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Effects of surface heterogeneity in adsorption from binary liquid mixtures. I. Adsorption from ideal solutions
Effects of Surface Heterogeneity in Adsorption from Binary liquid Mixtures I. Adsorption from Ideal Solutions J. OSCIK, A. D A B R O W S K I , AND M. J A R O N I E C
Department of Physical Chemistry, Institute of Chemistry UMCS, 20031 Lublin, Nowotki 12, Poland AND
W. R U D Z I N S K I
Department of Chemical Engineering, Queen's University Dupuis Hall, Kingston, Ontario, Canada Received March 31, 1975; accepted January 6, 1976 A new, theoretical description of adsorption from ideal liquid binary mixtures on patchwise heterogeneous solid surfaces has been proposed. The overall, individual adsorption isotherm was approximated by an exponential equation of the Dubinin-Radushkevich (DR) type. The customary DR and Freundlich equations, which were earlier used for description of adsorption from solutions on heterogeneous solid surfaces, appeared to be special cases of the exponential adsorption isotherm. Using the method of Stieltjes transforms, distribution functions of the difference of adsorption energies of liquid mixture components were found, which corresponded with the equation of the individual isotherm. Also an independent method for determination of the total number of moles in the adsorbed layer has been proposed. The theoretical considerations have been illustrated by using the data available from the literature for seven different adsorption systems. The results obtained were compared with those obtained on using the Everett isotherm equation of the model for a perfect adsorbed monolayer. INTRODUCTION Theories of adsorption from liquids are made difficult by the chemical heterogeneity and structural irregularity of the solid surface as well as the fact at least two components are involved in competitive adsorption at the solid surface (1). To avoid this difficulty in the majority of papers concerning adsorption from liquids on solid surfaces it was usually either assumed that the surface of the adsorbent was homogeneous, or the problem was given no attention. I n m a n y papers, however, deviations of experimental results from those theoretically expected have been associated with the heterogeneous character of a solid surface. Schuchowitzky (2), Hansen (3), as well as Delmas and Patterson (4) and Si~kova and
Erd6s (5, 6) saw in the surface heterogeneity of the adsorbent a source of imperfection of surface phases. T h e y suggested that the change of the sign of excess adsorption isotherms should be associated with surface heterogeneity. Their suggestions were supported by the studies of Coltharp (7), which showed that heterogeneity of the adsorbent surface could be the cause of preferential adsorption of one of the components of binary liquid mixture. The number of qualitative and semiquantitative descriptions known in the literature (7-12) that take into consideration the effect of surface structure on adsorption from solutions is unfortunately small. Hence little information is available on the quantitative estimation of heterogeneity of the adsorbent 403
Copyright ~) 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
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404
OgCIK ET AL.
surface and its influence on the adsorptive properties. Most realistic descriptions of heterogeneity effects of surfaces were initially developed in the theory of adsorption from the gaseous phase. Halsey and Taylor (13), in a paper of fundamental significance, proposed an integral equation for the overall adsorption isotherm on heterogeneous surfaces. A basic role was also played by the papers of Sips (14, 15). On the basis of this fundamental work, a number of theoretical approaches were developed by means of which heterogeneity of the adsorbent surface in adsorption from the gaseous phase can be quantitatively evaluated (16-22). The first quantitative attempts to describe the mechanism of adsorption from binary solutions on heterogeneous surfaces can be found in the papers (23-29), in which the patchwise topographical distribution of the adsorption sites introduced by Halsey and Taylor (13) was used. It can be supposed that the application of virial formalism in the future to the description of adsorption from solutions will make the use of surface models of the adsorbents more realistic (30-32). The comparison of the results obtained in the papers (23-26) with the independently evaluated effect of surface heterogeneity of the adsorbent in a manner similar to that used in treating adsorption from the gaseous phase showed a good coincidence of the results obtained (20-22). The so-called patchwise model of surface heterogeneity previously used by us (23-26) will also be a basis of the considerations presented below. We propose a quantitative description of adsorption from ideal binary solutions on highly heterogeneous solid surfaces. The overall individual adsorption isotherm on solid surfaces will be described by an exponential equation of the DR type. The application of the DR equation suggested by us leads to realistic forms of an energy distribution function (21, 22). At the same time this equation will allow us to evaluate the total number of moles in the adsorbed layer.
THEORETICAL
f. Ejects of Surface Heterogeneity The basis for our theoretical considerations will be the following assumptions: (a) A liquid mixture is ideal both in the surface and bulk phase. (b) The adsorbent surface consists of energetically homogeneous patches within which identical adsorption conditions exist; the surface of each patch is assumed to be large enough that interaction of molecules adsorbed on neighboring patches and also within patches can be neglected. (c) The molar volumes of both components of the liquid mixture are approximately the same. According to the results of Everett (8), Si~kova and Erd6s (5, 6), and Myers and Sircar (12), we assume that the source of competitive adsorption from a binary mixture is the difference of potential adsorption energy of both components: E = ~1 - e2. With regard to the conventionally accepted variation range E~(i = 1, 2) from zero to infinity, the difference in e depends on the interval ( - o o , + oo). In (28) we introduced the following integral equation for tile overall adsorption excess of component "1" n l J ~ (Xl) = f~ nl,l ~n~ (Xl, ~)X(~)d~ [1] where nl.t ~(') (Xl) is the overall excess adsorption isotherm on a heterogeneous solid surface (describes adsorption on whole surface of the adsorbent), nlS(n)(xl, ~) is the local excess adsorption isotherm of the component "1" that governs adsorption within the small area (which is called the homogeneous patch of the adsorbent surface) having the difference of adsorption energies equal to e, x(e) is the differential distribution function of adsorption sites with regard to the value e, A = (--oo, + oo) is the interval of possible changes in ~, and xl is the mole fraction of the component "1" in the bulk phase. Our first
Journal of Colloid and Interface Science, Vol. 56, No. 3, September 1976
ADSORPTION F R O M LIQUID MIXTURES
and third assumptions will allow us to take the known equation of Everett (8), Kiselev (33), and Schay (11) for the local adsorption excess of the component "1", r~l,Z*(n), which can be easily transformed into the form: ~1,l *(n)
01,,(x, ~) -- - -
+ xl
nie
=
[
1 + -- exp
[2]
X
where x = x l / ( 1 - xl) is the ratio of mole fractions of both components in the bulk phase in the adsorption equilibrium; n f is the total number of moles in the adsorbed layer on a given homogeneous surface patch; K is the constant connected with the ratio of molecular partition functions of both components in the bulk and surface phases (24-29). Consider that the expression ['(nl,d(~)/nF) + xl] describes the individual adsorption of the component "1" on the homogeneous patch of the adsorbent surface in the units of relative coverage. Denoting the left side of Eq. [2] by the symbol O~,i(x, e) we can treat as the local adsorption isotherm (individual adsorption) of the component "1". Analogous to [1], the equation of the overall individual adsorption 01,~(x) has, therefore, the following form: 01,,(x) = f a 01,~(x, e)X(e)de.
