Calorimetric Effects and Temperature Dependence of Simple Ion

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Langmuir 1997, 13, 483-495

483

Calorimetric Effects and Temperature Dependence of Simple Ion Adsorption at Oxide-Electrolyte Interface: A Theoretical Analysis Based on the Triple-Layer Complexation Model W. Rudzinski* and R. Charmas Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie Sklodowska University, Pl. M. Curie Sklodowskiej 3, 20-031 Lublin, Poland

J. M. Cases, M. Francois, F. Villieras, and L. J. Michot Laboratoire Environnement et Mineralurgie, ENSG and URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy Cedex, France Received June 28, 1996X A rigorous thermodynamic interpretation based on the triple-layer complexation model is given of the experimental heats of adsorption recorded in various types of calorimetric experiments. Also, theoretical expressions are developed describing temperature dependence of the measured surface charge. Our theoretical description is based on a model of an energetically homogeneous surface, i.e., the surface oxygen has identical adsorption properties. The developed equations are applied to reproduce the temperature dependence of some experimentally measured surface charge isotherms to prove to what extent the triplelayer model of adsorption on a homogeneous oxide surface is able to reproduce the behavior of the actual adsorption systems.

Introduction For decades, only the values of the free energy of ion adsorption were available from the experimental adsorption (titration) isotherms. But then it was increasingly realized that the knowledge of the entropy changes accompanying ion adsorption may provide answers to many fundamental questions concerning the mechanism of ion adsorption at the oxide-electrolyte interface. As the entropy changes cannot be measured, the only way is to measure the contribution to the free energy of adsorption from enthalpy changes. The first attempts to estimate the enthalpy changes were made 25 years ago, but it was only during the last decade that the data to constrain the energetics of the surface acid-base reactions were accumulated rapidly. The data were elucidated first from the temperature dependence of ion adsorption isotherms, but now they are much more frequently measured directly by using appropriate experimental calorimetric sets. It seems, however, that the progress on the experimental side has not been accompanied by 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 most interesting fundamental studies of ion adsorption are the nonconfigurational heats of single-ion adsorption. Meanwhile, their estimations were frequently carried out in such a way that the estimated heats of adsorption were, in fact, configurational ones. Such configurational heats of adsorption give detailed information about all of the combined heat effects accompanying coadsorption of all ions. That combined information can be decoded only by using rigorous expressions describing the configurational heats of adsorption. As the equations corresponding to the triple-layer complexation model are probably the most commonly * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, December 15, 1996.

applied to correlate the experimental data for ion adsorption, it was obvious to us that a theoretical description of the accompanying heat effects should be based on this model first. Thus, 3 years ago, we published our first theoretical papers concerning that problem.1-3 On the basis of the triple-layer model (TLM), we developed corresponding equations for the heats of adsorption on both homogeneous and heterogeneous oxide surfaces. Using these equations, we analyzed enthalpy changes measured directly by using immersional adsorption calorimetry. The temperature dependence of the isotherms and other types of calorimetric experiments were not considered by us. Meanwhile, a complete theoretical description of enthalpic effects of ion adsorption should take into account both directly measured calorimetric data and the information deduced from the temperature dependence of adsorption isotherms. With this purpose in mind, we have developed here a theoretical description that makes possible the interpretation of all the reported experimental data within the frame of one theory. Because the illustrative calculations in our previous papers concerned directly measured calorimetric data, we focus our attention in this paper on temperature dependence of the adsorption isotherms of ions. Theory 1. Principles of the Adsorption Model. A large number of models describing simple ion adsorption at the oxide-electrolyte interface are available.4-11 They differ in the formulation of the surface complexation reactions (1) Rudzinski, W.; Charmas, R.; Partyka, S. Langmuir 1991, 7, 354. (2) Rudzinski, W.; Charmas, R.; Partyka, S.; Foissy, A. New J. Chem. 1991, 15, 327. (3) Rudzinski, W.; Charmas, R.; Partyka, S.; Thomas, F.; Bottero, J. Y. Langmuir 1992, 8, 1154. (4) Hingston, F. J. In Adsorption of Ionorganics at Solid-Liquid Interfaces; Anderson, M. A., Rubin, A. J., Eds.; Ann Arbor Science: Ann Arbor, MI, 1981; Chapter 2.

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Langmuir, Vol. 13, No. 3, 1997

and/or in the description of the structure of the double electric layer. The arguments generally focus on surface charge-pH data via a 2-pK or 1-pK approach. The 2-pK approach seems to be preferred in a number of the papers that have been published so far. Though a 1-pK model also received much attention.12-17 Discriminating between the 2-pK and 1-pK approaches will require simultaneous consideration of a variety of experimental data obtained by using various adsorption techniques. In one of their most recent papers, Hiemstra and Van Riemsdijk wrote, “Unfortunately such extended data sets for one particular ion and for one surface are not available. This limits the possibility of discriminating between the various models.” 18 Wood et al.19 have stressed the present situation as follows: “In the absence of exact chemical knowledge of just how surface metal ions coordinate with surface O(H) or OH(H) ions and which specific groups are involved with the loss or gain of protons, deciding which model is most appropriate becomes a matter of personal taste”. It has been known for a long time that enthalpic effects are much more sensitive to the nature of a certain (true or model) physical system than are adsorption isotherms. We believe, therefore, that studying enthalpic effects of adsorption at the oxide-electrolyte interface may shed new light on the mechanism of the interface charging in these adsorption systems. In this paper, we begin our study by considering the most popular triple-layer model proposed by Davis et al.20-22 on the basis of the conceptualizations of the electric double layer discussed by Yates et al.23 and Chan et al.24 General reviews and representative applications of this model have been given by Davis and Leckie25 and by Morel et al.26

Rudzinski and Charmas

Thus, we will consider the following surface reactions:

SOH2+ T SOH0 + H+

(1a)

SOH0 T SO- + H+

(1b)

SOH2+A- T SOH0 + H+ + A-

(1c)

SOH0 + C+ T SO-C+ + H+

(1d)

where SO- is the outermost surface oxygen, H+ is the proton, and A- and C+ denote the accompanying anion and cation, respectively. The equilibrium constants int int int Ka1 , Ka2 , *Kint A , and *KC of these four reactions are defined, respectively, in the literature.20-22 The formation of the surface complexes can also be considered as a result of 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)

Using the notation,

∑ ) [SOH0] + [SOH2+] + [SOH2+A-] + [SO-C+] + [SO-] (3)

∑ ) θ+,

[SOH2+]/

∑)

[SOH0]/

∑ ) θC

θ0, [SO-C+]/ (5) James, R. O.; Parks, G. A. In Surfaces and Colloid Science; Matijevic, E., Ed.; Plenum: New York, 1982; Vol. 12, Chapter 2. (6) Sposito, G. The Surface Chemistry of Soils; Oxford University Press: New York, 1984. (7) Barrow, N. J. Reactions with Variable Charge Soils; Nijhoff: Dodrecht, The Netherlands, 1987. (8) Dzombak, D. A.; Morel, F. M. M. Surface Complexation Modelling: Hydrous Ferric Oxide; Wiley: New York, 1990. (9) Davis, J. A.; Kent, D. B. In Mineral-Water Interface Geochemistry: Reviews in Mineralogy; Hochella, M. F., White, A. F., Eds.; Minerelogical Society of America: Washington, DC, 1990; Vol. 23, p 177. (10) Goldberg, S. In Advances in Agronomy; Sparks, J. L., Ed.; Academic Press: New York, 1992; Vol. 47, p 233. (11) Stumm, W. Chemistry of the Solid-Water Interface; Wiley: New York, 1992. (12) Koopal, L. K.; Van Riemsdijk, W. H.; Roffey, M. G. J. Colloid Interface Sci. 1987, 118, 117. (13) Hiemstra, T.; Van Riemsdijk, W. H.; Bolt, G. H. J. Colloid Interface Sci. 1989, 133, 91. (14) Hiemstra, T.; Van Riemsdijk, W. H.; Bruggenwert, M. G. M. Neth. J. Agric. Sci. 1987, 35, 281. (15) Fokkink, L. G. J.; de Keizer, A.; Lyklema, J. J. Colloid Interface Sci. 1989, 127, 116. (16) Van Riemsdijk, W. H.; De Wit, J. C. M.; Koopal, L. K.; Bolt, G. H. J. Colloid Interface Sci. 1987, 116, 511. (17) Hiemstra, T.; De Wit, J. C. M.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1989, 133, 105. (18) Hiemstra, T.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1996 179, 488. (19) Wood, R.; Fornasiero, D.; Ralston, J. Colloids Surf. 1990, 51, 389. (20) Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1978, 67, 90. (21) Davis, J. A.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 480. (22) Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1980, 74, 32. (23) Yates, D. E.; Levine, S.; Hearly, T. W. J. Chem. Soc., Faraday Trans. 1 1974, 70, 1807. (24) Chan, D.; Perram, J. W.; White, L. R.; Hearly, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (25) Davis, J. A.; Leckie, J. O. In Chemical Modelling in Aqueous Systems; Jenne, E. A., Ed.; American Chemical Society: Washington, DC, 1979; Chapter 15. (26) Morel, F. M. M.; Yested, J. G.; Westall, J. C. In Adsorption of Inorganics at Solid-Liquid Interfaces; Anderson, M. A., Rubin, A. J., Eds.; Ann Arbor Science: Ann Arbor, MI, 1981; Chapter 1.

