Langmuir 1993,9, 2547-2554
2547
Evaluation of the Energy Distribution Function from Liquid/Solid Adsorption Measurements M. Heuchel, P. Brauer, M. v. Szombathely, U. Messow, and W.-D. Einicke Institute of Physical Chemistry, Leipzig University, Linnbstrasse 2, Leipzig 0-7010, Germany
M. Jaroniec* Department of Chemistry, Kent State University, Kent, Ohio 44242 Received September 30, 1992. In Final Form: February 5,199P Various problem associated with numerical evaluation of the adsorption energy distributionfunction, F(U21), from the experimental excess adsorption isotherms of binary nonelectrolytic liquid mixtures on heterogenoussolid surfaces have been discussed. Severalmethods known in the literature did not address the numerical ill-posed character of the integral equation, which has been used for calculating F(UZ1).In the current work two alternative approaches, which take into account the ill-posed nature of this integral equation,have been presented. In the first approacha regularization method has been employed,whereas in the second one an expansion of the Stieltjes' transformation hae been utilized. Also, the problem of choice of the local adsorptionmodel has been considered. A representation of the overall excess adsorption isotherm in the form of the Mt-plot has been found to be useful for selecting this model. Illustrative calculations have been carried out for various mixtures on the same adsorbent and for one mixture on differentadeorbenta. Suchas methodologicalapproachis especiallyuseful for analyzingdifferencesbetween the systems studied. By use of the regularization method, the distributionfunctionsF(U21)for n-hexanel 1-hexanol,n-hexaneltoluene,and n-hexanolltolueneon an active carbon have been calculatedand compared. Another comparison of the F(U21) functions have been presented for water/ethanol on ZSM-5zeolites of different Si/& ratios.
Introduction In 1973 Rudzinski et al.l formulated the fundamental integral equation for physical adsorption of binary nonelectrolyticliquid mixtures on energeticallyheterogeneous solid surfaces. In this formulation a continuous distriwas assumed; U2 and bution function of U21 = U2 - UI, U1 are respectivelythe adsorption energies of components 2 and 1. The energy distribution function, F(U21), is usually used to characterize heterogeneity effects in adsorption at the liquidlsolid interface. If molecules of both components of a binary liquid mixture occupy identical sizes on the surface, then the totalmole fraction, xan,of component 2 in the surface phase is given as follows
where xzt8 can be calculated from the experimental adsorption excess I'f(xJ) via rze= n8(x$ - xzl)/mA. Here n* is the total number of molecules in the surface phase, mA is the mass of adsorbent, xJ is the mole fraction of component 2 in the bulk phase, T is the absolute temperature, R is the universal gas constant, and x 2 2 = xJ/x$. The expression F(U21) dU21 is the fraction of surface sites with the differences in adsorption energies between U21 and Uzl+ dU21, and therefore the following normalization condition is satisfied:
The function C = C(x~~,xz') takes into account all molecular interactions between molecules in the surface and bulk phases, and it depends on the model of the surface and bulk phases as well as on the topography of adsorption Abstractpubliehedin Advance ACS Abstracts, August 15,1993. (1)Rudziiki, W.;Oe.cik, J.;Dabrowski,A.Chem.Phys.Lett. 1973,20, 5.
sites on the solid surface. If a lattice model is used to describe molecular interactions in both phases (i.e., the number of nearest neighbors, c', and the interaction energies between these neighbors, ellr, c22', and €12~for r = 1, s),then the function C can be represented by the ratio of the molecular partition functions for the surface and bulk solutions. The molecular partition function for the surface solution depends additionally on the distribution of adsorption sites on the surface (surface topography), which can be represented by two extreme models: (i) random distribution approximation (RDA) and (ii) homotattic patch approximation (HPAlq2The C function can be represented by3
where As,A', g",and g1are defined in Table I and specified for different models of the surface and bulk phases. From the mathematical point of view eq 1 is a linear Fredholm integral equation of the first kind, which can be written in a more general form as follows: gW = JabK(xy) f ( x )
dx
(4)
The integral kernel K(x,y) represents the local isotherm from eq 1,i.e., the physicochemical model of adsorption on the surface sites of the same energy, whereas sty) in eq 4 is a known function, which can be obtained from the measurable excessadsorptionisotherm. Many approaches have been proposed to calculate f ( x ) =- F(U21) from the liquid/solid excess adsorption isotherms.2J These approaches have been developed by extension of the numerical methods for solving the adsorption integral equation for gaslsolid adsorption. The earliest approaches are based on the so-called "condensation approximation" (2) Jaroniec, M.; Madey, R. Physical Adsorption on heterogeneolur Solids; Elsevier: Amsterdam, 1988. ( 3 ) Heuchel, M. Ph.D. Thesis, University Leipzig, 1988.