[-3]
Attention should be drawn to the fact that in this notation the individual adsorption isotherm O~,t(x) is expressed by the relative coverage of the surface O~,~(x)
~,,(x) =
-
405
which is formally identical with the integral equation for the overall adsorption isotherm from the gaseous phase (19-22). Therefore, the methods for solving Eq. [4] will be analogous to those used in the description of adsorption from the gaseous phase on heterogeneous surfaces. From Eq. [4], one may obtain 1. an analytic form of the isotherm
01,,(x) for analytical forms of the function x(E) assumed a priori; such solutions of Eq. [4-] for the exponential function are already known (28), 2. a numerical solution with regard to the function x(e)-based on the development of the functions {~l,t(g), 01,l(g, •), and X(e) in series in relation to the complete and orthonormal function system, and 3. an analytical form of the function x(e) for the adsorption isotherm 01,t(x) assumed a priori. The two last possibilities have not been used yet in the theory of adsorption from solutions on heterogeneous solid surfaces. In this paper the third possibility will be discussed. Studies of the adsorption process from the gaseous phase showed that the exponential form of equations of the overall adsorption isotherm is a very good approximation of the experimental data and is very useful for numerical calculations (21, 22, 34). For the case of adsorption from solutions studied by us, we shall propose the exponential form of the overall individual adsorption isotherm:
i=1
-
1¢ ~s
where nl.~(x) is the number of moles of component "1" in surface layer, nd is the total number of moles in the adsorbed layer on the whole heterogeneous surface. Equations [2-] and [3] lead to the following expression:
01,~(x) = f [1 + (K/x) exp (-- e/RT)]-lx (~)de d lA
[4]
where B~ are constants analogs to the parameter B in DR equation, and x0 is a constant corresponding to the plateau on the individual adsorption isotherm 01.t(x). The expression of the individual adsorption isotherm in the form of Eq. E5-] is general enough to encompass the known equations for adsorption isotherms from the gaseous phase on heterogeneous surfaces (21, 22), including those from the liquid phase (24, 25, 28).
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406
OSClK ET
AL.
TABLE I Parameters of Equations of Adsorption Isotherms for Systems Studied No. adsorption system at 30°C
Reference
Homogeneous surfaces parameters of Eq. [-23] n8 (mmole/g)
Benzene (1)-cyclohexane (2) mixtures on silica gel Benzene (1)--n-heptane (2) mixtures on silica gel Benzene (1)-cyclohexane (2) mixtures on active carbon Benzene (1)-ethyl acetate (2) mlxtures on active carbon Benzene (1)-cyclohexane (2) mixtures on Cab-O-Sil Benzene (1)-cyclohexane (2) mixtures on Hi-Sil Benzene (1)-cyclohexane (2) mixtures on graphon 2
(x/x0)-]q
n
N s
S
(mmole/g)
0.0588
2.78
0.220
4
2.88
0.031
12
0.0879
3.17
0.209
4
2.84
0.037
40
0.0559
2.16
0.083
6
2.15
0.028
40
0.1028
1.61
0.006
5
1.60
0.004
41
--0.0763
1.75
1
1.89
0.550
41
0.0546
2.85
4
2.81
0.145
41 3
0.3216 4
3.10 5
1 7
3.84 8
0.024 9
x <~ xo
x ( x0.
1.109 0.053 6
£
x ( ~ ) & = 1.
[12a-]
Inversion of the Stieltjes transform [-10-] leads to the distribution function X l ( e ) = nt"x(e), which fulfils the folloMng condition
[-7-]
An interesting problem is the determination of the function corresponding to Eq. [-5-]. We find an accurate, analytical form of this function from Eq. [4-] after obtaining Stieltjes transform. Denoting
14.63
Function x(e) fulfils the normalization condition
E6-]
whereas for Bz < 0, Bi -- 0 (i = 1 and i ) 3) it is reduced to the isotherm analogous to the D R equation 01,,(x) = e x p { B ~ [ l n
S
12
We notice, therefore, that Eq. [5-] for Bi = 0 (i/> 2) and 0 < B~ < 1 assumes the form of Freundlich isotherm: 01,,(x) = (x/xo) B1
Heterogeneous surfaces parameters of Eq. [17]
£ Xl(e)& =nd.
[-12b]
Using the known formula for inversion of Stieltjes transform [-10], we obtain (14, 22)
E8-I
= t
from Eq. [4-] we obtain
01.,(K/s) = £ EF(t)dt/(t + s)-] or
[9-]
~o
,~.,(K/s) =£ D,e.F(t)dW(~+ s)-J [10-] F(t) is expressed F(t) = RT.x(RT in t).
where the function
( 0-T
by:
[11-I
J o u r n a l of Colloid and Interface Science, gol. 56, No. 3, September 1976
[13]
ADSORPTION FROM LIQUID MIXTURES where eo = R T
ill
[14]
(K/xo).
Introducing analogous simplifications as in the case of adsorption from the gaseous phase (22), we obtain the following analytical form for the function Xl(e) : =
EE
-
i=I ~z
x
exp
EE
B?(
0
-
[-15]
where
B,' = Bi/(RT) q
El6]
An alternative method for obtaining Eq. [-15~ can be the variation procedure proposed by Cerofolini (35) and O~cik et al. (26).