[SOH2+A-]/

∑ ) θA,

∑

[SO-]/ ) 1 - θ0 - θ+ - θC - θA ) θ-

we can write the conditions for these reaction equilibria in the following way

{ }

eψ0 (aH)θ) kT θ0

int Ka2 exp

{ } { }

int int Ka2 exp Ka1

2eψ0 (aH)2θ) kT θ+

(4b)

eψβ (aC)θ) kT θC

(4c)

int /*Kint Ka2 C exp

int Ka2

*Kint A

(4a)

{

}

e(2ψ0 - ψβ) (aH)2(aA)θexp ) kT θA

(4d)

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

ψβ ) ψ0 -

δ0 c1

(5)

in which the surface charge δ0 is defined as follows:

δ0 ) B(θ+ + θA - θ- - θC), where B ) Nse (6)

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Ion Adsorption at Oxide-Electrolyte Interface

Langmuir, Vol. 13, No. 3, 1997 485

In eq 5, c1 is the first integral capacitance and Ns in eq 6 is the surface density (sites/m2). To evaluate the bulk activity of ions ai (i ) A or C), we assume that the activity coefficients γi’s are given by the equation proposed by Davies:27

[

log γi ) -Az2

]

xI - 0.3I , 1 + xI

(7)

charge δ0, we accept the relation used by Yates et al.23 and by Bousse et al.28,29 (PZC is point of zero charge)

2.303(PZC - pH) )

i ) A, C

A)

1.825 × 106 (rT)3/2

(8)

θi )

1+

∑i

, i ) 0, +, A, C

( )

(14)

int 2e2Ns Ka2 cDLkT Kint a1

1/2

In eq 14, cDL is the linearized double-layer capacitance. The value of cDL can be theoretically calculated (depending on the salt concentration in the solution) in the way described in Bousse’s work:28

1 2kT/e 1 ) + cDL (8 kTc)1/2 cStern

In eq 8, r is the relative permittivity of the solvent. Now, we rewrite eqs 4 to the Langmuir-like form

Kifi

(13)

in which β is given by

β)

where z is the valency of ion i, I (mol/dm3) is the ionic strength of the suspension, and A is given by

( )

eψ0 eψ0 + sinh-1 kT βkT

(15)

r 0

(9)

Kifi

where r is the relative permittivity of solvent, 0 is the permittivity of vacuum space, and c is the concentration of the electrolyte (ions/m3). The value of cStern is assumed to be 0.2 F/m2. The PZC is defined by the condition

δ0(pH)PZC) ) 0

where int

*KC 1 1 K0 ) int, K+ ) int int, KC ) int , KA ) Ka2 Ka1 Ka2 Ka2 1 (10) int Ka2 *Kint A and where fi (i ) 0, +, A, C) are the following functions of proton and salt concentrations:

{

}

eψ0 f0 ) exp - 2.3 pH , f+ ) f20 kT

{

{

}

∑i

A

A

For pH ) PZC, where δ0 ) 0 and ψ0 ) 0, eq 12 can be transformed to the following form:

H2aA *Kint C aC H2 + - 1 ) 0 (18) int int int int int Ka1 Ka2 Ka2 *KA Ka2

(19)

H ) 10-PZC

eψ0 eδ0 - 4.6 pH kT kTc1

K+f+ + KAfA - KCfC - 1 , 1 + Kifi

a1

where

Taking into account eq 6, the nonlinear eq 9 can be transformed into a nonlinear equation with respect to δ0,

δ0 ) B

int 1 + (*Kint 1 1 C /Ka2 )aC int int PZC ) (pKa1 + pKa2 ) - log (17) 2 2 1 + (Kint/*Kint)a

(11)

eψ0 eδ0 fC ) aC exp + kT kTc1 fA ) aA exp -

therefore, from eq 12 we have

U)

}

(16)

(12)

i ) 0, +, A, C This nonlinear equation for δ0 can be easily solved by means of the Mueller-S iteration method to give the value of δ0 for each pH value. Having calculated these values, one can easily evaluate the individual adsorption isotherms θi’s from eq 9. To express ψ0-pH dependence, which occurs in eqs 9-12 for individual adsorption isotherms θi’s and for the surface (27) Davies, C. W. Ion Association; Butterworths: London, 1962.

The experimental studies show that in the majority of the investigated systems the value of PZC does not practically depend upon the salt concentration in the bulk solution.20,21,30,31 It seems that this very interesting feature of these systems has not received enough attention thus far. We recently showed in our publications2,3 that, while drawing mathematically rigorous formal consequences of the existence of CIP (common intersection point), one can arrive at interesting physical conclusions. Namely, except for very low and very high PZC values and for very low salt concentrations, one can assume that aC ) aA ) a. Thus, the independence of PZC of the salt concentration can formally be expressed as follows: int

*KC ∂U H2 - int ) 0 ) int ∂a K *Kint K a2

A

(20)

a2

(28) Bousse, L.; de Rooij, N. F.; Bergveld, P. IEEE Trans. Electron Devices 1983, 30, 8. (29) Van der Vlekkert, H.; Bousse, L.; de Rooij, N. F. J. Colloid Interface Sci. 1988, 122, 336. (30) Sprycha, R. J. Colloid Interface Sci. 1984, 102, 173. (31) Foissy, A.; M’Pandou, A.; Lamarche, J. M.; Jaffrezic-Renault, N. Colloids Surf. 1982, 5, 363.

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Solving the set of eqs 18 and 20 we obtain

*Kint A )

H2 H2 int and K ) a2 int *Kint Ka1 C

(21)

int int , Ka2 and Correlations 21 between the parameters Ka1 int int *KC , *KA reduce the number of the true free parameterssintrinsic equilibrium constants to be found by fitting the experimental datasfrom four to two. This fact had not been realized until we published our three papers1-3 in which the consequences of the existence of CIP were subjected to a rigorous mathematical analysis. While accepting the relations of eq 21, we obtain the following simplified expressions for the value of PZC

1 int int PZC ) (pKa1 + pKa2 ) 2

(22)

1 int PZC ) (p*Kint C + p*KA ) 2

(23)

The relation 22 has already been discussed in the literature. As for eq 23, it can also be developed in another way, but only for the particular case of the triple-layer model considered here. We will soon show that, while considering the temperature dependence of PZC calculated from both of these equations, one may draw important conclusions about the mechanism of ion adsorption. 2. Thermodynamic Interpretation of Calorimetric Experiments and of the Temperature Dependence of Adsorption Isotherms. Immersion Calorimetry. Now, let us consider the calorimetric effects accompanying the formation of the surface complexes SOH0, SOH2+, SOH2+A-, and SO-C+. In the calorimetric experiments reported by Griffiths and Fuerstenau32 and by Foissy,33 the heat of immersion of an outgased solid sample into a solution of a certain pH, Qim(pH), was measured. That heat of immersion is a sum of the following three contributions: (i) the (integral) heat of water adsorption from zero surface coverage up to the (polymolecular) surface coverage corresponding to the saturated vapor pressure ps, represented by the integral

∫0

ps

Qst(p) dp

(24)

where Qst is the differential isosteric heat of adsorption; (ii) the heat of the formation of water-vapor interface, the area of which is equal to the solid surface area S (This heat effect is represented by the equation

(

)

