/2)(K2C2)3 + R12(K2C2)2(K1C1) + (R21/2)(K2C2)(K1C1)2 (7e) P ) 1 + K1C1 + K2C2 + (R20/2)(K1C1)2 + (R02/2)(K2C2)2 + R11(K1C1)(K2C2) + (R30/6)(K1C1)3 + (R21/2)(K1C1)2(K2C2) + 2
(8n)
The equilibrium in eqs 8a, 8b, and 8n is characterized by the association equilibrium constants K2a, K3a, and Kna, respectively. The equilibrium concentrations may be expressed by means of the surface coverages of the nonassociated molecules, θA, and of the molecular clusters, θA2, θA3, and θAn, using the following relationships
(7a)
1 + KC + (R2/2)(KC)2 + (R3/6)(KC)3
3
An-1 + A h An
for ternary associates
KC + R2(KC)2 + (R3/2)(KC)3
θ1 )
for n-dimensional associates
θA2 ) Kaθ2A
(11a)
θA3 ) K2aθ3A
(11b)
for ternary associates
3
(R12/2)(K1C1)(K2C2) + (R03/6)(K2C2) (7f) The parameters R20, R02, R30, and R03 are obtained from the correlation of single-component data. The parameters R11, R21, and R12 account for the interaction between both components in the mixed adsorbed phase and must be derived from a correlation of the competitive adsorption data. 4. Kiselev Model. This model was originally derived for single-component localized adsorption with specific lateral interactions in the monolayer.30 It was applied successfully to the description of the adsorption of ethanol on the surface of graphitized carbon black, a case in which large-scale association of the adsorbate molecules on the surface by means of hydrogen bonding is probable.31 Within this model, the interactions between adsorbed molecules are represented by a set of quasichemical reversible reactions. The adsorption isotherm is
for n-dimensional associates n θAn ) Kn-1 a θA
(11n)
The total surface coverage, taking into account the multiplicity of associates will be
θ ) θA + 2θA2 + 3θA3 + ... + nθAn
(12)
and taking into account eqs 11a to 11n, we obtain n
n θ ) θA + 2Kaθ2A + 3K2aθ3A + ... + nKn-1 a θA ) θA +
∑iK i)2
i -1 i a θA
(13)
5436 Langmuir, Vol. 12, No. 22, 1996
Quin˜ ones and Guiochon
If we consider the formation of all the associates of any possible dimension (infinite series in eq 13) and the condition KaθA < 1, eq 13 can be replaced by the sum of the derivative of a geometric progression, giving
θ)
θA
The surface coverages of the associates will be given by
for component A θA2 ) Ka1θ2A
(19a)
θB2 ) Ka2θ2B
(19b)
θAB ) KmθAθB
(19c)
(14)
(1 - KaθA)2
for component B
The coverage of single molecules can be obtained from the adsorbate-adsorbent equilibrium, using, for example, the Langmuir equation. So,
for the mixture θA ) K(1 - θ)C
(15) isotherm,34
In principle, other models, such as the Jovanovic could be used in eq 15 instead of the Langmuir model. Kiselev preferred to apply the Langmuir model.30 Combination of eqs 14 and 15 gives a model for monolayer single-component adsorption on homogeneous surfaces, with lateral specific interactions30
θ K ) (1 - θ)C [1 - KKa(1 - θ)C]2
(17a)
for associates of component B (17b)
for mixed associates A + B h AB
(17c)
The corresponding equilibrium constants are given by
θA2 θAθA
θ2 ) θB + 2θB2 + θAB ) θB + 2Ka2θ2B + KmθAθB (20b) Because the associations A2, AB, and B2 do not exist in the solution, only A and B equilibrate between the solution and the surface. From the competitive Langmuir adsorbate-adsorbent equilibrium for these single, nonassociated molecules (surface concentrations θA and θB), we have
θA ) K1(1 - θ1 - θ2)C1
(21a)
θB ) K2(1 - θ1 - θ2)C2
(21b)
The set of eqs 20 and 21 constitutes the Kiselev competitive isotherm. It cannot be solved algebraically for θ1 and θ2, although numerical solutions are easily calculated. In a similar manner, if is possible to consider the formation of both binary and ternary molecular associations. Then, we obtain the following total coverages for the first and the second component, respectively.
θ1 ) θA + 2Ka1θ2A + 3K2a1θ3A + KmθAθB + 2K2mθ2AθB + K2mθAθ2B (22a) θ2 ) θB + 2Ka2θ2B + 3K2a2θ3B + KmθAθB + K2mθ2AθB + 2K2mθAθ2B (22b) The isotherm is obtained by combination with eqs 21a and 21b. An analogous set of equations may be written for a binary mixture which may form associations of any specified maximum finite n-dimension.