407
evaluation of the mono-or multilayer character of adsorption from solutions. In spite of these drawbacks the methods of Kiselev (38), Schay (11), and Everett (8) apparently can be used for evaluation of surface phase composition in many simple adsorption systems. According to the suggestion of Schay (ll) and Everett (8), a separate method for determining surface phase composition (the total number of moles of both components in the adsorbed layer) is desired to be developed, in which a model of this phase would not be assumed a priori. The present method for determining the total number of moles in the adsorbed layer in the case of adsorption on heterogeneous surfaces seems to be applicable. The Eq. [5] can be rewritten in a somewhat different form : n
[1". Calculation of the Total Number of Moles in the Adsorbed Layer The papers of Kiselev (38), Schay (11), and Everett (8) are of fundamental importance in the theory of surface phases in general and in determination of the total number of moles in the adsorbed layer in particular. The methods for determination of the surface phase composition given by these authors are graphical ones. The first two are based on the condition that linear segments of excess adsorption isotherm exist; whereas Everett's method (8) worked out for a model of a perfectly adsorbed monolayer is valid in the case where the separation coefficient a = xlx2~/ x~x2 is constant. These methods being simple and effective are widely used both in determination of surface phase composition and in assumption of the monolayer model for determination of the surface area of the adsorbent. The papers of Kiselev (38), Schay (11) and Everett (8) mentioned above were often criticized (36, 37, 39) for their being contradictory to the laws of thermodynamics. As shown by Rusanov (36), these methods allow us to determine only the minimal thickness of the surface phase, not to mention the effective thickness. They are of little use in
nl,t(x) = exp { E Bfln(x/xo)-]q
x ~ Xo. [-17-]
i=0
Comparing the Eqs. [-5~ and F17~ we obtain n~~ = exp(B0).
[-18~
Let us analyze the Eq. [18] in detail. For x = x0 the relative coverage of surface 01.t(Xo) = 1, however, the function nl,t(x0) = exp(B0). Taking into account these physical conditions we obtain Eq. E18~. To evaluate the coefficients B~ and also the surface phase composition, the points of experimental isotherms were approximated by the polynomial :
y
[19]
=
i=0
in which y and s are given by: y = in nl,t(x)
z -- in (X/Xo).
[-20-1
Numerical calculations will be discussed in the next part of this paper in detail. RESULTS AND DISCUSSION The purpose of the numerical calculations Was :
1. to examine the usefulness of Eq. [-5] for description of adsorption from solutions
Journal of Colloid and Interface Science, V o l . 56, N o . 3, S e p t e m b e r 1976
408
OSCIK E T AL. nt ~, and the sum of deviation squares [21-]
are given in the seventh, eighth, and ninth column, respectively, of Table I. Knowledge of the parameters Bi obtained by the method described makes the calculation of distribution functions according to the formula [15] possible. Theoretical adsorption excesses were calculated according to the equation:
\ t
x~ OB
x.
i
%" '/
x
t
x
0.4 !
nl,t~('°(xl) = nt"[Ol,t(xl) -
Xx
0.2
0.4
06
(28
x~
FIO. 1. The theoretical excess adsorption isotherms nl., ~(~) of benzene (1)-cyelohexane (2) mixtures on silica gel at 30°C: the solid line denotes n~,,"(~) from Eq. 1-22] and dashed line denotes nl "c~) from Everett (8, Eq. [-23]). on heterogeneous solid surfaces and to determine the distribution function of the difference of adsorption energies e = E 1 - e2, which corresponds to the Eq. I-5]. 2. to determine the total number of moles in the adsorbed layer by the proposed method and to compare the results obtained with those obtained on assuming a homogeneous surface of the adsorbent. For realization of the latter purpose we used the known Everett Eq. E8] for a perfect adsorbed monolayer. The calculations were carried out by using the data available from the literature for seven different adsorption systems. These systems have been summarized in the second column of Table I. Our calculation procedure was: the experimental data were approximated by Eq. [19], for different values of the parameter n,< To test the choice of the best approximation, the authors assumed the minimum sum of deviation squares defined by the equation
""
-~
nl~(n)
Z [Y.°~P--
£ i=0
B,(z,,,)"-J 2
Xl -]-
•8
1
!:]
[23]
i
E x-
z"
"d.7:0.5
[21]
where (z,,, y,~) is the ruth pair of the experimental points z and y and M is the total number of the experimental points. The degree of the best approximation n, the total number of moles in the adsorbed layer
'E
- -
where nl ~(~) is the excess isotherm describing the adsorption for the whole surface, and n ~ is the total number of moles in the adsorbed layer calculated on assuming a homogeneous surface. The numerical parameters n ~ and a, and sum of deviation squares for Eq. [-23] are placed in the columns four, five, andsix, respectively, of Table I. The most interesting results of numerical
0.25
m=l
[22]
For comparative purposes, the parameters of Everett's equation calculated on assuming homogeneous surface of the adsorbent for perfect adsorbed monolayer, are summarized in Table I. These parameters were calculated according to the equation:
M
S =
xl].
050 x1
0.75
FIG. 2. T h e t h e o r e t i c a l excess a d s o r p t i o n i s o t h e r m s
nl,d (") of benzene (1)-cyclohexane (2) mixtures (white circles, experimental points) and benzene (1)-ethyl acetate (2) (black circles, experimental points) on active carbon at 30°C: the solid lines denote ni,d (") from our Eq. [22] and the dashed lines denote nl ~(~) from Everett (8, Eq. 1-23-]).