0743-7463/93/2409-2547$04.00/00 1993 American Chemical Society
Heuchel et al.
2548 Langmuir, Vol. 9, No. 10, 1993 Table I. Parameters in Equation 3 Characterizing Nonideality of Bulk and Surface Phaees. model
by means of the inverse operator, A-l f = A-lg (6) Usually the experimental quantity g is measured with an error [, i.e.
expression
QCA
g=gwe+t 81 = [l = 4x&-
thermodynamic definition
x & l - e201/kT)11/2
f = A-lg,,,
71
1
[
BWA, RDA QCA, RDA
Since operator A-' is linear, substitution of eq 7 into eq 6 gives
1 d~1)-1 ??.,yw"L4kT I
A,? q~~~e-@(*-t11')/2kT Surface Phase (8) g = l(w' = 0) ideal g t I e-l-b$)/kT BWA, HPA QCA, HPA x$(P + 1 - 2%;) gi' = (1- x z i 8 ) ( f l i ' - 1 + 21czi3) li2
p,? = [l- 4 x d ( l -x$)(l -e*/kq11/2 g e-cWl-bna)/kT x2t'(P+ 1- 2 x d ) I
1
g'= [(l - "$)(@t'- 1+ 2rz,')
St' = [I - 4rzt8(l- xzr')(1 - e-/kT)]1/2 a Abbreviations: BWA, Brag-Williams approximation; QCA, quasi-chemical approximation; RDA, random distribution approximation; HPA, homotattic patch approximation; ppl, difference in the standard chemical potentials of components 2 and 1in the bulk phase; 921 = qZ/p1, ratio of molecular partition functions, w = e12 (ell + en)/2, yj, activity coefficient of component; x$, local mole fraction of component 2 a t sites j .
(CA-method)introduced by R~ginskij.~ The F(U21) functions determined with this method are often used in other iterative methods as a first approximation of the energy distribution.s$6In calculating the F(U21) functions, many authors have assumed a priori its shape, e.g. quasiGaussian.7J With such analytical functions it is possible to carry out integration in eq 1analytically for some local adsorption isotherms (e.g., Langmuir-typeisotherms). The final equations for the overall excess isotherm contain free parameters, which can be evaluated by a best-fit procedure with minimization of the sum of least squares. The knowledge of these parameters permib calculation of the distribution function F(U21) according to the a priori assumed analytical expression. The above described analytical methods have serious disadvantages. In general, it is not known whether the assumed shape of F(U21) is the correct one. Also, many different analytical functions can describe a given experimental isotherm with a similar accuracy. This is associated with the ill-posed nature of the integral eq 1;that means that the small changes in xzts caused by experimental errors can influence the function F ( U d significantly.To discuss this problem eq 4 should be rewritten as a linear operator equation g = Af (5) where the experimental quantity g x ~ can s be formally represented as the result of action of the linear operator A (representing a model of the local adsorption) on the unknown function f = F(U21). Then, f can be evaluated
(7)
+ A-'[
(8)
For t = 0, an unambiguous relation exists between f and gbe. In the case of # 0 (whichis the case in experimental situations), the term A-l[ in eq 8 represents distortion of f by the experimental error [. Also, solution f can be perturbed by errors generated in numerical calculations, e.g., by the quadrature of the integral eq 1. The above described uncertainties in evaluation off have numerical nature. Additional uncertainties in evaluating f are associated with the assumption of the local adsorption model. Since the operator A represents a local adsorption model, which is assumed a priori, there exists an additional problem to decide whether the assumed A is in accordance with the experimental observations. Solution of numerical unstable problems goes back to the pioneering works by Tichonow?Jo who introduced the regulatization method. Application of this method to gas adsorption was first done by House" and Merz.12 Further contributions to this field have been done by Papenhuijzen and Koopal13 (to our best knowledge, their work is the only one, which presents application of the regularization method to the liquidlsolidadsorption) and Brown et al.14J5 A detailed study of the regularization method with a singular value decomposition was carried out by v. Szombathely,16who also elaborated the generalnumerical program INTEG for inverting eq 4 with respect to the distribution function f ( x ) . First, this method was applied successfully for calculating the energy distribution from gas/solid adsorption data.17 This program can be used for all types of the local isotherms. In the current paper, the program INTEG has been adapted to calculatetheF(U21)distribution from the liquidholid excess adsorption isotherms. Using this program, we calculated and compared the F(Uzl)-distribution functions for adsorption of three different liquid mixtures on the same active carbon and for one liquid mixture on isostructural ZSM-5 zeolites of different Si/A1 ratios. In addition, the regularization method has been compared with a method based on an expansion of the Stieltjes transformation. This method, applicable for Langmuirian local isotherms, has been formulated here for the liquid/solid adsorption by ex(9) Tichonow, A. N. Dokl. Akad. Nauk SSSR 1943,39,196; 1963,163, 49; Sou. Math. 1963,4, 1035, 1624. (10) Tichonow, A. N.; h e n i n , V. Ja. Solution Methods for ill-posed Problem (in Russian); Nauka: Moscow, 1979. (11) House, W. A. J. Colloid Interface Sci. 1978, 67, 166. (12) Merz, P. H.J. Comput. Phys. 1980, 38, 64. (13) Papenhuijzen, J.; Koopal, L. K. In Adsorption from Solutions;
Ottewill, R. H., Rochester, C. H., Smith, A. L.,E&.; Academic Press: London, 1983; p 211. (14) Britten, J. A.; Travis, B. J.; Brown, L.F. AIChE Symp. Ser. 1983, 79. 1., _ __.