for associates of component A Ka1 )
θ1 ) θA + 2θA2 + θAB ) θA + 2Ka1θ2A + KmθAθB (20a) for the second component
for associates of component A
B + B h B2
for the first component
(16)
The combination of eq 15 with equations such as eq 13 having a finite number of terms could, in principle at least, give isotherm models accounting for the formation of molecular associations of finite dimensions. However, this approach was not suggested originally by Kiselev.30 5. Extension of the Kiselev Model to Competitive Adsorption. The extension of eqs 13 and 16 to the case of multicomponent adsorption is straightforward. This extension is done below in the simplest, particular case of a binary mixture and of the formation of only binary associates. It could easily be further extended to multicomponent mixtures and to the formation of more complex associations. In the case of a binary mixture forming only binary associates, the set of quasichemical reactions will be
A + A h A2
The total coverages for both components, taking into account the multiplicity of associates, are
(18a)
for the first component n
θ1 ) θA +
for associates of component B
∑iK
n j-1
i-1 i a1 θA
+
i)2
Ka2 )
θB2 θBθB
(18b)
n
θAB θAθB
(18c)
∑
∑∑kK
j-1 k j-k m θBθA
(22d)
j)2 k)1
If associations of any dimension may be formed, the following expressions for the competitive binary isotherm are valid
θ1 ) (34) Jovanovic, D. S. Kolloid Z. 1969, 235, 1203.
(22c)
n j-1
i iKi-1 a2 θB +
i)2
Km )
j-1 k j-k m θAθB
j)2 k)1
for the second component θ2 ) θB +
for the mixed associates
∑∑kK
KmθAθB
θA (1 - Ka1θA)
2
+
(1 - KmθB)(1 - KmθA)2
(23a)
Localized Monolayers with Lateral Interactions θ2 )
KmθAθB
θB (1 - Ka2θB)
2
+
(1 - KmθA)(1 - KmθB)2
Langmuir, Vol. 12, No. 22, 1996 5437 (23b)
Experimental Section 1. Origin of Experimental Data. The experimental data used in this study were reported previously.6 They deal with the adsorption of 2-phenylethanol (PE) and 3-phenylpropanol (PP) on ODS-silica (Vydac, Hesperia, CA) with (50:50) methanol-water solution as the mobile phase. Frontal analysis was performed following the procedure described by Frenz et al.,35 with a Gilson (Middleton, WI) Model 302 pump, a ten-port Valco (Houston, TX) valve, and a Spectroflow (Applied Biosystems, Ramsey, NJ) 757 UV-detector. The composition at each intermediate plateau was determined using an online liquid chromatograph assembled with a Beckman (Berkeley, CA) Model 110B pump, a Valco four-port valve, a YMC (Wilmington, NC) cartridge column, and another Spectroflow detector. Both UV analog signals were acquired with a Gilson Model 621 interface box and monitored with a computer. 2. Nonlinear Least Squares Analysis. The fit of the experimental data points to the isotherm equations was accomplished using the Marquardt algorithm.36 For models which are implicit with respect to the solid phase concentration, such as the Fowler and Kiselev models, the numerical inversion was accomplished using a GaussNewton algorithm. The fitting procedure requires the identification of the common parameters in a system of nonlinear equations. This task may be classified as a multivariant, multiobjective, nonlinear programming problem. The procedure used was reported previously.22 It is relatively simple. Convergence is achieved in most cases, even for complex isotherm models containing a significant number of adjustable parameters. Because the response surface used in such parameter optimization methods could exhibit several minima (global and/or local), we carried out the identification process several times, using a different initial guess vector each time. This is a good empirical way to determine if the solution converges toward the global minima. On the other hand, the parameters already identified for simpler models like the Langmuir model, previously reported,22 were always used in one initial guess vector. The procedure calculates the values of the isotherm parameters which minimize the residual sum of squares (RSS) for both components n
RSS )
(qex,i - qt,i)2 ∑ i)1
(24)
where qex,i are the elements of the vector qex, containing the complete single-component and competitive experimental adsorbed phase concentrations for both phenyl alcohols (n data points), and qt,i are the corresponding theoretical values calculated by the given model. An initial set of parameter values is required. The selection of the most adequate model was performed using Fisher’s test. The model selected was the one which exhibited the highest value of the Fisher parameter:37 (35) Jacobson, J. M.; Frenz, J. H.; Horva´th, Cs. Ind. Eng. Chem. Res. 1987, 26, 43. (36) Marquardt, D. W. J. Soc. Appl. Math. 1963, 11, 431. (37) Ajnazarova, S. L.; Kafarov, V. V. Metodi optimisatsi eksperimenta v khimicheskoy tekhnologui; Vishaia Shkola: Moscow, 1985.