dournal of Colloid and Interface Science, Vol. 56, No. 3, September 1976
ADSORPTION FROM LIQUID MIXTURES
409
calculations have been illustrated in Figs. 1, 2, and 3. Figure 1 and 2 present the theoretical -& \\ adsorption excess, which were calculated from x Eq. E22] for the systems: benzene from cyclohexane on active carbon, benzene from ethyl xx acetate on active carbon, and benzene from cyclohexane on silica gel. These systems were previously studied in the papers (12, 40, 41). \\\Xx\ The liquid mixtures summarized in Table I djy \\ j-. ~ are nearly ideal; the activity coefficients at -2000 -1000 0 1000 infinite dilution are 1.5 (12). The mixture S (ca[/mo~e) liquid systems placed in Table I show the Fro. 4. The distributionfunctionsxl (e) for adsorption small deviations from Raoult's law. Therefore, systems: benzene-cyclohexane on active carbon at these systems should obey Eq. E5] if the 30°C (solid line) and benzene-ethylacetate on active investigated adsorbents are heterogeneous and carbon at 30°C (dashed line). may be characterized by distribution of adsorption energies [-15-]. two basic types of active centres occurring on Ill Figs. 1 and 2 the experimental points are the surface of silica gel (21-24). A considerable denoted by circles, theoretical isotherms calcu- part of the function is in the range of negative lated assuming surface homogeneity (12, 40) energy values, which is connected with preferby the dashed lines, and theoretical isotherms ential adsorption of benzene on silica gel. calculated from Eq. [-22] by the continuous There exists, however, a part of the surface lines. As follows from Figs. 1 and 2, our of silica gel where cyclohexane is preferentially theoretical isotherms describe the experimental adsorbed. This conclusion coincides with the data better than the theoretical adsorption results of other papers (10, 41). isotherms calculated according to Eq. [-23-] An interesting course is also shown by (see also Table I, columns six and nine). distribution functions on active carbons. In Fig. 3 the distribution function xl(e) Generally speaking, active carbons are adcalculated from Eq. El5-] and corresponding sorbents of less defined compositions, particuto the excess isotherm from Fig. 1 is shown. lary with regard to surface, than silica gels. Two distinct maxima in the energy distri- The degree of heterogeneity can vary here bution for system 1 (Table I) correspond to within wide ranges in relation to its preparation, i.e., preferential adsorption of the components of a liquid, binary mixture may change in relation to preparation of the adsorbent. It follows from Figs. 2 and 4 that benzene is o 6 preferentially adsorbed also on active carbon, the surface of which shows one basic type of active site. However, a small fraction of carbon ~4 surface preferentially adsorbs ethyl acetate (system 4) and cyclohexane (system 3), in ~2 that ethyl acetate is better adsorbed than cyclohexane. The preferential adsorption of benzene is certainly caused by its nonspecific interaction with the surface of active carbon. -~oo -doo -~'oo 4oo' E, (cal/mole) The comparison of numerical values of total number of moles in adsorbed layer, which FIG. 3. The distribution functionXl(e) for adsorption of benzene-cyclohexanemixtures on silica gel at 30°C. were found by the method proposed by us,
ii)
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OSCIK E T AL.
with those obtained b y the Everett method based on a constant value of the separation coefficient a also appears interesting. The results are shown in columns five and eight of Table I. Good agreement of the results of our method with those by Everett method was obtained for the following adsorption systems : 1, 3, 4, 5, and 6 from Table I. The comparison of the total number of moles in the adsorbed layer obtained b y both methods supports Everett's assumption (8) for the model of perfect adsorbed monolayer. Similar conclusions were drawn in other papers (10, 41). Another significant conclusion resulting from this comparison is the fact known the theory of adsorption from gaseous phase that involved heterogeneity of a solid surface does not greatly effect the numerical values of the total number of moles in the adsorbed layer (42, 43). The obtained results show that the method presented allows us to calculate the total number of moles in the adsorbed layer without its model being assumed a priori. A good agreement of the results obtained by Everett's method with those of our method probably results from the analysis of the systems approximate to perfect ones. I t can be supposed that in the case of nonperfect systems for which Everett's method cannot be used, the method proposed in this paper will be useful. The analysis of such systems will be the subject of the next paper. REFERENCES 1. ZETTLEMOYER, A. C. AND MICALE, F. J., Croat. Chem. Acta 42, 247 (1970). 2. ScHveHOWITZKY,A., Acta Physicochem. URSS 8,
531 (1938). 3. HANSEN,R. S. ANDMAI, U. H., J. Phys. Chem. 61, 573 (1957). 4. DELMAS,G. AND PATTERSON,D., f . Phys. Chem. 64, 1827 (1960). 5. SIgKOVA, M. AND ERD6S, E., Coll. Czech. Chem. Commun. 25, 1729 (1960). 6. SIgKOVA, M. AND ERD6S, E., Coll. Czech. Chem. Commun. 25, 2599 (1960). 7. COLTtIARP,M. T. ANDHACKERMAN,N., J. Colloid Interface Sci. 43, 176 (1973); 43, 185 (1973). 8. EVERE~:T, D. H., Trans. Faraday Soc. 60, 1803 (1964); 61, 2487 (1965).
9. KIPLING,J. J., "Adsorption from Solutions of Nonelectrolytes," Academic Press, New York 1965. 10. KAGIYA,T., SUMIDA,Y., ANDTAeHI,T., Bull. Chem. Soc. Japan 45, 6 (1972) ; 44, 1219 (1971). 11. SCItAY,G. AND NAGY, L., J. Chim. Phys. 58, 149 (1961); 3". Colloid Interface Sci. 38, 302 (1972). 12. SIR,CAR, S. AND MYERS, A. L., J. Phys. Chem. 74, 2828 (1970); 76, 3415 (1972). 13. HALSEY,G. AND TAYLOR,H., J. Chem. Phys. 15, 624 (1947). 14. SIPs, R., 3". Chem. Phys. 16, 490 (1948). 15. SIPs, R., J. Chem. Phys. 18, 1024 (1950). 16. ADAMSON,A. W., LING, I., DORTMANT,L., AND OPmM, M., J. Colloid Interface Sci. 21, 445 (1966). 17. VANDONGEN,R. H., Surface Sci. 39, 341 (1973). 18. DRAIN,L. E. ANDMORRISON,J. A., Trans. Faraday Soc. 48, 840 (1952). 19. RUDZ~SKI,W. ANDJARONIEC,M., Surface Sci. 42, 552 (1974). 20. JARONIEC,NI. ANDRUDZI~!SKI,W. Colloid Polymer Sci. 253 (1975). 21. JARONIEC, M., RUDZI~ISKI,W., SOKOLOWSI~,S., AND SMARZEWSKI,R., Colloid Polymer Sci. 253, 164 (1975). 22. JARONIEC,M., Surface Sci. 50 (1975). 23. RCDZI~SKI,W., WAKSMUNDZKI,A., JARONIE¢,M., ANDSOKO~OWSI~X,S., Ann. Soc. Chim. Polonorum 48, 1985 (1974). 24. O~CIK, J., RuDzI~srx, W., AND DABROWSKI,A., Ann. Soc. Chim. Polonorum 48, 1991 (1974). 25. 0SClK, J., RUDZI~SKI,W., ANDDABROWSKI,A., On the formation of multilayer in adsorption from binary liquid mixtures, AIChE 67th Annual Meeting, Washington, D. C. December 1-5, 1974. 26. OSCIK, J., DABROWSKI,A., SOICOLOWSlCI,S., AND JARONIEC, M., d. Catal. Hokkaido Univ. 23 (1975). 27. OffClK,J., RUDZI~SKI,W., ANn DABROWSKI,A., IV Konf. Teor. Vopr. Adsorbcii, Leningrad 1974 February. 28. RuDzI~SlCI, W., 0~ClX, J., AND DABROWSKI,A., Chem. Phys. Lett. 20, 5 (1973). 29. WAKSMUNDZKI,A., DAWIDOWICZ,A., SOKOLOWSKI, S., JARONIEC,M., Chromatographia 8, 234 (1975). 30. RITDZI~SKI,W., Chem. Phys. Let& 10, 183 (1971). 31. SAMS, J. R., Progr. Surface Membrane Sci. 8, 1 (1974). 32. RADKE,C. J. ANDPRAUSNITZ,J. M., J. Chem. Phys. 57, 714 (1972). 33. KISELEV, A. V. AND PAVLOVA,L. F., Izv. Ahad. Nauk SSSR Otd. Khim. Nauk 1, 18 (1965). 34. KNOWLES,A. J. ANDMOFFAT,J. B., J. Colloid Sci. 41, 116 (1972). 35. CEROFOLINI,G. F., Surface Sci. 24, 391 (1971).
