Langmuir 1993,9, 2641-2651
2641
On the Nature of the Energetic Surface Heterogeneity in Ion Adsorption at a Water/Oxide Interface: Theoretical Studies of Some Special Features of Ion Adsorption at Low Ion Concentrations W. Rudzinski,*J R. Charmas,? S. Partyka,* and J. Y. Botterog Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Skbdowska University, P1. M.Curie Skbdowskiej 3, Lublin 20-031, Poland, Laboratoire de Physico-Chimie des Systemes Polyphases, LA 330, U.S.T.L., Place Eugene Bataillon, 34060 Montpellier Cedex, France, and Centre de Recherche 8ur la Valorisation des Minerais, B.P. 40,54501 Vandoeuvre Cedex, France Received November 16, 1992. In Final Form: July 2 9 , 1 9 9 5 At low surface concentrations the log-log plots of the experimental adsorption isotherms of ions on oxides show a transition from a linear (Henry's) plot with a tangent equal unity to a Freundlich plot with tangent smaller than unity. A theory has been developed reproducing that striking behavior of these adsorption isotherms. That theoretical treatment was based on a picture of a heterogeneoussolid surface, with different adsorption sites-the outermost surface oxygens. Our theoretical study indicates that the binding-to-surface energies of different surface complexes,formed on different surface oxygens, vary in an independent way from one surface oxygen to another.
Introduction
It is now generally recognized that the surfaces of the oxides are geometrically distorted. A variety of experimental techniques have been used to study the nature of these surface imperfections, and dozens of papers have been published on this subject. The reported results have already been a subject of a number of reviews.' The experimentalstudies were stimulated mainly by the widely spread view that these are the surface imperfections creating catalytic centers for many important catalytic reactions. Bakaev's computhr simulations of oxide surfaces26 suggestthat even in the case of oxides having a well-defined bulk crystal structure, the degree of the surface disorder may be larger than generally believed. The surface imperfections (disorder) of the oxide surfacesmust affect the properties of the electrical double layer formed at a water/oxide interface. This important factor has been ignored in hundreds of papers treating on ion adsorption within the electrical double layer. At the end of the 1970s Garcia-Miragaya and Page697 and Street et ale8reported a successful correlation by the Freundlich equation of the data of trace Cd2+adsorption
* Author to whom correspondence should be addressed. t
Maria Curie-Sklodowska University.
Laboratoire de Phyeico-Chimie des Syetames Polyphasee. I Centre de Recherche sur la Valorisation des Miner&. Abstract published in Adoance ACS Abstracts, September 15,
1993. (1) See for instance Chapter I in the monograph by Nowotny, J.; Wepper, W. Non-Stoichiometric Compounds. Surfaces, Grain Boundaries and Structural Defects; NATO AS1 Series, vol. 276; Kluwer Academic Publinhere: Dordrecht, 1989. (2) Bakaev, V. A. Surf. Sci. 1988,198, 671. (3) Bakaev, V. A. h o c . Int. Workshop;Adsorption on Microporous Adsorbents 1987,2, 33. (4)Bakaev, V. A.; Dubinii, M. M. Dokl. Acad. Nauk SSSR 1987,296, 369. (5) Bakaev, V. A.; Chelnokova, 0. V. Surf. Sci. 1989,216,521. (6) Garcia-Miagaya,J.; Page, A. L.Soil Sci. SOC.Am. J. 1976,40,668. (7) Garcia-Miragaya, J.; Page,A. L. Water, Air, Soil Pollut. 1977,9, 289. (8) Street, J. J.; Lideay, W. L.;Sabey, B. R. J. Enuiron. Quul. 1977, 6,72.
by both clay minerals and soils. Benjamin and Leckieg found the same for the trace adsorption of Cu2+,Zn2+, Cd2+,and Pb2+onto amorphous iron oxyhydride. The use of Freundlich equation was also suggested in the works by van BemmelenlO and Sposito.'l In the theories of gas adsorption the applicability of the Freundlich equation was long ago associated with the energeticheterogeneity of the adsorptionsites on the actual solid surfaces. It was also known, that the Freundlich equation is a simplified form of a more general isotherm equation which is now commonly called the LangmuirFreundlich isotherm. Thus, in 1980 Sposito12suggested that it was probably time to use it also in the case of ion adsorption at waterloxide interfaces. Benjamin and Leckie were among the first who in 1981 initiated the studies of the surface heterogeneity effects? They reported that the adsorption of Me2+metal ions onto oxyferrihydridecould be described only by assuming a large dispersion of adsorption site affinities. Two years later Kinninburgh et al.I3 tried to correlatesuch adsorption isotherms by using other empirical equations employed earlier to correlate experimental adsorption isotherms for single gas adsorption onto heterogeneous solid surfaces. Theoretical studies of surface heterogeneity effects on ion adsorption within the electricaldoublelayer were much advanced by Koopal, van Riemsdijk, and co-workers.16'8 (9) Beniamii, M. M.; Leckie, J. 0.J. Colloid Interface Sci. 198c 79, 209. (10) Van Bemmelen, J. M. Landwirtsch. Ver. Stn. 1888, 36,69. (11) Spoeito, G.Soil Sci. Sac. Am. J. 1979,43,197. (12) Spoeito,G. Cation exchangein mils: An historical and theoretical perspective. In Chemietry in the soil enuironment; Dowdy, R. H., Baker, D., Volk, V., Ryan, J., Ede.; SSSA Spec.Publ.; Soil Science Society of America: Madieon, WI, 1980. (13) Kdburgh,M.M.:Barkea..J.A.:Whitfield.M. J . Colloidlnterface . Sci; I ~ S S95,376. , . (14) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L.K.; Blaakmeer, J. J. Colloid Interface Sci. 1986.109. 219. (16) Van Riekdijk, W. H.;de Wit, J. C. M.; Koopal, L. K.; BoltiG. H. J. Colloid Interface Sci. 1987,116, 611. (16) Van Riemsdijk, W. H.; Koopal, L.K.; de Wit, J. C. M. J. Agric.
----.--.
