Langmuir 2002, 18, 4809-4818
4809
1pK and 2pK Protonation Models in the Theoretical Description of Simple Ion Adsorption at the Oxide/ Electrolyte Interface: A Comparative Study of the Predicted and Observed Enthalpic Effects Accompanying Adsorption of Simple Ions Wojciech Piasecki* Laboratory for Theoretical Problems of Adsorption, Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Ul. Niezapominajek, Krako´ w, 30-239 Poland Received February 8, 2002. In Final Form: March 26, 2002 There is developed a quantitative theoretical description of enthalpies of surface protonation of the (hydr)oxide/electrolyte solution interface assuming the 1pK charging mechanism. The predictions obtained by adopting the 1pK basic Stern model (BSM) of the interface are compared with the experimental data (potentiometric titration curves and heats of proton adsorption) and with the results obtained recently by adopting the 2pK triple-layer model (TLM) of the interface. It is found that the 1pK BSM gives similar results to the 2pK TLM, but in some cases the latter appears to be more flexible than the 1pK BSM. The conditions for existence of the common intersection point were consistently applied in the calculations to reduce the number of parameters used. The obtained values of heats of proton adsorption are compared with the values available in the literature. The found heats are exothermic and range between +17 kJ/mol (silica) and +46 kJ/mol (alumina) in the case of 1pK BSM. The calculated values of the proton adsorption entropy (T∆S) are almost constant for the investigated systems and equal to +3 kJ/mol. This means that enthalpic effects play a main part in the process of proton adsorption on metal (hydr)oxides and silica.
Introduction The metal (hydr)oxide/electrolyte interface and the silica/electrolyte interface play a important role in the environment and technology. A complete physicochemical description of these interfaces will require good understanding of many important phenomena occurring in these systems. Among the most essential ones is the process of the electric charge formation on the surfaces of metal (hydr)oxides and silica being in contact with an electrolyte solution. The most popular approaches describing this process are the two models called 2pK and 1pK. In a theoretical description of these interface systems, both 2pK and 1pK charging models can be combined with a variety of the mechanistic models of the electric double layer existing at the solid/solution interface. This may result in a number of different theoretical descriptions of the interface under consideration. So far, the combination of the 2pK charging model and the triple-layer model (TLM) of the interface has been most frequently applied. It was developed by Yates et al.,1 Chan et al.,2 and Davis et al.3,4 This approach assumes that there are neutral groups SOH on a metal oxide surface which are amphoteric (where S represents the complex surface group -MeOH25). They can attach or release protons to evolve into SOH2+ or SO-, respectively. Thus, * Correspondence should be forwarded to the following address: Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie Skłodowska University, Pl. Marii Curie Skłodowskiej 3, 20-031 Lublin, Poland. Ph: (48) (81) 5375519. Fax: (48) (81) 5375685. E-mail:
[email protected]. (1) Yates, D. E.; Levine, S.; Hearly, T. W. J. Chem. Soc., Faraday Trans. 1 1974, 70, 1807. (2) Chan, D.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (3) Davis, J. A.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 480. (4) Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1978, 67, 90. (5) Borkovec, M. Langmuir 1997, 13, 2608.
two dissociation steps should be observed. These charged groups are able to form ion pairs with anions and cations present in the solution. More recently, the 1pK charging model has been proposed by Bolt, van Riemsdijk, and Hiemstra.6-10 It is based on Pauling’s principle of local neutralization of electric charge in ionic crystals.11 According to that principle, the protonated surface oxygens should have fractional values of charge. In the simplest case, two surface groups SOH1/2- and SOH21/2+ are considered, so only one dissociation step should be observed. In the more advanced version of the 1pK model, various surface oxygens may have different fractional values of charge depending on their local coordination to the crystal cations. This approach is known as the multisite complexation model (MUSIC).7,9,10 The features of the metal oxide/electrolyte interface are investigated by using a variety of experimental techniques. The most frequently used technique is potentiometric titration which allows determination of surface charge as a function of pH. Other common experiments are electrokinetic and radiometric measurements of electrolyte ion adsorption. The recent theoretical study by Borkovec has revealed that the 2pK approach should be somewhat more accurate than the 1pK approach in describing the features of these systems.5 Lutzenkirchen has compared both these models (6) Bolt, G. H.; Van Riemsdijk, W. H. Physicochemical Models. In Soil Chemistry B, 2nd ed.; Bolt, G. H., Ed.; Elsevier: Amsterdam, 1982. (7) Hiemstra, T.; Van Riemsdijk, W. H. Colloids Surf. 1991, 59, 7. (8) Hiemstra, T.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1996, 179, 488. (9) Hiemstra, T.; Van Riemsdijk, W. H.; Bolt, G. H. J. Colloid Interface Sci. 1989, 133, 91. (10) Hiemstra, T.; De Wit, J. C. M.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1989, 133, 105. (11) Pauling, L. The Nature of the Electrostatic Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1967.