Figure 1. Comparison between experimental adsorption data of 2-phenylethanol and values calculated using the competitive Fowler model (eqs 2a and 2b) and single-component parameters. For mixtures, the mobile phase concentration is expressed via the relative composition of PE and PP. All concentrations are on milligrams per milliliter. Symbols: data for single component, O; data for 3:1 mixtures (three parts PE and one part PP), ×; data for 1:1 mixtures (equal parts of both components), +; data for 1:3 mixtures (one part PE and three parts PP), *. On the graph are also indicated the RSS calculated for each set of experiments performed with a constant relation of the mobile phase concentrations of the phenyl alcohols. n
(qex,i - qex)2 ∑ i)1
(n - l) Fcalc )
n
(qex,i - qt,i) ∑ i)1
(n - 1)
(25)
2
where qex is the mean value of the vector qex and l is the number of adjusted parameters of the model. The procedure was applied to each of the isotherm models, using the same set of experimental data6 to determine the best values of the parameters for the singlecomponent isotherm model. Then, the procedure was applied to the corresponding competitive model, using the best values of the parameters of the single-component isotherms previously determined and adjusting the remaining interaction parameters when needed. The empirical fit of the models to the whole set of experimental data was not accomplished, in account of the fact that we had already solved this task satisfactorily, using an 11parameter quadratic model.22 Results and Discussion The results of the regression analysis of the models evaluated are reported in Table 1 and in Figures 1 and 2. The corresponding values of the RSS are listed separately for each component, in the cases of the singlecomponent and of the competitive isotherm data. The values between parentheses correspond to the competitive data. The global value of the RSS for the whole set of single and competitive data (∑RSS) is also given, as well as the value calculated for the Fisher parameter. Note that the two compounds studied, 2-phenylethanol (PE) and 3-phenylpropanol (PP) are homologs. They are not expected to differ strongly, and the cross-coefficients of interaction could be safely assumed to differ little from an average (arithmetic or geometric) of the two coefficients of self-interaction.
5438 Langmuir, Vol. 12, No. 22, 1996
Quin˜ ones and Guiochon
Table 1. Summary of the Nonlinear Regression Analysisa RSS no.
model
parameters
1
eqs 1, 2a, and 2b; Pb
qs1 ) 131 K ) 0.0175 χ1 ) -0.1752 qs2 ) 275 χ2 ) 1.6100 qs1 ) 131 K ) 0.0175 χ1 ) -0.1752 χ12 ) 0.2478 qs2 ) 275 χ2 ) 1.6100 χ21 ) 4.8453 qs1 ) 159 K ) 0.0144 χ1 ) 0 qs2 ) 335 χ2 ) 2.2317 qs1 ) 85 K1 ) K2 ) 0.0268 b11 ) 1.6653 qs2 ) 176 b22 ) b12 ) 0.5774 qs1 ) 85 K1 ) K2 ) 0.0268 b11 ) 1.6653 qs2 ) 176 b22 ) 0.5774 b12 ) -0.9726 qs1 ) 84 K1 ) 0.0272 b11 ) 1.3073 qs2 ) 207 K2 ) 0.0230 b22 ) b12 ) 0.6322 qs1 ) 84 K1 ) 0.0272 b11 ) 1.3073 qs2 ) 207 K2 ) 0.0230 b22 ) 0.6322 b12 ) -0.1419 qs1 ) 154 K1 ) 0.0150 R20 ) 0 qs2 ) 88 K2 ) 0.0537 R02 ) R11 ) 0.1447 qs1 ) 154 K1 ) 0.0150 R20 ) 0 qs2 ) 88 K2 ) 0.0537 R02 ) 0.1447 R11 ) -0.3577 qs1 ) 154 K1 ) 0.0150 R20 ) 0 R30 ) 0 qs2 ) 30 K2 ) 0.1884 R02 ) R11 ) 0.3486 R03 ) 0.4564 R21 ) R12 ) 0.4564 qs1 ) 154 K1 ) 0.0150 R20 ) 0 R30 ) 0 qs2 ) 30 K2 ) 0.1884 R02 ) 0.3486 R11 ) 1.9163 R03 ) 0.4564 R21 ) -3.9792 R12 ) -2.0057
2
3
4
5
6
7
8
9
10
11
eqs 1, 3a, and 3b; Cb
eqs 1, 2a, and 2b; P
eqs 4a-4c; P
eqs 4a-4c; C
eqs 5a-5f; P
eqs 5a-5f; C
eqs 6a-6c; P
eqs 6a-6c; C
eqs 7a-7f; P
eqs 7a-7f; C
PE 0.0036 (3.26) 0.0036 (2.99)
0.0022 (3.72) 0.0011 (2.73) 0.0011
PP
∑RSS
Fcalc
0.24
124.61
46.07
46.35
120.46
126.05
46.17
146.48
39.19
81.03
74.73
124.51
45.48
69.89
86.64
280.04
20.50
262.67
23.05
390.36
14.51
337.72
17.46
(121.11) 0.24 (43.12)
0.22 (122.1) 0.36 (143.4) 0.36
(23.17)
(57.5)
0.001
0.32
(2.49)
(121.7)
0.001
0.32
(5.57)
(54.0)
0.0018 (6.58)
0.0018 (30.32)
0.0018 (30.25)
0.0018 (38.41)
0.36 (273.1)
0.36 (232.0)
0.0051 (360.1)
0.0051 (299.3)
Localized Monolayers with Lateral Interactions
Langmuir, Vol. 12, No. 22, 1996 5439
Table 1 (Continued) RSS no.