Sci. ... 1987. 36.241. ---
(17) Koopal, L. K.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1989,128,188. (18) Gibb, A. W. M.; Koopal, L.K. J. Colloid Interface Sci. 1990,134, 122.
0743-7463/9312409-2641$04.00/00 1993 American Chemical Society
Rudzinski et al.
2642 Langmuir, Vol. 9, No. 10, 1993 Rudzinski, Partyka, and c o - w ~ r k e r s lhave ~ * ~ subjected calorimetriceffects accompanying ion adsorption to their theoretical analyses. They showed that surface heterogeneity of oxides affected enthalpies of ion adsorption. A similar effect has been known for a long time in gas adsorption onto solid surfaces. In another paper, Rudzinski et alSz1 showed that titration curves were much less sensitive to the surface heterogeneity than individual adsorption isotherms of ions measured by using radiometric methods. Titration curves probably involve a certain mutual cancellationof heterogeneityeffects. This deserves some further experimental and theoretical studies. The adsorption of Me2+ metal ions at watedoxide interfaces at low ion concentration is a subject of continuously growing interest. There are three problems of great practical importance which stimulate that growing interest: (1) the adsorption in soil of highly poisoning cations of some heavy elements like Cd2+or Pb2+;(2) the adsorption in soil of radioactive ions in the areas where nuclear plants are located; (3)the adsorption of radioactive ions on corroded parts of nuclear plant installations. In all these systems, concentrations of the ions are low, and their adsorption characteristics are strongly affected by heterogeneity in the adsorption mechanism. The logarithm of the adsorbedamount, plotted vs the logarithm of the ion concentration in solution is always a Freundlich linear plot with a tangent much smaller than unity. However, at a sufficiently low ion concentration, a transition into a Henry's plot occurs,typical for a homogeneous solid surface, with the tangent equal to unity. This special behavior cannot be deduced from adsorption equations developed in previous publications,lkZ1 except the paper by Kinninburgh et al.13 They used an equation obtained by replacing bulk gas pressure by bulk ion concentration in the Toth empirical equation used to correlate experimental isotherms for single gas adsorption on solid surfaces. Van Riemsdijk et al.14J5 criticized Kinninburgh for ignoring the electrical properties of the interface. As the Toth equation is an empirical equation, one cannot know ita rigorous extension for a simultaneous, multicomponent, competitive adsorption. This means, the Toth equation cannot be used to describe the competition of metal ionswith other surfacecomplexes. Thus, the effect of pH or electrolyte concentration on metal adsorption cannot be studied. There is, therefore, a need to construct a theoretical description of the adsorption of ions within the electrical double layer, which could explain the special features of ion adsorption at low ion concentrations. This is the purpose of the present publication.
as surface reactions which cause the formation of the following surface complexes,
Theory 1. The Homogeneous Surface Model. Several theoretical approaches have been proposed to describe the equilibrium in ion adsorption at a water/oxide interface. The most commonlyused is the triple layer model proposed by Davis et al.2225 on which we will base our consideration. Thus, we considerthe dissociationreaction of the surface proton and the coadsorption of anions A- and cations C+
[SOH;l/I:
we obtain the following set of equilibrium equations,lS corresponding to the set of reactions 2a-e
(19)Rudzinski, W.; Charmas, R.; Partyka, S . Langmuir 1991,7,354. '(20) Charmas, R.;Bull. Pol. Acad. Sci. Chem. 1991,39,411. (21)Rudzinski, W.; Charmas, R.; Partyka, S.;Thomas,F.; Bottero, J. Y.Langmuir 1992,8,1154. (22)Davis, J. A.; Leckie, J. 0. J. Colloid Interface Sci. 1978,67,90. (23)Davis, J. A.;James, R. 0.; Leckie, J. 0. J. Colloid Interface Sci. 1978,63,480. (24)Davis, J. A.;Leckie, J. 0. J. Colloid Interface Sci. 1980,74, 32. (25)Davis, J. A.; Leckie, J. 0. In Chemical Modelling in Aqueolls S y s t e m ; Jenne, E. A., Ed.; American Chemical Society: Washington, DC, 1979,Chapter 15.
(26)Hayes, K.F.;Leckie,J. 0.J. ColloidlnterfuceSci. 1987,116,664. (27)Hayes, K.F.;Leckie,J. 0.ACSSympSer. 1986,No.323(Geochem. Process Miner. Surf.), 114. (28)Chisholm-Brause, C. J.; Brown, G. E., Jr.; Parka, G.A. Phyeica B 1989,158,616. (29)Chisholm-Brause, C. J.; Roe,A. L.; Hayes, K. F.; Brown, G. E., Jr.; Parka, G.A.; Leckie, J. 0. Physic0 E 1989,168,674.