(4) Roginskij, S. S. Dokl. Akad. Nauk SSSR 1944,45,61, 194. (5) Jaroniec, M. Adu. Colloid Interface Sci. 1983, 18, 149. (6) Jaroniec, M.; Br&uer,P. Surf. Sci. Rev. 1986, 6, 65. (7) Dabrowski, A.; Jaroniec, M. Mater. Chem. Phys. 1986, 12, 339. (8) Rudzinski,W.; Lajtar,L.; Zajac, J.; Wolfram, E.; Pazli, I. J. Colloid Interface Sci. 1983, 96, 33.
(15) Brown, L. E.; Travis,B. J. InFundamentaLp ofddsorption; Myere, A. L.,Belfort, G., Eds.; American Institute of Chemical Engineers: New York, 1984, p 123. (16) v. Szombathely, M. Ph.D. Thesis, University Leipzig, 1988. (17) v. Szombathely,M.; Brtiuer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17.
Adsorption Energy Distribution Function
Langmuir, Vol. 9,No. 10,1993 2549
tending the EDCAIS algorithm elaborated previously for the gaslsolid adsorption data.18
0.50,
I
Numerical Procedures for Evaluating the F(U21) Distribution Regularization Method. The regularization method requires discretizationof the integral eq 4 by a quadrature. The integral eq 4 has to be transformed to the following system of linear equations" g=Af (9) where g and fare one-dimensional matrices representing respectively the functions g and f and A is the twodimensional matrix representing kernel K(x,y). Regularization consists of replacing the ill-posed problem of minimizing the functional llAf - gJI2by a well-posed one, which smoothes the distribution function and distorts it insignificantly. It can be done by addition of a second term to the minimization functional Min = llAf - AI2 + -yll)W(f)l12 (10) The regularization parameter y is a measure for the weighting of both terms. Usually, llW(f)112 is defined by one of the following expression^:^^*^^
0 . 5 -
0.00
0.25 0.50
0.75 1.00 1
Mole Fraction, x2
Figure 1. Overall excess adsorption isotherm (solid line) generated for two Gaussian peaks (dotted lines) shown in Figure 2. Points represent the overall isotherm with 5% random error.
by the following expression:21 1
p(t) = s [ q ( t
- i r ) - q(t + i r ) ]
(18)
The function p ( t ) can be expanded in a series
(11)
F(U21) can be obtained from the function p ( t ) as follows 1
F(U2,) = "U21)1
(20)
where The parameter y could be chosen either a priori or determined with an automatic strategy. In the current work, the regularizationmethod has been studied by using the program INTEG developed by v. Szombathely.lB This program employs the singular value decomposition (SVD) of the matrix A representing the discretized operator A in eq 5. Combination of regularization and SVD posses the following advantages: (i) minimizes numerical errors through application of the orthogonaltransformations, which are numericallystable; (ii)allows simplication of the numerical algorithm already on the analytical level by making possible only one calculation of SVD and, after that, a fast optimization of the final result by choosing different regularization parameters. Method of Stieltjes Transformation. Let us define the new variables t and z K = et + Kmh;
[-I-'
= e'- K,
(14)
where K = exp(U21lRT) and Kmin = exp(U2iminlRT). Expression of eq 1 in terms of the variables t and z gives
where q ( z ) and p ( t ) are given by q(z) = xJx&))
(16)
p ( t ) = RTF[U,,(t)I
(17) The distribution function p ( t ) in eq 15 can be calculated (18) Briluer, P.; Fassler, M.; Jaroniec, M. Thin Solid Films 1986,123, 245; Chem. Phys. Lett. 1986,125, 241. (19) Golub,G. H.; Reinsch, C. Numer. Math. 1970, 14, 403. (20) Eckhart, U. Computing 1976,17,193.