10.1021/la025614r CCC: $22.00 © 2002 American Chemical Society Published on Web 05/02/2002
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Langmuir, Vol. 18, No. 12, 2002
by analyzing the quality of fit of surface charge data for (hydr)oxides.12 In his opinion, combination of the 1pK charging model with the basic Stern model of the oxide/ electrolyte interface (1pK BSM) should be considered as the first choice model with respect to the quality of fit and uniqueness of the estimated parameters. Recently, there has been published a more detailed analysis of both 1pK and 2pK approaches by comparing the quality of fits of three kinds of experimental data,13 that is, the surface charge isotherm, electrokinetic potential, and the individual isotherms of electrolyte ion adsorption measured for the TiO2/NaCl system. The 1pK and 2pK models yielded equally good fits when combined with an appropriate model of the interface. This was the TLM model in the case of the 2pK approach and the basic Stern model in the case of the 1pK approach, in agreement with the earlier Lutzenkirchen findings. Thus, the choice of one of these theoretical descriptions still remains a matter of personal choice. To get more information about the features of ion adsorption at the investigated interface, not only free energy of adsorption as a whole but also enthalpic and entropic effects accompanying this process should be analyzed. The heats of ion adsorption can be measured in appropriate calorimetric experiments or deduced from temperature dependence of adsorption data. The first studies of the heats accompanying proton adsorption were based on recording the change of PZC (point of zero charge) with temperature and were started by Berube and de Bruyn.14 The investigations of that kind were next conducted by Tewari et al.,15,16 Fokkink et al.,17,18 Subramanian et al.,19 Kuo and Yen,20 Akratopulu et al.,21,22 Blesa et al.,23 and Kosmulski et al.24 The influence of temperature on the surface charge isotherm was studied by Blesa et al.,25 Regazzoni,26 Brady,27 Kosmulski,28 and Machesky et al.29 Valuable information concerning that problem has been collected in the review by Machesky.30 The first direct calorimetric experiments were reported by Griffiths and Fuerstenau31 and by Foissy32 who (12) Lutzenkirchen, J. Environ. Sci. Technol. 1998, 32, 3149. (13) Piasecki, W.; Rudzin´ski, W.; Charmas, R. J. Phys. Chem. B 2001, 105, 9755. (14) Berube, Y. G.; De Bruyn, P. L. J. Colloid Interface Sci. 1968, 27, 305. (15) Tewari, P. H.; McLean, A. W. J. Colloid Interface Sci. 1972, 40, 267. (16) Tewari, P. H.; Campbell, A. B. J. Colloid Interface Sci. 1976, 55, 531. (17) Fokkink, L. G. J.; De Keizer, A.; Lyklema, J. J. Colloid Interface Sci. 1989, 127, 116. (18) Fokkink, L. G. J. Ph.D. Thesis, Agricultural University, Wageningen, The Netherlands, 1987. (19) Subramanian, S.; Schwarz, J. A.; Hejase, Z. J. Catal. 1989, 117, 512. (20) Kuo, J. F.; Yen, T. F. J. Colloid Interface Sci. 1989, 121, 220. (21) Akratopulu, K. Ch.; Vordonis, L.; Lycourghiotis, A. J. Chem. Soc., Faraday Trans. 1 1986, 82, 3697. (22) Akratopulu, K. Ch.; Kordulis, C.; Lycourghiotis, A. J. Chem. Soc., Faraday Trans. 1990, 86, 3437. (23) Blesa, M. A.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1990, 140, 287. (24) Kosmulski, M.; Matysiak, J.; Szczypa, J. J. Colloid Interface Sci. 1994, 164, 280. (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 1994, 83, 237. (29) Machesky, M. L.; Wesolowski, D. J.; Palmer, D. A.; IchiroHayashi, K. J. Colloid Interface Sci. 1998, 200, 298. (30) Machesky, M. L. In Chemical Modelling in Aqueous Systems II; Melchior, D. C., Bassett, R. L., Eds.; ACS Symposium Series, Vol. 416; American Chemical Society: Washington, DC, 1990. (31) Griffiths, D. A.; Fuerstenau, D. W. J. Colloid Interface Sci. 1981, 80, 271.