-+ -
SOH;
S O P + H+
(la)
SO- + H+
(1b)
SO@ SOHCA-
+ H+ + ASOT+ + H+
SOH'
(IC)
SOH' C+ (Id) The equilibrium constants Kdht, Kdht, *KAmt,and *Kcht of the above reactions were defined in the l i t e r a t ~ r e . ~ " ~ ~ Then we consider the coadsorption of another cation (a strongly adsorbed bivalent ion), which is introduced to the system in the form of a M A 2 salt. Its addition can cause the formation of two kinds of surface complexes: inner-sphere complexes SOM+ formed at the O-plain of the triple layer and of outer-sphere complexes SO-M2+ formed at the @-plain.Some recent studies by Hayes and L e ~ k isuggest e ~ ~ that ~ the formation of the inner-sphere complexesis more probable for divalent cations like Cd2+, Pb2+, etc. than the formation of outer-sphere surface complexes. So, in generalZ927
-
SOH' + M2+ SOM' + H+ (le) where the equilibrium constant for this reaction, K&t, has the following form:
Formation of other possible surface complexes SOM+Aand SOMA connected with the formation of the surface complex SOM+ is not considered here, for the reasons explained by Hayes and Leckie.*% For our purpose, it will be useful to consider also the following equivalent reactions:
--
SO- + H+
(24
SO- + 2H+
(2b)
so-c+ so- + c+
(2c)
SOH' SOH; SOH;A-
- + - + SO-
SOM' Introducing the notation
Z = [SOH']
+ [SOH:] = e+,
SO-
2H+ + A-
(2d)
M2+
(2e)
+ [SOH;A-I + [SO-C'I + [SO-]+ [SOM'I
[SOH']/Z = Bo, [SO-C+]/Z = ,e, [SOM+]/I: = ,0
[SOH2+A-I/2= eA,
(3)
[so-]/Z = 1-eo - e+ - e, - eA-
e, = e-
Ion Adsorption at a WaterlOxide Interface
Langmuir, Vol. 9, No. 10, 1993 2643 where the values of the activity coefficient yi for z-valent ion 3” can be calculated from the ionic strength of the suspension, I (mol/dm3), and where A is a temperature dependent constant?l In A = -0.8484 + (0.6724 X lO*)p (11) To express +,(pH) dependence, which occurs in the equations for the individual adsorption isotherms, we accept here the relation used by Yates et al.32and by Bousse et al.B*a
where U H is the proton activity in the equilibrium bulk phase and UA, ac, and U M are the bulk activities of anion, cation, and the bivalent cation, respectively. Further, is the surface potential and +@ is the mean potential at the plane of specifically adsorbed counterions, which is given by (5) where the surface charge 60 is defined as follows 6,=B[0++0~+0~-0~-0~ and ]
B=N,e (6)
c1 in eq 5 is the first integral capacitance and N,in eq 6 is the surface density (sites/m2). One can solve numerically the system of the nonlinear eqs 4 to obtain the individual adsorption isotherms of ions, Oi (i = 0, +, A, C, M). For this purpose it is convenient torewrite the equation system 4 to the followingLangmuirlike form,
Kifi
0: =
(7)
1+ C K i f i a
2.303(PZC - pH) =
m)
sinh-’( e+0
(12)
where /3 is given by
In eq 13, CDL is the linearized double-layer capacitance. The value Of C D L C be ~ ~calculatedtheoretically (depending on the salt concentration in the solution) in the way described in Bousse’s workS3
1- 2kTle
112
: 1
(14) (8qeokTc) Catem where e, is the relative permittivity of the solvent, eo is the permittivity of the free space, and c is the concentration of the electrolyte (ions/m3). The value of the c,hm is assumed to be 0.2 F/m2. The nonlinear equation system 7 can be transformed into one nonlinear equation, taking into account eq 6 and until the concentration of the bivalent cation is so small that the surface complex, OM, does not affect the surface charge 60 CDL
6, = B
where
2+
K+f++ KAfA
- KCfC
1+ C K i f i
-
(15)
I
i = 0, +, A, C
and where fi (i = 0, +, A, C, M) are the following functions of proton and salt concentrations: f, = exp{-
{
fc = ac exp - -
fM
f+ = f:
- 2.3pH),
= aMexp{-
;kj
+-
$1
The activity of ions ai (i = A, C, M) can be calculated by considering their activity coefficient yj and the concentrations of ions. We assume that yi is given by the equation proposed by Davies30 log yi = -Azf
1’12
- 0.311 1+ 1’12
i = A, C, M (30) Davies, C.W.Ion Association; Butterworths: London, 1962.
As we are dealing here with adsorption at low divalent ion concentrations, we will accept eq 15. This nonlinear equation for 60 can be solved by means of the Mueller-S iteration method, to give the value of 60 for each pH. Having these values, we can evaluate the individual adsorption isotherms from eq 7. 2. Adsorption on Heterogeneous Surfaces. According to our previous paperl9the “intrinsic” constant Kj can be written as follows Kj = K: exp
{LT)
i = 0, +, A, C, M where ti is the adsorption (binding)energyof the ith surface complex, and Kj’ is related to ita molecular partition function. The surface heterogeneity will cause the variation of ti acrossthe surface, from one to another site. Then, for the reasons explained in our previous publication,21we accept (31) Rudzinski, W.; Charmas, R.; Partyka, 5.;Foieey,A.New J. Chem. 1991, 15, 327. (32) Yates, D.E.;Levine, S.; Hearly, T. W. J. Chem. SOC.,Faraday Trans. 1 1974, 70,1807. (33) B o w e , L.; de Rooij, N. F.; Bergveld, P. ZEEE Trans.Electron Deuices 1983,30, 8. (34)Vander Vlekkert,H.;Bousse,L.;de Rooij,N. F. J. Colloidlnterface Sci. 1988, 122, 336.
Rudzinski et al.