t(u2,)= ln[eWRT - e"2-/RTl
(21) Extensive numerical studies18have shown that the series given by eq 19 converges rapidly; consequently the first three terms of eq 19 are sufficient for calculating F(U21). Brauer et al.18 used the smoothing spline functions for describing the experimental functions q(z) and therefore the abbreviation EDCAIS (energy distribution computation from adsorption isotherms utilizing smoothing spline functions) was used for their method. The ill-posed character of the problem has been considered by specification of a smoothing parameter. Numerical Testing of the Regularization Method. Figure 1 shows an excess isotherm of a hypothetical binary liquid mixture adsorbed at 300 K on a solid surface characterized by an adsorption energy distribution with two isolatedGaussian peaks (seethe dotted lines in Figures 2 and 3). The solid line denotes the isotherm calculated from eq 1 with C = 1 by integration with 5 digits accuracy. The points in Figure 1 correspond to a quasi-experimental isotherm with a mean random error equal to 5 % of the maximum value of the adsorption excess. In Figures 2 and 3 are shown, along with the true adsorption energy distribution function, the reproduced distributions calculated with the EDCAIS and INTEG algorithms. In Figure 2 these distributionswere calculated from the exact isotherm data, whereas Figure 3 shows results calculated from the data taken with 5 % error. Already in the case of the "exact" data, the resolution of two peaks is worse when EDCAIS is used and the regularization method (INTEG)gives better results. The EDCAIS distributions show stronger oscillations than those calculated by the INTEG program. Therefore, further calculations have been performed by using the regularization method. The (21) Titchmarsh, E. C. Introduction to the Fourier Integrals; Oxford University Preea: London, 1959.
2550 Langmuir, Vol. 9,
5
n
fl
B 031
I
Heuchel et al.
No.10,1993
-
-0.1 -10
-5
5
0
10
Energy Dimerenee, V,, [kl/mol]
Figure 2. Comparison of the energy distributions F(&) calculated by using INTEG (solid line) and EDCAIS (dashed line) programs with the true distribution (dotted line). These distributionswere calculatedfrom the exact isotherms(solidline in Figure 1). 0.4
:
-
-0.1 -10
-5
Energy DiEeren-
5
0
10
V, [kl/mol]
Figure 3. Comparison as in Figure 2 of the F(Uzl) functions calculatedfrom theexceasisothermwith5% randomerror (points in Figure 1).
program INTEG has been tested also for theoretical adsorption isotherms data used by Papenhuijzen and K00pal.l~
The Choice of Local Isotherm The local isotherm represents the physicochemical model of adsorption on sites of the sameadsorption energy. This model should describe the real situation as well as possible in order to obtain physically realistic information about adsorbent heterogeneity. Since the local isotherm is assumed a priori, any criterion facilitating ita selection is desirable. In the theory of the gas/solid adsorption, two criteria have been proposed: (i) the “concavity criterion” by Men12 and (ii) the “Mt-plot” by Briiuer et al.22p2sIn this paper, the Mt-plot concept has been extended to the case of adsorption from solutions on solid surfaces. All models of liquid/solid adsorption with C-functions given by eq 3 and parameters from Table I can be represented by the following integral equation
where (23) Briiuer et 81.2012~suggested the empirical function Mt@) y
~~
~
x2;c(X;,xJ ~
(22) Briuer, P.; Dunken, H.; Hiifer, R. Wies. Z . Wedrich-SchillerUniv. J e w , Math.-Naturwiee. Reihe 1981,30, 505. (23) Brliuer, P.; Jaroniec, M. J. Colloid Interface Sci. 1986, 108, 60.
I
heterogeneous
Variable Iny
Figure 4. comparison of the Mt plots for homogeneous and heterogeneous surfaces.
to estimate qualitatively the energetic heterogeneity of an adsorbent from experimental gas adsorption isotherm. Extension of this function to the liquid/solid adsorption is
The mathematical properties of Mt are given in the appendix of ref 23. Figure 4 shows general behavior of the function Mt@). For homogeneous surfaces In Mt@) is constant, but for heterogeneous surfaces it is a monotonic decreasing function. It should be emphasized here that such a plot provides information about possible heterogeneity only if lateral interactions in the surface and bulk phases are correctly incorporated into the interaction function C = C(xJ,xzs). The generalized function Mt is useful to carry out an initial analysis of experimental data in order to obtain information about energetic heterogeneity of the solid surface. This analysis can be done as follows: First, In Mt should be plotted as a function of In y with y = a&a?(C = yJ/y?) by using smoothed experimental data. Here, a) and y)are respectively the activity and activity coefficient of the ith component in the bulk phase. With y defined as azl/a? (all interactions in the bulk phase have been taken into account) the shape of the Mt function depends on the adsorbent heterogeneity and on the lateral interactions in the surfacephase. In the appendixof this paper, the influence of lateral interactions in the surface phase on the local adsorption isotherm is discussed in detail. Figure 5 shows selected curves of In Mt vs In (aJ/a?) for systemswith lateral interactions of positive [w8> 0 where wB= ~ 1 -20.5(tnB ~ + ~ 2 2 ~ and ) l negative (w8< 0) deviations from Raoult’s law. For the surface phase with positive deviations from Raoult’s law, attraction between molecules of the same kind is stronger than that between molecules of different kind. If the Mt plot with y = aJ/aJ does not give lines like both curves in Figure 4 (e.g. a decreasing function) one should select a model for the local isotherm, which takes into account lateral interactions. It is recommended to try, at first, the local isotherm basing on BWA and HPA. Figure 5 shows that special attention should be paid to interpretation of the Mt plot with w8 < 0, i.e. binary mixtures having negative deviations from Raoult’s law. Then, the Mt plot for a homogeneous surface looks like the typical curve in Figure 4 for the “heterogeneous” case, and a distinction of heterogeneity and interaction effecta on the basis of the Mt plot is difficult. An application of the Mt plot to experimental examples is shown later.