modelb
parameters
PE
12
eqs 13, 20a and 20b (n ) 2); P
qs1 ) 80 K1 ) K2 ) 0.0285 Ka1 ) 0.3845 qs2 ) 166 Ka2 ) Km ) -0.1609 qs1 ) 135 K1 ) 0.0170 Ka1 ) 0.0731 qs2 ) 546 K2 ) 0.0089 Ka2 ) Km ) -1.9459 qs1 ) 97 K1 ) K2 ) 0.0237 Ka1 ) 0.2428 qs2 ) 200 Ka2 ) Km ) -0.3294 qs1 ) 106 K1 ) K2 ) 0.0217 Ka1 ) 0.1977 qs2 ) 218 Ka2 ) Km ) -0.4052 qs1 ) 97 K1 ) K2 ) 0.0236 Ka1 ) 0.2397 qs2 ) 201 Ka2 ) Km ) -0.3339 qs1 ) 154 K1 ) 0.0150 Ka1 ) 0 qs2 ) 546 K2 ) 0.0089 Ka2 ) Km ) -1.946 qs1 ) 135 Ki ) 0.0170 Ka1 ) 0.0731 qs2 ) 546 K2 ) 0.0089 Ka2 ) -1.9459 Km ) -3.4486
0.0015
13
14
15
16
17
18
eqs 13, 22a and 22b (n ) 3); P
eqs 13, 22c and 22d (n ) 4); P
eqs 13, 22c and 22d (n ) 5); P
eqs 16, 23a, and 23b (n ) ∞); P
eqs 13, 22a, and 22b (n ) 3); P
eqs 13, 22a, and 22b (n ) 3); C
(1.95) 0.0017 (2.42)
0.0011 (1.56) 0.0013 (1.47) 0.0011 (1.55) 0.0018 (2.2)
0.0017 (3.62)
PP
∑RSS
Fcalc
0.38
160.47
35.77
117.08
48.36
172.67
33.25
177.59
32.33
172.95
33.19
135.59
42.34
92.89
65.19
(158.15) 0.20 (114.46)
0.34 (170.77) 0.35 (175.78) 0.34 (171.06) 0.20 (133.2)
0.20 (89.07)
a Competitive models which use single-component identified parameters. b P means a purely predictive calculation of the competitive data based solely on single-component identified parameters, and C means a correlation of the cross interaction parameters present on the competitive model using experimental binary adsorption data.
Figure 2. Same as Figure 1 but data for 3-phenylpropanol.
When fitted to a Langmuir isotherm, the singlecomponent adsorption data gave a reasonably good correlation, with initial slopes (i.e., Henry constants) equal to 2.3 and 4.82, respectively, and hence a separation factor of 2.10.6 The values of the second or curvature coefficients were 0.015 and 0.039 mg/mL, respectively.6 The ranges of concentrations investigated for PE and PP were 0-7.5
and 0-14.5 mg/mL, respectively. Accordingly, the maximum values of the experimental products bCi were 0.225 for PE and 0.565 for PP. These values are low, especially the first one, and not quite sufficiently high to give an accurate representation of the curvature of the isotherm. 1. Fowler Model. 1.1. Correlation of Single-Component Data (SCD) by the Single Component Isotherm Model. The fit of the single-component data (SCD) to eq 1 was accomplished taking into account that the LCEC should be the same for both components within the framework of the Zhu et al. extension of the Fowler model (eqs 2a and 2b). It is noteworthy that we obtained better results in the fit of the overall SCD for both components when considering equal values of K instead of allowing for different values, in spite of the relatively large separation factor at infinite dilution (R ) 2.10). It seems that in terms of the Fowler model both components have a marked tendency to a similar adsorbate-adsorbent interaction, as expected from their closely similar structures. The agreement of the model with the data is good, as shown in Table 1 (entry 1) and Figures 1 and 2. In general, for all the models evaluated, the quality of the fit of the PP data determines the overall quality of the fit of the data for both compounds. This is because the value of the RSS for PE is almost always less by several orders of magnitude than the similar value for PP. This is probably a consequence of the selection of a narrower range of
5440 Langmuir, Vol. 12, No. 22, 1996
Quin˜ ones and Guiochon
Table 2. Summary of the Nonlinear Regression Analysis22 RSS model Langmuir and LeVan-Vermeulena quadratic Jovanovic for zero dispersion of settling times Jovanovic for random distribution of settling times Jovanovic for a Heaviside distribution of settling times
PE
PP
∑RSS
Fcalc
0.002 (3.08) 0.002 (2.74) 0.002 (4.04) 0.002 (2.94) 0.001 (2.60)
0.444 (249.61) 0.444 (27.65) 0.7 (46.72) 0.444 (66.49) 0.44 (43.65)
253.14
22.99
30.84
181.05
51.46
116.14
69.88
86.66
46.69
129.68
a Prediction based solely on the use of single-component identified parameters. For the rest of the models, binary data were used to accomplish the correlation.