∂γg S γg - T ∂T

(25)

where γg is the surface tension of water-vapor interface); and (iii) the heat of the formation of the electrical double layer, after bringing the saturated solid surface into the contact with the bulk liquid. The first of the above listed contributions can be measured in an independent adsorption (calorimetric) experiment. That measurement, as well as the subsequent one of the heat of immersion of the saturated surface into pure water, was the basis for the Harkins-Jura (HJ) method for the determination of the surface area S. The third contribution, which was probably one of the sources of the observed discrepancies between the surface areas (32) Griffiths, D. A.; Fuerstenau, D. W. J. Colloid Interface Sci. 1981, 80, 271. (33) Foissy, A. Ph.D. Thesis, Universite` de Franche-Comte` Bescanc¸ on, France, 1985.

determined by Brunauer-Emmett-Teller (BET) and HJ methods, has been ignored. As a matter of fact, a theoretical interpretation of this third contribution seems to be extremely difficult. A first interpretation was proposed by Healy and Fuerstenau34 who assumed that the charging of an oxide surface is due solely to the formation of the SOH2+ complexes. Starting from that assumption, they developed the relation between the heat of immersion of an oxide into pure water Qim(pH ) 7) and the PZC value of that oxide,

Qim(pH)7) ) 4.6kT PZC + Qim(pH)0)

(26)

Although a large scatter in the experimental data exists, there is no doubt that a linear relationship can roughly be seen. In view of our present knowledge of the mechanism of the charging process, the assumption made by Healy and Fuerstenau34 can no longer be accepted. Therefore, the linear relation between Qim(pH)7) and PZC is an intriguing problem which deserves further investigation. However, one can easily interpret the difference between the heats of immersions Qim(pH1) and Qim(pH2) in two solutions of pH1 and pH2, respectively,

Qim(pH2) - Qim(pH1) )

[

M

∫pHpH

2

1

]

∂γg(pH1) ∂γg(pH2) + +T ∂T ∂T ∂θ0 ∂θ+ ∂θC Q0 + Q+ + QC + ∂(pH) T ∂(pH) T ∂(pH) T ∂θA d(pH) (27) QA ∂(pH) T

S γg(pH2) - γg(pH1) - T

[( )

( ) ( ) ( )]

where Q0, Q+, QC, and QA are the molar differential heats of the formation of the SOH0, SOH2+, SOH2+A-, and SO-C+ complexes and M is the number of adsorption sites in appropriate units. The expressions for (∂θi/∂(pH))T for the triple-layer model have already been published by us,2 but they are given here in Appendix for the reader’s convenience. The molar differential heats of the formation of these surface complexes are given by the expressions1,2

[

]

b ∂ µSOH - µH Q0 ) -k kT ∂(1/T)

[

{θ0,+,A,C}

{θ0,+,A,C}

{θi}

(28a)

( )

)k

∂F+

∂(1/T)

{θi}

(28b)

]

b ∂ µSO-C+ - 2µC QC ) -k kT ∂(1/T)

[

∂F0

∂(1/T)

)

b ∂ µSOH2+ - 2µH Q+ ) -k kT ∂(1/T)

[

( )

)k

{θ0,+,A,C}

( )

)k

∂FC

∂(1/T)

] ( )

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

k

{θ0,+,A,C}

(28c)

)

∂FA

∂(1/T)

{θi}

{θi}

(28d)

where µSO-C+, µSOH2+A-, µSOH, and µSOH2+ are the chemical potentials of the surface complexes. The analytical form of the derivatives ∂θi/∂(pH) and of the functions Fi (i ) 0, +, A, C) will depend on the assumed model (34) Healy, T. W.; Fuerstenau, D. W. J. Colloid Interface Sci. 1965, 20, 376.

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Langmuir, Vol. 13, No. 3, 1997 487

of the water-oxide interface. Similarly, as in the theories of gas adsorption, we introduce molar differential isosteric heats of adsorption Qi’s. The theoretical interpretation of the heats of immersions (their differences) still requires a measurement or a calculation of γg, not only as a function of temperature but also as a function of pH. Such measurements and calculations are not easy at all. Therefore, in our previous interpretations1,2 of these heats of immersion, we simply neglected the first term on the rhs of eq 27. From this point of view, titration calorimetry is a better experiment for a theoretical interpretation. It was reported first by De Kaizer, Fokkink, and Lyklema,35 and by Machesky and Anderson.36-38 Titration Calorimetry. After an outgased solid sample was introduced into a solution, the pH of that solution was measured and sometimes also the concentration of other ions in the equilibrium bulk electrolyte. Then, a titration step was carried out (from the base or acid side), and the evolved heat was recorded. Also, the new concentration of the equilibrium bulk electrolyte was measured. The recorded heat effect is described by the second term on the rhs of eq 27. The accompanying measurements of ion (proton, cation, anion) adsorption are very useful in further theoretical interpretation. This is the way in which Machesky and Anderson36 carried out their experiment. They measured the heat evolved on changing the pH by ∆pH and ascribed it solely to the proton adsorption (consumption) accompanying that change of pH. Therefore, as the final output of their experiment, they considered the so-called “heat of proton adsorption” Qpr,

Qpr )

( ) [( ) ( ) ( )] ∫pH

pH+∆pH

∑iQi ∂(pH) T d(pH)

∂θ

∂θ

∂θ0

∂(pH)

d(pH)

T

(29)

In the limit of small ∆pH changes we may write

Qpr ) ∂θ0 Q0 ∂(pH)

( )

( ) ( ) ( ) ( ) ( ) ( ) + Q+

∂θ+

∂θC

+ QA

∂(pH) T ∂θ+ ∂θA ∂θ0 2 +2 + ∂(pH) T ∂(pH) T ∂(pH)

T

∂(pH)

+ QC

T

∂θA

∂(pH)

T

T

(30)

From the equation, it follows that Qpr must be dependent on pH, and it was observed in the experiments reported by Machesky and Jacobs.37,38 De Kaizer, Fokkink, and Lyklema35 have monitored the heat effects accompanying ion adsorption as the function of the surface charge δ0. For the purpose of the present consideration we will write δ0 as follows:

δ0 ) B[2θ+ + 2θA + θ0 - 1]

Fx ) δ0 - B[2θ+ + 2θA + θ0 - 1] ) 0

(32)

An incremental increase in δ0 and dδ0, caused by changing pH by d(pH),

∂Fx/∂(pH) d(pH) ) ∂Fx/∂δ0 ∂θ+ ∂θA ∂θ0 B 2 +2 + ∂(pH) T ∂(pH) T ∂(pH) T d(pH) (33) ∂θ+ ∂θA ∂θ0 1-B 2 +2 + ∂δ0 T ∂δ0 T ∂δ0 T

dδ0 ) -

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

will be accompanied by the heat effect dQ,

[( )

dQ ) M Q0

∂θ0

∂(pH)

( ) ( ) ( )] ∂θ+

+ Q+ T

∂(pH)

QA

T

∂θC

+ QC

∂(pH)

∂θA

∂(pH)

+

T

d(pH) (34)

T

where the derivatives (∂θi/∂δ0) from eq 33 are to be evaluated in a manner similar to that for the derivatives (∂θi/∂(pH)). De Keizer, Fokkink, and Lyklema35 considered the cumulative heat of adsorption Qcum(δ0), where

Qcum(δ0) )

∫δδ

0

0,initial

( ) ∂Q ∂δ0

dδ0

(35)

T

and where, according to eqs 33 and 34,

∂θi

+ A +2 + ∫pHpH+∆pH 2 ∂(pH) ∂(pH) T T

nonlinear equation.

(31)

which is equivalent to eq 6. One must remember that θi is dependent on δ0, therefore, we must treat eq 31 as a (35) De Keizer, A.; Fokkink, L. G. J.; Lyklema, J. Colloids Surf. 1990, 49, 149. (36) Machesky, M. L.; Anderson, M. A. Langmuir 1986, 2, 582. (37) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 297. (38) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 315.

dQ M ) × dδ0 B ∂θ0 ∂θ+ ∂θC ∂θA + Q+ + QC + QA Q0 ∂(pH) T ∂(pH) T ∂(pH) T ∂(pH) T × ∂θ+ ∂θA ∂θ0 2 +2 + ∂(pH) T ∂(pH) T ∂(pH) T ∂θ+ ∂θA ∂θ0 1-B 2 +2 + (36) ∂δ0 T ∂δ0 T ∂δ0 T

( )

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

De Kaizer et al.35 reported that Qcum(δ0) is a linear function of δ0 for both rutile and hematite. This means that the function (dQ/dδ0) does not depend upon δ0. They called the derivative (dQ/dδ0) the “heat of proton adsorption” for the titration going from the base side. Kallay et al.39 have recently proposed that, in order to compensate for accompanying heat effects, the heat of proton adsorption should be represented by the integral PZC-∆pH ∫PZC+∆pH ∑Qi i

( ) ∂θi

∂(pH)

d(pH)

(37)

T

In contrast to the results reported by De Kaizer et al.,35 Casey40 has reported recently that in the case of silica the “heat of proton adsorption” (dQ/dδ0) varies with the varying charge of surface δ0. In all of the equations for the heats of adsorption there will appear expressions for the temperature dependence of PZC, which was monitored experimentally by several authors.41-45 Therefore, the description of the calorimetric (39) Kallay, N.; Zalac, S.; Stefanic, G. Langmuir 1993, 9, 3457. (40) Casay, W. H. J. Colloid Interface Sci. 1994, 163, 407. (41) Tewari, P. H.; McLean, A. W. J. Colloid Interface Sci. 1972, 40, 267.