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ADSORPTION FROM LIQUID MIXTURES 36. RUsANov, A. I., Progr. Su~Jace Membrane Sci. 4, 57 (1971). 37. tLA•IONOW, O. G., "Physical Adsorption from Mulficomponents Phases," (M. M. Dubinin, Ed.), p. 85. Pub. Nauka, Moscov 1972. 38. KISELEV, A. W. AND SCHERBAKOVA,K. D., Acta Physicochim. URSS 21, 539 (1946). 39. BERING, ~B. P., fgv. Akad. Nauk S S S R Otd. Khim. Nauk 6, 1232 (1970).
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40. MINI,:-A,C. AND MYERS, A. L., AIChE Y. 19, 453 (1973). 41. MATAYO,D. R. AND WIO~I:r~.N, J. P., .1. Colloid Interface Sci. 44, 162 (1973). 42. WAKSM~JNI)ZKI,A., RUDZlNSKI, W., SOKOLOWSKI, S., AND~ARONIEC,M., Ann. Soc. Chim. Polonorum 48, 1741 (1974). 43. CERO:~OLINI, G. F., J. Low Temp. Phys. 6, 473 (1972).
Journal of Colloid and Interface Science, Vol. 56. No. 3. September 1976
Department of Physical Chemistry, Institute of Chemistry UMCS, 20031 Lublin, Nowotki 12, Poland AND
W. R U D Z I N S K I
Department of Chemical Engineering, Queen's University Dupuis Hall, Kingston, Ontario, Canada Received March 31, 1975; accepted January 6, 1976 A new, theoretical description of adsorption from ideal liquid binary mixtures on patchwise heterogeneous solid surfaces has been proposed. The overall, individual adsorption isotherm was approximated by an exponential equation of the Dubinin-Radushkevich (DR) type. The customary DR and Freundlich equations, which were earlier used for description of adsorption from solutions on heterogeneous solid surfaces, appeared to be special cases of the exponential adsorption isotherm. Using the method of Stieltjes transforms, distribution functions of the difference of adsorption energies of liquid mixture components were found, which corresponded with the equation of the individual isotherm. Also an independent method for determination of the total number of moles in the adsorbed layer has been proposed. The theoretical considerations have been illustrated by using the data available from the literature for seven different adsorption systems. The results obtained were compared with those obtained on using the Everett isotherm equation of the model for a perfect adsorbed monolayer. INTRODUCTION Theories of adsorption from liquids are made difficult by the chemical heterogeneity and structural irregularity of the solid surface as well as the fact at least two components are involved in competitive adsorption at the solid surface (1). To avoid this difficulty in the majority of papers concerning adsorption from liquids on solid surfaces it was usually either assumed that the surface of the adsorbent was homogeneous, or the problem was given no attention. I n m a n y papers, however, deviations of experimental results from those theoretically expected have been associated with the heterogeneous character of a solid surface. Schuchowitzky (2), Hansen (3), as well as Delmas and Patterson (4) and Si~kova and
Erd6s (5, 6) saw in the surface heterogeneity of the adsorbent a source of imperfection of surface phases. T h e y suggested that the change of the sign of excess adsorption isotherms should be associated with surface heterogeneity. Their suggestions were supported by the studies of Coltharp (7), which showed that heterogeneity of the adsorbent surface could be the cause of preferential adsorption of one of the components of binary liquid mixture. The number of qualitative and semiquantitative descriptions known in the literature (7-12) that take into consideration the effect of surface structure on adsorption from solutions is unfortunately small. Hence little information is available on the quantitative estimation of heterogeneity of the adsorbent 403
Copyright ~) 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.
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404
OgCIK ET AL.
surface and its influence on the adsorptive properties. Most realistic descriptions of heterogeneity effects of surfaces were initially developed in the theory of adsorption from the gaseous phase. Halsey and Taylor (13), in a paper of fundamental significance, proposed an integral equation for the overall adsorption isotherm on heterogeneous surfaces. A basic role was also played by the papers of Sips (14, 15). On the basis of this fundamental work, a number of theoretical approaches were developed by means of which heterogeneity of the adsorbent surface in adsorption from the gaseous phase can be quantitatively evaluated (16-22). The first quantitative attempts to describe the mechanism of adsorption from binary solutions on heterogeneous surfaces can be found in the papers (23-29), in which the patchwise topographical distribution of the adsorption sites introduced by Halsey and Taylor (13) was used. It can be supposed that the application of virial formalism in the future to the description of adsorption from solutions will make the use of surface models of the adsorbents more realistic (30-32). The comparison of the results obtained in the papers (23-26) with the independently evaluated effect of surface heterogeneity of the adsorbent in a manner similar to that used in treating adsorption from the gaseous phase showed a good coincidence of the results obtained (20-22). The so-called patchwise model of surface heterogeneity previously used by us (23-26) will also be a basis of the considerations presented below. We propose a quantitative description of adsorption from ideal binary solutions on highly heterogeneous solid surfaces. The overall individual adsorption isotherm on solid surfaces will be described by an exponential equation of the DR type. The application of the DR equation suggested by us leads to realistic forms of an energy distribution function (21, 22). At the same time this equation will allow us to evaluate the total number of moles in the adsorbed layer.
