Calorimetric Effects Accompanying Ion Adsorption at the Charged

5210

Langmuir 1998, 14, 5210-5225

Calorimetric Effects Accompanying Ion Adsorption at the Charged Metal Oxide/Electrolyte Interfaces: Effects of Oxide Surface Energetic Heterogeneity W. Rudzin´ski,*,† R. Charmas, and W. Piasecki Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie Skłodowska University, Pl. Marii Curie Skłodowskiej 3, 20-031 Lublin, Poland

F. Thomas, F. Villieras, B. Prelot, and J. M. Cases Laboratoire Environnement et Mineralurgie, ENSG and URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy Cedex, France Received January 5, 1998. In Final Form: May 27, 1998 A quantitative theoretical analysis of the enthaplic effects accompanying ion adsorption at the oxide/ electrolyte interface, based on a model of energetically heterogeneous surface oxygens, is presented. The triple layer complexation model is accepted, along with the 2-pK charging mechanism. For the purpose of illustration a set of experimental data is subjected to that quantitative analysis including titration curves, radiometrically measured individual iostherms of ions, and calorimetric titration data for the alumina/NaCl electrolyte system. Two models of energetic heterogeneity were taken into consideration. One of them assumes that the binding-to-oxygen energies of the surface complexes vary but are highly correlated when going from one to another surface oxygen. The other model of surface heterogeneity assumes that these correlations are very small. Our numerical simultaneous analysis of the titration data, of the individual isotherms of Na+ and Cl- adsorption, and of the accompanying heat effects advocates strongly for the model of surface heterogeneity assuming small correlations to exist. A good simultaneous fit of all three kinds of experimental data is obtained, with a small uncertainty as for the values of the estimated adsorption parameters. A simultaneous fit of the measured enthalpic effects appears to be an especially strong criterion for a proper choice of adsorption parameters.

Introduction Studies of surface heterogeneity of oxide surfaces have a long history in the area of adsorption and catalysis. In catalysis, the surface defects in oxide surfaces are believed to be the catalytic centers for various catalytic reactions. For many other reactions the variation of the acid-base strength on oxide adsorption sites is of crucial importance. So, it is no surprise that the studies of surface heterogeneity of oxides were the subject of dozens of papers published by the scientists working on catalysis. When a metal oxide is brought into contact with an electrolyte, the outermost surface oxygens adsorb one or two protons, a cation, or an aggregate composed of two protons and an anion. In that way various surface complexes are formed and an electrically charged interface is developed. Because oxides are one of the most important components of soils, the above described adsorption of protons, anions, and cations influence several processes of environmental concern such as transport of contaminants and nutrient availability. So, the ion adsorption at the oxide/ electrolyte interfaces has been a subject of an extensive study by many scientists active in the areas of “soil science” and “environmental science”. The actual oxide surfaces are, as a rule, more or less geometrically distorted. This causes a variation, from one surface oxygen to another, of the binding-to-surface energy for each of these surface complexes. That energetic heterogeneity of the actual oxide surfaces must affect the * To whom the correspondence should be addressed. † Fax: +48 81 5375685. E-mail: Rudzinsk@ hermes.umcs.lublin.pl.

adsorption of ions within the electrical double layer formed at the oxide/electrolyte interfaces. In the late 1970s, Davis and Leckie1 and Benjamin and Leckie2 reported that the adsorption of Me2+ ions onto ferrihydride cannot be described by theories of ion adsorption onto a homogeneous solid surface. An agreement between theory and experiment could only be obtained by assuming a large dispersion of site affinities. Two years later Kinniburgh et al.3 demonstrated that Toth’s4 isotherm equations for adsorption on heterogeneous surfaces apply best to these adsorption systems. This problem was next considered by van Riemsdijk et al.,5,6 who criticized Kinninburgh et al.3 for ignoring the electrical properties of the interface. They demonstrated that by taking the coulombic interactions between adsorbed ions into account, the agreement between theory and experiment was greatly improved. Finally, they concluded that their ion adsorption isotherms were rather insensitive to the surface heterogeneity. Their conclusion deserves a certain explanation, in view of the well-established image of the heterogeneity of the really existing oxide surfaces. This is true not only in the case of the amorphous oxides. Even if the interior of an (1) Davis, J. A; Leckie, J. O. J. Colloid Interface Sci. 1978, 67, 90. (2) Benjamin, M. M.; Leckie, J. O. J. Colloid Interface Sci. 1981, 79, 209. (3) Kinniburgh, D. G.; Barkes, J. A.; Whitfield, M. J. Colloid Interface Sci. 1983, 95, 370. (4) Toth, J.; Rudzin´ski, W.; Waksmundzki, A.; Jaroniec, M.; Sokolowski, S. J. Acta Chim. Hung. 1974, 82, 11. (5) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, L. K. J. Colloid Interface Sci. 1986, 109, 219. (6) Van Riemsdijk, W. H.; De Wit, J. C. M.; Koopal, L. K.; Bolt, G. H. J. Colloid Interface Sci. 1987, 116, 511.

S0743-7463(98)00043-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 08/12/1998

Calorimetric Effects Accompanying Ion Adsorption

oxide exhibits a regular crystallographic structure, its surface shows usually a much less degree of organization, discussed in our previous publications.7-10 The various theoretical approaches were mainly tested by fitting the related expressions to the experimental titration curvessthe most commonly measured adsorption characteristics. This surface charge isotherm is a composite one, so there are many factors affecting its behavior. As it is to be expected in such a case, there will then occur several compensating effects. Also, some of the physical parameters may be highly correlated. In other words, the surface charge isotherms, i.e. the titration curve, will not be sensitive to a physical model underlying the corresponding computer fit. This fact bothered some of the researchers like Sposito11 who wrote: “... the surface complexation models are, in a sense, too successful, i.e., several different models can represent the same set of adsorption data equally well with corresponding chemical parameters in the model taking quite different values”. In some instances where single (individual) isotherms of surface complexes could be measured, the failure to fit them by isotherm equations corresponding to a model of a homogeneous oxide surface was reported long ago. These were the studies of trace adsorption of such ions as Cu2+, Zn2+, Cd2+, and Pb2+, carried out as early as the beginning of the seventies. However, except for the measurements of the adsorption of the poisoning ions of heavy metals, studies of individual adsorption isotherms of ions were rarely reported. Then, it has been also known for a long time in adsorption science that calorimetric effects of adsorption are much more sensitive to surface energetic heterogeneity than adsorption isotherms. The reported experimental heat effects accompanying single gas adsorption could not, as a rule, be reproduced by theoretical expressions corresponding to various models of adsorption on a homogeneous solid surface.12 Thus, it is expected that calorimetric studies of ion adsorption at the oxide/electrolyte interface may put more light on the role played in these systems by the energetic heterogeneity of oxide surfaces. The first studies of enthalpic effects of ion adsorption, based on temperature dependence of adsorption isotherms, started 25 years ago. It was Berube and de Bruyn13 who studied the effect of temperature on PZC. Studies of that kind were next conducted by Tewari et al.,14,15 Fokkink et al.,16,17 Schwarz and co-workers,18 Kuo and Yen,19 Akratopulu et al.,20,21 Blesa et al.,22 and Kosmulski et al.23,24 The influence of temperature on the surface charge vs pH (7) Rudzin´ski, W.; Charmas, R.; Partyka, S. Langmuir 1991, 7, 354. (8) Rudzin´ski, W.; Charmas, R.; Partyka, S.; Foissy, A. New J. Chem. 1991, 15, 327. (9) Rudzin´ski, W.; Charmas, R.; Partyka, S.; Thomas, F.; Bottero, J. Y. Langmuir 1992, 8, 1154. (10) Rudzin´ski, W.; Charmas, R.; Partyka, S.; Bottero, J. Y. Langmuir 1993, 9, 2641. (11) Sposito, G. J. Colloid Interface Sci. 1993, 91, 329. (12) Rudzin´ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London/New York, 1992. (13) Berube, Y. G.; De Bruyn, P. L. J. Colloid Interface Sci. 1968, 27, 305. (14) Tewari, P. H.; McLean, A. W. J. Colloid Interface Sci. 1972, 40, 267. (15) Tewari, P. H.; Campbell, A. B. J. Colloid Interface Sci. 1976, 55, 531. (16) Fokkink, L. G. J.; De Keizer, A.; Lyklema, J. J. Colloid Interface Sci. 1989, 127, 116. (17) Fokkink, L. G. J. Ph.D. Thesis, Agricultural University, Wageningen, The Netherlands, 1987. (18) Subramanian, S.; Schwarz, J. A.; Hejase, Z. J. Catal. 1989, 117, 512. (19) Kuo, J. F.; Yen, T. F. J. Colloid Interface Sci. 1989, 121, 220.