2644 Langmuir, Vol. 9, No. 10, 1993 the random model of surface topography. According to this model, adsorption sites characterized by different adsorption energies are distributed on a solid surface at random. Also variations in the local Coulombicforce fields $0 and $,936 will occur, but we will neglect them because Coulombic interactions are long-ranged. In the case of random topography, it will cause a “smoothing”of these variations over the local structure of the outermost surface oxygens. Thus, as in the model of a homogeneous oxide surface, we will consider $0 and $@ to be functions of the average compositionof the adsorbed phase. Physisorption interactions, (chemical binding forces), are short-ranged, so, the local variations in ei must be taken into account. The experimentally measured adsorption isotherms have to be related to the following averages, Oit
ij = 0, +, A, C, M where fj’s are the same as in eq 9, and the averagedquantity Oj, reads
where {a)is the set of the bulk concentrations UH,ac, UA, and UM, {e) is the set of the adsorption energies eo, e+, (A, ec, and EM, 0 is the physical domain of (e), and x((e))is the multidimentional differential distribution of the number of adsorption sites among various sets (e), normalized to unity
Above, xi(ci) is the differential distribution of the number of adsorption sites, among various values of ei, normalized to unity,
J-;- ~ ( { e ) )deode+d€Adec deM= 1
The kernel under the integral sign is Langmuir-like,so, the problem formulated above is identical with the one considered in dozens of the papers treating gas adsorption on heterogeneous solid surfaces. There, most of the adsorption equations used to correlate experimental data, were obtained by accepting the condensation approximation to perform the integration in eq 21. It is usually assumed, for the purpose of mathematical convenience, that Ai in eq 21 is the infinite interval (-=,+a). Equations developed by accepting such integration limits never reduce to Henry’s law. On the contrary, they usually work very well at moderate surface coverages. While investigatingsinglegas adsorption,Rudzinski and Everett36 have shown, that, in order to arrive at isotherm equations which would reduce correctly to Henry’s (linear) log-log plot, one must accept physically correct finite domain of ei. As the condensation approximation leads to isotherm expressions working well at moderate surface coverages, we tried first to apply this approximation, together with some finite integration limits e j and epl. This can be doneby applyingthe Rudzinski-Jagiello approach.” For that purpose we write the kernel of the integral in eq 21 in the following form
(18)
We now have to consider the fundamental physical question of whether the variables ei are totally independent. In other words, whether the value of ei is correlated in a way with ejzi (iJ = 0, +, A, C, M)when they change from one to another site. The degree of the correlation between ei and ej+i will affect the result of the integration in eq 17. So far, only two extreme models have been considered. The first model assumes, that for all adsorption sites, the difference between the adsorption energies t i and ej is constant and equal to Aij. The other extreme model assumes that the energies ei and tj+i are not correlated at all. An argument for the second model is that it has led us to predict serious discrepanciesbetween PZC and IEP values which are often observed.21 These two models represent somewhat extreme views, and the truth probably lies somewhere in between. We felt, that it was still too early to provide a definite answer for the fundamental question of which one of these two models is closer to reality. There might be still some important factors which were not taken into account, and consequences of accepting some approximate solutions seemed to deserve further study. 2A. The Model Assuming High Correlations between Adsorption Energies of Various Surface Complexes. We begin with considering the first model assuming high correlations between the adsorption energies ei and ej+i, Le.,
+ Aji
(19a) So, the intrinsic equilibrium constant Kj can be written as follows ej
ei
where (19c)
where
When adsorption runs at very small,or very high surface coverages,then it is necessary to consider the existence of minimum and maximum values of ti: e j and qm in real physical systems. Next, the integral in eq 21 is evaluated by using the integration by parts
The equation system 7 can then be written as follows (36)Barrow,N. J.; Gerth,J.; BnZmmer,G. W. J.Soil Sci. 1989,40,437.
(36) Rudzinski, W.; Everett, D. H. Adsorption of Cases on Heterogeneous Surface; Academic Press: New York, 1991.
Ion Adsorption at a WaterlOxide Interface
Langmuir, Vol. 9, No. 10,1993 2645
and might be thought to be generally valid, until the surface is strongly heterogeneous. The next step accordingto the Rudzinski-JagieUo procedure would, thus, be using the analytical form of function 29 in eq 33 in which the condition of continuity of xi(ti) is already coded. When til becomes largely negative, and tim becomes largely positive, Le., when e) --m and tim +-m, then Bje(ttm) 1and Ojz(t)) 0. So, reducesto the expression obtained by applyingthe condensation approximation and assuming infinite integration limits
ejt = -
ti,
X i ( t i 1 ) , ti
Xi(ti) =
Xi(fi.),
t; ti
Iti Iti.
kjfj
~
K:fj [xK:fjlkTlei Xi(tiC)= TKjofi 1+ [CK:fjlkT/ci Zkjfj J
< ti1
-
- -
-
(34)
J
J
where (28)
> tim
where xi and Xi on the right-hand side of eqs 27 and 28 are given by appropriate analytical expressions. As in our previous publication,21we will represent Xi(ti) by the following Gaussian-like function
(35) Equation 34 was first published by Van Riemsdijk et al.14J5 and it was used as a starting point in our previous publication21to investigate the features of the model assuming high correlations to exist between ti and t j # i . Thus, eq 33 representathe generalizationof this expression for the case of finite energylimita. As it should be expected, Oj, in eq 33 does not depend upon the integration constant in eq 26. Now, let us remark that
where FNis the normalizationfactor given by the condition Thus, for the function xi(ci) defined in eq 29 FiN
[
iy)] [ +
= 1 + exp -
- 1 e x p l q ) ] (31)
The function xi(ti) defined in eq 29 is discontinuous at the points E ) and elm, but the actual functions Xi(ti) in the real adsorption systems will be continuous at these points with all their derivatives. Therefore, one can evaluate the integral on the right-hand side of eq 25 by expanding Xi into ita Taylor series around the point ti = tic, at which (aBje/&i) reaches its sharp maximum
g) I
4fi = exp(
and the normalization factor F i N reads
FiN= [l + (4fi)kT’Ci]-1 - [l + (4im)kT/ci]-1(37) where C i s are appropriate expansion coefficients. The properties of this expansion were investigated by Jagiello,97 in the case of the infinite integration limits (--OD,+-). When the surface is strongly heterogeneous, one can neglect safely all the terms on the right-hand side of eq 32, except the first leading one. Assuming that this is true also in the regime of low surface concentrations of ions, Ojt is given by
Further
+
-1 [l (4im)kT/”’1-’ Xi(ti.) = FN -1 X.(e‘) (4n)kT/”]-’ I t = - [l
+
FiN
(39)
kf; and
e&9[xi(tic)- xi(t;)ij(33) Expression 33 is not related to a particular form of Xi(ti) (37) Jagidlo, J. Ph.D. Thesis, Department of Theoretical Chemistry, Maria Curie Sklodoweka University, Lublin, Poland, 1984. (38) E h h i d i , M. A.;O’Connor, G. A. Soil Sci. SOC.Am. J. 1982,46, 1163.
tic
= -kT ln[xK:fjl
+ ti0
J
This notation creates links to our previous publication,l where KjO was introduced. Now, we have three more constants, e), tam, and qo, to be considered in our investigation.
Rudzimki et ai.