concentrations for the acquisition of the PE data than for the PP data. For the Fowler model, the value of the RSS for PE is slightly lower than the value obtained with the Langmuir model6,22 (Table 2) while for PP this value is approximately two times lower. So, the Fowler model better accounts for the PP data than the Langmuir model and the consideration of possible lateral interactions in the adsorbed phase has a stronger effect on the fit of the PP data than on the fit of the PE data. The absolute value and (more importantly) the sign of the lateral interaction parameters for both components are different, possibly (but improbably) reflecting a different nature of this interaction. The value of χ2 for PP has a higher absolute value. However, as stated earlier, the two components are homologs, and there is no reason to think that their adsorbate-adsorbate interactions should be very different in pure or mixed adsorbed phases. To clarify this issue, we eliminated the adsorbateadsorbate interaction parameter for PE (by letting χ1 ) 0), thus considering that its adsorption may be described by the ideal Langmuir model. The results, presented in Table 1 (entry 3), show that the RSS for PE is only slightly increased, while the same value for PP is slightly decreased. So, the elimination of the adsorbate-adsorbate interactions for PE does not significantly affect either the fit of the PE SCD or the global fit of the SCD for both components. The derivation of single-component isotherm models within the theory of lattice statistics is commonly performed with the assumption that adsorbed molecules occupy only one site on the surface.3,29,32 In the case of competitive adsorption, equal adsorption capacities for all the components present in the mixture is also assumed. However, for the system analyzed here, the monolayer capacity obtained for PP in terms of the Fowler model is approximately twice as large as that obtained for PE, a result different from the one obtained with the Langmuir and the Jovanovic models. These last two models predict almost the same adsorption capacity for both components but different values of the LCEC, approximately in the ratio of 1 to 2.22 This departure from the theoretical assumption of equal saturation capacities for both components within the framework of the Fowler model could be a consequence of the application of a relatively simple model to a rather complex experimental system. Although the two phenyl alcohols studied have a similar size, their geometry is not simple and they may have different areas of contact with the solid surface. On the other hand, in the experimental case studied, the separation factor at infinite dilution, by definition equal to the ratio of the Henry constants of the two components, is R ) 2.10. When the LCEC’s of the two components are equal, the ratio of the monolayer capacities must then be close to the separation factor. Similarly, when the monolayer capaci-
ties are nearly the same,22 the ratio of the LCEC’s is close to the separation factor. However, it is difficult to derive from the mere adsorption data what are the relative contributions of the differences (1) in the adsorbateadsorbent energies of interaction and (2) in the areas of contact of the molecules with the solid surface. Additional data, such as the isosteric heats of adsorption, would be necessary to settle this issue.3 1.2. Prediction of Competitive Adsorption Data Using the Parameters of the Single-Component Isotherm in a Competitive Isotherm Model. Introduction of the SCD parameters into the competitive model (eqs 2a and 2b) gives results which are better than those of other purely predictive models (e.g., Langmuir and LeVan-Vermeulen),22 as exposed in Table 1 (entry 1), Table 2, and Figures 1 and 2. The main gain resides in the prediction of the competitive PP data. The value of the corresponding RSS decreased twofold with respect to the values obtained with either the Langmuir or the LeVan-Vermeulen models.22 The value of the RSS for the competitive data of PE is only slightly increased with respect to the value obtained with the Langmuir model.22 It is worth noting that the elimination of the adsorbate-adsorbate interaction for PE did not affect significantly the prediction of either the PE or the PP data (Table 1, entry 3). The deviations observed for the competitive data increase with increasing PP concentration in the mixture, as seen in Figures 1 and 2. It seems that there exists a nonideal behavior of the mixture which becomes more and more pronounced with increasing PP concentration. 1.3. Identification of the Cross-Coefficients of the Mixed Isotherm Using the Competitive Data. Using the best values of the SCD parameters of the Fowler isotherms for PE and PP, the whole set of competitive isotherm data were fitted to the Jaroniec competitive version of the Fowler model (eqs 3a and 3b). This made possible the determination of the best values of the interaction parameters (Table 1, entry 2). Although this model allows for a different value of the LCEC constant of each component, this additional flexibility did not permit a significant decrease of the RSS. Other models with crossparameters gave better or similar results.22 It is interesting to note that the value of χ21, the coefficient accounting for the influence of PE over PP, has a higher absolute value than the similar coefficient χ12, which characterizes the opposite influence. A similar behavior was obtained previously with other models.22 1.4. Conclusions Regarding the Fowler Models. Unfortunately, the Fowler isotherm models suffer from two major drawbacks. The first one is theoretical but specific to the case in point studied. The second one is practical but general. First, the difference between the values of the monolayer capacities of PE and PP renders the competitive model thermodynamically inconsistent. The
Localized Monolayers with Lateral Interactions
parameters obtained have no physical sense. The use of this model for an empirical representation of the experimental data and for their interpolation is suspicious.2 The second drawback has especially serious consequences in chromatography. It is not possible to inverse analytically the Fowler models with respect to the concentration in the adsorbed phase. This makes their use much more cumbersome for the calculation of multicomponent band profiles using models of nonlinear chromatography.2 The run times are much longer and the extensive numerical calculations required cause additional numerical errors. 2. Moreau et al. Model. 