Langmuir, Vol. 14, No. 18, 1998 5211

profiles was studied by Blesa and co-workers,25,26 Brady,27 and Kosmulski.28 Valuable information concerning that problem has been collected in the review by Machesky et al.29 First direct calorimetric experiments were reported by Griffiths and Fuerstenau30 and by Foissy,31 who measured the heat of immersion of an outgassed solid sample into the solutions of changing pH. The titration calorimetry is a better experiment for theoretical interpretation. It was reported first by De Kaizer, Fokkink, and Lyklema,32 Machesky et al.,33-35 Mehr et al.,36 and Casey.37 After introduction of an outgassed solid sample into a solution, the pH of that solution is measured and sometimes also the concentration of other ions in the equilibrium bulk electrolyte. Then, a titration step is carried out (from base or acid side), and the evolved heat is recorded. The accompanying measurements of ion (proton, cation, anion) adsorption are very useful for further theoretical interpretation. And this is the way in which Machesky and Anderson33 and Machesky and Jacobs34,35 carried out their experiments. De Kaizer, Fokkink, and Lyklema32 monitored the heat effects accompanying ion adsorption as the function of the surface charge. Kallay et al.38-40 have recently proposed a titration experiment aimed at determining the experimental conditions under which the obtained experimental data could be free of the coulombic contribution to the measured heat of adsorption. It seems, however, that the progress on the experimental side has not been accompanied by a suitable progress in the theoretical description of the enthalpic effects accompanying ion adsorption. Interpretation of the experimental data was carried out mostly on a qualitative level. The first quantitative interpretation was proposed by De Keizer et al.32 in 1990. One year later, we published our first theoretical papers concerning that problem.7,8 On the basis of the triple layer model, we developed corresponding equations for the heats of adsorption on both a homogeneous and a heterogeneous oxide surface. (20) Akratopulu, K. Ch.; Vordonis, L.; Lycourghiotis, A. J. Chem. Soc., Faraday Trans. 1 1986, 82, 3697. (21) Akratopulu, K. Ch.; Kordulis, C.; Lycourghiotis, A. J. Chem. Soc., Faraday Trans. 1990, 86, 3437. (22) Blesa, M. A.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1990, 140, 287. (23) Kosmulski, M.; Matysiak, J.; Szczypa, J. J. Colloid Interface Sci. 1994, 164, 280. (24) Kosmulski, M.; Matysiak, J.; Szczypa, J. Presented at the 8th International Conference on Surface and Colloid Science; Adelaide, Feb 13-18, 1994. (25) Blesa, M. A.; Figliolia, N. M.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1984, 101, 410. (26) Regazzoni, A. E. Ph.D. Thesis, Universidad National de Tucuman, Tucuman, Argentina, 1984. (27) Brady, P. V. Geochim. Cosmochim. Acta 1992, 56, 2941. (28) Kosmulski, M. Colloids Surf. A: Physicochem. Eng. Aspects 1994, 83, 237. (29) Machesky, M. L. In Chemical Modelling in Aqueous Systems II; Melchior, D. C., Bassett, R. L., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990; Vol. 416. (30) Griffiths, D. A; Fuerstenau, D. W. J. Colloid Interface Sci. 1981, 80, 271. (31) Foissy, A. Ph.D. Thesis, Universite de Franche-Comte: Bescancon, France, 1985. (32) De Keizer, A.; Fokkink, L. G. J.; Lyklema, J. Colloid Surf. 1990, 49, 149. (33) Machesky, M. L.; Anderson, M. A. Langmuir 1986, 2, 582. (34) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 297. (35) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 315. (36) Mehr, S. R.; Eatough, D. J.; Hansen, L. D.; Lewis, E. A.; Davis, J. A. Thermochim. Acta 1989, 154, 129. (37) Casay, W. H. J. Colloid Interface Sci. 1994, 163, 407. (38) Kallay, N.; Zˇ alac, S.; Stefanic, G. Langmuir 1993, 9, 3457. (39) Kallay, N.; Zˇ alac, S. Croat. Chem. Acta 1994, 67, 467. (40) Zˇ alac, S., Kallay, N. Croat. Chem. Acta 1996, 69, 119.

5212 Langmuir, Vol. 14, No. 18, 1998

Using these equations, we analyzed enthalpy changes measured directly by using immersional adsorption calorimetry. Very recently, an exhaustive theoretical study of these enthalpic effects was published by us.41,42 That study also takes into consideration the data obtained by using titration calorimetry. A theoretical study based on phenomenological thermodynamics has also been recently published by Hall.43 The already published experimental and theoretical studies of the enthalpic effects accompanying ion adsorption at the oxide/electrolyte interface have shown that these studies may contribute to our understanding of the mechanism of adsorption in these systems. These studies are also very important for the scientists active in the area of soil science and environmental science. The reason for that is the following. As we have shown in our previous publications,41,42 the theoretical description of temperature dependence of ion adsorption involves introducing the same functions and parameters which appear in the theoretical description of the enthalpic effects accompanying ion adsorption. In large areas of our planet, which are important for growing plants for food, the changing seasons are a source of temperature changes in soil as much as by 40 K. Of course, such temperature changes must affect strongly the adsorption of ions. So, calorimetric studies of ion adsorption should be an important source of information about the above discussed temperature effects. As surface energetic heterogenity is known to affect strongly the calorimetric effects of adsorption, it is essential that the relevant theoretical studies should be based on a realistic model taking into account the surface energetic heterogeneity. The theoretical description of surface heterogeneity effects on ion adsorption from an electrolyte solution onto surface oxygen atoms involves three characteristics of the surface energetic heterogeneity,12 to be considered: (1) the differential distribution of the number of adsorption sites among corresponding values of adsorption energy, for every kind of the formed surface complexes; (2) the topography of the surface, i.e. the way in which various adsorption sites are distributed on a solid surface; (3) the correlations between adsorption energies of various surface complexes, when going from one to another surface oxygen. In our first paper,7 we accepted a simple model by assuming high correlations to exist. Unfortunately, that model of oxide surface heterogeneity failed, later on, to represent bivalent ion adsorption.10 On the contrary, their adsorption could, successfully, be described by accepting a model by assuming small correlations between adsorption energies of various surface complexes.10 That experience has encouraged us to use that model for developing a new theoretical description of the enthalpic effects accompanying ion adsorption at the oxide/electrolyte interface. That new theoretical description is the subject of the present publication.

Rudzin´ ski et al.

proposed by Davis et al.44-46 and showing it schematically in the paper by Davis et al.44 or in our previous paper.8 Below, we are going to discuss briefly the principles and definitions which will be necessary for our further consideration. Thus, we consider the adsorption of protons and the coadsorption of anions A- and cations C+ as surface reactions leading to the formation of the following surface complexes int K a1

SOH2+ 798 SOH0 + H+ int K a2

SOH0 798 SO- + H+ *K int A

SOH2+A- 798 SOH0 + H+ + A*K int C

SOH0 + C+ 798 SO- C+ + H+

(41) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Cases, J. M.; Franç ois, M.; Villieras, F.; Michot, L. J. Colloids Surf. 1998, 137, 57. (42) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Kallay, N.; Cases, J. M.; Franç ois, M.; Villieras, F.; Michot, L. J. Adsorption, in press. (43) Hall, D. G. Langmuir 1997, 13, 91.

(1b) (1c) (1d)

int int int The equilibrium constants K int a1 , K a2 , *K A , and *K C of the above reactions were defined in the literature.44-46 For our purpose, it will be useful to consider also the following equivalent reactions:

SOH0 T SO- + H+

(2a)

SOH2+ T SO- + 2H+

(2b)

SO-C+ T SO- + C+

(2c)

SOH2+A- T SO- + 2H+ + A-

(2d)

Introducing the notation

N ˜ s ) [SOH0] + [SOH2+] + [SOH2+A-] + [SO-C+] + [SO-] [SOH2+]/N ˜ s ) θ+ [SOH0]/N ˜ s ) θ0 [SO-C+]/N ˜ s ) θC (3) ˜ s ) θA [SOH2+A-]/N [SO-]/N ˜ s ) 1 - θ 0 - θ + - θC - θA ) θwe arrive at the following set of equilibrium equations, corresponding to the set of reactions 2a-d:

K int a2 exp

{ } { } { }

int K int a1 K a2 exp

(aH)θeψ0 ) kT θ0

int K int a2 *K A exp

{

(4a)

2eψ0 (aH)2θ) kT θ+

(4b)

(aC)θeψβ ) kT θC

(4c)

int K int a2 /*K C exp

Theory Principles of the Adsorption Model. The starting point of our consideration will be the triple layer model

(1a)

}

e(2ψ0 - ψβ) (aH)2(aA)θ) kT θA

(4d)

(44) Davis, J. A.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 480. (45) Davis, J. A.; Leckie, J. O. In Chemical Modelling Aqueous Systems; Jenne, E. A., Ed.; American Chemical Society: Washington, DC, 1979; Chapter 15. (46) Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1980, 74, 32.


Langmuir, Vol. 14, No. 18, 1998 5213

where aH is the proton activity in the equilibrium bulk solution and aA and aC are the bulk activities of anion and cation, respectively. Further, ψ0 is the surface potential and ψβ is the mean potential at the plane of specifically adsorbed counterions, which is given by

ψβ ) ψ 0 -

δ0 c1

(5)

where the surface charge δ0 is defined as follows:

δ0 ) B[θ+ + θA - θ- - θC]

B ) Ns e

(6)

c1 in eq 5 is the first integral capacitance, and Ns 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, θi (i ) 0, +, A, C). For this purpose it is convenient to rewrite the equation system 4 to the following Langmuirlike form

θi )

K if i 1+

i ) 0, +, A, C

∑i Kifi

where β is given by

β)

( )

2e2Ns K int a2 ctkT K int a1

1/2

(11)

In eq 11, ct is the linearized double-layer capacitance. The value of ct can be calculated theoretically (depending on the salt concentration in the solution) in the way described in the Bousse’s work.49 The way of solving the nonlinear equation system 7 to calculate the individual adsorption isothems θi’s and the surface charge δ0 as a function of pH was shown in our previous papers.8,9 From the condition that the experimentally observed PZC is practically independent of the salt concentration, int the relations between the intrinsic constants K int a1 , K a2 , int int *K C , and *K A are established in the way discussed in our previous publication.9 Thus, putting aC ) aA ) a and solving the set of equations9

δ0(pH ) PZC) ) 0

(7)

∂ [δ (pH ) PZC)] ) 0 (12) ∂a 0

for the model of a homogeneous oxide surface, one obtains

where

K0 )

KC )

1 1 K+ ) int int int K a2 K a1 K a2

*K int C K int a2

KA )

*K int A )

1 int K int a2 *K A

(8)

and where fi, (i ) 0, +, A, C) are the following functions of proton and salt concentrations:

{

f0 ) exp -

eψ0 - 2.3pH kT

{

fC ) aC exp -

{

fA ) aA exp -

}

f+ ) f02

}

eδ0 eψ0 + kT kTc1

(9a,b)

(9c)

eψ0 eδ0 - 4.6pH kT kTc1

}

(9d)

The activity of ions ai (i ) A, C) can be calculated by considering their activity coefficient γi and the concentrations of ions. We assume8 that γi is given by the equation proposed by Davies.47 To express ψ0(pH) dependence, which occurs in the equations for the individual adsorption isotherms, we accept here the relation used by Yates et al.,48 by Bousse et al.,49 and by Van der Vlekkert et al.,50

2.303(PZC - pH) )

( )

eψ0 eψ0 + sinh-1 kT βkT

(10)

(47) Davies, C. W. Ion Association; Butterworths: London, 1962. (48) Yates, D. E.; Levine, S.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1974, 70, 1807. (49) Bousse, L.; De Rooij, N. F.; Bergveld, P. IEEE Trans. Electron Devices 1983, 30, 8. (50) Van der Vlekkert, H.; Bousse, L.; De Rooij, N. F. J. Colloid Interface Sci. 1988, 122, 336.