2646 Langnuir, Vol. 9, No. 10, 1993
[K?fj]k T / c j
According to the definitions 28 and 33, Bjt)s are defined as follows
ojt =
(48)
+
1 c[Kjofi]kT/cj J
KjOfj
for cic < c)
ejt = -8:(cim),
(41b)
CKjOfj
ejt = (1- Ceit)rejc~jm)rxj(~jm) - xj(tjci,)i+
I
and when c) IciC Ielm, Bjt(cic) is given by eq 33. At intermediate values of eic far from e,J and tim, ejt reduces practically to eq 34. 2B. The Model Assuming Lack of Correlations between Adsorption Energies. Now, let us consider the second model of adsorption assuming that no correlations exist between the adsorption energies ei, cj#i. To show in a simple way how the equations for 0 j t ) S are now developed,we consider for the moment the simpler case, when €1 -m, cIm +m. When ci and tj#i are not correlated at all,the adsorption of the complex “i”is influenced by the presence of another complex “j”,only through arandom blocking of the surface sites SO-. Thus
- -
where accordingto the Rudzinski-Jagiello30 approach, the function t j c is found from the condition
(3)
=o
(43)
fj’t,C
The derivative (d26j/&j2) is evaluated from the equation system 4. While evaluating that derivative, one has to remember that $0 and $b are not longer functions of Oj’s, but of 0 j t ) S . This is the consequenceof acceptingthe model of random surface topography, in which the adsorption sites characterizedby different values of cj’s are distributed on an oxide surface at random. Consequently, Coulombic interactions depend on the averaged values 6jt)s. While applying the condition 43 to the equation system 4, we obtain cjc
Equation 48 was a starting point in our previous studies of the model assuming lack of correlations between the adsorption energies of different surface complexes.’ In the general case, when one takes into account the physical limits, mcI, c j , instead of equation 33, the condensation approximation leads us to the following expression for ejt
= -kT In K / f i
i#I
-
ejct,’)[xjctjc) x j ( t , ’ ) i ~(49)
where
Equation 49 denotes symbolically a system of nonlinear equations to be solved by computer with respect to Bjis. However, ita approximate but sufficiently accurate analytical solution is also possible, by making a certain assumption which is fully justified in our case. Except for the bivalent metal ion, the surface coverages by other surface complexes reach moderate values. For such values the right-hand side of eq 49 reduces practically to eq 42. Then, because the contribution from the bivalent ion complex to CBit is small, one can evaluate the terms (1- C i Z j B i t ) in eq 49 by accepting the relation 48 for all &t)s (i = 0, +, A, C). Then
While doing so, we obtain
ejt =
1
+ (K;fj)hT/cj kT/c/ e I. ( cI . “ ) [ x j ( c j m ) - Xj(tjC)l+ 0
1+ -
f$
-
e j ( t / ) [ X j ( t j c )Xj(c,’)lJ
1
(44)
for c) Itic Itim (52a)
+ (K?fJkT/‘j
where K j = K / e x p (dT)
(45)
ejt =
When x(tj)is the function 29 (when FIN = l),eq 42 takes the form
.
ej(cjm)
J ‘J’
for ciC < e! (52c)
+
1 x(K:fj)rT/cj I
where
And because
equ 46 takes finally the form
Xj(tjC) =
-1
(Kiofi)kT/cj
F ~ 1N+ (Kiofi)kT/cj
(54)
and expressions for F J ~Xj(cIm), , and X j ( b ) i.e. eqs 37 and 38ab remain unchanged. In the limit of very low concentrations of the bivalent ion j = M, Le., when f M 0, ita presence does not affect practically the term 1- C i + j B i t ) and eq 51 becomes correct.
-
Zon Adsorption at a WaterlOxide Interface
Langmuir, Vol. 9, No. 10,1993 2647
When, on the contrary,the surface coverage by the bivalent ions reaches moderate values, equation system 48 applies to all surface complexes, and eq 51 becomes correct too. In the region of intermediate coverages (the transition from Henry's to Freundlich's plot) the terms (Kjofj)kTlcj for j = M in the numerator and denominator on the righthand side of eq 52 may affect the behavior of Ojt (j = M) to some extent. However, it is obvious that it is the expression within the big brackets on the right-hand side of eq 52 which governs mainly the behavior of 8jt (j = M). 2C. Consequencesof Accepting the Condensation Approximation To Describe the Ion Adsorption at Low Ion Concentrations. In the previous sections, we obtained two different sets of equations for 8jt'~,i.e. eq 33 and eq 52, corresponding to two different correlation models. These equations contain the same parameters, Kio,ci/kT, etm, e,; tio, and the same variables f i , SO one can compare easily the differences in the behavior of these two sets of the solutions for Ojt'S. Of course, in the second model, instead of one set of parameters, ci/kT, e,=, e?, and eio, we will have at least four sets, or even five, when the coadsorptionof the accompanyingcation M2+is considered (i = 0, +,C, A, M). However, for the purpose of comparison we made all of them equal. Then, because the surface coverageby the bivalent metal ions is small compared to all others, we will neglect its effectson the PZC value. Thus, the problem of establishing the relations between the intrinsic constants Kalint,Kdht, *Kcht, and * K A ~from ~ , the condition that the experimentally observed PZC is practically independent upon the salt concentration, is essentially the same as the one considered in our previous publication.21 Thus, putting ac = UA = a, and solving the set of equations21 Go(pH=PZC)= 0
and
z[Go(pH=PZC)l a =0
(55)
for the model of a homogeneous oxide surface one obtains
where
H
(57)
In the case of the model assuming high correlations between the adsorption energies ej, ci#j (section 2A), we have
whereas in the model assuming lack of correlations between the energies ei, ci#j (section 2B), these interrelations are obtained by solving the following system of nonlinear equations
The correlations 66,58,and 59 between the parameters K@, Kdbt and *Kcht, *KAit reduce the number of the
-'I
I d 1 M
I
I
I
25-C p H =5.10
a -2-
u
L
I
'
Cu / Fc203. H20 l a m )
-
- -3 -4n -2 b
I
I
1
(i
L
-
o?