2.1. Correlation of SCD by the Single-Component Isotherm Model. The SCD were fitted to the quadratic version of this model (eq 4a). Because of the similarity of the LCEC obtained for both compounds, a unified value was estimated, as in the case discussed earlier with the Fowler isotherm (Table 1, entry 4). The fit of the SCD for both compounds is better than the fit given by the Langmuir model22 (Table 2), although the improvement is not great. For the quadratic Moreau et al. model, the adsorbate-adsorbate interaction parameters for both components have the same sign and their absolute values are of the same order of magnitude. This result is consistent with the fact that the two components are similar homologs which should exhibit, in a first approximation, the same degree of adsorbateadsorbate interaction. Within this model, the monolayer capacity of PP is higher than that of PE, a result similar to the one observed for the Fowler model. Then, the SCD were fitted to the triple-cell interaction model (eq 5a); see Table 1, entry 6. The Ki values derived were close but not exactly equal. The fit of the SCD to the model is not much better than the same fit to the quadratic model (eq 4a, Table 1, entry 4) or to the Langmuir model22 (Table 2). The sign and the order of magnitude of the adsorbate-adsorbate interaction parameters are similar to those derived from the fit of the SCD to the quadratic model. The PP monolayer capacity is also greater than that for PE, the same behavior as exhibited by the Fowler model. 2.2. Prediction of Competitive Adsorption Data Using the Parameters of the Single-Component Isotherm in a Competitive Isotherm Model. The purely predictive calculation of competitive equilibria was done using eqs 4b and 4c. Here we consider that the adsorbate-adsorbate interactions are dominated by the PP molecular interactions in the mixed monolayer and that the value of the cross-parameter b1,2 is equal to that of the singlecomponent parameter, b2,2; see Table 1, entry 4. For single components, the quality of the prediction is better than those of the Langmuir and the LeVan-Vermeulen models6,22 (Table 2). For PE, the prediction is even better than the one obtained with the Fowler model (Table 1, entry 1). The quality of the prediction of the competitive data, however, is not as good as that obtained with the Fowler model. The value of the global RSS is mainly determined by the value of the PP competitive RSS. The quality of the global prediction of the double-cell Moreau et al. (Table 1, entry 4) model is worse than that of the Fowler isotherm (Table 1, entry 1). The quality of the purely predictive calculation of the competitive data done using the single-component parameters with the condition b12 ) b22 and eqs 5b and 5c (Table 1, entry 6) is better than that of the one obtained with either the Langmuir, the LeVan-Vermeulen (see Table 2), or the rest of the models tested in this study, with the exception of the Kiselev model for n ) 3 (Table 1, entry 13); see below. It should be noted that the quality of this prediction is only slightly better than that of the one obtained with the Fowler model (Table 1, entry 1).
Langmuir, Vol. 12, No. 22, 1996 5441
2.3. Identification of the Cross-Coefficients of the Mixed Isotherm Using the Competitive Data. If one considers the cross-coefficient of the double-cell model as an adjustable parameter (eqs 4b and 4c), we obtain a rather low value of the global RSS (Table 1, entry 5). However, this value is higher than the similar values obtained with other models previously tested22 (Table 2). The adsorbateadsorbate interaction cross-coefficient thus identified has a different sign than the similar parameters derived for both components from the correlation of the SCD. This observation, however, does not seem to have a theoretical justification. Derivation of the value of the cross-coefficient by fitting the global binary data to the triple-cell model gave the lowest value of the RSS (Table 1, entry 7) obtained in this study. The result, however, is not better than the one obtained with other models reported previously22 (Table 2). The value of the cross-parameter of the adsorbateadsorbate interactions obtained by identification has the same sign and an order of magnitude comparable to the value obtained from the similar fit accomplished using the quadratic model; see Table 1, entry 5. 2.4. Conclusions Regarding the Moreau Models. The cubic Moreau et al. model, which considers also the possibility of triple-cell interactions in addition to doublecell interactions, accounts better for the competitive data than the corresponding quadratic model. The Moreau et al. models are explicit with respect to the mobile phase concentrations of the adsorbates, which is a great advantage for their use in the calculations of nonlinear chromatographic band profiles using the equilibriumdispersive models.2 3. Ruthven Model. 3.1. Correlation of SCD by the Single-Component Isotherm Model. The fit of the SCD of both components to the quadratic version of the Ruthven model (eq 6a) gave only a slight improvement of the correlations obtained previously with the Langmuir model22 (Table 2), as seen in Table 1 (entry 8). The improvement is more pronounced for PP, as obtained previously with the Fowler and the Moreau models. The elimination from the model of the adsorbate-adsorbate interaction parameter for PE did not significantly affect the correlation of the SCD for either PE or PP. For this model, the regression gave a value of the monolayer capacity for PE which is higher than that for PP, an obvious result if one considers that for PE only one molecule is allowed per cage while for PP two molecules can fit on the cage. So, if one considers that the monolayer capacity of PP is twice the identified monolayer capacity of the cages, which accommodate two molecules each, we obtain a higher corrected capacity for PP, a result similar to the one obtained for the Fowler and the Moreau models. The identified values of the LCEC constants are clearly different within the framework of this model. On the other hand, a very good fit of the PP SCD was obtained with the cubic version of the Ruthven model (eq 7a, Table 1, entry 10). The value of the RSS obtained for PP SCD is the lowest one obtained in this study and in the previous one.22 The improvement to the fit of PE SCD is not significant, however, as also observed with the other models previously discussed. We also tried a version of the Ruthven model which considers up to four interacting molecules in a cage, but the improvement in the fit of the SCD was not significant with respect to the fit obtained with the model which considers only binary and ternary interactions in the cages. So, interactions involving a number of molecules greater than three are also not probable in terms of the Ruthven model. 3.2. Prediction of Competitive Adsorption Data Using
5442 Langmuir, Vol. 12, No. 22, 1996