Piasecki
measured the heat of immersion of an outgassed solid oxide sample into solutions of changing pH. However, the best experiment for theoretical interpretation is so-called titration calorimetry. It was developed by De Keizer et al.,33 Machesky et al.,34-36 Mehr et al.,37 and Casey.38 Here, the course of measurement is as follows: after introducing a solid sample into a solution, the pH of that solution is measured. Then, a titration step is carried out, and the evolved heat and the equilibrium pH are recorded. Kallay et al.39-41 have recently designed an experiment of that kind aimed at determining the conditions under which the obtained experimental data could be free of the electrostatic contribution to the measured heat of ion adsorption. According to Kallay, the electrostatic contributions to the enthalpy will cancel if the difference between initial pH of the suspension and point of zero charge equals the difference between final pH and PZC.39 At the beginning, however, development of this experimental technique was not accompanied by suitable progress in the theoretical interpretation of the results obtained. This was caused by the fact that the measured heats of adsorption are due to a number of simultaneously occurring surface reactions. So, their theoretical interpretation is not as easy as in the case of the gas/solid interface. In 1990, the first theoretical analysis of the heat of proton adsorption on the metal oxide surface was proposed by De Keizer et al.33 The next year, Rudzin´ski and coworkers started publishing papers where they proposed a rigorous thermodynamic description of enthalpic effects accompanying ion adsorption at the metal oxide/electrolyte interface.42-53 Their calculations based on the combination of the 2pK model of surface charging with the triple-layer model of the interface led to reproducing quantitatively the recorded calorimetric data. The first attempts to interpret the enthalpic effects by using the 1pK model were undertaken by Machesky.29,36 This was a semiquantitative analysis of experimental data. So far, nobody has developed a rigorous quantitative description of the heats of ion adsorption based on the (32) Foissy, A. Ph.D. Thesis, Universite de Franche-Comte, Bescancon, France, 1985. (33) De Keizer, A.; Fokkink, L. G. J.; Lyklema, J. Colloids Surf. 1990, 49, 149. (34) Machesky, M. L.; Anderson, M. A. Langmuir 1986, 2, 582. (35) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 297. (36) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 315. (37) Mehr, S. R.; Eatough, D. J.; Hansen, L. D.; Lewis, E. A.; Davis, J. A. Thermochim. Acta 1989, 154, 129. (38) Casey, W. H. J. Colloid Interface Sci. 1994, 163, 407. (39) Kallay, N.; Zalac, S.; Stefanic, G. Langmuir 1993, 9, 3457. (40) Kallay, N.; Zalac, S. Croat. Chem. Acta 1994, 67, 467. (41) Zalac, S.; Kallay, N. Croat. Chem. Acta 1996, 69, 119. (42) Rudzin´ski, W.; Charmas, R.; Partyka, S. Langmuir 1991, 7, 354. (43) Rudzin´ski, W.; Charmas, R.; Partyka, S.; Foissy, A. New J. Chem. 1991, 15, 327. (44) Rudzin´ski, W.; Charmas, R.; Partyka, S.; Thomas, F.; Bottero, J. Y. Langmuir 1992, 8, 1154. (45) Rudzin´ski, W.; Charmas, R.; Cases, J. M.; Francois, M.; Villieras, F.; Michot, L. J. Langmuir 1997, 13, 483. (46) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Cases, J. M.; Francois, M.; Villieras, F.; Michot, L. Colloids Surf., A 1998, 137, 57. (47) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Kallay, N.; Cases, J. M.; Francois, M.; Villieras, F.; Michot, L. Adsorption 1998, 4, 287. (48) Charmas, R. Langmuir 1998, 14, 6179. (49) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Thomas, F.; Villieras, F.; Prelot, B.; Cases, J. M. Langmuir 1998, 14, 5210. (50) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Prelot, B.; Thomas, F.; Villieras, F.; Cases, J. M. Langmuir 1999, 15, 5977. (51) Rudzin´ski, W.; Panas, G.; Charmas, R.; Piasecki, W.; Kallay, N.; Proecanin, T. J. Phys. Chem. B 2000, 104, 11912. (52) Rudzin´ski, W.; Panas, G.; Charmas, R.; Piasecki, W.; Kallay, N.; Proecanin, T. J. Phys. Chem. B 2000, 104, 11923. (53) Rudzin´ski, W.; Piasecki, W.; Charmas, R.; Panas, G. Adv. Colloid Interface Sci. 2002, 95, 95.