H2 *K int C

K int a2 )

H2 K int a1

(13a,b)

where H ) 10-PZC. Titration Calorimetry. The theoretical interpretation of the “batch adsorption calorimetry” is rather difficult as we discussed in our previous publication.51 Much easier for interpretation are the heat effects measured in the experiment called “titration calorimetry”.33-40 Here after introduction of an outgassed solid sample into a solution, the pH of that solution is measured and sometimes also the concentration of other ions in the equilibrium bulk electrolyte. Then, a titration step is carried out (from the base or acid side), and the evolved heat is recorded. That heat effect is given by the expression

M

[

∫pHpH+∆pH Q0(∂pH0 )T + Q+(∂pH+ )T + QC(∂pHC )T + ∂θ

∂θ

∂θ

( )]

QA

∂θA ∂pH

T

dpH (14)

where Q0, Q+, QC, and QA are the molar differential heats of the formation of SOH0, SOH2+, SO-C+, and SOH2+Asurface complexes and M is expressed in moles. Depending on the reported experiments, the consumption (adsorption) of protons and the ions of the inert electrolyte is monitored, accompanying the change of pH by ∆pH. And this is the way in which Machesky and Jacobs34 carried out their experiment. They ascribed the heat evolved on changing pH solely to proton adsorption (consumption) also monitored in their experiment. The heat effect, Qpr, considered by these authors is described by the following equation:41,51 (51) Rudzin´ski, W.; Charmas, R.; Cases, J. M.; Franç ois, M.; Villieras, F., Michot, L. J. Langmuir 1997, 13, 483.

5214 Langmuir, Vol. 14, No. 18, 1998

Rudzin´ ski et al.

( ) [( ) ( ) ( )] ∂θi

Table 1. Surface Reactions, Equilibrium Constants, and Heats of Adsorption

∫pHpH+∆pH∑Qi ∂pH T dpH i

Qpr )

∂θ+

∂θA

∂θ0

∫pHpH+∆pH 2 ∂pH T + 2 ∂pH T +

∂pH

dpH

T

(15)

i ) 0, +, A, C Here Qi is the molar heat of formation of the ith complex. Three years ago, Kallay and co-workers38-40 started a theoretical-experimental study aimed at determining the experimental conditions under which the obtained experimental data could be free of the coulombic contributions to the measured heats of adsorption and refer only to the nonconfigurational heat values. This would eliminate the necessity of carrying out a complicated theoretical-numerical analysis. In one of their papers the authors postulated that the “average” heat of proton adsorption should correspond to the heat of titration from (PZC ∆pH ) to (PZC + ∆pH ). The consumption of protons from the bulk solution was monitored in the Kallay’s experiment; so, the “average” molar heat of proton adsorption, QAv is given by42,51

( ) [( ) ( ) ( )] ∂θi

equil constants

heat of reacn

-pK int a1 -pK int a2 -p*K int A -p*K int C int -pK a1 - pK int a2 int -pK int a2 - p*K A int int p*K C - pK a2 int pK int a1 - p*K A

Qa1 Qa2 QaA - Qa2 Qa2 - QaC Qa1 + Qa2 QaA QaC QaA - Qa1 - Qa2

defined by

∂θ+

∂θA

PZC+∆pH 2 +2 + ∫PZC-∆pH ∂pH T ∂pH T

∂θ0

∂pH

T

(16)

The conditions under which Kallay’s method may lead to a reliable estimation have been discussed in our recent publication.42 The molar differential heats of adsorption Qi’s are “configurational”, as they must depend on the concentrations of the surface complexes θi’s. Their detailed explicit form can be developed by applying the appropriate thermodynamic relations,51

[

]

{θi}

]

b ∂ µSO-C+ - µC QC ) -k kT ∂(1/T)

[

( ) ∂F0

∂(1/T)

{θi}

] ( )

[

b ∂ µSOH2+ - 2µH Q+ ) -k kT ∂(1/T)

[

)k

{θi}

{θi}

∂F+

)k

)k

b b ∂ µSOH2+A- - µA - 2µH QA ) -k kT ∂(1/T)

∂(1/T)

( )

{θi}

∂FC

∂(1/T)

]

eδ0 eψ0 + ln aA - 4.6pH kT kTc1 θA ln ) 0 (18d) θ-

(17a)

Then neglecting the temperature dependence of pH 50 and taking into account that the constant set of θi’s also means a constant value of δ0, from eqs 18 we arrive at the following expressions for configurational heats of formation of different surface complexes, Qi:51

Q0 ) Qa2 - eψ0 -

QC ) QaC - eψ0 -

( )

QA ) QaA - eψ0 -

∂FA

∂(1/T)

{θi}

2e ∂ψ0 T ∂(1/T)

{θi}

(19a) (19b)

( )

( )

( )

δ0 e ∂ψ0 -e T ∂(1/T) {θi},pH c1 ∂ln aA eδ0T ∂c1 +k (19d) 2 ∂T ∂(1/T) {θi} (c )

( )

( )

where Qa1, Qa2, QaC, and QaA are the nonconfigurational heats of appropriate reactions presented in Table 1,

)

( )

{θi}

δ0 e ∂ψ0 +e + T ∂(1/T) {θi},pH c1 ∂ln aC eδ0T ∂c1 +k (19c) 2 ∂T ∂(1/T) {θi} (c ) 1

(17b)

(17c)

( ) ( )

e ∂ψ0 T ∂(1/T)

Q+ ) Qa2 + Qa1 - 2eψ0 -

1

{θi}

k

{θi}

eδ0 eψ0 θC + + ln aC - ln ) 0 1 kT θkTc (18c)

int FC ) -ln(K int a2 /*K C ) -

dpH

i ) 0, +, A, C

b ∂ µSOH - µH Q0 ) -k kT ∂(1/T)

θ+ 2eψ0 - 4.6pH - ln ) 0 kT θ(18b)

int F+ ) -ln(K int a1 K a2 ) -

i

QAv )

θ0 eψ0 - 2.3pH - ln ) 0 (18a) kT θ-

F0 ) -lnK int a2 -

int FA ) -ln(K int a2 *K A ) -

∫PZC-∆pH ∑Qi ∂pH T dpH PZC+∆pH

reacn type SOH0 + H+ T SOH2+ SO- + H+ T SOH0 SOH0 + H+ + A- T SOH2+ASO-C+ + H+ T SOH0 + C+ SO- + 2H+ T SOH2+ SO- + 2H+ + A- T SOH2+ASO- + C+ T SO-C+ SOH2+ + A- T SOH2+A-

(17d)

where µSO-C+, µSOH2+A-, µSOH, and µSOH2+ are the chemical potentials of these surface complexes whereas µbj (j ) H, C, A) are those in the bulk phase of proton, cation, and anion of the basic electrolyte, respectively. In particular cases for a homogeneous oxide surface model considered in this section, the functions of Fi (i ) 0, +, A, C) are

dln K int ai

Qai ) -k

d(1/T)

i ) 1, 2

int dln(K int a2 /*KC )

QaC ) -k

d(1/T) int dln(K int a2 *K A )

QaA ) -k

d(1/T)

(20a)

(20b)

(20c)


Langmuir, Vol. 14, No. 18, 1998 5215

While calculating the derivative (∂c1/∂T){θi}.pH, we assumed52,53 that there are two different values of the c1 parameter, depending on the pH value, one for the acidic region (pH < PZC ) and another one for the basic region (pH > PZC ). Then, following Blesa’s recommendation,54 we treated c1 as the linear functions of temperature,

c1 )

cL1

cL,0 1

)

+

RL1 ∆T

pH < PZC

(21a)

R c1 ) cR1 ) cR,0 1 + R1 ∆T pH > PZC

(21b)

which may be considered as the formal Taylor expansions R R,0 around T ) T0 so that cL1 (T0) ) cL,0 1 and c1 (T0) ) c1 . 8 that the We have shown in our previous publication derivatives (∂ ln ai/∂(1/T)){θi}, i ) C and A, occurring in eqs 19 can be expressed as follows:

∂ln ai ∂(1/T)

) -2.0172 × 10-8T4 ln γi i ) A, C (22)

Let us consider finally the derivatives (∂ψ0/∂(1/T))pH occurring in eqs 19. After evaluating them from eq 10, we arrive at the following explicit expression:

( ) ∂ψ0

∂(1/T)

pH

[

) -ψ0T

]

t Qa2 - Qa1 β + + β+t β+t 2kT kT β ∂PZC 2.3 (23) e β + t ∂(1/T)

Here

t)

1

x( ) eψ0 βkT

2

(23a) +1

In eq 23, the derivative (∂PZC/∂(1/T)) is replaced by the term (Qa1 + Qa2)/4.6k, according to the result of the formal differentiations of PZC in eq 13a and the definition of Qa1 and Qa2 in eq 20a,

∂PZC Qa1 + Qa2 ) 2.3 2k ∂(1/T)

(24a)

Similarly to eq 13a, the same differentiation with respect to (1/T) in eq 13b leads to the result

∂PZC QaA - QaC ) 2.3 2k ∂(1/T)

(24b)

in view of the definitions of QaC and QaA in eqs 20b,c. From eqs 24,b we have

QaA - (Qa1 + Qa2) ) QaC

(25)

Except for the functions Qi, the derivatives (∂θi/∂pH)T from eqs 15 and 16 for the homogeneous model of TLM were developed earlier,8 but for the reader’s convinience they are given in the Appendix. Adsorption on Heterogeneous Surfaces. According to our previous paper7 the “intrinsic” constant Ki can be (52) Sprycha, R. J. Colloid Interface Sci. 1984, 102, 173. (53) Blesa, M. A.; Kallay, N. Adv. Colloid Interface Sci. 1988, 28, 111. (54) Blesa, M. A.; Figliolia, N. M.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1984, 101, 410.