2 -3-
-4
-2 c N
L
Pb/FeZ% H 2 0 lam) I
-
-3-
,
25-t pH.6 4 Zn/FeZO3HZ0(aml
I
-75
I
I O I M
-70
- 6 5
/-' ---slope
I
-60
log ( M e
1
I
-55 2+
= I O
-50
I
-45
-40
'
)
Figure 1. Experimental adsorption isotherms of the bivalent ions Cu2+, Pb2+,and Zna+ adsorbed on the amorphous iron oxyhydroxide,reported by Benjamin and Leckie (Figure7 in ref 9).
parameters; intrinsic equilibrium constants can be found by fitting experimental data, from four to two.21 In the case of a heterogeneous surface, eqs 55 are solved by putting in e4 6 expressions for Bit's instead of q s . Thus, in a first step, the solution of eqs 55 leads us to appropriate interrelations of kiO's. The expressions 58 and 59 are, therefore, now the relations between the intrinsic constants, corresponding to the most probable energies eio's. All the adsorption parameters appearing either in eq 33 or in eq 52, except for those related to the bivalent ion adsorption, can be found by an appropriate analysis of titration data, radiometric measurements, and other independent experimentalmeasurementa. It is, therefore, recommended that fundamental studies of bivalent metal ion adsorption be accompanied by suitable additional experiments. Such a fortunate situation can be found in the case of the data published by Benjamin and Leckie (see Figure 1) on Cu2+, Pb2+, Zn2+, and Cd2+ adsorption on an amorphous iron oxyhydroxide.9 The corresponding titration data were analyzed by Davis and Leckie22in terms of the homogeneous surface model, and the estimated intrinsic equilibrium constants and other parameters were collected in Table I of their paper.= Although the tabulated values are expected to differ to some extent from appropriate values which would be found by applying a more realistic model of a heterogeneousoxide surface,we treated them as a good starting point in a certain model investigation. Thus, we took these values and then tried to find such values of the bivalent ion parameters that they could fit the data presented in Figure l(A,C), either by accepting eq 33 or eq 52. After numerous numerical exercises, we came to the conclusion that this goal can never be achieved. Figures 2 and 4 show some of the ineffective results. The model investigation based on eq 33 (Figure 2) shows that the application of the condensation approximation and the assumption of the high correlations between the adsorption energies of ions does not suggest even two different linear regions to exist on the log-log plots.
2648 Langmuir, Vol. 9, No.10,1993 O
Rudzimki et
41.
4r
/
h
. "
J
Part(1)
-1
3.. h
Ea
I:
-1-
ul
2 -2-
2-
I 2
v
161
E . x
0"
_I
0-
-2 -1
J
-5-
-3
-G
-8
-7
-6
-5
-4
QM
Figure 2. Results of our model investigation based on eqs 33 and 41by assuminghigh correlationsto exist between adsorption energiesof ions. The numbersat the theoretical lines correspond to the numbers of the parameter sets presented in Table 11. Changing the parameter kT/c does not practically affect the calculated theoretical isotherms. That approach predicts only Henry's region to exist, up to high surface coverages. (The adsorption isotherms for other surface complexes Bit, i = 0, +, A, C, are oalculated by accepting the infinite integration limits (--,+a)). Our model investigation based on eq 52 is presented in Figure 4. One can see here a transition region between Henry's region (slopeequal to unity) and Freundlich region (slopesmaller than unity), but the tangent in this transition region reaches values larger than unity. Such behavior has never been observed in an experiment (see Figure 3). Thus, our computer exercises based on eqs 52 led us to the conclusion that the condensation approximation cannot be applied to the region of low surface coverages by some species, where the transition occurs from Henry to Freundlich behavior. This will be somewhat striking in view of the general belief that the condensation approximation is a very good approach for such strongly heterogeneous surfaces (kT/cj smaller than 0.9). However, the different behavior of the log-log plots presented in Figures 2 and 4 suggests that the degree of the correlations between the adsorption energies of ions may affect strongly the behavior of these adsorption systems at low ion concentrations. This seems to create hope that the studies of the adsorption at low ion concentrations may provide the answer to the fundamental question, which correlation model is to be accepted. Therefore we have decided that in equation system 42, the Bjt function related to the metal ion adsorption (Cu2+, Pb2+,Zn2+, Cd2+, ...I will be represented by the exact formula
and we will look for a possibility of performing the integration of eq 60 in an exactly analytical way. Such a possibility arises when one assumes that x(ej)in eq 60 is
-2
-3
0
-1
3
2
1
Log C (ppm )
-3
-L
Log
-5
Figure 3. Experimental adsorption isotherms of zinc adsorbed on Glendale and R-28soils,reported by Elrashidi and O'Connor (Figure 1 in ref 38).
-1
I
-8
/
-7
I
I
I
-6
-5
-L -Log
I
-3 OM
Figure 4. Results of our model investigation bawd on eqs 52 by assuming lack of correlations between adsorption energies of ions. The numbers at the theoretical lines correspond to the numbers of the parameter sets presented in Table 111. the rectangular energy distribution,
The rectangular distribution is a good approximation in the case of strongly heterogeneous surfaces.% And this is just the case of the bivalent metal ion adsorption (Freundlich's plots with cj/kTsmaller than 0.9). At the same
Zon Adsorption at a WaterlOxide Interface
time, the term (1 - CizjOit) will still be represented by the expression 51, according to our discussion at the end of the previous section. In this way, one arrives at the following expression for the adsorption isotherm of the bivalent ion
ejt =
+ 1+ c(K?fj)kT/cjejm - e) 1 (KjOfj)kT/cj kT
+
1 exp[(ejm- cjc)/kT]
In 1+ exp[(e) - ejc)/kT]
(62) = 0, +, A, C) values are still calculated as in
Other Ojt 0’ eq 48. And now we face the situation that the same physical function xj(tj) is represented by two analytical functions (29) and (62). However, we have already known, that in the case of a strongly heterogeneous surface, these functions should affect 0, in a similarway, if the parameters cj,, :E and cJm are suitably chosen. So, we took the integration limits ,:e e,m as the best-fit parameters and adjusted C j in eq 62 in such a way that at higher surface concentrations Ojt, the tangent of the Freundlich plot for Ojt is equal to kT/cj. The corresponding isotherm equation for the model assuming high correlations between the adsorption energies of different surface complexes, reads
Langmuir, Vol. 9, No. 10, 1993 2649
Table I. Values of the Parameters Obtained by Davis and Lgckie,P Who Analyzed their Titration Data for the Amorphous Iron Oxyhydroxide in Terms of the Homogeneous surface Model pKarht 5.10 p*Kc”t = 9.00 P K , =~10.70 ~ p*K~”t 6.90 N,= 10 siteslnmz PZC = 7.90 I = 0.1 M c1 = 1.4 F/m2 T=25OC
Table 11. Heterogeneous Model (High Correlations). (2) 6.4 0.7 4.0 -8 8 0
(1)
PH
6.4 0.7 4.0
kTlc
PK&~ t1
(kJ)
-m
em (kJ)
+m
€0 (kJ)
0.7 3.5 -10 10
(5) 6.4 0.7 4.0 -10 10
0
0
0
6.4
(6) 6.4 0.7 4.5 -10 10 0
The seta of the parametersused by ua in our model investigation shown in Figure 2. Other parameters are those presented in Table I, except that pKdht and p * K ~are ~ t now equal to 11.90 and 6.80, respectively. These new values were calculated from the values of pKalht,p*Kc”t, and PZC a~reported in Table I using eq 58 and the value of kTlc shown in this table. -1 r
0
I
-zt
Cu/ Fe,O,.