THEORETICAL
f. Ejects of Surface Heterogeneity The basis for our theoretical considerations will be the following assumptions: (a) A liquid mixture is ideal both in the surface and bulk phase. (b) The adsorbent surface consists of energetically homogeneous patches within which identical adsorption conditions exist; the surface of each patch is assumed to be large enough that interaction of molecules adsorbed on neighboring patches and also within patches can be neglected. (c) The molar volumes of both components of the liquid mixture are approximately the same. According to the results of Everett (8), Si~kova and Erd6s (5, 6), and Myers and Sircar (12), we assume that the source of competitive adsorption from a binary mixture is the difference of potential adsorption energy of both components: E = ~1 - e2. With regard to the conventionally accepted variation range E~(i = 1, 2) from zero to infinity, the difference in e depends on the interval ( - o o , + oo). In (28) we introduced the following integral equation for tile overall adsorption excess of component "1" n l J ~ (Xl) = f~ nl,l ~n~ (Xl, ~)X(~)d~ [1] where nl.t ~(') (Xl) is the overall excess adsorption isotherm on a heterogeneous solid surface (describes adsorption on whole surface of the adsorbent), nlS(n)(xl, ~) is the local excess adsorption isotherm of the component "1" that governs adsorption within the small area (which is called the homogeneous patch of the adsorbent surface) having the difference of adsorption energies equal to e, x(e) is the differential distribution function of adsorption sites with regard to the value e, A = (--oo, + oo) is the interval of possible changes in ~, and xl is the mole fraction of the component "1" in the bulk phase. Our first
Journal of Colloid and Interface Science, Vol. 56, No. 3, September 1976
ADSORPTION F R O M LIQUID MIXTURES
and third assumptions will allow us to take the known equation of Everett (8), Kiselev (33), and Schay (11) for the local adsorption excess of the component "1", r~l,Z*(n), which can be easily transformed into the form: ~1,l *(n)
01,,(x, ~) -- - -
+ xl
nie
=
[
1 + -- exp
[2]
X
where x = x l / ( 1 - xl) is the ratio of mole fractions of both components in the bulk phase in the adsorption equilibrium; n f is the total number of moles in the adsorbed layer on a given homogeneous surface patch; K is the constant connected with the ratio of molecular partition functions of both components in the bulk and surface phases (24-29). Consider that the expression ['(nl,d(~)/nF) + xl] describes the individual adsorption of the component "1" on the homogeneous patch of the adsorbent surface in the units of relative coverage. Denoting the left side of Eq. [2] by the symbol O~,i(x, e) we can treat as the local adsorption isotherm (individual adsorption) of the component "1". Analogous to [1], the equation of the overall individual adsorption 01,~(x) has, therefore, the following form: 01,,(x) = f a 01,~(x, e)X(e)de.
[-3]
Attention should be drawn to the fact that in this notation the individual adsorption isotherm O~,t(x) is expressed by the relative coverage of the surface O~,~(x)
~,,(x) =
-
405
which is formally identical with the integral equation for the overall adsorption isotherm from the gaseous phase (19-22). Therefore, the methods for solving Eq. [4] will be analogous to those used in the description of adsorption from the gaseous phase on heterogeneous surfaces. From Eq. [4], one may obtain 1. an analytic form of the isotherm
01,,(x) for analytical forms of the function x(E) assumed a priori; such solutions of Eq. [4-] for the exponential function are already known (28), 2. a numerical solution with regard to the function x(e)-based on the development of the functions {~l,t(g), 01,l(g, •), and X(e) in series in relation to the complete and orthonormal function system, and 3. an analytical form of the function x(e) for the adsorption isotherm 01,t(x) assumed a priori. The two last possibilities have not been used yet in the theory of adsorption from solutions on heterogeneous solid surfaces. In this paper the third possibility will be discussed. Studies of the adsorption process from the gaseous phase showed that the exponential form of equations of the overall adsorption isotherm is a very good approximation of the experimental data and is very useful for numerical calculations (21, 22, 34). For the case of adsorption from solutions studied by us, we shall propose the exponential form of the overall individual adsorption isotherm:
i=1
-
1¢ ~s
where nl.~(x) is the number of moles of component "1" in surface layer, nd is the total number of moles in the adsorbed layer on the whole heterogeneous surface. Equations [2-] and [3] lead to the following expression:
01,~(x) = f [1 + (K/x) exp (-- e/RT)]-lx (~)de d lA
[4]
where B~ are constants analogs to the parameter B in DR equation, and x0 is a constant corresponding to the plateau on the individual adsorption isotherm 01.t(x). The expression of the individual adsorption isotherm in the form of Eq. E5-] is general enough to encompass the known equations for adsorption isotherms from the gaseous phase on heterogeneous surfaces (21, 22), including those from the liquid phase (24, 25, 28).
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OSClK ET
AL.
TABLE I Parameters of Equations of Adsorption Isotherms for Systems Studied No. adsorption system at 30°C
Reference
Homogeneous surfaces parameters of Eq. [-23] n8 (mmole/g)
Benzene (1)-cyclohexane (2) mixtures on silica gel Benzene (1)--n-heptane (2) mixtures on silica gel Benzene (1)-cyclohexane (2) mixtures on active carbon Benzene (1)-ethyl acetate (2) mlxtures on active carbon Benzene (1)-cyclohexane (2) mixtures on Cab-O-Sil Benzene (1)-cyclohexane (2) mixtures on Hi-Sil Benzene (1)-cyclohexane (2) mixtures on graphon 2
(x/x0)-]q
n
N s
S
(mmole/g)
0.0588
2.78
0.220
4
2.88
0.031
12
0.0879
3.17
0.209
4
2.84
0.037
40
0.0559
2.16
0.083
6
2.15
0.028
40
0.1028
1.61
0.006
5
1.60
0.004
41
--0.0763
1.75
1
1.89
0.550
41
0.0546
2.85
4
2.81
0.145
41 3
0.3216 4
3.10 5
1 7
3.84 8
0.024 9
x <~ xo
x ( x0.