written as follows:

Ki ) K′i exp

{ } i kT

i ) 0, +, A, C

(26)

Here i is the adsorption (binding) energy of the ith surface complex and K′i is related to its molecular partition function. The surface heterogeneity will cause the variation of i across the surface, from one site to another. Then, for the reasons explained in our previous publication9 we accept 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 coulombic force fields ψ0 and ψβ55 will occur, but we will neglect them because coulombic interactions are long-ranged. In the case of random topography, this will cause “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 ψ0 and ψβ to be functions of the average composition of the adsorbed phase. Physisorption interactions (chemical binding forces) are short-ranged, so the local variations in i must be taken into account. The experimentally measured adsorption isotherms have to be related to the following averages, θit,

θit({a}, T) )

∫‚‚‚∫ θi({}{a},T)χ({}) d0 d+ dA dC Ω

(27)

where {a} is the set of the bulk concentrations, aH, aC, and aA, {} is the set of the adsorption energies, 0, +, A, and C, Ω is the physical domain of {}, and χ({}) is the multidimensional differential distribution of the number of adsorption sites among various sets {}, normalized to unity:

∫‚‚‚∫ χ({}) d0 d+ dA dC ) 1 Ω

(28)

Now we have to consider the fundamental physical question of whether the variables i are totally independent. In other words, whether the value of i is correlated in a way with j*i (i, j ) 0, +, A, C) when they change from one site to another. The degree of the correlation between i and j*i will affect the result of the integration in eq 27. So far, only two extreme models have been considered. The first one assumes that, for all the adsorption sites, the difference between the adsorption energies i and j is constant and equal to ∆ij. The other extreme model assumes that the energies i and j*i are not correlated at all. These two models represent somewhat extreme views, and the truth probably lies somewhere in between. The Model Assuming High Correlations among Adsorption Energies of Various Surface Complexes. We begin with considering the first model assuming high correlations to exist between the adsorption energies i and j*i, i.e.,

j ) i + ∆ji

(29)

In our previous publications7,8,10 we used the following function to represent the differential distribution of the (55) Barrow, N. J.; Gerth, J.; Bru¨mmer, G. W. J. Soil Sci. 1989, 40, 437.

5216 Langmuir, Vol. 14, No. 18, 1998

Rudzin´ ski et al.

number of adsorption sites, among various values of i, normalized to unity, χi(i),

χi(i) )

{ } { }]

i - 0i 1 exp ci ci

[

(30)

This is a Gaussian-like, fully symmetrical function, centered around i ) 0i . The heterogeneity parameter ci is proportional to the variance of χi(i), i.e. πci/x3. Then, after rigorous thermodynamic treatment7,8,10 it follows that θjt in this model takes the explicit form

θjt )

K 0j fj

∑ 1 + [∑j K 0j fj]kT/c 0 kT/c j K j fj]

[

∑

0 j K j fj

{

(

}

)

(32)

kT/c-1

*K int A )

H2 *K int C (33a,b)

Now, let us consider the calorimetric effects predicted by this model. The explicit expressions for Fi’s functions are obtained from the eqs 31,

F0 ) -ln K int a2 -

eψ0 - 2.3pH - ln φ0 ) 0 kT

int F+ ) -ln(K int a1 K a2 ) -

int FC ) -ln(K int a2 /*K C ) -

int FA ) -ln(K int a2 *K A ) -

(34a)

2eψ0 - 4.6pH - ln φ+ ) 0 kT (34b)

eδ0 eψ0 + + ln aC kT kTc1 ln φC ) 0 (34c)

eδ0 eψ0 + ln aA - 4.6pH kT kTc1 ln φA ) 0 (34d)

where

φi )

(

)

θit 1 - θ-t 1 - θ-t θ-t

c/kT

(37)

∂PZC ) 1 ∂ T Qa1 + Qa2 Qa1 kT Qa2 1 kT H2 - T - 1+ rˇ ln int int 2k c 2k 2k 2 c K a1 K a2 rˇ 1 kT 1- 1[1 + rˇ] 2 c (38) 2.3

0i + ∆ji kT

H2 H H2 + int int int int K a1 K a2 K a1 K a2

1 - θ-t ∂ln φi ) -c ln i ) 0, +, A, C 1 θ-t ∂ T

-k

From eq 33a we obtain

The problem of establishing the relations between the int int int intrinsic constants K int a1 , K a2 , *K C , and *K A using the Rudzin´ski-Charmas criterion leads in this model to the following equations:8,9

K int a2 )

(36)

where the superscript (c) stands for high correlations and Qi is the heat for the model of a homogeneous oxide surface. Thus, surface energetic heterogeneity is the source of an additional configurational heat effect defined by the second term on the right-hand side of eq 36, which has the following form:

j ) 0, +, A, C (31)

where c0 ) c+ ) cC ) cA ) c, and

˜ j exp Kj0 ) K

∂ln φi i ) 0, +, A, C 1 ∂ T

Qi(c) ) Qi - k

2

i - 0i 1 + exp ci

complexes, the molar differential heats of adsorption Qi(c) are given by

i ) 0, +, A, C

(35)

and where θ-t ) 1 - ∑iθit is the function of the free surface oxygens. Thus, for the considered model assuming high correlations to exist between the adsorption energies of surface

( )[

]

(

(

)

)

where

rˇ )

H +H

Ka1int

(38a)

Considering the denominator in eq 38a one can see that ˇ e 0.01, and we can put safely if PZC - pK int a1 g 2, then r in the denominator of eq 38 1 + rˇ = 1. An inspection into the data reported in the literature must bring one to the conclusion that it is usually true. Then, the denominators in eq 38 can be well approximated by (1 + kT/c)/2. Next from eq 33b we have

2.3

∂PZC QaA - QaC ) 1 2k ∂ T

(39)

From eqs 38 and 39 one can eliminate one of the four parameters, Qa1, Qa2, QaC, and QaA, similarly as we did it previously in the case of homogeneous surface model. To calculate Qpr defined in eq 15, we still need to know the form of the derivatives (∂θit/∂pH)T for this model. Their explicit form is given in the Appendix. So far, calorimetric titrations have not been considered in terms of this model. The Model Assuming Small Correlations among Adsorption Energies of Various Surface Complexes. In one of our previous publications9 we considered a model in which the adsorption of the complex “i” is influenced by the presence of another complex “j”, only through a random blocking of the surface sites SO-. Thus,

θjt ) -(1 -

θit)χi(ic) ∑ i*j

(40)

where, according to the Rudzin˜ski-Jagiello approach,12 the function jc is found from the condition


( ) ∂2θj ∂j2

)0

Langmuir, Vol. 14, No. 18, 1998 5217

(41)

j)jc

The derivative (∂2θj/∂j2) is evaluated from the equation system 4. While evaluating that derivative, one has to remember that ψ0 and ψβ are not longer functions of θj’s but of θjt’s. This is the consequence of accepting the model of random surface topography, in which the adsorption sites characterized by different values of i’s are distributed on an oxide surface at random. Consequently, coulombic interactions depend on the averaged values θjt’s. While applying the condition (41) to the equation system 4, we obtain

jc ) -kT ln K′jfj

(42)

{ }

(43)

Kj ) K′j exp

j kT

When χj(j) is the function (30), eq 40 takes the form

[K 0j fj]kT/cj 1+

∑j[K 0j fj]kT/c

j

j ) 0, +, A, C

( ) (

( )(

a2

A

)

2eψ0 - 4.6pH kT θ+t c+ ln ) 0 (46b) kT θ-t

int FC ) - ln(K int a2 /*K C ) -

eδ0 eψ0 + + ln aC kT kTc1 θCt cC ln ) 0 (46c) kT θ-t

eδ0 eψ0 + ln aA - 4.6pH kT kTc1 θAt cA ln ) 0 (46d) kT θ-t

and the molar heats of adsorption Qi(s) take now the form

(44)

)

kT kT/cC cC *K int H C a + 1 - 1 ) 0 (45a) int kT K int K int a1 K a2 a2 cA int kT/cC kT/cA 2 kT H a kT *K C a ) 0 (45b) cA K int*K int cC K int kT/c+

int F+ ) - ln(K int a1 K a2 ) -

Qi(s) ) Qi - ci ln

The equation of that type has been known for a long time in the theories of mixed-gas adsorption on solid surfaces and was called the “multicomponent LangmuirFreudlich isotherm”. It was used as an empirical equation at the beginning, and its first derivation based on eq 40 was proposed by Rudzin´ski and co-workers.56,57 The assumption 40 about a random mutual blocking of the surface by competing components led Rudzin´ski to postulate that the special edition of the master eq 40 represented by eq 44 reflects the situation when no correlations exist between adsorption energies of various components. Recently, however, the multicomponent LangmuirFreundlich isotherm has been subjected to a rigorous analysis by Rush et al.,58 who showed that the analytical form of eq 44 corresponds, in fact, to the situation when small correlations exist between adsorption energies of various components (complexes). While applying the Rudzin´ski-Charmas criterion for CIP to exist, we arrive at the following interrelation of the equilibrium constants:9 2

c0 θ0t eψ0 - 2.3pH ln ) 0 (46a) kT kT θ-t

int FA ) - ln(K int a2 *K A ) -

where

θjt )

F0 ) - ln K int a2 -

( ) a2

The related expressions for the calorimetric effects of adsorption have never been considered. The functions Fi’s to be used in calculating the heats effects predicted by this model are given by (56) Rudzin´ski, W. In Chromatographic Theory and Basis Principles; Jonsson, J. A., Ed.; Marcel Dekker: New York, 1988; p 245. (57) Rudzin´ski, W.; Nieszporek, K.; Moon, H.; Rhee, H.-K. Heterogeneous Chem. Rev. 1994, 1, 275. (58) Rusch, U.; Borkovec, M.; Daicic, J.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1997, 191, 247.