H,O l a m i
Other ejt 0’ = 0, +, A, C) functions for this model are still calculated as in eq 34, because other ions appear at moderate concentrations. So, the contribution from the nonphysical intervals (-=,E:) and (eJm,+-) does not affect much the calculated Ojt functions.
Results and Discussion First we have carried out some model investigations to study the behavior of the ejt functions, predicted by eqs 61 and 62. Then, our model investigation showed that Ojt functions calculated by using eq 62 never exhibited the transition from Henry’s to Freundlich’splot. They exhibit Henry’s behavior up to quite high surface coverages. We have postponed presenting these curves, because their behavior is very similar to that observed in Figure 4. Thus, it was not the condensation approximation responsible for that nonphysical behavior but the assumption of the high correlations between the adsorption energies of different surface complexes expressed in eq 19a. As the transition from Henry’s to Freundlich’s plot has been observed in adsorption on so different materials, we arrive at the following important conclusion. The model assuming high correlations between the adsorption energies of various surface complexes is to be abandoned in the studies of ion adsorption within the electrical double layer formed at water/ oxide interfaces. On the contrary, our model calculations showed that eq 61 corresponding to the assumption that no correlations exist between the adsorption energies of different surface complexes, could reproduce very well the behavior observed in Figures 1and 3. This finding corresponds well to another one published in our previous paper.21 There, only the equations obtained by assuming lack of the correlations could explain the commonly observed differences between the PZC and IEP values. Rather than to show an extensive model investigation based on eq 61, we are going to demonstrate its utility to fit the experimental data. We took the parameters obtained by Davis and Leckie analyzing their titration curves (Table I),calculated the corresponding pKaht and
(4)
(3) 6.4 0.7 4.0 -12 12
I
-1
/
I
I
I
I
i1 -3t
-&I/ \slope=,
-8
-7
0
I
I
,
-6
-5
-L log
1
-3 %I
Figure 5. Results of our best-fit exercises based on eq 6 2 (A) The comparison of the experimentaladsorptionisothermof Cu2+ ions adsorbed at pH = 5.1, reported by Benjamin and Leckie? with our theoretical curves calculated by wing eq 62 and the parameters collected in Tables I and IV. (B) The comparison of the experimental adsorption isotherm of Zn2+ions adsorbed at pH = 6.4, reported by Benjamin and Leckie? with our theoretical curves calculated by using eq 62and the parameters collected in Tables I and IV. p*KAmtvalues from relation 59, and then adjusted the three other parameters p K ~ h t ad, , and e# to fit the experimental isotherms of Cu2+ and Zn2+ions shown in Figure 1. The results of these numerical exercises are shown in Figure 5, and the obtained best-fit parameters are collected in Table IV. This was a pretty raw fit, and we suppose that we could arrive a t a similar agreement with slightly different values of these parameters. We must, however, realize that the Davis and Leckie values in Table I were obtained by accepting the rather raw assumption of a homogeneous solid surface and only the parameters pKdht and ~ * K A ~ ~
Rudzinski et 01.
2650 Langmuir, Vol. 9,No. 10,1993 -2 r
Table 111. Heterogeneous Model (Lack of Correlations). PH kT/@ kT'cM PK~P cd(kJ) cp(kJ) t ~ (kJ) '
(1) 6.4
(2) 6.4
0.7 0.1
0.7 0.7
4.0
4.0 -8
-m
+=
(3) 6.4 0.1
(4) 6.4
(5) 6.4
0.7
0.7
(6) 6.4 0.1
0.7
8
4.0 -12 12
0.1 3.5 -10 10
0.7 4.0 -10 10
4.5 -10 10
0
0
0
0
0
/ 1
Cd/Fe,O,.H,O(om) 0,
0
-3
0.7
a The seta of the parameters used by us in our model investigation shown in Figure 4. Other parameters are those presented in Table I, excep that pKaht and p*K~htare now equal to 10.70 and 6.80, respectively. These new values were calculated from the values of pKalht,p*Kcht, and PZC as reported in Table I using eq 59 and the values of kT/q shown in this table. b For i = 0, +, C, A.
Table IV. The Sets of the Parameters Used by Us in Our Model Investigation Shown in Figure 5. _____ (A) Cu/FezOs.HzO (am) pK@ 2.40 cd -16 kJ PP = 12 kJ with these parameters the slope of the Freundlich log-log plot is 0.4, so we took kT/cM = 0.4 to calculate the isotherm in Henry's and in the transition regions
-8
-7
-
-6
-5
-3
-4
log a H
Figure 7. Comparison of our theoretical adsorption isotherm calculated by using eq 62 with the experimental data of Cd*+ adsorption measured by Benjamin and Leckie. (Data from Figure 5 of ref 9). The parameters are the same as those used to fit the data in Figure 6 and the two additional parameters ed and c p appearing in eq 62 are equal to -8 and 8 kJ/mol, respectively.