1.109 0.053 6
£
x ( ~ ) & = 1.
[12a-]
Inversion of the Stieltjes transform [-10-] leads to the distribution function X l ( e ) = nt"x(e), which fulfils the folloMng condition
[-7-]
An interesting problem is the determination of the function corresponding to Eq. [-5-]. We find an accurate, analytical form of this function from Eq. [4-] after obtaining Stieltjes transform. Denoting
14.63
Function x(e) fulfils the normalization condition
E6-]
whereas for Bz < 0, Bi -- 0 (i = 1 and i ) 3) it is reduced to the isotherm analogous to the D R equation 01,,(x) = e x p { B ~ [ l n
S
12
We notice, therefore, that Eq. [5-] for Bi = 0 (i/> 2) and 0 < B~ < 1 assumes the form of Freundlich isotherm: 01,,(x) = (x/xo) B1
Heterogeneous surfaces parameters of Eq. [17]
£ Xl(e)& =nd.
[-12b]
Using the known formula for inversion of Stieltjes transform [-10], we obtain (14, 22)
E8-I
= t
from Eq. [4-] we obtain
01.,(K/s) = £ EF(t)dt/(t + s)-] or
[9-]
~o
,~.,(K/s) =£ D,e.F(t)dW(~+ s)-J [10-] F(t) is expressed F(t) = RT.x(RT in t).
where the function
( 0-T
by:
[11-I
J o u r n a l of Colloid and Interface Science, gol. 56, No. 3, September 1976
[13]
ADSORPTION FROM LIQUID MIXTURES where eo = R T
ill
[14]
(K/xo).
Introducing analogous simplifications as in the case of adsorption from the gaseous phase (22), we obtain the following analytical form for the function Xl(e) : =
EE
-
i=I ~z
x
exp
EE
B?(
0
-
[-15]
where
B,' = Bi/(RT) q
El6]
An alternative method for obtaining Eq. [-15~ can be the variation procedure proposed by Cerofolini (35) and O~cik et al. (26).
407
evaluation of the mono-or multilayer character of adsorption from solutions. In spite of these drawbacks the methods of Kiselev (38), Schay (11), and Everett (8) apparently can be used for evaluation of surface phase composition in many simple adsorption systems. According to the suggestion of Schay (ll) and Everett (8), a separate method for determining surface phase composition (the total number of moles of both components in the adsorbed layer) is desired to be developed, in which a model of this phase would not be assumed a priori. The present method for determining the total number of moles in the adsorbed layer in the case of adsorption on heterogeneous surfaces seems to be applicable. The Eq. [5] can be rewritten in a somewhat different form : n
[1". Calculation of the Total Number of Moles in the Adsorbed Layer The papers of Kiselev (38), Schay (11), and Everett (8) are of fundamental importance in the theory of surface phases in general and in determination of the total number of moles in the adsorbed layer in particular. The methods for determination of the surface phase composition given by these authors are graphical ones. The first two are based on the condition that linear segments of excess adsorption isotherm exist; whereas Everett's method (8) worked out for a model of a perfectly adsorbed monolayer is valid in the case where the separation coefficient a = xlx2~/ x~x2 is constant. These methods being simple and effective are widely used both in determination of surface phase composition and in assumption of the monolayer model for determination of the surface area of the adsorbent. The papers of Kiselev (38), Schay (11) and Everett (8) mentioned above were often criticized (36, 37, 39) for their being contradictory to the laws of thermodynamics. As shown by Rusanov (36), these methods allow us to determine only the minimal thickness of the surface phase, not to mention the effective thickness. They are of little use in
nl,t(x) = exp { E Bfln(x/xo)-]q
x ~ Xo. [-17-]
i=0
Comparing the Eqs. [-5~ and F17~ we obtain n~~ = exp(B0).
[-18~
Let us analyze the Eq. [18] in detail. For x = x0 the relative coverage of surface 01.t(Xo) = 1, however, the function nl,t(x0) = exp(B0). Taking into account these physical conditions we obtain Eq. E18~. To evaluate the coefficients B~ and also the surface phase composition, the points of experimental isotherms were approximated by the polynomial :
y
[19]
=
i=0
in which y and s are given by: y = in nl,t(x)
z -- in (X/Xo).
[-20-1
Numerical calculations will be discussed in the next part of this paper in detail. RESULTS AND DISCUSSION The purpose of the numerical calculations Was :
1. to examine the usefulness of Eq. [-5] for description of adsorption from solutions
Journal of Colloid and Interface Science, V o l . 56, N o . 3, S e p t e m b e r 1976
408
OSCIK E T AL. nt ~, and the sum of deviation squares [21-]
are given in the seventh, eighth, and ninth column, respectively, of Table I. Knowledge of the parameters Bi obtained by the method described makes the calculation of distribution functions according to the formula [15] possible. Theoretical adsorption excesses were calculated according to the equation:
\ t
x~ OB
x.
i
%" '/
x
t
x
0.4 !