Adsorption Energy Distribution Function I ?
Langmuir, Vol. 9, No.10, 1993 2661 Table 11. Adrorption Parameters for Three Liquid Mixturw on Active Carbon R2S hexanol(l)/ hexane (1)/ hexane (1)/ toluene (2) toluene (2) hexanol(2)
I
I
w>o we0
we0
n1,' ~mmoVg1 nB'
b"mUg1
t d
[kJ/moll
en' [kJ/moll t d
[kJ/moll
w' [kJ/moll
Variable hy
Figure 5. The Mt curves for different adsorption modela calculated accordingto equations given in Table I. These curvea were plotted for the nonideal bulk phase and ideal surfacephase formed on a solid with random topography, Le., y = uJ/a?.
3.00 3.87
3.37 3.87
3.37 3.00
-9.40 -6.10 -7.49 0.26
-6.30 -6.10 -6.68 0.12
-6.30 -9.40 -7.06 0.30
C'
6
6
6
r m E (Points)
0.30 (30)
0.09 (62)
0.28 (48)
4
2
1,51
hexane(l)/toluene(2)
1
-
20
hexanol(l)/toluene(2)
Y
10
Interpretation of Adsorption Energy Distribution Functions The calculated distribution function depends on the a priori assumed model of local adsorption on surface sites of the same energy. In the case of adsorption at the liquid/ solid interface the exchange of molecules of components 2 and 1between the surface and bulk phases takes place. During this exchange process the energetic states (rotation, vibration) of molecules change and these changes should be incorporated into the local adsorption model. Since selection of a proper local adsorption model is difficult, there is an uncertainty in the evaluated distribtion function F(&I). Therefore, evaluation of the heterogeneity effects from the liquid/solid adsorption data should be done under special conditions in order to eliminate the above mentioned uncertainty, e.g., (i) comparative analysis of differences in the F(U21)functions for various mixtures on the same adsorbent or (ii) analysis of relative changes in theF( U21)functions for the same liquid mixture on various samples of solid obtained via different surface modifications (e.g., temperature treatment, chemical modification, etc.). Experimental Examples Binary Mixtures of Hexane, Toluene, and Hexanol on Active Carbon R23. First, the distribution functions
F(U21)calculated from the excess adsorption isotherms of n-hexane (l)/l-hexanol(2),n-hexane (l)/toluene (2), and 1-hexanol (l)/toluene (2) on active carbon R23 at 273 K will be compared. Experimental details concerning these systems are given in ref 24. Active carbon R23 is a microporous solid with the BET specific surface area of 780 m2 gl.Its micropore volume is equal to 0.47 cm3g-l. Elementary analysis gives 88.9% of C, 3.9% of H, 1.3% of N, 0.1% of S, and 6.7% of 0. Table I1 contains the surface phase capacities for toluene, 1-hexanol, and n-hexane on this active carbon. The experimental excess adsorption isotherms are shown in Figure 6. Except for very high hexanol concentrations,hexanol is preferentially adsorbed in comparison to n-hexane. Also, toluene is adsorbed stronger than n-hexane. The hexanol/toluene mixture shows an interesting behavior. At low toluene concentrationstoluene is adsorbedpreferentially,whereas at the high concentrationrange hexanol shows preferential adsorption. The wetting enthalpies of toluene, n-hexane, and 1-hexanol on the R23 carbon are respectively -26.4, (24) Kind, B.Diploma Thesis, University Laipzig, 1988.