Ion Adsorption at the Oxide/Electrolyte Interface
Langmuir, Vol. 18, No. 12, 2002 4811
The above reactions leading to the formation of these surface complexes onto free sites (SOH1/2- in this model) have the following equilibrium constants: K1+ ) 1/1Kint H , 1 1 int 1 int K1C ) 1Kint C , and KA ) KA / KH for adsorption of protons, cations, and anions, respectively. The last equilibrium constant corresponds to the combination of reactions 1a and 1c, which leads to the following reaction: K1A
SOH1/2- + H+ + A- 798 SOH21/2+AK1A
where
Figure 1. Diagrammatic presentation of the 1pK model with two charged planes (basic Stern model). ψ0, δ0: the surface potential and the surface charge density in the 0-plane; ψd, δd: the diffuse layer potential and its charge; c1: the electrical capacitance of the rigid layer, constant within the region between the planes.
1pK approach. The main aim of this paper is to derive such a description and to compare it with the observed experimental data and with the theoretical results obtained by applying the 2pK model. Following the results obtained in two previous papers,12,13 the combination of the 1pK charging mechanism and the basic Stern model of the interface will be adopted for that purpose. Theory 1. Theoretical Description of the Oxide/Electrolyte Interface Based on the 1pK Charging Model and the Basic Stern Model of the Double Layer. The 1pK model was proposed by Bolt and Van Riemsdijk,6 and its general reviews and applications were given by Hiemstra and Van Riemsdijk.7-10 The schematic picture of this surface charging model combined with the basic Stern model (BSM) of the double layer is shown in Figure 1. In their works based on the 1pK charging model, Hiemstra and Van Riemsdijk placed cations and anions of the basic electrolyte in the diffuse layer but assumed that they form ionic pairs with surface sites. Protons, along with the anions and cations of the basic electrolyte, form the following surface complexes: SOH21/2+, SOH21/2+A-, and SOH1/2-C+ onto the sites SOH1/2-. According to the mass action law, 1/1Kint H
SOH1/2- + H+ 798 SOH21/2+ where 1Kint H ) SOH
1/2-
+
1Kint C
[SOH
1/2-
](aH)
1/2+
[SOH2 1/2-
+ C 798 SOH
where 1Kint C )
]
+
C
[SOH1/2-C+] 1/2-
[SOH
](aC)
1Kint A
SOH21/2+ + A- 798 SOH21/2+A1/2+
where 1Kint A )
1/2+
[SOH2
-
A ]
[SOH2
{ } {} { }
exp -
](aA)
exp
eψ0 kT
eψd kT
eψd
exp -
kT
(1a)
)
1/2-
[SOH
](aH)(aA)
exp
{
1 δ0 ) B(θ+ + θA - θ- - θC) 2
kT
(1d)
where B ) eNs
(2)
where Ns is the surface site density (sites/m2). Surface coverages, θi, can be expressed in the following Langmuir-like form:13
θi )
K1i fi 1+
∑i
i ) +, A, C
(3)
K1i fi
where fi (i ) +, A, C) are the functions of proton and salt activities,
{
}
f+ ) exp -
eψ0 - 2.3pH kT
{
eδ0 eψ0 + kT kTc1
fC ) aC exp -
{
fA ) aA exp -
}
(4a) (4b)
}
eδ0 - 2.3pH kTc1
(4c)
Taking into account eq 3, we can write eq 2 as follows: 1 1 1 1 K+f+ + KAfA - KCfC - 1 δ0 ) B 2 1 + K1i fi
∑i
i ) +, A, C
(5)
As has been shown in a recently published paper,13 in the 1pK model the surface potential ψ0 is expressed by the following equation obtained from eq 5:
(
1 B(aHK1AX - 1) - δ0(1 + aHK1AX) 2 aK1C aK1C 1 B HK1+ δ0 1 + 1 2 HK1 X HK1 X
[(
+
) (
(
where X ) exp (1c)
}
e(ψ0 - ψd)
where ψ0 is the surface potential and ψd is the mean potential at the onset of the diffuse layer. Further, aH, aC, and aA are the bulk activities of protons, anions, and cations, respectively. The surface charge can be calculated as follows:
kT ln ψ0 ) e (1b)
[SOH21/2+A-]
+
)
)
)]
eδ0 (6) kTc1
One can establish the relations between the intrinsic 1 int 1 int constants 1Kint H , KC , and KA from the condition that the
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Langmuir, Vol. 18, No. 12, 2002
Piasecki
experimentally observed PZC is practically independent of the salt concentration.43,44 From the set of equations
∂δ0(pH ) PZC) ) 0 (7) ∂a
δ0(pH ) PZC) ) 0 and
the following relations are obtained:
Q1+ ) Q1aH - eψ0 -
Kint C 1 int KA
)1
(8a)
or
( ) ( ) ( )
δ0 eδ0T ∂c1 c1 (c )2 ∂T
Q1A ) Q1aA - e PZC ) p
1
Kint H
and p
1
Kint C
)p
1
Kint A
(8b)
(11a)
{θi},pH
1
{θi},pH
+k
( (
) )
∂ ln aA ∂(1/T)
(11b)
pH
(11c)
pH
where Q1aH, Q1aC, and Q1aA denote the nonconfigurational heats of reactions 1a,b,d,
where 1 int (i ) H, C, A) and H ) 10-PZC p1Kint i ) - log Ki (8c)
Thus, for the 1pK model eqs 8a,b reduce the number of the unknown equilibrium constants determined from fitting suitable experimental data from three to one. The authors using the 1pK model in their analysis of experimental data have not employed this feature yet. 2. The Calculation of the Heats of Ion Adsorption in the 1pK Model. The molar differential heats accompanying the surface reactions 1a,b,d are defined as follows:
[
[
] ( ) ] ( )
[
]
b ∂ µSOH1/2- - µH Q1+ ) -k kT ∂(1/T)
b ∂ µSOH1/2-C+ - µC Q1C ) -k kT ∂(1/T)
Q1A
( )
e ∂ψ0 T ∂(1/T)
e ∂ψ0 + T ∂(1/T) {θi},pH ∂ ln aC δ0 eδ0T ∂c1 +k e + 2 c1 (c ) ∂T {θi},pH ∂(1/T) 1
Q1C ) Q1aC - eψ0 -
1
H ) 1Kint and H
Differentiation of the above equations with respect to 1/T yields
∂F+
)k
{θi}
{θi}
∂(1/T)
)k
b b ∂ µSOH21/2+A- - µA - µH ) -k kT ∂(1/T)
∂FC
∂(1/T)
{θi}
k
(9a)
{θi}
(9b)
{θi}
)
( ) ∂FA
∂(1/T)
{θi}
(9c)
where µSOH2 , µSOH C , and µSOH2 A are the chemical potentials of the corresponding surface complexes whereas µbj (j ) H, C, A) are those of proton, cation, and anion in the bulk electrolyte, respectively. The functions Fi (i ) +, A, C) are defined in the following way: 1/2+
1/2- +
F+ ) -ln(1Kint H ) -
FC ) -ln(1/1Kint C ) -
1/2+ -
θ+ eψ0 - 2.3pH - ln ) 0 kT θ-
(10a)
eδ0 eψ0 θC + + ln aC - ln ) 0 kT kTc1 θ(10b)
eδ0 1 int FA ) -ln(1Kint - 2.3pH + ln aA H / KA ) kTc1 θA ln ) 0 (10c) θ-
Q1aH
d ln(1Kint H )
) -k
d(1/T)
dPZC ) 2.3k d(1/T)
d ln(1/1Kint C )
Q1aC ) -k
d(1/T)
1 int d ln(1Kint H / KA )
Q1aA ) -k
d(1/T)
(12a)
(12b)
(12c)
The molar differential heats of adsorption, Q1i , defined in eqs 11a-c are configurational since they depend on the concentrations of the surface complexes θi. To achieve the best fit of data, we have to assume that the electrical capacitance of the double layer can change its value at PZC.54,55 So, while calculating the derivative ((∂c1/∂T)){θi},pH it has been assumed that there are two different values of the c1 parameter: one for the acid region (pH < PZC) and another one for the basic region (pH > PZC). The simplest manner to take into consideration the temperature dependence of c1 is treating it as the linear function of temperature,25 L c1 ) cL1 ) cL,0 1 + R1 ∆T
pH < PZC
(13a)
R c1 ) cR1 ) cR,0 1 + R1 ∆T
pH > PZC
(13b)
The above equations may be considered as the formal Taylor expansion for c1 around T ) T0 so that cL1 (T0) ) cL,0 1 L R and cR1 (T0) ) cR,0 1 . The coefficients R1 and R1 play an important role in fitting the experimental values of heats. The derivative (∂ψ0/∂(1/T))pH is calculated by differentiation of eq 6 by 1/T. During this operation, one should remember that δ0 has to be treated as a constant value because all the surface coverages θi are assumed to be constant. The full form of the derivative (∂ψ0/∂(1/T))pH is presented in the Appendix. 1 int In eqs 12b,c, 1Kint C ) KA , from which it follows that