θit i ) 0, +, A, C θ-t

(47)

where the superscript (s) stands for small-correlated adsorption energies and Qi’s are still the expressions for a homogeneous solid surface. The second term on the right-hand side of eq 47 represents now the additional configurational contribution due to the surface energetic heterogeneity. From eqs 45a,b we can eliminate H to obtain the following interrelation of the adsorption equilibrium constants:

[ [ ( )( ) ] ] [ [( ) ] ]

c+/kT kT int kT/cC cC *K C a int -1 G ) ln K int a1 K a2 1 kT K int a2 cA cA/kT kT int int int kT/cC K a2 *K A cC *K C a ) 0 (48) ln a kT K int a2 cA

Now, we calculate the derivative ∂G/∂(1/T) to arrive at the following interrelation of the nonconfigurational heats of adsorption:

[ ( )( ) ]

kT cC Qa1 + Qa2 T ln 1 -1 k kT kT c+ cA

( )(

)

kT cC *K int C a -1 kT K int a2 cA

QaA -

kT/cC

T ln

*K int C a

( ) *K int C a

(

K int a2

kT cC × kT c+

kT/cC

K int a2 -

QaC 1 ∂a a ∂(1/T) k

)(

)

kT/cC *K int c a 1-1 kT/cA K int a2

kT/cC Q kT/cA aC T + ln k kT cA

()( )

kT/cC

-

kT kT cC cC 1 ∂a )0 - 1kT kT a ∂(1/T) cA cA (49)

5218 Langmuir, Vol. 14, No. 18, 1998

Rudzin´ ski et al.

And, again, from eqs 49 we can eliminate one of the four parameters, Qa1, Qa2, QaC, and QaA. The derivative ∂PZC/∂(1/T) is to be evaluated from equation system 45a,b. This can also be done by solving first this equation system to eliminate one of the equilibrium constants, *K int C , for instance,

(

H

2

)

kT/c+

int K int a1 K a2

( )(

kT cC + -1 kT cA

kT cA H2a int kT K *K int a2 A cC

)

kT/cA

-1)0 (50)

From eq 50 we evaluate now ∂PZC/∂(1/T),

4.6

{

(

kT H2 ∂PZC ) 1 c+ K int K int a1 a2 ∂ T

T ln

(

H

2

int K int a1 K a2

)]

) [ kT/c+

Qa1 + Qa2 k

( )( ) [ ( )]}/[( ) ( )( ) ]

kT cA kT H2a + 1int cA kT K *K int a2 A cC

H2a 1 ∂a - T ln int int a ∂(1/T) K a2 *K A

kT/cA

H2 int K int a1 K a2

kT/c+

kT cA H2a 1kT K int*K int a2 A cC

QaA + k

+

kT/cA

(51)

To calculate Qpr defined in eq 15, we still need to know the form of the derivatives (∂θit/∂pH)T for this model. Their explicit form is given in the Appendix. Discussion So far only a few experiments have been reported where, in the course of titration, also the accompanying heat effects were measured upon the addition of an acid or base. The first experiments of that kind were performed by Machesky and Anderson33 in 1986, by Mehr et al.36 in 1989, De Keizer et al.32 in 1990, Machesky and Jacobs34,35 in 1991, Kallay et al.38 in 1993, and Casey37 in 1994. While selecting the experimental data for our analysis, we looked for those which could be easily read either from the tables or from the good quality figures. As our present theoretical analysis is focused on the effects of surface energetic heterogeneity, we tried to find which of the adsorption systems studied so far by titration calorimetry, might exhibit the strongest heterogeneity effects. That led us to select the data reported by Machesky and Jacobs,34,35 who used titration calorimetry to study heats of proton adsorption in the γ -alumina/NaCl electrolyte system. On the alumina surfaces, five isolated -OH groups stretching frequencies have been identified, in addition to a band associated with the hydrogen-bonded SOH groups.35,59,60 Of course, the proton (ion) binding constants (59) Peri, J. B. J. Phys. Chem. 1965, 69, 220.

of the various groups should vary. Moreover, even within one kind of -OH group, a certain dispersion of adsorption properties is to be expected. All that may suggest that alumina is a good candidate for studying effects of energetic heterogeneity of surface oxygens on proton adsorption accompanying heats. Besides the reasons related to soil science and environmental sciences, studying ion adsorption on alumina is extremely important for catalysis. Alumina is one of the most frequently used carriers in the preparation of supported catalysts. That preparation is done by adsorption on alumina, from the concentrated solutions of metals ions having desirable catalytic properties, and next drying or filtration. These processes were described and analyzed in the numerous papers by Lycourghiotis and co-workers.20 Machesky and Jacobs34,35 carried out their experiments at 25 °C, over a pH range 4-10, and at three, 0.001, 0.01, and 0.1 M, concentrations of NaCl. They investigated two kinds of alumina samples, named by them alumina I and alumina II. The value of PZC for alumina II was determined to be 8.5. Other experimental details are available in the original papers by Machesky and Jacobs.34,35 Unfortunately, only some of the reported heats of adsorption data can be subjected to our analysis. These are the cases where heat measurements were accompanied by potentiometric titration. Thus, of all the reported data, only calorimetric titration of alumina II suspension at two salt concentrations 0.001 and 0.1 M NaCl could be subjected to our analysis. Those two calorimetric titrations were carried out from the base side, whereas the potentiometric titrations were carried out from the acid side. Thus, for the reason described below only the calorimetric and potentiometric titrations corresponding to 0.1 mol/dm3 salt concentration can be subjected to our quantitative analysis. The reason for that is the following: Both calorimetric and potentiometric titrations exhibit hysteresis; i.e., two sets of data are obtained, one for the titration made from the base side and another one when titration is carried out from the acid side. That kind of hysteresis was reported by Mehr et al.,36 but its origin is still not well understood. Therefore, Mehr et al.36 have proposed to consider the arithmetic average of the data obtained in the course of the two titrations carried out from the base and acid sides (and corresponding to the same pH). As mentioned above, only the base branches of calorimetric titration were reported by Machesky and Jacobs34,35 for alumina II and the salt concentrations 0.1 mol/dm3 and 0.001 mol/dm3. As for the calorimetric titration data, a large hysteresis was observed for the salt concentration of 0.001 mol/dm3 and alumina I sample. A similar serious hysteresis must, of course, be assumed in the case of the corresponding calorimetric titration of alumina II. So having only the base branch of that calorimetric titration curve at disposal does not allow one to carry out a quantitative, simultaneous analysis of both potentiometric and calorimetric titration data. Meanwhile, an inspection into the data corresponding to the salt concentration 0.1 mol/dm3 shows small hysteresis between the acid and base titrations for alumina I. Of course, it is to be expected that something similar would be observed in the case of alumina II. Thus, of all the numerous data reported by Machesky and Jacobs,34,35 only the calorimetric and potentiometric titrations of alumina II, corresponding to the salt concentration 0.1 (60) Boehm, H. P.; Knozinger, H. In Catalysis-Science and Technology; Anderson, J. R., Boudart, M., Eds.; Springer: Berlin, 1983; Vol. 4, p 39.


mol/dm3, have been subjected here to our quantitative analysis. That quantitative analysis will include not only the potentiometric and calorimetric data reported by Machesky and Jacobs.34,35 We will follow their idea incorporating also into such an analysis Sprycha’s radiometric measurements of the individual adsorption isotherms of Na+ and Cl- ions on alumina.61 Although it was not an identical alumina sample, Sprycha’s data should be very helpful for monitoring the behavior of the theoretical individual isotherms of Na+ and Cl- adsorption, corresponding to the calculated theoretical heats of adsorption. That monitoring is very important for the following reasons. We will show it soon that the experimetal δ0(pH) titration curves can be fitted equally well by using a variety of adsorption parameters sets. However, the simultaneous fit of the experimental δ0(pH) curves, and the corresponding individual adsorption isotherms, eliminates all other sets, except a certain class of parameter sets in a rather narrow range of values. Having fitted simultaneously the experimental δ0(pH) curves, and the individual isotherms of Na+ and Cl- adsorption, we have only four parameters Qa1, Qa2, RL1 , and RR1 which can be freely chosen to fit the experimental heats of proton adsorption Qpr(pH). In our previous publications41,42 we launched the hypothesis that the enthalpic effects of the purely coulombic adsorption of alkali cations on the SO- are negligible and we put QaC ) 0. Such a conclusion was, in fact, drawn by Kosmulski,28 who used the radiometric methods to study the individual adsorption isotherms of ions of the inert electrolyte. Although it was not said expressis verbis, such a point of view was also expresed by Machesky and Jacobs34,35 who called Qpr the heat of proton adsorption. The above remarks are true no matter which of the three adsorption models considered in the theoretical section is taken into consideration. As mentioned before, our previous studies9,10 favored much the model of surface heterogeneity assuming small correlations between the adsorption energies of the four surface complexes SOH, SOH2+, SOH2+A-, and SO-C+. So, we begin here our study by using the equations developed by us for that model of energetic heterogeneity of surface oxygens. While fitting simultaneously the experimental titration curve δ0(pH), and the individual isotherms of ions, we can int L choose freely the following parameters: pK int a1 , p*K C , c1 , R c1 , kT/c0, kT/c+, kT/cC, and kT/cA. int The other parameters pK int a2 and p*K A are automatically calculated through appropriate relations developed by applying the Rudzin´ski-Charmas criterion for CIP to exist. In the case of the adsorption model assuming small correlations between the adsorption energies of various surface complexes, these are interrelations (45). Before we start presenting the results of our computer exercises, it is necessary to explain the precision of the experimental data. In Figure 8 of their paper, Machesky and Jacobs34 present bars showing the precision of all the experimental Qpr points, corresponding to different pH values. The last point corresponding to the highest value of pH has, in the original Figure 8 by Machesky and Jacobs,34 an exceptionally large bar indicating a high degree of uncertainty for that point. So, in our further best fit exercises, we did not pay much attention to that point. Another reason is, as stated above, that this last (61) Sprycha, R. J. Colloid Interface Sci. 1989, 127, 12.