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488



measurements and of the temperature dependence of PZC are closely related problems. 3. Explicit Expressions Developed for the Homogeneous Surface Model. For the model of the homogeneous solid surface, assumed in this paper by us, the functions Fi are defined by eqs 4: int F0 ) -ln Ka2 -

eψ0 θ0 )0 - 2.3pH - ln kT θ-

int int F+ ) -ln(Ka1 Ka2 ) -

(38a)

2eψ0 θ+ ) 0 (38b) - 4.6pH - ln kT θ-

int /*Kint Fc ) -ln(Ka2 C ) -

eψ0 θC eδ0 + ln aC - ln )0 + kT kTc1 θ(38c)

eψ0 eδ0 int *Kint + ln aA FA ) -ln(Ka2 A ) kT kTc1 θA 4.6pH - ln ) 0 (38d) θVan der Vlekkert et al.29 have shown that (∂(pH)/∂T) is small for two kinds of investigated buffers. Thus, neglecting the temperature dependence of pH, from eqs 28 we arrive at the following expressions for Q0, Q+, QC, and QA:

Q0 ) Qa2 - eψ0 -

( ) ( )

e ∂ψ0 T ∂(1/T)

2e ∂ψ0 Q+ ) Qa2 + Qa1 - 2eψ0 T ∂(1/T)

{θi},pH

{θi},pH

( ) ( ) ( ( ) ( ) (

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

(39b)

{θi},pH

(39c)

δ0 e ∂ψ0 -e T ∂(1/T) {θi},pH c1 ∂(δ /c ) ∂(ln aA) e 0 1 +k (39d) T ∂(1/T) {θi},pH ∂(1/T) {θi},pH

e ∂ψ0 T ∂(1/T)

{θi},pH

Q+ ) Qa2 + Qa1 - 2eψ0 -

2e ∂ψ0 T ∂(1/T)

{θi},pH

where the nonconfigurational values are

∂(ln Kint ai ) Qai ) -k , i ) 1, 2 d(1/T) int /*Kint d[ln(Ka2 C )]

QaC ) -k

d(1/T) int *Kint d[ln(Ka2 A )]

QaA ) -k

d(1/T)

(40a)

( ) ( ) ( ( ) ( ) (

δ0 e ∂ψ0 +e + T ∂(1/T) {θi},pH c1 eδ0T dc1 ∂(ln aC) +k 2 dT ∂(1/T) {θi},pH (c )

QC ) QaC - eψ0 -

1

1

∂γi ∂(1/T)

(42)

where

∂(ln A) ∂(ln A) ) -3T4(0.6724 × 10-8) ) -3T4 3 ∂(1/T) ∂T

(43)

Since

) ci

∂γi

, i ) A, C

∂(1/T)

(44)

therefore,

∂ai

(42) Tewari, P. H.; Campbell, A. B. J. Colloid Interface Sci. 1976, 55, 531. (43) Blesa, M. A.; Figliolia, N. M.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1984, 101, 410. (44) Akratopulu, K. Ch.; Vordonis, L.; Lycourghiotis, A. J. Chem. Soc., Faraday Trans. 1 1986, 82, 3697. (45) Akratopulu, K. Ch.; Kordulis, C.; Lycourghiotis, A. J. Chem. Soc., Faraday Trans. 1 1990, 86, 3437.

(41c)

)

∂(ln A) ) γi ln γi , i ) A, C ∂(1/T)

∂(1/T)

(40c)

{θi},pH

(41b)

While taking into account the temperature dependence of γi, (i ) A, C), we assumed that γi was given by eq 7 and that the value of the parameter A depends on temperature in an explicit way and in an indirect way through the relative permittivity of the solvent r, which also depends on temperature. Our analysis2 of the values of A(T) published by Davies27 showed that ln A varies almost linearly with the third power of T, i.e., ln A ) -0.8484 + 0.6724 × 10-8 T3, where the coefficients -0.8484 and 0.6724 × 10-8 were determined by us from the appropriate computer regression

∂ai (40b)

)

(41a)

δ0 e ∂ψ0 -e T ∂(1/T) {θi},pH c1 eδ0T dc1 ∂(ln aA) +k (41d) ∂(1/T) {θi},pH (c )2 dT {θi},pH

QA ) QaA - eψ0 -

QA ) QaA - eψ0 -

)

( ) ( )

Q0 ) Qa2 - eψ0 -

(39a)

QC ) QaC - eψ0 -

)

In addition to the above mentioned findings reported by Van der Vlekkert et al.,29 there is still another fundamental reason for the partial derivatives in eqs 28 to be taken not only at the constant θi’s but also at the constant pH. Namely, looking at the form of the functions (∂θi/∂(pH)) in Appendix, we can see that they are functions of θi’s and of pH. Then, a constant set of θi’s, (i ) 0, +, A, C) also means a constant value of δ0. Therefore, the partial differentiations outlined in eqs 39c,d have to be carried out by assuming that δ0 is a constant. This leads us to the following expressions:

∂(1/T)

) -(2.0172 × 10-8) T4ai ln γi, i ) A, C

(45)

where ci is the concentration of cation and anion in the equilibrium bulk solution. As for the derivative ∂ψ0/∂(1/T) which occurs in eqs 39, Van der Vlekkert et al.29 took equation

ψ0 )

β 2.3kT (PZC - pH) β+1 e

(46)

+

+


Langmuir, Vol. 13, No. 3, 1997 489

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

heat of reaction

no.

reaction type

1 2 3 4

SOH0 + H+ T SOH2+ SO- + H+ T SOH0 SOH0 + H+ + A T SOH2+ASO-C+ + H+ T SOH0 + C+

5 6 7

int int SO- + 2H+ T SOH2+ Qa1 + Qa2 -pKa1 - pKa2 SO- + 2H+ + A- T SOH2+A- -pKint - p*Kint QaA a2 A int SO- + C+ T SO-C+ QaC p*KCint - pKa2

8

SOH2+ + A- T SOH2+A-

Qa1 Qa2 QaA - Qa2 Qa2 - QaC

int -pKa1 int -pKa2 -p*KAint -p*KCint

int pKa1 - p*KAint

∂ψ0

∂(1/T)

) pH

[

QaA - Qa1 - Qa2

)]

(

2.3kT2 β 1 Qa2 - Qa1 1+ -1 e β+1 β+1 2kT k β ∂(PZC) (pH - PZC) + 2.3T (47) e β + 1 ∂(1/T)

which was used in one of our previous publications.1 However, our later studies have shown that this simplification has a serious effect on the accuracy of the calculated heats of adsorption.2 Therefore, we used the exact form of this derivative, resulting from the accurate eq 13, which is

( )

[

]

β t Qa2 - Qa1 ) -ψ0 T + + β+t β+t 2kT ∂(1/T) pH k β ∂(PZC) (48) 2.3T e β + t ∂(1/T) ∂ψ0

( ) ∂ψ0

∂(1/T)

which is valid in the vicinity of PZC, where (eψ0/kT) < β, and which is a simplified version of the accurate eq 13. In doing so, they arrived at the following expression (as ψ0 is only a function of pH and T):

( )