Q1aA ) Q1aH + Q1aC
(14)
The last relation makes it possible to eliminate from further best-fit calculations one of the three parameters Q1aH, Q1aC, or Q1aA. (54) Sprycha, R. J. Colloid Interface Sci. 1984, 102, 173. (55) Blesa, M. A.; Kallay, N. Adv. Colloid Interface Sci. 1988, 28, 111.
Ion Adsorption at the Oxide/Electrolyte Interface
Langmuir, Vol. 18, No. 12, 2002 4813
reactions, equivalent to the above reactions:
SO- + H+ T SOH0
(17a)
SO- + 2H+ T SOH2+
(17b)
SO- + C+ T SO-C+
(17c)
SO- + 2H+ + A- T SOH2+A-
(17d)
Let Qi (i ) 0, +, A, C) denote the molar differential heats of formation of these surface complexes. Their detailed explicit form can be developed by applying the appropriate thermodynamic relations.42 As a final result, we obtain
Figure 2. Diagrammatic presentation of the 2pK model with three charged planes (TLM). ψ0, δ0: the surface potential and the surface charge density in the 0-plane; ψβ, δβ: the potential and the charge of the β-plane formed by the adsorbed ions (cations C+ and anions A- ) of the basic electrolyte; ψd, δd: the potential and the charge diffuse layer; c1 and c2 are the electrical capacitances, assumed to be constant in the regions between the planes.
As was explained in the Introduction, the experimental calorimetric data suitable for the theoretical analysis are those obtained from titration calorimetry. In this experiment, heat of the proton adsorption is measured. As follows from eqs 1a,b,d, the molar heat of proton adsorption is a composite quantity that can be expressed by the following equation:
Qpr )
( ) [( ) ( ) ] ∂θi
i
∫pH
∂θ+
∂pH
+
T
∂θA
∂pH
i ) +, C, A (15)
δ0 e ∂ψ0 -e T ∂(1/T) {θi},pH c1 ∂ ln aA eδ0T ∂c1 +k 2 ∂T ∂(1/T) {θi},pH (c )
pH
+e
(
(
int d ln Kai
int Ka1
-
+
SOH 798 SO + H *Kint C
SOH0 + C+ 798 SO-C+ + H+ SOH2+A- 798 SOH0 + H+ + A-
(16a) (16b) (16c) (16d)
int int for which the equilibrium constants are Kint a1 , Ka2 , *KC , int and *KA , respectively. We consider the enthalpic effects accompanying the formation of the surface complexes SOH0, SOH2+, SO-C+, and SOH2+A- as being due to the following surface
)
(18c)
)
(18d)
where Qa1, Qa2, QaC, and QaA are the nonconfigurational heats related to reactions 16,
T
int Ka2
pH
Qai ) -k
SOH2+ 798 SOH0 + H+
*Kint A
QA ) QaA - eψ0 -
(18b)
{θi},pH
δ0 + c1 {θi},pH ∂ ln aC +k ∂(1/T) {θi},pH
e ∂ψ0 T ∂(1/T) eδ0T ∂c1 (c )2 ∂T 1
dpH
where heats Qi are given by eqs 11a-c, and the derivatives (∂θi/(∂pH))T can be calculated numerically by applying eq 3. 3. Theoretical Description of the Enthalpic Effects Based on the 2pK Charging Model Combined with the Triple-Layer Model of the Interface. A schematic diagram of the 2pK triple-layer model is presented in Figure 2. According to Davis et al.,3,4 the 2pK TLM assumes occurrence of the following surface reactions:
0
( ) ( ) ( ) ( )
(18a)
{θi},pH
2e ∂ψ0 T ∂(1/T)
1
∫pHpH+∆pH ∑Q1i ∂pH T dpH pH+∆pH
Q+ ) Qa1 + Qa2 - 2eψ0 QC ) QaC - eψ0 -
( ) ( )
e ∂ψ0 T ∂(1/T)
Q0 ) Qa2 - eψ0 -
d(1/T)
i ) 1, 2
int d ln(Kint a2 /*KC )
QaC ) -k
d(1/T) int d ln(Kint a2 *KA )
QaA ) -k
d(1/T)
(19a)
(19b)
(19c)