' . -0.5 0.00 0.25
0.50
0.75
1.00
-0.5
000
0.25
Mole Fraction, x i
-0.5
OW
025
050
075
0.50
0.75
1 00
Mole Fractlon, x i
100
Mole Fraction, xz1
Figure 6. Experimental excesa adsorption isotherms for n-hexane (l)/toluene (2), n-hexane (l)/l-hexanol(21, and n-hexane (l)/l-hexanol(2) on active carbon R23 at 273 K . M
-24.8, and -37.6 kJ/mol.= According to these values the stronger adsorption of both toluene and 1-hexanol in mixtures with n-hexane is expected. The next step of analysis of the above experimental isotherms was construction of the Mtplots in order to get information about the local adsorption model. The Mt curves (lines without points in Figure 7) calculated from the smoothed experimental isotherms withy = u&/a$,i.e. with lateral interaction in the bulk phase only, show a drastically different behavior than that predicted by the model studies presented in Figure 5. Therefore, the Mt curves (lines with points) presented in Figure 7 were construct8d by also taking into account lateralinteractions in the surface phase according to the BWA-RDA model (y = A@g%J/a$).These curves are physically realistic but still show a compensation effect: their initial negative slope may be caused by energetic heterogeneity and further increase in the Mtvalues can be attributed to lateral interactions with a positive deviation fiom Raoult's law (tu8 > 0). Since all three bulk m i x t w s show positive deviation from Raoult's law, positive deviation in the surface phase is expected. The adsorbata-adsorbate interadions in the lattice model (BWA)were incorporated by using the interaction energies elf, c d , and €12. in the (25) Kohler, F.Monotsh. Chem. 1967,88,857.
Heuchel et al.
2562 Langmuir, Vol. 9, No.10,1993 0.8,
1
heunolholuene b hexmettoluene
c hexmetheunol
-0.2
a
-2.0
-1.0
0.0
1.0
2.0
Variable Lay
Figure 7. The M Icurves calculated from the excess adsorption isothermsshown in Figure 6 with (lineswith point4 and without (lines)the surface-phaselateral interactionsincorporated to the variable y.
-20 -10
0
10
20
Energy Difference, V, [kl/mol]
Figure 8. Energy distributions F(U21) characterizing heterogeneity of active carbonR23 with respect to three differentliquid mixtures calculated by INTEG. surface layer and the number of nearest neighbors given in Table 11. The interaction energies ells and ~ 2 were 2 ~ determined according to the method by Kohlerw from the heats of evaporation and e2pwas estimated from the excess free enthalpy of the equimolar mixture. A determination of the number of nearest neighbors,cs,is difficult. It should be smaller than that in the bulk liquid. We used c' = 6 for all three systems. Figure 8 shows the F(U21) functions calculated with INTEG by assumingBWA for the surface phase and taking into account the activity Coefficients in the bulk phase at T = 273 K. The regularization parameter y was the same for each system. As criterion for finding the optimal regularization parameter, the root mean square of the experimental data was used (cr. Table 11). The parameter y was changed until the calculated root mean square was in the range of the experimental one. If assumptions concerning interactions in the surface phase are correct, the following information can be obtained from theF(U21) functions: For n-hexaneltoluene mixture the mean value of U21 is 6 kJ/mol,it indicates that toluene is preferentially adsorbed. For n-hexanell-hexanolmixture the mean value U21 is about 15 kJ/mol and it indicates preferential adsorption of l-hexanol. The same tendency is observed for l-hexanoVtoluenemixture. The calculated distribution function shows a shoulder at -11 kJ/mol and a maximum at -7 kJ/mol. These values indicate that at low concentrations hexanol is more strongly adsorbed than toluene. (26) Einicke, W.-D.; Heuchel, M.; v. Szombathely, M.; Briiuer, P.; Sch6llner,R.; Rademacher, 0.J. Chem.SOC.,Faruday Tru~.11989,86, 4217.