Taking eq 50a into account, we write the derivative ∂ψ0/∂(1/T) in the form

pH

[

) - ψ0 T

Bousse et al.28,29 neglected the temperature dependence of c1, which occurs in eqs 41c,d. To the contrary, Blesa et al.43 argue that the temperature dependence of c1 exists and that the function c1(T) is almost linear. While carrying out the partial differentiation with respect to ∂/∂(1/T), we follow the recommendation by Blesa43 to treat c1 as the following linear functions:

c1 ) c1+ ) c01+ + R+∆T, for pH < PZC (53a) c1 ) c1- ) c01- + R-∆T, for pH > PZC (53b) While accepting Blesa’s assumption, the expressions for the differential isosteric heats of adsorption Qi’s, developed in eqs 41, take the following form:

Q0 ) Qa2 - eψ0 -

QC ) QaC - eψ0 -

(49)

For t ) 1 (see eq 46 for ψ0) eq 48 reduces to eq 47. For the reader’s convenience, Table 1 summarizes the types of surface reactions, along with the corresponding equilibrium constants, and the nonconfigurational heats of adsorption. While calculating the derivative ∂(PZC)/∂(1/T) from eqs 22 and 23, we obtain

∂(PZC) Qa1 + Qa2 ) 2.3 2k ∂(1/T)

(50a)

∂(PZC) QaA - QaC ) 2.3 2k ∂(1/T)

(50b)

Now, let us consider a consequence of expression ∂(PZC)/ ∂(1/T) in eqs 47 and 48 arising from the existence of CIP. From eqs 50 we arrive at the following equation:

QaA - (Qa1 + Qa2) ) QaC

(51)

which is surely an intriguing result. This means that the heat effect related to cation adsorption on an empty oxygen atom (QaC) is the same as the heat relaxed to the attachment of the anion to the already existing complex SOH2+. That intriguing result will be discussed in the next section.

( ) ( )

e ∂ψ0 T ∂(1/T)

Q+ ) Qa2 + Qa1 - 2eψ0 -

( )

e ∂ψ0 T ∂(1/T)

where

t ) 1/x(eψ0/βkT)2 + 1

]

β t Qa2 - Qa1 + + β+t β+t 2kT T β Qa1 + Qa2 (52) e β+t 2

QA ) QaA - eψ0 -

2e ∂ψ0 T ∂(1/T)

{θi},pH

(54a)

(54b)

δ0 eδ0TR( + + c1 {θi},pH (c1)2 ∂(ln aC) (54c) k ∂(1/T) {θi},pH

( )

e ∂ψ0 T ∂(1/T)

{θi},pH

+e

(

)

δ0 eδ0TR( + c1 {θi},pH (c1)2 ∂(ln aA) (54d) k ∂(1/T) {θi},pH -e

(

)

where R( is equal to R+ for pH < PZC and to R- for pH > PZC. Because PZC is commonly known, through the application of eqs 22 and 23 one can reduce the number of the independent “intrinsic” constants to two. Therefore, we int and *Kint will later operate only by the parameters Ka2 A treated as the best-fit parameters in our numerical exercises. Of course, we will have two more best-fit parameters, i.e., the c01+ and c01- values, to be found by fitting the experimental adsorption isotherms (titration curves). 4. Estimating Adsorption Enthalpies from Temperature Dependence of Adsorption Isotherms. The apst direct measurements of the calorimetric effects accompanying ion adsorption at the oxide-electrolyte interface required solving complicated technical problems. Therefore, almost all of the body of the reported experimental data has been measured only during the last decade. However, the first studies of the enthalpic effects of ion adsorption based on temperature dependence of adsorption isotherms started 25 years ago. It was Berube and de

+

490

+



Bruyn46 who studied the effect of temperature on PZC. Studies of that kind were conducted next by Tewari et al.,41,42 Fokkink et al.,15,47 Schwarz and co-workers,48 Kuo and Yen,49 Akratopulu et al.,44,49 Blesa et al.,50 and Kosmulski et al.51,52 The influence of temperature on the δ0 vs pH profiles was studied by Blesa and co-workers43,53 and Brady54 and Kosmulski.55 Valuable information concerning that problem has been collected in the review by Machesky et al.56 The calorimetric effects of ion adsorption were sometimes estimated from the relation56,57

(

d(1/T)

)

(

, i ) A, C

)

d(ln ai) d(1/T)

∂δ0 ∂T

)pH

( )

1 ∂δ0 T2 ∂(1/T)

)

pH

1 (∂G/∂(1/T))δ0,pH T2 (∂G/∂δ0)T,pH

K+f+ + KAfA - KCfC - 1

G ) δ0 - B

∑i Kifi

) 0,

(58)

(55a) i ) 0, +, A, C

, i ) A, C

(55b)

δ0

Though it was not explicitly said, there was a tendency to treat Qion as a nonconfigurational quantity, related only to the single act of the adsorption of the considered ion. Therefore, usually only some values were reported for the Qion. Meanwhile, it should be realized that Qion is a function of pH (or δ0) and its interpretation involves the necessity of considering enthalpic effects caused by the coadsorption of other ions. The theoretical interpretation of the derivatives in eqs 55a,b appears to be pretty complicated and will be the subject of a separate publication. Exact measurements of PZC and, therefore, of the temperature dependence on PZC create some experimental problems. The shift in PZC with the temperature was sometimes estimated from the shift of the whole titration curve δ0(pH), whose shift was assumed to be a number. This procedure must be treated with caution, and in view of the equations developed in this paper, a better way would be to analyze the shift of the whole titration curve, assumed to be a function of δ0. Therefore, let us say for this purpose that having measured the titration curve at a temperature T1, one can easily calculate the titration curve at another temperature T2 from the obvious relation

δ0(T2,pH) ) δ0(T1,pH) +

(57)

where

pH

or from:47,55

Qion ) -k

( )

1+

d(ln ai)

Qion ) -k

The derivative (∂δ0/∂T)pH is evaluated from eqs 11 and 12 as follows:

∫TT ( ∂T0)pH dT 2

1

∂δ

and fi’s are given by eq 11. Then, it is to be noted that, while taking appropriate derivatives, the pH in the functions fi’s is to be treated as a constant. After performing the differentiation outlined in eq 56, we arrive at the following explicit expression for (∂δ0/∂T)pH

( )

{( )

∂δ0 B ∂K0f0 )- 2 (1 - K+f+ - KAfA + KCfC) + ∂T pH T ∂(1/T) pH ∂K+f+ ∂KAfA (2 + K0f0 + 2KCfC) + (2 + K0f0 + ∂(1/T) pH ∂(1/T) pH ∂KCfC (K f + 2K+f+ + 2KAfA) × 2KCfC) ∂(1/T) pH 0 0

( )

( )

( )

{

}

∑iKifi)2 + kTc1[KAfA(2 + K0f0 + 2KCfC) + eB

(1 +

}

-1

KCfC(K0f0 + 2K+f+ + 2KAfA)]

(59)

because

( ) ∂K0f0 ∂δ0

( ) ∂KCfC ∂δ0

( ) ( ) ∂K+f+ ∂δ0

)0 T,pH

∂KAfA ∂δ0

e ) KCfC kTc1 T,pH

)0

(60a)

T,pH

T,pH

e ) -KAfA kTc1 (60b)

where

(56)

(46) Berube, Y. G.; de Bruyn, P. L. J. Colloid Interface Sci. 1968, 27, 305. (47) Fokkink, L. G. J. Ph.D. Thesis, Agricultural University, Wageningen, 1987. (48) Subramanian, S.; Schwarz, J. A.; Hejase, Z. J. Catal. 1989, 117, 512. (49) Kuo, J. F.; Yen, T. F. J. Colloid Interface Sci. 1989, 121, 220. (50) Blesa, M. A.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1990, 140, 287. (51) Kosmulski, M.; Matysiak, J.; Szczypa, J. J. Colloid Interface Sci. 1994, 164, 280. (52) Kosmulski, M.; Matysiak, J.; Szczypa, J. In Proceedings of the 8th Int. Conf. Surf. and Colloid Sci., Adelaide, Australia; 13-18 Feb, 1994. (53) Regazzoni, A. E. Ph.D. Thesis, Universidad National de Tucuman, Argentina, 1984. (54) Brady, P. V. Geochim. Cosmochim. Acta 1992, 56, 2941. (55) Kosmulski, M. Colloids Surf., A 1994, 83, 237. (56) Machesky, M. L. In Chemical Modelling in Aqueous Systems II; Melchior, D. C., Bassett, R. L., Eds.; ACS Symposium Series 416; American Chemical Society: Washington, DC, 1990. (57) Bru¨mmer, G. W.; Gerth, J.; Tiller, K. G. J. Soil Sci. 1988, 39, 37.