the Parameters of the Single-Component Isotherm in a Competitive Isotherm Model. The prediction of the competitive data using the parameters obtained by identification of the SCD was initially accomplished considering that the binary interactions between the molecules inside a cage are dominated by the PP molecules and that the interaction parameter R1,1 in eqs 6a and 6b can be considered as equal to R02. The results presented in Table 1, entry 8, show that the prediction obtained is worse than the one given by the Langmuir model22 (Table 2). The prediction of the binary data using the parameters of the single-component isotherm, with the condition that the cross-coefficients are considered equal to the binary and ternary interaction parameters determined from the PP SCD, is poor (Table 1, entry 10), worse than the predictions afforded by the competitive Langmuir model (Table 2). 3.3. Identification of the Cross-coefficients of the Mixed Isotherm Using the Competitive Data. The fit of the parameter R1,1 (in eqs 6b and 6c) to the whole set of competitive dat (Table 1, entry 9) resulted in a value of the RSS which was also worse than the one supplied by the purely predictive calculation of the multicomponent equilibria by the Langmuir model (Table 2). If one considers the values of the parameters R11, R21, and R12 as floating parameters, adjustable to the binary data, the value of the RSS obtained (Table 1, entry 11) is also worse than in the case of the Langmuir model.22 3.4. Conclusions Regarding the Ruthven Models. In general, the performance of the Ruthven models was poor in the particular case under discussion. The Ruthven models were derived for stationary phases having cages of nearly constant dimensions and accommodating the same number of adsorbate molecules per cage. The adsorbent used in this study is better described by a pore distribution function, so the parameters obtained are valid for the “cage” of average dimension. 4. Kiselev Model. 4.1. Correlation of SCD by the Single-Component Isotherm Model. The SCD were fitted to the Kiselev models, which consider the formation of associations of either finite or infinite dimensions. Equation 13 was employed for considering the formation of finite size associations having the following maximum dimensions: two, three, four, and five (n ) 2, 3, 4, 5). The results are exposed in Table 1 (entries 12-15). Equation 16 was applied to account for the possible formation of associations of infinite dimension (n ) ∞); the results obtained are also summarized in Table 1 (entry 16). The fit of the SCD of PE is slightly better than the one obtained with the Langmuir model22 (Table 2). The behavior was found to be similar to the one observed with the Fowler model. Thus, in the case of the Kiselev models, consideration of specific lateral interactions in the monolayer improves markedly the fit of the PP SCD, while it does not greatly affect the fit of the PE SCD. A more pronounced improvement was observed for PP, especially with the model which considers the formation of both binary and ternary associations on the surface (n ) 3); see Table 1, entry 13. It is noteworthy to remark on the similarity of this result with the result obtained from the Ruthven model which considers up to three molecules of adsorbate per cage (Table 1, entry 10) and with the Moreau model which considers both binary and ternary interactions on the surface (Table 1, entry 7). Quite unexpectedly and as in the case of the Fowler and quadratic Moreau models, the best fit obtained for the models with n ) 2, 4, 5, and ∞ is achieved with equal values of the LCEC (Table 1, entries 12, 14, 15, and 16, respectively). The only exception to this observation is in the case of the
Quin˜ ones and Guiochon
model with n ) 3, and in this respect the behavior is similar to the one observed with the cubic Moreau model. The values of the parameters Ka1 and Ka2 for n ) 2, 4, 5, and ∞ have different signs but absolute values which are comparable (Table 1, entries 12, 14, and 15). It is interesting to note that, in this respect, the exception is also the model with n ) 3 (Table 1, entry 13). In this last case, the behavior is similar to the one observed with the Fowler model; the absolute value of the interaction constant for PE is lower than that for PP. The two signs are different, as for other models. For the best model (n ) 3), we also carried out the fitting while eliminating the adsorbate-adsorbate interaction parameter for PE (by setting Ka1 ) 0); see Table 1, entry 17. The correlation of the data showed that the RSS of the individual components and the global RSS are practically unchanged. Also for the Kiselev models, the monolayer capacity for PP is higher than that for PE, as was the case for the Fowler, Moreau, and Ruthven isotherms. 4.2. Prediction of Competitive Adsorption Data Using the Parameters of the Single-Component Isotherm in a Competitive Isotherm Model. A purely empirical calculation of the competitive equilibria using the parameters identified from the SCD was done, assuming that the formation of mixed associations is governed by the behavior of the molecules of PP in the adsorbed phase, as described above (i.e., Km ) Ka2). The results are presented in Table 1 (entries 12-17). They show that the model which considers the formation of binary and ternary associations best accounts for the competitive data (Table 1, entry 13). The result is better than the comparable ones obtained with the Fowler model (Table 1, entry 1) and the cubic Moreau model (Table 1, entry 6), although the differences are not large. The result is also better than the one given by the Langmuir model for the SCD and the competitive data of both compounds;22 see Table 2. The elimination of the adsorbate-adsorbate interactions for PE also gives a good prediction of the competitive data (Table 1, entry 17), although the RSS is higher than the one obtained with the Kiselev model which considers these PE interactions (Table 1, entry 13). The Fowler model also gives a value of the RSS which is lower than the one obtained with this modified version of the Kiselev model, although the difference is not significant (Table 1, entry 1). The predictions given by the Kiselev models with n ) 4, 5, and ∞ (Table 1, entries 14-16) are similar. On the other hand, they are worse than the predictions obtained with either the Kiselev model for n ) 3 (Table 1, entry 1) or both versions of the Moreau model (Table 1, entries 4-7). It would be reasonable to assume that the formation of molecular associations of more than three members is difficult in a chemically bonded C18 solution, which constitutes the region where the monolayer is adsorbed. The model which considers only the formation of binary associations (Table 1, entry 12) predicts the competitive equilibria worse than the model assuming n ) 3 and slightly better than the other models (n ) 4, 5, and ∞). 4.3. Identification of the Cross-coefficients of the Mixed Isotherm Using the Competitive Data. If we consider the value of Km for the model assuming n ) 3 as a crosscoefficient for binary interactions and fit its value to the whole set of competitive data, we observe in Table 1, entry 18, that the improvement of the fit is not significant. Other models with cross-coefficients analyzed in this study, like eqs 3a and 3b (Table 1, entry 2), 4b and 4c (Table 1, entry 5), and 5a, 5b, and 5c (Table 1, entry 7), and those tested previously22 (Table 2) account better for the whole set of competitive data. The value of Km identified in such a way has the same sign and order of magnitude as Ka2.