- Log (Me'') 6
5
4
3
2
1
(B)Zn/FezOs.HzO (am) c M =~ -2.0 kJ
pK@' 3.60 = 2.5 kJ wth these parameters the slope of the Freundlich log-log plot is 0.65,so we took kT/cM = 0.65 to calculate the isotherm in Henry's and in the transition regions a Other parameters are those collected inTable I, except that and ~ * Kare Anow ~ equal ~ to 10.10 and 6.80, respectively. These new values were calculated from the values of pK.,lht, p*K@t, and PZC as reported in Table I using the relation 59 and the values of kT/ci reported in this table. A
-
I
25OC 4.0 7.5
I
7.0
6.5 6.0
,
5.0 -log [CdI,q
5.5
I
4.5
1 I
1
4.0 3.5
I 3.0
Figure 6. Effects of pH on the isotherms of cadmium ions, adsorbed on the amorphous iron oxyhydroxide investigated by Benjamin and Leckie.s The solid l i e s were drawn by Benjamin and Leckie as the best linear regression for their experimental points. Our theoretical lines were calculated from eq 48 by using the parameters pK.~,~t, p+Kcht, PZC, N,,c1, I , and T collected in Table I, kT/c = 0.7 ( i = 0, +, A, C) and the parameters pKaht = 10.7 and ~ K = A 6.8 calculated ~ ~ by us from eqs 69. Our theoreticallinea calculatedin thie way match exactlythe Benjamin and Leckie solid lines if the best fit parameter pK#t = 3.9.
were calculated by us using the correct relation 59 corresponding to the assumption that the oxide surface is heterogeneous. The philosophy lying behind that best-fit exercise arose from our earlier observation that surface heterogeneity does not affect much titration curves and the estimated surface complexation parameters.21 As we have already discussedz1there must exist some compensation effects in these composite adsorption isotherms. On the contrary, we found, that surface heterogeneity affects strongly the individual adsorption isotherms of the surface complexes, measured radiometrically or in another way. This, of course,is also true in the case of the individually measured isotherms of adsorption of the bivalent ions.
9
8
7
6
5
4
-tog IZn")
Figure 8. (A) The set of the theoretical adsorption isotherms calculated by Dzombak and Morel (Figure 2.4 in ref 39). rMe denotes the sorption density expressed in mol/mol. (B) The experimental Freundlich log-log plots reported by Barrow et al. for the zinc sorption on goethite. (Figure 4 from ref 36). Unite for the zinc sorption rb are pmol/g.
Using eq 48, we also fitted the isotherms of adsorption of cadmium ions reported by Benjamin and Leckieg for three different pH values. These isotherms are shown in Figure 6 (original Figure 3 in the work of Benjamin and Leckieg). This time we required that with the same set of the best fit parameters the Calculatedisotherms matched the measured ones €orall the three pH values. The reason why we used eq 48 instead of eq 61 was that the isotherms for cadmium adsorption were reported only for the coverage region corresponding to Feundlich behavior. In this region simple eq 48 can safely be used. As the slope of all these isotherms can easily be determined to be equal to 0.66 (kT/cM = 0.66) for all the investigated pH values, we have only one unknown parameter, pK&t. Then, with only one best fit parameter, pKr,Pt = 3.9,we arrive at an excellent agreement with the experimental data. These calculated adsorption isotherms match ideally the solid lines in this figure, drawn by Benjamin and Leckie as the best linear regression for their experimental points.
Ion Adsorption at a WaterlOxide Interface
For pH = 5.8 Benjamin and Leckies measured the adsorption at much smaller concentrations of Cd2+ions. For such small concentrationsthey observed the transition to Henry’s log-log plot with tangent equal to unity. According to our theory the adsorption in that region is affected by the physical energy limits ad and a p . So, no surprise, that in order to fit these data we were forced to take into account that physical condition, i.e. to use the more general eq 62. While fitting these data we required that the theoretical log-log plot must have atangent equal to 0.66 in the region where it becomes a Freundlich log-log plot (like in Figure 6). Then we took the same parameters which were used to fit the data in Figure 6, and adjusted only in addition of cd and c p . Figure 7 shows the fit obtained when ad - -8 kJ/mol and tp = 8 kJ/mol. Finally, we would like to comment on the works by Dzombak and Morels-1 dealing with the transition from Henry’s log-log plots to the Freundlich ones. They argue that the heterogeneity of ferric oxides may well be represented by assuming that there are two kinds of sites: the strongly adsorbing surface oxygenscoordinated to two (39) Dzombak, D. A.; Morel, M. M. Surface Complexation Modeling, Hydrous Ferric Oxide; John Wiley I%Sons,Inc.: New York, 1990. (40)Farley, K . J.; Dzombak, D. A.; Morel, M. M. J. Colloid Interface Sci. 1985, 106, 226. (41) Dzombak, D. A.; Morel, M. M. J. Colloid Interface Sci. 1986,112, 588.
Langmuir, Vol. 9, No. 10, 1993 2651
metal atoms, and the weakly adsorbing ones coordinated to one metal atom. The ratio of the strong to weak adsorption sites should be equal to 0.005/0.2. Figure 8A shows the theoretical adsorption isotherms calculated by Dzombak and Morel30by accepting a certain reasonable set of adsorption parameters. One can see that these theoretical isotherms do not reproduce the two following features observed in the experiments: (1)The Freundlich log-log plots must be linear over several orders of ion concentrations. (2) The Freundlich log-log plots corresponding to different pH values must be parallel. These important features of the experimental Freundlich log-log plots were studied thoroughly by Barrow et aLs6 Figure 8B presents the experimental plots reported by Barrow et al.% for goethite. They found that their log-log plots presented in Figure 8B could be correlated by the following linear equation log S = -4.17 + 0.55 log c + 0.765pH (64) where S is the amount adsorbed and c is the zinc ion concentration expressed in ~mol/dm3. It is therefore concluded that the dual-site model is not adequate enough to represent the surface heterogeneity of the actual oxide surfaces. On the contrary, our approach assuming a large variety of existing surface sites can reproduce well the behavior of the actual adsorption systems described by Barrow’s eq 64.