( ) ∂K0f0

∂(1/T)

( )

pH,δ0

∂K+f+

∂(1/T)

( )

( )]

Qa2 eψ0 e ∂ψ0 ) k k kT ∂(1/T) pH K0f0Q0 (61a) k

[

Qa1 + Qa2 2eψ0 k k K+f+Q+ 2e ∂ψ0 ) (61b) kT ∂(1/T) pH k

) K+f+ pH,δ0

∂KCfC

∂(1/T)

[

) K0f0

[

) KCfC pH,δ0

( )]

( ) ( ) ]

QaC eψ0 e ∂ψ0 k k kT ∂(1/T)

( )

∂c1 eδ0 eδ0 2 kc1 kT(c ) ∂(1/T) 1

pH,δ0

+

pH

∂(ln aC)

+

∂(1/T)

pH,δ0

)

KCfCQC k (61c)

+

+


( ) ∂KAfA

∂(1/T)

[

) KAfA pH,δ0

Langmuir, Vol. 13, No. 3, 1997 491

( ) ( ) ]

QaA eψ0 e ∂ψ0 k k kT ∂(1/T)

( )

∂c1 eδ0 eδ0 + 2 kc1 kT(c ) ∂(1/T) 1

-

pH

∂(ln aA)

+

∂(1/T)

pH,δ0

)

pH,δ0

KAfAQA k (61d)

and because taking partial derivatives at constant and δ0 is equivalent to taking these partial derivatives at constant pH and θi, (i ) 0, +, A, C), as was done in eqs 39. Because the derivative [(∂ψ0/∂(1/T)]pH in eqs 61 was evaluated at constant pH, one can use the expression 52 to represent [(∂ψ0/∂(1/T)]pH in eqs 61. The derivative (∂δ0/∂T)pH includes all of the calorimetric int 0 parameters of interest: Ka1 , *Kint C , Qa2, QaA, c1+, R+, 0 c1-, R-, and Ns. Equation 56 enables one to generate the theoretical δ0(pH) curves for the temperatures at which the experimental curves δ0(T) were measured. Therefore, int one can try to determine the parameters Ka1 , *Kint C , Qa2, 0 0 QaA, c1+, R+, c1-, and Ns as those leasing to a best fit of the experimental δ0(pH) curves by those generated theoretiint cally. (The remaining parameters Ka2 , *Kint A , Qa1, and QaC are next calculated from the relations 22, 23, and 50a,b.) It is important to realize that, as long as the amount of the bulk solution being in contact with an oxide sample is large, the change in the ln aC or ln aA due to adsorption or desorption of the cations and anions will be negligible. Therefore, one can safely accept that in most of the adsorption systems of a practical interest the relation 45 can be used to represent

k

(

)

∂(ln ai) ∂(1/T)

{θi},pH

In other words, the changes in ai’s caused by changing temperature will be due to the changes in the properties of the bulk electrolyte. To calculate δ0(pH) at different temperatures one must have the following parameters calculated for different temperatures:

[ [ [ [

] ] ] ]

int int pKa1 (T2) ) pKa1 (T1) +

Qa1 1 1 2.3k T2 T1

(62a)

int int (T2) ) pKa2 (T1) + pKa2

Qa2 1 1 2.3k T2 T1

(62b)

int p*Kint C (T2) ) p*KC (T1) +

Qa2 - QaC 1 1 2.3k T2 T1

(62c)

int p*Kint A (T2) ) p*KA (T1) +

QaA - Qa2 1 1 2.3k T2 T1

(62d)

[

]

(63a)

[

]

(63b)

and

PZC(T2) ) PZC(T1) +

Qa1 + Qa2 1 1 4.6k T2 T1

PZC(T2) ) PZC(T1) +

QaA - QaC 1 1 4.6k T2 T1

or

Illustrative Calculations and Discussion The theoretical expressions developed in the previous sections make it possible to analyze both the temperature

dependence and the calorimetric effects accompanying simple ion adsorption at the oxide-electrolyte interfaces. There are a number of questions which still remain open. For instance, (i) why the heat of immersion of oxides is a linear function of PZC; (ii) why the cumulative heat of adsorption Qcum(δ0) is a linear function of δ0 for some oxides, (TiO2 and Fe2O3), but it is not for others (SiO2); (iii) why the temperature “congruency” of the δ0(pH) curves is observed in some systems, but not observed in others. We start by discussing the intriguing eq 51. Equations 23 and 50b have been discussed in our papers1,2 as the result of us drawing rigorous mathematical consequences of the existence of CIP. Equation 51 states that the enthalpic effects accompanying anion adsorption on an already existing complex SOH2+ and the effects accompanying adsorption of cation on an empty surface oxygen atom are the same. Our present hypothesis is that these enthalpic effects are the same because they are simply nonexistent. Such a conclusion was drawn 2 years ago by Kosmulski55 who studied the temperature dependence of adsorption of ions of an inert electrolyte. In such a case, the number of the best-fit parameters introduced by the appearance of the nonconfigurational heats of adsorption: Qa1, Qa2, QaC, and QaA reduces to one. In our forthcoming illustrative calculations, we will choose Qa2 as the free parameter to be found by computer. For the purpose of illustration, we have chosen the data reported by Blesa et al.43 These are their titration curves δ0(pH) for magnetite, measured at three concentrations of the inert electrolyte KNO3: 10-3, 10-2, and 10-1 M. Every titration curve was measured at three temperatures: 30, 50, and 80 °C. As a matter of fact, the experimental data of that kind have rarely been reported, and Blesa’s data are probably those containing the largest number of experimental points. Blesa and co-workers43 also proposed an interesting theoretical interpretation of their data which was a partial inspiration for us to publish this paper. Blesa’s investigation trategy was to determine the int int int , Ka2 , *Kint values of Ka1 C , and *KA for every temperature by using the popular graphical extrapolation method. They also assumed that the capacitance parameters c01+, c01-, R+, and R- do not depend on the concentration of the inert int int electrolyte. The determined parameters Ka1 , Ka2 , int int *KC , and *KA well satisfied the relations 22 and 23, and the theoretical δ0(pH) curves well fitted the experimental data at the highest concentration of the inert KNO3 electrolyte, 10-1 M. The least successful was the fit in the case of the smallest concentration of the inert electrolyte, 10-3 M. While commenting on the discrepancies between theory and experiment, they ascribed them to the existence of sites of different reactivities (energetic heterogeneity of surface oxygens) and to the existence of a porous double layer. We generally share the opinion that the above mentioned factors may influence the agreement between theory and experiment. As a matter of fact, we analyzed the role of the energetic heterogeneity of surface oxygens in our recent publications.1,3 However, our recent theoretical works seem to suggest that there is still another physical factor which may influence the behavior of these adsorption systems. Namely, that the electric capacitancies may be affected by the concentration of the inert electrolyte.2,3 Although this effect does not follow conceptually from the hitherto theoretical treatments of the triple-layer model, we decided to include it into our analysis by assuming that the (best-fit) parameters c01+, c01-, R+, and R- may be different for different electrolyte concentrations. Next, we decided to check how we, considering this effect alone, may improve the

+

+

492



Table 2. Temperature Dependence of PZC in the Magnetite-KNO3 System Reported by Blesa et al.43 T (K)

PZC

303 323 353

6.80 6.45 6.00

Table 3. Values of the Parameters r+ and r- Found by Us while Fitting the Blesa’s Titration Curves for the System Magnetite-KNO3 at Three Temperatures c (M)

R+ (F/m2 deg)

R- (F/m2 deg)

10-3 10-2 10-1

0.060 0.005 0.003

0.0 0.0 0.0

Table 4. Values of the Parameters c01+ and c01- at 303 K Found by Us while Fitting Blesa’s Titration Curves for the Magnetite-KNO3 System 10-3 M

10-2 M

10-1 M

c01+

c01-

c01+

c01-

c01+

c01-

1.40

1.80

0.95

1.15

1.10

1.30

Figure 1. Comparison of the theoretical titration curves calculated by us using the best-fit and calculated parameters collected in Table 5 with the experimental data found by Blesa et al.13 from the KNO3-magnetite system at three temperatures: (b) 30 °C, (+) 50 °C, and (0) 80 °C when the concentration of the inert electrolyte KNO3 is 10-3 M.

agreement between theory and experiment, without taking into account the energetic heterogeneity of surface oxygens. Table 2 shows the temperature dependence of PZC reported by Blesa et al.43 From that temperature dependence and eqs 50a,b the following relations are found:

Qa1 + Qa2 ) QaA - QaC ) 65.5 kJ/mol Then, according to the hypothesis that QaC ) 0, we choose Qa2 as the only best-fit parameter introduced into our bestfit exercises by the appearance of the four nonconfigurational heats of adsorption Qa1, Qa2, QaA, and QaC. For its physical meaning, the parameter Qa2 should be temperature independent and should not depend on the concentration of the inert electrolyte. While anticipating further discussion, we report that the value of Qa2 we found was 45 kJ/mol. At the same time, the parameters pKint should depend on temperature. However, for their physical meaning, they should not depend on the concentration of the inert electrolyte. According to eqs 22 and 23, only two of the four pKint parameters can be chosen as free best-fit parameters in appropriate best-fit calculations. We have int and p*Kint chosen pKa1 C as the best-fit parameters, while fitting our theoretical expressions to Blesa’s experimental int and p*Kint data. The values of pKa1 C found by us for the magnetite-KNO3 system at 303 K are 4.40 and 6.60, respectively. We would like to mention that following Blesa’s consideration43,58 we accepted a priori that Ns ) 6 sites/nm2. Now we are approaching an essential point by saying that the capacitances c1+ and c1- have appeared to be not only temperature dependent. To fit Blesa’s experimental data, we were forced to assume that the parameters c01+, R+, c01-, and R- are also functions of the concentration of the inert electrolyte. Tables 3 and 4 show the values of these parameters found in our computer best-fit exercises. That the capacitances c1+ and c1- must be temperature dependent can easily be understood while considering the principles of the three-layer surface complexation model. However, this model does not clearly predict the dependence of the capacitances c1+ and c1- on the concentration of the basic electrolyte. (58) Regazzoni, A. E.; Blesa, M. A.; Maroto, A. J. G. J. Colloid Interface Sci. 1983, 91, 560.

Figure 2. Comparison of the theoretical titration curves calculated by us using the best-fit and calculated parameters collected in Table 5 with the experimental data found by Blesa et al.43 from the KNO3-magnetite system at three temperatures: (b) 30 °C, (+) 50 °C, and (0) 80 °C when the concentration of the inert electrolyte KNO3 is 10-2 M.

Nevertheless, the necessity of considering this dependence has already been reported in some works.31,59,60 Taking this into account, we have seen substantial improvement of the agreement between theory and experiment for the lowest concentration of the inert electrolyte, 10-3 M. This is shown in Figure 1. Figures 2 and 3 show good agreement between theory and experiment for the two higher electrolyte concentrations. Of course, one may ask immediately whether the better fit here may not be due to a larger number of best-fit parameters which could appear in our theoretical treatment. Therefore, let us remark that the number of the best-fit parameters in Blesa’s analysis was equal to 16. These were four pKint values estimated at three temperatures, plus four values of the parameters c01+, R+, c01-, and R-. In our theoretical analysis, the number of the free best-fit parameters is equal to 15. These are three values of the int , and p*Kint parameters Qa2, pKa1 C , six values of the parameters R+ and R-, and six values of the parameters c01+ and c01-. (59) Sprycha, R. Habilitation Thesis, Maria Curie Sklodowska University, Lublin, Poland, 1986. (60) Blesa, M. A.; Kallay, N. Adv. Colloid Interface Sci. 1988, 28, 111.

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Langmuir, Vol. 13, No. 3, 1997 493

Figure 3. Comparison of the theoretical titration curves calculated by us using the best-fit and calculated parameters collected in Table 5 with the experimental data found by Blesa et al.43 from the KNO3 magnetite system at three temperatures: (b) 30 °C, (+) 50 °C, and (0) 80 °C when the concentration of the inert electrolyte KNO3 is 10-1 M.

Figure 4. Individual adsorption isotherms of ions θi (i ) 0, A, C) calculated for three temperatures: (s) 30 °C, (- -) 50 °C, and (‚‚) 80 °C when the concentration of the inert electrolyte KNO3 is 10-2 M.

Table 5. Collection of the Best-Fit and Calculated Parameters, Which Were Used by Us To Fit Our Equations to Blesa’s Titration Curves Measured at Three Temperatures and for Three Concentrations of the Inert Electrolyte Ns (sites/nm2)

Qa1 (kJ/mol)

Qa2 (kJ/mol)

QaC (kJ/mol)

QaA (kJ/mol)

6

20.5

45.0

0.0

65.5

T (K)

pKint a1

pKint a2

p*Kint C

p*Kint A

303 323 353

4.40 4.18 3.90

9.20 8.72 8.10

6.60 6.12 5.50

7.00 6.78 6.50

c (M)

R+ (F/m2 deg)

R1 (F/m2 deg)

10-3

0.060 0.005 0.003

0.0 0.0 0.0

10-2 10-1 10-3 M

10-2 M

Figure 5. Individual adsorption isotherms of ions θi (i ) +, -) calculated for three temperatures: (s) 30 °C, (- -) 50 °C, and (‚‚) 80 °C when the concentration of the inert electrolyte KNO3 is 10-2 M.

10-1 M

T (K)

c1+

c1-

c1+

c1-

c1+

c1-

303 323 353

1.40 2.60 4.40

1.80 1.80 1.80

0.95 1.05 1.20

1.15 1.15 1.15

1.10 1.16 1.25

1.30 1.30 1.30

For the purpose of convenience of the reader who would like to repeat our calculations, or to calculate other physical values of interest, Table 5 is the collection of all of the parameters which were used in our calculations. The lack of temperature dependence of the electrical capacitancies for the alkaline branch is very interesting, suggested by the data presented in Table 3. Table 4 shows that the parameters c01+ and c01- are much higher for the lowest electrolyte concentration, 10-3 M. They are pretty much the same for the two higher electrolyte concentrations. Much higher values of c01+ and c01- parameters for the lowest electrolyte concentrations, along with the high temperature sensitivity of the electrical capacitance (high value of R+), suggest that another factor may play an important role at such low electrolyte concentrations. This is obviously the region where the adsorption of cations and anions plays the smallest role, i.e., the properties of the adsorption system may be much more strongly influenced by pure adsorption of protons. Their adsorption can be more strongly influenced by the energetic heterogeneity of surface oxygens, and this has been ignored in the present analysis. It may well be ture that the relatively high values of c01+, c01-, and R+ in

this region of low electrolyte concentrations simulate to some extent the effects due to energetic heterogeneity of surface oxygens. This problem will be the subject a future study. Meanwhile, we would like to draw the reader’s attention to the calculated individual adsorption isotherms of ions, presented in Figures 4 and 5. They show that the adsorption isotherm of anions θA is pretty much temperature independent, whereas the individual adsorption isotherm of cations θC is affected strongly by temperature. This property seems to be a common feature of these systems. Figure 6 shows the same effect found in adsorption studies where the isotherms of cations and anions were directly monitored experimentally by using the radiometric method.55 One can see that the adsorption of cations Na+ on alumina is strongly affected by temperature, whereas the adsorption of Cl- ions is pretty much temperature independent. Our theoretical calculations and the above discussed experimental findings show the risk of using the simple formulas 55 to estimate heats of adsorption. The temperature dependence of cation adsorption might suggest the existence of considerable heat effects accompanying cation adsorption. Meanwhile, that behavior of cation adsorption isotherm was found by us by assuming that the nonconfigurational heat effect accompanying cation adsorption is nonexistent. Figure 4 shows that the temperature dependence of cation adsorption is due to the temperature dependence of the (first) proton adsorp-

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494

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Figure 6. Individual adsorption isotherms of Na+ and Clions on alumina measured by Kosmulski55 the radiometric methods. Here (4) and (9) are the individual (experimental) isotherms of Cl- ions measured at 15 and 35 °C, respectively. Also, (2) and (0) are the individual (experimental) adsorption isotherms of Na+ ions measured respectively at 15 and 35 °C. The solid and broken lines are the second-older polynomials approximating best experimental data and drawn here to help the eye.