Langmuir, Vol. 14, No. 18, 1998 5219

Figure 1. Comparison of the experimental potentiometric titration data δ0(pH) reported by Machesky and Jacobs34 for the salt concentration 0.1 mol/dm3 (b) with the theoretical ones calculated by using the parameters from the first rows of Tables 2-4. The solid line (s) is for kT/ci ) 0.9, i ) 0, +, A, and C, pK int a1 L R 2 2 ) 5.00, p*K int C ) 9.30, c1 ) 0.80 F/m , and c1 ) 0.90 F/m ; the int slighty broken line (- - -) is for kT/ci ) 0.8, pK a1 ) 5.00, p*K int C ) 9.55, cL1 ) 0.80 F/m2, and cR1 ) 0.90 F/m2; and the strongly int broken line (‚‚‚) is for kT/ci ) 0.7, pK int a1 ) 5.00, p*K C ) 9.80, cL1 ) 0.78 F/m2, and cR1 ) 0.90 F/m2.

point does not fit the tendency in the experimental data in the large, preceding pH region, from 7 to 9. A large uncertainty also characterizes the Qpr value corresponding to the lowest investigated pH value, as can be seen in the original Figure 8 by Machesky and Jacobs.34 However, as the position of that point corresponds to the trend in the experimental data in the region of pH from 4 to 7, we will treat that point with remarkable confidence. According to the currently considered model, assuming small correlations between adsorption energies of various surface complexes, we may treat kT/c0, kT/c+, kT/cC, and kT/cA as independent, free parameters. However, we also realize that decreasing the number of such freely chosen parameters would result in growing confidence of our best fit exercises. So, we decreased the number of the free parameters from 4 to 2 by accepting the following rationalization. The energetic heterogeneity of surface oxygens should affect in a similar way the surface complexes containing two protons. So, we put in our best-fit calculation c+ ) cA. Then, we assumed also that the energetic heterogeneity will affect, in a pretty similar way, the adsorption of the first proton and the adsorption of cations carrying the same charge. So, we put c0 ) cC. In that way we reduced the number of freely chosen heterogeneity parameters from 4 to 2. Figures 1-7 show the results of our computer best-fit exercises, carried out by accepting that kT/c0 ) kT/cC and kT/c+ ) kT/cA. As for Figure 1, it shows the agreement between the experimental titration curve δ0(pH) and the theoretical one calculated by using the parameters collected in Table 2. By using the parameters collected in Tables 3-5, we arrive practically at the same fit of the experimental titration curves. The theoretical lines overlap so much that it could not be possible to show them as the separate lines even in a much bigger figure. This again confirms what has been known in literature for a long time. Namely, that experimental titration curves can be fitted by using a variety of adsorption models and sets of parameters. Thus, until no other experimental data (individual adsorption isotherms of ions, heats of adsorption) are taken into analysis, the titration curves

5220 Langmuir, Vol. 14, No. 18, 1998

Rudzin´ ski et al.

Figure 2. Comparison between the experimental (b) and theoretical heats of proton adsorption Qpr(pH), calculated by assuming that kT/c+ ) kT/cA ) 0.9 (s), kT/c+ ) kT/cA ) 0.8 (- - -), and kT/c+ ) kT/cA ) 0.7 (‚‚‚), and assuming that kT/c0 ) kT/cC ) 0.9. Other related parameters are collected in Table 2.

Figure 5. Comparison of the individual adsorption isotherms of Na+ (b) and Cl- (9) adsorption and the theoretical ones calculated by using the same sets of parameters as those used to prepare Figure 2. Also the meanings of the solid (s), dashed (- - -), and strongly broken (‚‚‚) theoretical lines are the same.

Figure 3. Comparison between the experimental (b) and theoretical heats of proton adsorption Qpr(pH), calculated by assuming that kT/c+ ) kT/cA ) 0.8 (s), kT/c+ ) kT/cA ) 0.7 (- - -), and kT/c+ ) kT/cA ) 0.6 (‚‚‚) and assuming that kT/c0 ) kT/cC ) 0.8. Other related parameters are collected in Table 3.


Figure 4. Comparison between the experimental (b) and theoretical heats of proton adsorption Qpr(pH), calculated by assuming that kT/c+ ) kT/cA ) 0.7 (s), kT/c+ ) kT/cA ) 0.6 (- - -), and kT/c+ ) kT/cA ) 0.5 (‚‚‚), and assuming that kT/c0 ) kT/cC ) 0.7. Other related parameters are collected in Table 4.


alone are not a much reliable source of information. Looking at Figures 1-7, one may draw the following conclusions: 1. By acceptance of kT/c0 ) kT/cC > kT/c+ ) kT/cA, an impressive improvement between the experimental and theoretical heats of proton adsorption is achieved, compared to the situation when kT/c0 ) kT/cC

) kT/c+ ) kT/cA. 2. Until kT/c0 - kT/c+ ) 0.1, a substantial improvement is also observed in the agreement between the experimental and theoretical individual adsorption isotherms. When kT/c0 - kT/c+ ) 0.2, that agreement becomes even worse than in the case when kT/c0 ) kT/cC ) kT/c+ ) kT/cA.


Langmuir, Vol. 14, No. 18, 1998 5221

Table 2. Parameters Used To Calculate the Theoretical Values Presented in Figures 2 and 5a best-fit parameters cL1

cR1

calcd parameters

kT/c+ ) kT/cA

pK int a1

p*K int C

(F/m2)

(F/m2)

Qa1 (kJ/mol)

Qa2 (kJ/mol)

RL1 (F/(m2‚deg))

0.9 0.8 0.7

5.00 5.00 5.00

9.30 9.40 9.30

0.80 0.90 0.90

0.90 0.90 0.90

19.0 26.0 27.0

67.0 70.0 68.0

-0.0030 -0.0035 -0.0038

RR1 (F/(m2‚deg))

pK int a2

p*K int A

QaA (kJ/mol)

-0.0030 -0.0035 -0.0038

12.00 11.34 10.92

7.70 7.77 8.00

86.0 88.4 86.5

a The kT/c ) kT/c values are taken 0.9. The first columns report the values found from the best fit, whereas the other columns shows 0 C the values calculated automatically through relations (45) and (49).

Table 3. Parameters Used To Calculate the Theoretical Values Presented in Figures 3 and 6a best-fit parameters kT/c+ ) kT/cA

pK int a1

p*K int C

cL1 (F/m2)

0.8 0.7 0.6

5.00 5.00 5.00

9.55 9.60 9.40

0.80 0.90 0.90

a

calcd parameters

cR1 (F/m2)

Qa1 (kJ/mol)

Qa2 (kJ/mol)

RL1 F/(m2‚deg))

RR1 F/(m2‚deg))

pK int a2

p*K int A

QaA (kJ/mol)

0.90 0.90 0.90

17.0 17.0 15.0

72.0 73.0 68.0

-0.0030 -0.0040 -0.0045

-0.0030 -0.0040 -0.0045

12.00 11.47 10.96

7.45 7.59 7.96

89.0 87.3 80.6

The kT/c0 ) kT/cC values are taken as 0.8. Table 4. Parameters Used To Calculate the Theoretical Values Presented in Figures 4 and 7a best-fit parameters cR1

calcd parameters

kT/c+ ) kT/cA

pK int a1

p*K int C

(F/m2)

(F/m2)

Qa1 (kJ/mol)

Qa2 (kJ/mol)

RL1 F/(m2‚deg))

0.7 0.6 0.5

5.00 5.00 5.00

9.80 9.90 9.60

0.78 0.90 0.90

0.90 0.90 0.90

10.0 9.0 -2.0

84.0 77.0 67.0

-0.0030 -0.0045 -0.0060

a

cL1

RR1 (F/(m2‚deg))

pK int a2

p*K int A

QaA (kJ/mol)

-0.0030 -0.0045 -0.0060

12.00 11.59 11.03

7.20 7.31 7.82

94.0 86.4 68.0

The kT/c0 ) kT/cC values are taken as 0.7. Table 5. Parameters Used To Calculate the Theoretical Values Presented in Figure 13 best-fit parameters cL1

cR1

calcd parameters

kT/c

pK int a1

p*K int C

(F/m2)

(F/m2)

Qa1 (kJ/mol)

Qa2 (kJ/mol)

RL1 (F/(m2‚deg))

0.9 0.8

5.00 5.00 5.00

9.20 9.15 9.15

0.80 0.80 0.75

0.90 0.90 0.90

29.0 48.0 48.0

57.0 59.0 59.0

-0.0030 -0.0031 -0.0035

To our personal feeling, the best compromise for a good behavior of both heats of proton adsorption and individual adsorption isotherms of ions is represented by kT/c0 ) kT/cC ) 0.8 and kT/c+ ) kT/cA ) 0.7. This is the dashed line (---) in Figures 3 and 6. If, however, we put less confidence in the individual iostherms of Na+ and Cladsorption, measured by Sprycha61 using a nonidentical alumina sample, we would prefer the situation when kT/ c0 ) kT/cC ) 0.9 and kT/c+ ) kT/cA ) 0.7. The theoretical heat of proton adsorption is then a strongly broken line in Figure 2. We do, in fact, believe this situation more, because the previous situation leads to very small values of Qa1 parameters. For the latter situation, we are going to show how the parameters Qa1, Qa2, RL1 , and RR1 , which are related only to the heat of proton asorption, affect its behavior. This is shown in Figures 8-11. Now, we are going to show how the heats accompanying the formation of various surface complexes contribute to the total heat of proton adsorption Qpr(pH) monitored experimentally. These are the following contributions:

∫pHpH+∆pHQi(∂pHi )T dpH ∂θ

Qipr )

[

]

∫pHpH+∆pH 2(∂pH+ )T + 2(∂pHA )T + (∂pH0 )T ∂θ

∂θ

∂θ

dpH

i ) 0, +, A, C (52)

RR1 (F/(m2‚deg))

pK int a2

p*K int A

QaA (kJ/mol)

-0.0030 -0.0031 -0.0035

12.00 12.39 12.88

7.80 7.85 7.85

86.0 85.4 83.6

Figure 8. Effect of Qa1 on the behavior of heat of proton adsorption Qpr(pH). The calculations correspond to the situation when kT/c+ ) kT/cA ) 0.7 and kT/c0 ) kT/cC ) 0.9. The solid line (s) is for Qa1 ) 27 kJ/mol, the dashed line (- - -) is for Qa1 ) 30 kJ/mol, and the strongly broken line (‚‚‚) is for Qa1 ) 24 kJ/mol. Other related parameters are the same as in Table 2. A C Here the heats Q0pr, Q+ pr, Qpr, and Qpr are those accompanying formation of the surface complexes SOH, SOH2+, SOH2+A-, and SO-C+, respectively. A C The sum Q0pr + Q+ pr + Qpr + Qpr ) Qpr. For the parameters kT/c0 ) kT/cC ) 0.9 and kT/c+ ) kT/cA ) 0.7 preferred here by us, these contributions are shown in Figure 12.