The maxima of the F(U21) functions satisfy the thermodynamic consistencytest, i.e., (Uh-1- Uheme) + (Ut,,luene - Uhewol) - (Ut,,lue,e - U h e w e ) = I, which is below 10% of the highest value of the maximum. A comparison with the experimental heats of adsorption could give some information about accuracy of the method. The adsorption energy differences calculated from the calorimetry enthalpydata Uh-1- & m e = +12.8 (+15) kJ/mol, (Ut,,lum - &mol) = -12.2 (-9) kJ/mol, and (Ut,,luene- Uherane) = +0.6 (+6) kJ/mol. The values given in parentheses represent maxima or center in the F(Uzl)functions,which show a satisfactoryagreement with the calorimetricvalues. By changing slightly the lateral interactions in the surface phase or the parameter c8, it would be possible to change position of the F(U21) functions and improve agreement between the adsorption energy data. Water/Ethanol on ZSM-5 Zeolites with Different Si/ A1 Ratio. Another experimental illustration has been done for water/ethanol adsorption on different zeolites. Experimentaldata were published in ref 26. An advantage of these zeolites is the knowledge of their structure from X-ray diffraction patterns. The pentasil zeolites studied are isostructural. They differ only in their Si/Al ratio and in the sodium content per unit cell. The aluminum free member of the pentasil series, the so-called silicalite, can only interact with molecules of the mixture via dispersive forces. Introduction of aluminum atoms into unit cell increases the possibility of specific interactions and pentasil becomes more energetically heterogeneous. Figure 9 shows the excess adsorption isotherms for watedethanol on pentasil zeolites of different Si/Al ratio. For each zeolite studied, ethanol adsorbs preferentially in the whole range of concentrations of ethanol in water. For small Si/Al ratios the excess isotherm has the type I11 according to Schay-Nagy classification,and with increasing Si/A1ratio its shape changes gradually to type I. This change in the shape of the water/ethanol excess isotherms for zeolites with different Si/& ratios already indicates that energetic heterogeneity of these zeolites decreases with increasing Si/Al ratio. The Mt plots confirm this observation. Figure 10presents theMt curves for all water/ ethanol excess isotherms calculated from nonsmoothed experimental adsorption data. These curves were plotted by incorporating lateral interactions in the surface and bulk phase to the variable y. This variable was calculated according to expressions given in Table I by using the bulk phase activity coefficients and the following surfacephase parameters: cs = 3, e l l s = -20.1 kJ/mol, e22' = -23.0 kJ/mol, and €12~ = ( ~ 1 1 % 2 2 ~ ) ~ Although /~. the Mt curves calculated from the nonsmoothed isotherms overlap, it is possible to see that they show a tendency to decrease with increasing y. At the high values of y their behavior does not agree with the model studies shown in Figure 5, and this range of data can be omitted in calculation of F(U21). Figure 11shows the Mt plots calculated from the smoothed isotherms in the range of In y up to -0.5. Analysis of the Mt plots presented in this figure shows a gradual change in their shape with increasing Si/Al ratio. For small values of the Si/Al ratio the Mt curves are decreasing, and this decrease indicates a relatively strong energetic heterogeneity of zeolites with high content of Al. When the Si/Al ratio increases, the initial part of the Mt plot still decreases (although this decrease becomes smaller and smaller), whereas the other part of this plot increases slowly. This increase in the Mt plot for samples with the high Si/& ratio indicates that the contribution of the lateral interactions to the global nonideality of the surface phase
Langmuir, Vol. 9, No.10, 1993 2553
Adsorption Energy Distribution Function 2.51
0
.........................
3
2.0
B
...................
..........
Y
Y
'tr' PJ
2.5,
I
Y
UPI
k
f
0.5
0. 0.00
1 Mole Fraction, x2
1 Mole Fraction, x2
0.25
0.50
0.71
1.00
Mole Fraction, x21
,. . . . . . . . . . . .
6 1.1 U N
k
1.0
. . . . .:. . . . .
..........
...
.....
...
w 0.0 0.00
0.25
0.50
0.75
1.00
Mole Fraction, x21
0.
0.00
0.25
0.50
0.75
1.00
Mole Fraction, x21
Figure 9. Experimental excess adsorption isotherms for water (l)/ethanol(2)on ZSM5 zeolites of different Si/AI ratio at 273 Kaas (AlOd)-Na+ with water molecules is stronger than with ethanol, and the peak attributed to these interactions should lie in the negative range of V2l. This could be fulfilled with the assumption that these interactions take place inside channel pores. The position and height of the right peak is nearly independent of the Si/Al ratio of the zeolites studied and can be attributed to the difference in dispersive interactions of the zeolitic oxygen walls with the components of the mixture.
-0.0
-2.0
-1.o
0.0
Variable h y
Figure 10. The Mt curves calculated from the nonsmoothed excess isotherms shown in Figure 9 by wing y = adla?,Le., only the bulk-phase lateral interactions incorporated to the variable Y.
I
SUM = 13
t
0'5 O.OL8
" " "
-2.0
'
-1.o
I
'
" J
0.0
Variable Iny
Figure 11. The Mt curves aa in Figure 10 but calculated from the smoothed excess isotherms for In y below -0.5. becomes comparable with the effect of adsorbent heterogeneity. Figure 12showsthat for the greater A1content in pentasil zeolite the peak at the lower Uzl values is higher. This peaks reflects the difference in interactions of ethanol and water with the zeolitic dipoles (AlO*)-Na+. From a theoretical point of view, interaction of the dipoles
Conclusions The current work is the first contribution utilizing the presently most advanced numerical methods for evaluating energeticheterogeneity of solids from the excess adsorption isotherms. Since the heterogeneity effects in competitive adsorption at the liquid/solid interface are smaller than those in the gas/solid adsorption, their evaluation requires a very good numerical method. For example, the model studies showed that the INTEG program basing on the regularization method gives better results than the EDCAIS algorithm. Also, presentation of the Mt plot for some model systems showed that this plot is useful for initial analysis of the excessadsorption data. This analysis can give some suggestions concerning the local adsorption model as well as the range of adsorption data, which can be used for evaluating the energy distribution function. The Mt analysis indicates also difficulties in interpretation of adsorption systems with adsorbent heterogeneity and/ or negative deviation of the surface solution from Raoult's law, because both factors cause decreases in the Mt plot. Finally, it is shown that the energy distribution F(V21) is especiallyuseful for comparing energetic heterogeneities of different solids with respect to one selected liquid mixture, and for studying the influence of absolute heterogeneity of a solid on its interactions with different liquid mixtures. In such cases uncertainties in evaluating F(U21)are analogous for all systems studied and then the F(U21) functions can be compared in order to detect differences in heterogeneity effects on these systems. Appendix Figure 5 shows the curves lnMt vs ln(aJla2) for different models of adsorption, where x2ts is given by the following
Heuchel et al.