tion isotherm θ0, which is the adsorption process strongly competitive to cation adsorption. Here is the point where results of our analysis differ from the results of the analysis by Blesa and co-workers. They have found the adsorption of cations to be an exothermic process characterized by the value QaC ) 17.4 kJ/mol. The same heat effect was found for anion adsorption on the already existing surface complex SOH2+, QaA - Qa1 - Qa2 ) 17.4 kJ/mol. Although the above values found by Blesa and coworkers satisfy relation 51, we still support the view that these values should, in fact, be equal to zero. Otherwise it is difficult to understand why many different pairs of cations and anions should satisfy relation 51. In trying to understand why the two heat effects are equal to zero, we state the following argument. The relatively big alkali cations are probably adsorbed in a purely electrostatic way. The same is probably true for the adsorption of anions on the already existing complexes SOH2+. The large alkali cations and anions are not hydrated in the bulk solution, and their electrostatic adsorption does not involve “melting” phenomena. This may not be true in the case of small Cd2+ ions, for instance. In addition, protons and small cations may form additionally real chemical bonds to the surface oxygens. Thus, Fokkink et al.61 found experimentally considerable heat effects accompanying Cd2+ adsorption on rutile and hematite. As we mentioned at the beginning of this section, the developed equations may be used to study a number of interesting phenomena which are still not well understood. Because we focus our attention here on the temperature dependence of adsorption isotherms, we are going to consider the problem of the “temperature congruency” of the δ0(pH) curves as well. Fokkink et al.15 have suggested that changing temperature causes only a shift of the whole titration curve δ0(pH) toward the pH scale without changing its shape. They argued next that this may be a universal feature of these adsorption systems. Thus, we have decided to check the existence of such a congruence in the system investigated here, magnetite-KNO3. (61) Fokkink, L. G. J.; de Keizer, A.; Lyklema, J. J. Colloid Interface Sci. 1990, 135, 118.


Figure 7. Congruency test for the KNO3-magnetite system studied by Blesa and co-workers.43 The solid lines are the functions ∆δ0 defined in eqs 64 drawn for the values δin 0 given at the lines. The initial temperature T0 ) 303 K.

Provided that the above mentioned congruence exists, the function ∆δ0,

∆δ0 ) δin ˜ H,T) 0 (pH,T0) - δ0(p

(64a)

p˜H ) pH + [PZC(T0) - PZC(T)]

(64b)

should be equal to zero for any pair of values δin 0 andT.In eq 64a, δin is a certain value of the surface charge 0 corresponding to a certain value of pH and T0 is the initial temperature of the system. Figure 7 shows a number of ∆δ0(δin 0 ,T) functions calculated for both acidic (positive δln 0 values) and alkaline branches. One can see in Figure 7 that the congruency does not exist in this system and that the lack of congruency will be more clearly seen in the acidic branch of the titration curve. While concluding the results of our present work, we say that the studies of the temperature dependence and of the calorimetric effects accompanying the adsorption of simple ions at the oxide-electrolyte interface provide new interesting information about the mechanism of adsorption in these systems. The conclusion that the adsorption of cations on the surface oxygens and anions on the already existing complex SOH2+ does not involve noticeable heat effects will surely simplify further theoretical studies. Our present study seems to suggest that electrical capacitancies may noticeably be affected by the concentration of the inert electrolyte. This should be understood rather as the dependence of the electrical capacitancies on the composition of the adsorbed phase. How much more important this effect is cannot be definitely stated without studying the effects of the energetic heterogeneity of surface oxygens simultaneously. In spite of a frequent criticism, the triple-layer model is probably that most frequently used, because it is very flexible and can be applied over a large range of ionic strengths. The theoretical principles of TLM still arouse a strong interest. We mention here the extremely interesting fundamental study by Lutzenkirchen et al.62 and the paper by Katz and Hayes.63 As we have already mentioned, these fundamental studies must, in most cases, include the more or less pronounced effect of the energetic heterogeneity of oxide surfaces. Their modeling can be done in a number of ways. (62) Lutzenkirchen, J.; Magnico, P.; Behra, P. J. Colloid Interface Sci. 1995, 170, 326. (63) Katz, L. E.; Hayes, K. F. J. Colloid Interface Sci. 1995, 170, 477.

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Langmuir, Vol. 13, No. 3, 1997 495

In many cases a dual-site model may constitute a sufficiently satisfactory approximation.8,64 Then, in certain cases of very heterogeneous surfaces, an assumption of a continuous (“diffuse”) adsorption energy distribution may be necessary.3,65 A generalization of the present theoretical treatment for the case of energetically heterogeneous surface oxygens will be the subject of a future publication.

∂F0 ∂F0 ∂F0 ∂F+ ∂F+ ∂F+ ∂FA 1 ) ) ) ) ) ) )- , ∂θ+ ∂θC ∂θA ∂θ0 ∂θC ∂θA ∂θC θ∂FC ∂FC ∂FC 1 1 )+ w, ) )+ 2w, ∂θ0 θ∂θ+ ∂θA θ∂FA ∂FA 1 1 )- w, )- 2w ∂θ0 θ∂θ+ θ-

Appendix The derivatives (∂θi/∂(pH)) (i ) 0, +, A, C) from eq 27 are to be evaluated from eqs 38. For instance,

∂θ0 ∂(pH)

where

w) ) (-1) ×

|

∂F0/∂(pH) ∂F0/∂θ+ ∂F0/∂θC ∂F0/∂θA ∂F+/∂(pH) ∂F+/∂θ+ ∂F+/∂θC ∂F+/∂θA ∂FC/∂(pH) ∂FC/∂θ+ ∂FC/∂θC ∂FC/∂θA ∂FA/∂(pH) ∂FA/∂θ+ ∂FA/∂θC ∂FA/∂θA ∂F0/∂θ0 ∂F0/∂θ+ ∂F0/∂θC ∂F0/∂θA ∂F+/∂θ0 ∂F+/∂θ+ ∂F+/∂θC ∂F+/∂θA ∂FC/∂θ0 ∂FC/∂θ+ ∂FC/∂θC ∂FC/∂θA ∂FA/∂θ0 ∂FA/∂θ+ ∂FA/∂θC ∂FA/∂θA

|

|

|

t ) -2.3 t+β ∂(pH) ∂FC ∂(pH)

β ) 2.3 t+β

∂F+ ∂(pH)

∂θi )∂(pH)

(A.1)

2.3θi Xi , i ) 0, +, A, C t+β Y

(A.4)

where

X0 ) t[θ- + θC - θ+ - θA] + 2wt[θ-θA - θ+θC] + β[θC - θA] (A.5a)

t ) -4.6 (A.2) t+β

∂FA

(A.3a)

After solving the determinants, we arrive at the following explicit form of the derivatives (∂θi/∂(pH)),

While evaluating the partial derivatives (∂Fi/∂(pH)) occurring in the denominator of eq A.1, one has to consider ψ0 as a function of pH defined in eq 13. Then,

∂F0

eB kTc1

2t + β t+β

X+ ) t[θ0 + 2θ- + 2θC] + wt[θ0θC + θAθ0 + 4θAθ- + 4θAθC] + β[θC - θA] (A.5b) XC ) -t[θ0 + 2θ+ + 2θA] + wt[θ0θ+ + θ-θ0 + 4θ-θ+ + 4θ-θA] - β[θ0 + θ+ + θ- + 2θA] (A.5c)

) -2.3 ∂(pH)

Furthermore, while evaluating the derivatives (∂Fi/∂θj), (i, j ) 0, +, A, C), one still has to consider eq 31 to obtain

θ0 + θ- ∂F+ θ+ + θ∂F0 ), ), ∂θ0 θ0θ∂θ+ θ+θ-

XA ) t[θ0 + 2θ- + 2θC] - wt[θ0θ+ + θ-θ0 + 4θ-θ+ + 4θ+θC] + β[θ0 + θ+ + θ- + 2θC] (A.5d) Y ) 1 + w[θA(2θ- + 2θC + θ0) + θC(2θ+ + 2θA + θ0)] (A.5e)

(A.3)

∂FC θC + θ- ∂FA θA + θ), )- 2w, ∂θC θCθ∂θA θAθ(64) Dzombak, D. A.; Morel, M. M. J. Colloid Interface Sci. 1986, 112, 588. (65) Rudzinski, W.; Charmas, R.; Partyka, S.; Bottero, J. Y. Langmuir 1993, 9, 2641.

where t is given by eq 49 and w by eq A.3a. For t ) 1, eqs A.5a-e reduce to the expressions which we developed in our first publication dealing with the influence of surface heterogeneity on the calorimetric effects of ion adsorption.1 However, we found out later that applying the full form of eq 13 yields theoretical heats of adsorption exhibiting much different behavior from those calculated by accepting the appropriate relation 16.2 LA960649A

Calorimetric Effects and Temperature Dependence of Simple Ion

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