5222 Langmuir, Vol. 14, No. 18, 1998

Figure 9. Effect of Qa2 on the behavior of heat of proton adsorption Qpr(pH). The calculations correspond to the situation when kT/c+ ) kT/cA ) 0.7 and kT/c0 ) kT/cC ) 0.9. The solid line (s) is for Qa2 ) 68 J/mol, the dashed line (- - -) is for Qa2 ) 72kJ/mol, and the strongly broken line (‚‚‚) is for Qa2 ) 64 kJ/mol. Other related parameters are the same as in Table 2.

Figure 10. Effect of RL1 on the behavior of heat of proton adsorption Qpr(pH). The calculations correspond to the situation when kT/c+ ) kT/cA ) 0.7 and kT/c0 ) kT/cC ) 0.9. The solid line (s) is for RL1 ) -0.0038 F/(mol‚deg), the dashed line (- - -) is for RL1 ) -0.0030 F/(mol‚deg), and the strongly broken line (‚‚‚) is for RL1 ) -0.0045 F/(mol‚deg). Other related parameters are the same as in Table 2.

Finally we are going to answer the question which many readers are sure to raise: why did we not investigate the situation when kT/c0 ) kT/cC < kT/c+ ) kT/cA? Yes, we did investigate such a situation, but the related theoretical curves showed highly unrealistic behavior for all such cases. As mentioned in the Introduction, our previous studies10 of bivalent ion adsorption at different pH values and small ion concentration favored strongly the model of surface heterogeneity assuming small correlations between adsorption energies of various surface complexes. Our present theoretical studies of the heats of proton adsorption provide new impressive arguments in favor of that model of surface heterogeneity. Figure 13 shows the behavior of the theoretical Qpr curves, calculated by using the model of surface heterogeneity assuming high correlations between the adsorption energies of various surface complexes. The theoretical curves presented in Figure 13 the represent the best-fits which were obtained by ignoring the existence of the Qpr value corresponding to the highest investigated pH. In such a case, all these theoretical curves must be viewed as fitting the experimental data poorly.

Rudzin´ ski et al.

Figure 11. Effect of RR1 on the behavior of heat of proton adsorption Qpr(pH). The calculations correspond to the situation when kT/c+ ) kT/cA ) 0.7 and kT/c0 ) kT/cC ) 0.9. The solid line (s) is for RR1 ) -0.0038 F/(mol‚deg), the dashed line (- - -) is for RR1 ) -0.0020 F/(mol‚deg), and the strongly broken line (‚‚‚) is for RR1 ) -0.0050 F/(mol‚deg). Other related parameters are the same as in Table 2.

Figure 12. Contributions Qipr(pH), i ) 0, +, A, and C, to the experimentally monitored heat of proton adsorption at the alumina II/NaCl electrolyte interface, when the salt concentration is equal to 0.1 mol/dm3. These theoretical contributions have been calculated by assuming that kT/c+ ) kT/cA ) 0.7 and kT/c0 ) kT/cC ) 0.9. Other related parameters are collected in Table 2.

Our best-fit exercises based on the model of an energetically homogeneous solid surface showed that the fit of the experimental data is even poorer than in the case of the model assuming high correlations between the adsorption energies of various complexes. Finaly, we would like to draw the reader's attention to one intriguing observation. Namely, no matter which heterogeneity model is accepted, and the set of parameters used, the numerically estimated pK int a1 values (see Tables 2-5), are close to 5, in accordance with the theoretical calculations by Sverjensky and Sahai.62-64 According to their calculations, the oxides having PZC values close to 8 should be characterized by pK int a1 values close to 5. It seems, therefore, that Sverjensky’s theoretical values may be used as a good estimation of average values of parameters in the case of energetically heterogeneous oxide surfaces. (62) Sverjensky, D. A.; Sahai, N. Geochim. Cosmochim. Acta 1996, 60, 3773. (63) Sahai, N.; Sverjensky, D. A. Geochim. Cosmochim. Acta 1997, 61, 2801. (64) Sahai, N.; Sverjensky, D. A. Geochim. Cosmochim. Acta 1997, 61, 2827.


Langmuir, Vol. 14, No. 18, 1998 5223

Conclusions

Figure 13. Comparison of the experimental heats of proton adsorption Qpr(pH) measured by Machesky and Jacobs34 for the salt concentration 0.1 mol/dm3 (b) with theoretical ones calculated by using the model of surface heterogeneity assuming high correlations between the adsorption energies of various surface complexes. The strongly broken line (‚‚‚) is for kT/c ) 0.8 and the dashed line (- - -) is for kT/c ) 0.9, whereas the solid line (s) was calculated by assuming a fully homogeneous solid surface. The other parameters used in these calculation are collected in Table 5.

Now, we would like to refer, again, to the papers by Machesky and Jacobs,34,35 where they attempted to draw some qualitative or semiqualitative conclusions concerning the enthalpic effects of ion adsorption. Their considerations were based on somewhat different model of charging the surface oxygens, being a generalization of the 1-pK charging model for the case when a number of different surface oxygens exists on an oxide surface (MUSIC). Quantitive conclussions were drawn concerning the free energies of adsorption, but qualitative conclusions were drawn only about enthalpies of adsorption predicted by this model of charging the surface oxygens. This is because the related equations for enthalpic effects, corresponding to this charging mechanism, have not been developed yet. Our theoretical considerations have been based on the 2-pK charging model which is still the most commonly accepted in literature. The answer for which of these models represents the charging mechanism better requires futher extensive fundamental studies. Because the definition of enthalpic effects depends on the accepted charging mechanism, the semiquantitative conclusions by Machesky and Jacobs cannot be directly compared with the quantitative conclusions drawn by us, accepting the 2-pK charging mechanism. Finally, we mention that not only the triple layer model of the oxide/electrolyte interface could be accepted to study these enthalpic effects. Recently one of us (R.C.) has made attempts to apply the fourth-layer model along with the 2-pK assumption about the charging mechanism.65-67 The obtained results look very promising and are now prepared for publication. It seems, however, that the many scientists using still the triple-layer model will appreciate having at their disposal a theoretical description of the enthalpic effects, based on that popular triple-layer model. Even if more refined models might gain a large popularity in the future, it is worthy of knowing the full possibilities offered by the classical triple-layer model.

Experimental and theoretical studies of the enthalpic effects accompanying ion adsorption at the oxide/electrolyte interface are crucial for our better understanding of most fundamental features of these adsorption systems. In particular, they are essential for predicting effects of changing temperature on these systems. That problem is essential for the soil and environmental sciences. Our theoretical studies of the enthalpic effects accompanying ion adsorption, which have been published in a series of our recent papers, are extended here to take into account the energetic heterogeneity of the actual oxide surfaces. It has been known for a long time that surface oxygens exibit a much smaller degree of organization than the oxygens in the interior of metal oxides. This is a source of energetic heterogeneity of the oxide surfaces. The adsorption features of surface oxygens vary from one to another surface oxygen. So, the surface complexes formed on various surface oxygens have different binding-tosurface energies. That phenomenon is called the “surface energetic heterogeneity”. Two different models of that surface energetic heterogeneity have been considered by us. One of them assumes that the binding-to-surface energies of different surface complexes vary from one to another surface oxygen but in the pretty same way; i.e., these energies are highly correlated. The other model of surface heterogeneity assumes that only small correlations exist between the energies of adsorption of various surface complexes, when going from one to another surface oxygen. For both models of surface energetic heterogeneity the corresponding equations were developed for the isotherms of ion adsorption and for the enthalpic effects accompanying ion adsorption. The choice of alumina as a good example to study the effects of surface heterogeneity was dictated by the fact that five kinds of surface oxygens may exist on the alumina surfaces, so the effects of surface heterogeneity should be clearly detected in this system. As auxiliary experimental information, the radiometrically measured isotherms of Na+ and Cl- adsorption on alumina, reported by Sprycha,61 were also taken into consideration. Thus the theoretical expressions developed by us were used to fit simultaneously, applying the same set of parameters, the three sets of experimental data: the titration curves δ0(pH);34 the enthalpy changes accompanying the titration steps;34 the individual isotherms of Na+ and Cl- adsorption on alumina, reported by Sprycha.61 Our numerical exercises showed that the best simultaneous fit of all those experimental data is achieved by using the expressions developed for the model of surface heterogeneity assuming small correlations between the adsorption energies of various surface complexes. Acknowledgment. This research was supported by KBN Research Grant No. 3 T09A 078 11. One of the authors (R.C.) wishes to express his thanks and gratitude to the Foundation for Polish Science for the grant making his one-year stay in the Laboratoire Environnement et Mineralurgie, ENS Geologie, CNRS-URA 235, Vandoeuvre les Nancy, France, possible. Appendix

(65) Charmas, R.; Piasecki, W.; Rudzin´ski, W. Langmuir 1995, 11, 3199. (66) Charmas, R.; Piasecki, W. Langmuir 1996, 12, 5458. (67) Charmas, R. Langmuir, in press.