Localized Monolayers with Lateral Interactions
4.4. Conclusions Regarding the Kiselev Models. It is important to remark on the similarities exhibited by the Fowler, Moreau, and Kiselev competitive isotherm models, although the former consider nonspecific lateral interactions and the latter considers specific ones, such as the possible formation of hydrogen bonds between the molecules of the two phenyl alcohols at the surface of the adsorbent. For all these models, the main deviations from the model predictions are observed for mixtures in which the concentration of PP is high and the pattern of Figures 1 and 2 representing the behavior of the Fowler model is close to the patterns depicted by both the Kiselev and Moreau models. The Kiselev models exhibit the same limitations as the Fowler model. These are related with the impossibility of their closed-form inversion with respect to the concentrations in the adsorbed phase. Conclusions The prediction of multicomponent adsorption equilibria using only the parameters of single-component equilibrium isotherms identified by fitting SCD to the corresponding model parameters is a complex task, as shown by the results presented here and by previous studies.6,22 This approach, although highly desirable from an experimental point of view, is complicated and not entirely justified from a theoretical viewpoint. The results reported here and in previous publications are not completely satisfactory, although they are restricted to the specific, relatively simple case of two immediate homologs, taken from the whole set of possible problems encountered in nonlinear chromatography. In this paper, the prediction of the competitive isotherm data was attempted, using models which assume the surface to be homogeneous and consider the possible nonspecific and specific lateral interactions in the adsorbed phase. A new extension of the classical Kiselev model to multicomponent mixtures which considers the formation of both binary and ternary associations as a result of specific lateral interactions was found to account better for the competitive data in a purely predictive way. This result is illustrated by the comparison between the results obtained with this model and those obtained with other models tested here and in previous studies.6,22 The Kiselev model, however, exhibits the same drawback as the Fowler model. It is impossible to inverse it analytically, as required for the calculations
Langmuir, Vol. 12, No. 22, 1996 5443
of the adsorbed phase concentrations in the algorithms used to model nonlinear chromatography,2 and its numerical inversion increases prohibitively the run time of the programs. In this respect, the Moreau et al. models are better, since they are explicit with respect to the stationary phase concentration, thus avoiding the need of a numerical inversion. The models considered in this study exhibited several common features, like (i) close values of the lowconcentration adsorbate-adsorbent interaction parameters of both components; (ii) a greater improvement of the fit of the SCD of PP as a result of the consideration of adsorbate-adsorbate lateral interactions in the adsorbed phase, while a comparable degree of improvement was not observed for PE; (iii) a greater value of the monolayer capacity for PP than for PE; (iv) the impossibility to obtain a much better fit of the competitive data using adjustable cross-interaction parameters; and (v) the great probability that only binary and ternary adsorbateadsorbate interactions take place in the adsorbed phase. Adsorbate-adsorbate interactions are not by far the only source of nonideal behavior which should be taken into account in adsorption studies. Other sources are (i) adsorbent heterogeneity; (ii) differences in adsorbate sizes; (iii) irreversible adsorption; (iv) interactions in the bulk phase; and (v) the combinations of these different contributions.38 Other models which can account for these additional nonideal effects should be tested, such as the multicomponent heterogeneous models with and without lateral interactions39 and the adsorbed solution theory models.40 Acknowledgment. This work was supported by UNDP project CUB/91/001. I.Q. thanks the Laboratoire de Ge´nie Chimique (URA CNRS 192 of INP-ENSIGC, Toulouse, France) and especially A. M. Wilhelm for the support extended to him. He acknowledges Ulises Ja´uregui from CQF (Habana, Cuba) for his comments and for helpful discussions. LA9603333 (38) de Kock, F. P.; van Deventer, J. S. J. Min. Enger. 1995, 8, 473. (39) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (40) Scholl, S.; Schachtl, M.; Sievers, W.; Schweighart, P.; Mersmann, A. Chem. Eng. Technol. 1991, 14, 311. (41) Bakaev, V. Dokl. Akad. Nauk 1966, 167, 369.