nl,t~('°(xl) = nt"[Ol,t(xl) -
Xx
0.2
0.4
06
(28
x~
FIO. 1. The theoretical excess adsorption isotherms nl., ~(~) of benzene (1)-cyelohexane (2) mixtures on silica gel at 30°C: the solid line denotes n~,,"(~) from Eq. 1-22] and dashed line denotes nl "c~) from Everett (8, Eq. [-23]). on heterogeneous solid surfaces and to determine the distribution function of the difference of adsorption energies e = E 1 - e2, which corresponds to the Eq. I-5]. 2. to determine the total number of moles in the adsorbed layer by the proposed method and to compare the results obtained with those obtained on assuming a homogeneous surface of the adsorbent. For realization of the latter purpose we used the known Everett Eq. E8] for a perfect adsorbed monolayer. The calculations were carried out by using the data available from the literature for seven different adsorption systems. These systems have been summarized in the second column of Table I. Our calculation procedure was: the experimental data were approximated by Eq. [19], for different values of the parameter n,< To test the choice of the best approximation, the authors assumed the minimum sum of deviation squares defined by the equation
""
-~
nl~(n)
Z [Y.°~P--
£ i=0
B,(z,,,)"-J 2
Xl -]-
•8
1
!:]
[23]
i
E x-
z"
"d.7:0.5
[21]
where (z,,, y,~) is the ruth pair of the experimental points z and y and M is the total number of the experimental points. The degree of the best approximation n, the total number of moles in the adsorbed layer
'E
- -
where nl ~(~) is the excess isotherm describing the adsorption for the whole surface, and n ~ is the total number of moles in the adsorbed layer calculated on assuming a homogeneous surface. The numerical parameters n ~ and a, and sum of deviation squares for Eq. [-23] are placed in the columns four, five, andsix, respectively, of Table I. The most interesting results of numerical
0.25
m=l
[22]
For comparative purposes, the parameters of Everett's equation calculated on assuming homogeneous surface of the adsorbent for perfect adsorbed monolayer, are summarized in Table I. These parameters were calculated according to the equation:
M
S =
xl].
050 x1
0.75
FIG. 2. T h e t h e o r e t i c a l excess a d s o r p t i o n i s o t h e r m s
nl,d (") of benzene (1)-cyclohexane (2) mixtures (white circles, experimental points) and benzene (1)-ethyl acetate (2) (black circles, experimental points) on active carbon at 30°C: the solid lines denote ni,d (") from our Eq. [22] and the dashed lines denote nl ~(~) from Everett (8, Eq. 1-23-]).
dournal of Colloid and Interface Science, Vol. 56, No. 3, September 1976
ADSORPTION FROM LIQUID MIXTURES
409
calculations have been illustrated in Figs. 1, 2, and 3. Figure 1 and 2 present the theoretical -& \\ adsorption excess, which were calculated from x Eq. E22] for the systems: benzene from cyclohexane on active carbon, benzene from ethyl xx acetate on active carbon, and benzene from cyclohexane on silica gel. These systems were previously studied in the papers (12, 40, 41). \\\Xx\ The liquid mixtures summarized in Table I djy \\ j-. ~ are nearly ideal; the activity coefficients at -2000 -1000 0 1000 infinite dilution are 1.5 (12). The mixture S (ca[/mo~e) liquid systems placed in Table I show the Fro. 4. The distributionfunctionsxl (e) for adsorption small deviations from Raoult's law. Therefore, systems: benzene-cyclohexane on active carbon at these systems should obey Eq. E5] if the 30°C (solid line) and benzene-ethylacetate on active investigated adsorbents are heterogeneous and carbon at 30°C (dashed line). may be characterized by distribution of adsorption energies [-15-]. two basic types of active centres occurring on Ill Figs. 1 and 2 the experimental points are the surface of silica gel (21-24). A considerable denoted by circles, theoretical isotherms calcu- part of the function is in the range of negative lated assuming surface homogeneity (12, 40) energy values, which is connected with preferby the dashed lines, and theoretical isotherms ential adsorption of benzene on silica gel. calculated from Eq. [-22] by the continuous There exists, however, a part of the surface lines. As follows from Figs. 1 and 2, our of silica gel where cyclohexane is preferentially theoretical isotherms describe the experimental adsorbed. This conclusion coincides with the data better than the theoretical adsorption results of other papers (10, 41). isotherms calculated according to Eq. [-23-] An interesting course is also shown by (see also Table I, columns six and nine). distribution functions on active carbons. In Fig. 3 the distribution function xl(e) Generally speaking, active carbons are adcalculated from Eq. El5-] and corresponding sorbents of less defined compositions, particuto the excess isotherm from Fig. 1 is shown. lary with regard to surface, than silica gels. Two distinct maxima in the energy distri- The degree of heterogeneity can vary here bution for system 1 (Table I) correspond to within wide ranges in relation to its preparation, i.e., preferential adsorption of the components of a liquid, binary mixture may change in relation to preparation of the adsorbent. It follows from Figs. 2 and 4 that benzene is o 6 preferentially adsorbed also on active carbon, the surface of which shows one basic type of active site. However, a small fraction of carbon ~4 surface preferentially adsorbs ethyl acetate (system 4) and cyclohexane (system 3), in ~2 that ethyl acetate is better adsorbed than cyclohexane. The preferential adsorption of benzene is certainly caused by its nonspecific interaction with the surface of active carbon. -~oo -doo -~'oo 4oo' E, (cal/mole) The comparison of numerical values of total number of moles in adsorbed layer, which FIG. 3. The distribution functionXl(e) for adsorption of benzene-cyclohexanemixtures on silica gel at 30°C. were found by the method proposed by us,
ii)
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OSCIK E T AL.
with those obtained b y the Everett method based on a constant value of the separation coefficient a also appears interesting. The results are shown in columns five and eight of Table I. Good agreement of the results of our method with those by Everett method was obtained for the following adsorption systems : 1, 3, 4, 5, and 6 from Table I. The comparison of the total number of moles in the adsorbed layer obtained b y both methods supports Everett's assumption (8) for the model of perfect adsorbed monolayer. Similar conclusions were drawn in other papers (10, 41). Another significant conclusion resulting from this comparison is the fact known the theory of adsorption from gaseous phase that involved heterogeneity of a solid surface does not greatly effect the numerical values of the total number of moles in the adsorbed layer (42, 43). The obtained results show that the method presented allows us to calculate the total number of moles in the adsorbed layer without its model being assumed a priori. A good agreement of the results obtained by Everett's method with those of our method probably results from the analysis of the systems approximate to perfect ones. I t can be supposed that in the case of nonperfect systems for which Everett's method cannot be used, the method proposed in this paper will be useful. The analysis of such systems will be the subject of the next paper. REFERENCES 1. ZETTLEMOYER, A. C. AND MICALE, F. J., Croat. Chem. Acta 42, 247 (1970). 2. ScHveHOWITZKY,A., Acta Physicochem. URSS 8,
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Journal of Colloid and Interface Science, Vol. 56. No. 3. September 1976