The derivatives (∂θi/∂pH)T (i ) 0, +, A, C) in eqs 15 and 16 are to be evaluated from the equation systems 18, 34, or 46, depending on the accepted model. For instance,

| | | |

5224 Langmuir, Vol. 14, No. 18, 1998

∂F0 ∂pH ∂F+ ∂pH ∂FC ∂pH ∂FA ∂θ0 ∂pH ) (-1) ∂F ∂pH T 0 ∂θ0 ∂F+ ∂θ0 ∂FC ∂θ0 ∂FA ∂θ0

( )

∂F0 ∂θ+ ∂F+ ∂θ+ ∂FC ∂θ+ ∂FA ∂θ+ ∂F0 ∂θ+ ∂F+ ∂θ+ ∂FC ∂θ+ ∂FA ∂θ+

∂F0 ∂θC ∂F+ ∂θC ∂FC ∂θC ∂FA ∂θC ∂F0 ∂θC ∂F+ ∂θC ∂FC ∂θC ∂FA ∂θC

∂F0 ∂θA ∂F+ ∂θA ∂FC ∂θA ∂FA ∂θA ∂F0 ∂θA ∂F+ ∂θA ∂FC ∂θA ∂FA ∂θA

Rudzin´ ski et al.

account the fact that, according to eqs 3 and 6, δ0 is the function of θi’s,

δ0 ) B[2θ+ + 2θA + θ0 - 1]

A special computer subroutine was used to find the analytical form of the derivatives (∂θi/∂pH)T,

X0 ) t[θ- + θC - θ+ - θA] + 2wt[θ-θA - θ+θC] + β[θC - θA] X+ ) t[θ0 + 2θ- + 2θC] + wt[θ0θC + θAθ0 + 4θAθ- + 4θAθC] + β[θC - θA]

(A.2)

∂F0 1 )∂θ+ θ∂F+ θ+ + θ )∂θ+ θ+θ∂FC 1 )+ 2w ∂θ+ θ∂FA 1 )- 2w ∂θ+ θ-

∂F0 1 )∂θA θ∂F+ 1 )∂θA θ∂FC 1 )+ 2w ∂θA θθ A + θ∂FA )- 2w ∂θA θAθ-

XC ) -t[θ0 + 2θ+ + 2θA] + wt[θ0θ+ + θ-θ0 + 4θ-θ+ + 4θ-θA] - β[θ0 + θ+ + θ- + 2θA] XA ) t[θ0 + 2θ- + 2θC] - wt[θ0θ+ + θ-θ0 + 4θ-θ+ + 4θ+θC] + β[θ0 + θ+ + θ- + 2θC] Y ) 1 + w[θA(2θ- + 2θC + θ0) + θC(2θ+ + 2θA + θ0)]

On the contrary, the derivatives ∂Fi/∂θj, i, j ) 0, +, A, and C in the determinants and (A.1) will acquire different forms depending on the adsorption model. Below, the explicit form of these derivatives is given for all the models under consideration. The Homogeneous Surface Model.

∂F0 1 )∂θC θ∂F+ 1 )∂θC θθ C + θ∂FC )∂θC θCθ∂FA 1 )∂θC θ-

(A.6)

where

∂F+ ∂F0 t t ) -2.3 ) -4.6 ∂pH t + β ∂pH t+β

θ 0 + θ∂F0 )∂θ0 θ0θ∂F+ 1 )∂θ0 θ∂FC 1 )+w ∂θ0 θ∂FA 1 )-w ∂θ0 θ-

2.3θi Xi ∂θi )i ) 0, +, A, C ∂pH t+β Y

(A.1)

While evaluating the partial derivatives (∂Fi/∂pH) occurring in the numerator of eq A.1, one has to consider ψ0 as a function of pH defined in eq 10. Then, for all the models being under the consideration we have the following equations

∂FC ∂FA β 2t + β ) 2.3 ) -2.3 ∂pH t + β ∂pH t+β

(A.5)

(A.3)

where t is given by eq 23a and w defined in (A.4). The Model Assuming Hight Correlations among Adsorption Energies of Various Surface Complexes. The derivatives ∂Fi/∂θj, i, j ) 0, +, A, and C, in the determinants (A.1) take now the form

∂F0 1 ) - + x′ - y′ ∂θ0 θ0 ∂F+ ) x′ - y′ ∂θ0 ∂FC ) x′ - y′ + w ∂θ0 ∂FA ) x′ - y′ - w ∂θ0 ∂F0 ) x′ - y′ ∂θC ∂F+ ) x′ - y′ ∂θC ∂FC 1 ) - + x′ - y′ ∂θC θ0 ∂FA ) x′ - y′ ∂θC

∂F0 ) x′ - y′ ∂θ+ ∂F+ 1 )+ x′ - y′ ∂θ+ θ+ ∂FC ) x′ - y′ + 2w ∂θ+ ∂FA ) x′ - y′ - 2w ∂θ+

∂F0 ) x′ - y′ ∂θ+ ∂F+ ) x′ - y′ ∂θA ∂FC ) x′ - y′ + 2w ∂θA ∂FA 1 )+ x′ - y′ - 2w ∂θA θA

(A.7)

where w is defined in eq A.4 and

x′ )

1 c 1 y′ ) 1 - θ-t kT θ-t(1 - θ-t)

(A.8)

Finally, we arrive at the expressions

where

w)

eB kTc1

2.3θit Xi ∂θit )i ) 0, +, A, C ∂pH t+β Y

(A.4)

While calculating these derivatives, one has to take into

where

(A.9)


Langmuir, Vol. 14, No. 18, 1998 5225

X0 ) t - (x′ - y′)[t(θCt - θ+t - θAt) 2wtθ+tθCt + β(θCt - θAt)] + 2wtθAt X+ ) 2t - (x′ - y′)[t(θ0t + 2θCt) + wt(θ0tθCt + θAtθ0t + 4θAtθCt) + β(θCt - θAt)] + 4wtθAt XC ) (x′ - y′)[t(θ0t + 2θ+t + 2θAt) - wtθ0tθ+t + β(θ0t + θ+t + 2θAt)] + wt(θ0t + 4θ+t + 4θAt) - β XA ) 2t - (x′ - y′)[t(θ0t + 2θCt) - wt(θ0tθ+t + 2θ+tθCt) + β(θ0t + θ+t + 2θCt)] - wt(θ0t + 4θ+t) + β Y ) 1 - (x′ - y′)[1 - θ-t + w(θ0tθCt + θ0tθAt + 2θ+tθCt + 4θCtθAt)] + 2wθAt and t is defined in eq 23a. The Model Assuming Small Correlations among Adsorption Energies of Various Surface Complexes. The derivatives ∂Fi/∂θj, i, j ) 0, +, A, and C, in the determinants (A.1) now take the form

c0 θ0 + θ∂F0 )∂θ0 kT θ0θc+ 1 ∂F+ )∂θ0 kT θcC 1 ∂FC )+w ∂θ0 kT θcA 1 ∂FA )-w ∂θ0 kT θc0 1 ∂F0 )∂θC kT θc+ 1 ∂F+ )∂θC kT θcC θC + θ∂FC )∂θC kT θCθcA 1 ∂FA )∂θC kT θ-

c0 1 ∂F0 )∂θ+ kT θc+ θ+ + θ∂F+ )∂θ+ kT θ+θ∂FC cC 1 )+ 2w ∂θ+ kT θ∂FA cA 1 )- 2w ∂θ+ kT θc0 1 ∂F0 )∂θA kT θc+ 1 ∂F+ )∂θA kT θcC 1 ∂FC )+ 2w ∂θA kT θcA θA + θ∂FA )- 2w ∂θA kT θAθ(A.10)

(

2.3θit Xi ∂θit )∂pH t+β Y and

i ) 0, +, A, C

(A.11)

(

)

) ]

[

( ( ) ] [ ( ) ) ] [ ( ) ) ( ) ] [

)

[

]

c+ cC c0 c A cA c0 (θ-t + θCt) + X+ ) t 2 2 θ kT kT kT kT kT kT 0t c+ c+ c 0 cA cA c 0 2 θ + wt 2 θ θ + kT kT kT At kT kT kT 0t Ct c+ c C c0 c0 cC c 0 cC θAtθ-t + 4 + 2 θAtθ0t + 4 kT kT kT kT kT kT kT c+ c0 c+ cA cC cA θCt θ θAtθCt + β kT kT kT kT kT kT At

(

(

[

]

]

cC c+ cA c0 cA c0 c+ θ +2 θ +2 θ + XC ) -t kT kT kT 0t kT kT +t kT kT At c+ c+ cA cA c 0 θ θ + wt 2 θ θ + kT kT kT 0t +t kT kT -t 0t cC c0 cA c0 c+ c+ c0 θ-tθ+t + 4 θ-tθAt + 4 2 kT kT kT kT kT kT kT c 0 c + cA cA cC (θ0t + θ+t + θ-t) + + θ0tθAt - β kT kT kT kT kT cA θ kT At

[ (

)

) ]

[

[ (

)

(

(

) ]

cA c C c+ c 0 c0 c+ (θ + θCt) + XA ) t 2 θ0t + 2 kT kT kT kT kT kT -t cA c+ cC c 0 c 0 c+ 2 θ+t - wt 2 θ θ + kT kT kT kT kT kT 0t +t cC c0 c C c0 cA c+ cC θ θ +4 θ θ +4 + kT kT -t 0t kT kT -t +t kT kT kT c+ cA c0 c+ cC + -2 θ θ + θ θ + kT +t Ct kT kT kT kT 0t Ct cA cC c 0 c + cC (θ0t + θ+t + θ-t) + + β θ kT kT kT kT kT Ct

(

)

Y) where ∂θit/∂pH is now given by the equations

[

c+ cC c+ cA cA c0 X0 ) t (θ-t + θCt) 2 θ kT kT kT kT kT kT +t cA c + c0 cA c0 c+ c C θ-tθAt 2 θAt + 2wt 2 kT kT kT kT kT kT kT c+ cA cC c+ c0 θ θ 2 θ θ + kT +t Ct kT kT kT kT Ct At cC c 0 c + cA θCt θ β kT kT kT kT At

c0c+ (kT)

w 2

[

) ] [ (

[

(

)

( ) ] ( ) ]

cC cA θAt(2θ-t + 2θCt + θ0t) + θ (2θ + kT kT Ct +t c0c+cCcA 2θAt + θ0t) + (kT)4

]

where t is defined in eq 23a and w is defined in eq A.4. When kT/c0 ) kT/c+ ) kT/cC ) kT/cA ) 1, eqs A.11 reduce to eqs A.6 developed for the homogeneous surface model. LA980043H

Calorimetric Effects Accompanying Ion Adsorption at the Charged

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