2554 Langmuir, Vol. 9, No.10,1993
B
0 3 . . . . . . . . . .. . L
Y
02
. . . . . . .:. . . _:. :, . . . .. . . . . . . . .
R
0.1
. . . :. . . . . . . . .
l a .
g
.....................
...........
.
.
n
I
.
.....
, ,
. . . .
. .
0
0
.
.
-0.1 -15 -10
-5
,
.
0
.
.
5
10
I
,
15
-0.1 -15 -10
.
,
.
,
-5
0
5
10
15
U, [kJ/moll
0.4
p
Y
-0.1'
-I5
' -10
' -5
' 0
'
5
'
10
J
15
.......
........
0.3 . . . . . . . . . . . . . . . . .
-0.1' -15
' -10
ua, [kJ/moll
' -5
'
0
'
5
. I . .
' 10
.
'
15
U ,[kJ/mol]
Figure 12. Energy distributionsF(&) for isostructural ZSM of different Si/Al ratio calculated by INTEG. C HomogeneousSurfacewith Lateral Interactions According to BWA. For this model k = 1andgl*is given in Table I. The Mt curve is given by M~= ~ 8 ~ ~ 4 1 - 2 W (27)
and a22 = A1a&all. In eq 25 hj is the fraction of adsorption sites of type j . If we assume q21f = 1then all Af = As = constant. To get an idea what can be expected from the Mt analysis, the Mt plots are discussed for five adsorption models denoted from A to E. A: Homogeneous Surface without Lateral Interactions. In this case k = 1,hl = 1,and gls = 1. Then Mt = AaK1and it is independent on a22. Thus, the Mt plot is represented by the line parallel to the x axis. B: Heterogeneous Surface without Lateral Interactions. For this model k > 1 and gf = 1,where j = 1, -., k. The Mt plot is monotonously decreasing. If 0 Ef-lhju. defines the mean value, then this plot satisfies the fodowing conditions: lim M~= KA" an-
-
-
1 lim M t = az+(KAS)-~
(26)
The slope d In Mt/d In a211is always 10. Its limits for xJ 0 and xJ 1are zero. Existance of both limits of this slope can be used as a criterion for completeness of experimental data. If the initial limit is not reached, then information about stronger sites, preferentially occupied by component 2, is not available. The lack of the final limit indicates that information about strongest sites with respect to component 1is missing. Also, if the adsorption energiesof various sites differ significantly, then one could expect the existence of steps on the Mt plot. Calculations of the Mt plots for gas adsorption showed23 that these steps can be more visible at very low temperatures.
where E* = cawa/(kT). The limites of this plot are lim Mt = ABKe+* aalQ
lim Mt = A'Ke"
(28)
a214-
The shape of the Mt plot depends on the sign of E*. For negative deviation from Raoult's law (e* C 01,the Mt plot decreases, whereas for positive deviation (E* > 0) it increases. The slopes of the final parts of the Mt plot are equal to zero. Special attention should be paid to the case with negative deviation from Raoult's law. In this case the Mt curve behaves similarly as for the model with heterogeneous surface and then it is difficult to make a distinction between effects of lateral interactions and adsorbent heterogeneity. D: HeterogeneousSurface with HPA Topography and Lateral Interactions According to BWA. The limits of the Mt plot are lim M~ = ABKe+* az+J
e" lim Mt = - (29) aZ+' (ASK)-'
The slopes of the initial and final parts of Mt are zero. Analysis of the Mt plot shows that In Mt = f ( l n az2) decreases for E* < 0, but in the initial and final ranges the positive slopes could occur also (see ref 23). E: Heterogeneous Surface with RDA Topography and Lateral Interactions According to BWA. Discussion of this model gives the same results as for model D, but in this case the positive slopes of the initial and final parts of the Mt curves are very small and practically not visible.