Applying the Rudzin´ski-Charmas criterion for the common intersection point (CIP)43,44 to the 2pK model, we arrive at the following relations between the intrinsic int int int constants Kint a1 , Ka2 ,*KC , and *KA :
Kint a2 ) *Kint A )
H2 1 int or PZC ) (pKint a1 + pKa2 ) (20a) 2 Kint a1
H2 1 int or PZC ) (p*Kint C + p*KA ) (20b) int 2 *KC
The above equations reduce the number of unknown equilibrium constants from four to two, which is very important for practical calculations. This is because these unknown parameters are found by fitting theoretical expressions to experimental data, and such a best-fit procedure becomes quickly less and less reliable as the number of the best-fit parameters increases.
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From eqs 20a,b, the derivative (∂PZC/[∂(1/T)]) can be calculated. Taking into account definitions 19a,b,c, we obtain
∂PZC Qa1 + Qa2 2.3 ) 2k ∂(1/T)
(21a)
∂PZC QaA - QaC 2.3 ) 2k ∂(1/T)
(21b)
Comparing the right-hand sides of eqs 21a,b, we finally get
QaA - (Qa1 + Qa2) ) QaC
(22)
Some important general conclusions may be drawn from eq 22. Equation 22 suggests that the heat QaC related to the cation adsorption on the free surface site SO- is the same as the heat of the attachment of the anion to the already existing complex SOH2+. In the case of nonspecific adsorption of electrolyte ions, the above conclusion should be correct for many different oxides and various inert electrolytes. Rudzin´ski et al.45 have launched the hypothesis that considered heats are always the same because they are equal to zero or very small. Such a conclusion was also drawn by Kosmulski28 who used radiometric methods to study the individual adsorption isotherms of ions of the inert electrolyte. Although it was not directly pointed out, such a point of view was also expressed by Machesky et al.35,36 who assigned the measured calorimetric effects solely to proton adsorption. In such a case, of all the parameters Qa1, Qa2, QaC, and QaA only one can be chosen freely while fitting the experimental heat of proton adsorption by the theoretical functions.49 This is also true when one fits simultaneously the experimental titration curves measured at different temperatures, as was shown in the paper by Rudzin´ski et al.45 In this case, the assumption that QaC ) 0 and that QaA - (Qa1 + Qa2) ) 0 led to a good agreement between the theory and experiment. In the 2pK model considered now, molar heat of proton adsorption is calculated using the following equation:46
( ) [( ) ( ) ( )] ∂θi
∫pHpH+∆pH ∑Qi ∂pH T dpH i
Qpr )
∫pHpH+∆pH
∂θ+
+2
2
∂pH
T
∂θA
∂pH
+
T
∂θ0
∂pH
dpH
T
i ) 0, +, C, A (23) Heats Qi are defined by relations 18a-d. Expressions for (∂θi/∂pH)T have already been presented in earlier papers.42,49 Discussion In this section, the comparison is presented of the results obtained by adopting the 1pK model with those obtained by adopting the 2pK approach and with some published experimental data. Analysis of the enthalpic effects based on the 2pK approach was already developed by Rudzin´ski and co-workers.42-53 In their papers, a few kinds of experimental data were considered. As follows from their considerations, the set of data suitable for analysis of enthalpic effects should consist at least of surface charge
isotherms and heats measured by means of titration calorimetry. To compare the results given by the 1pK model with the earlier findings in this field based on the 2pK model, we examine the same experimental data as those analyzed by Rudzin´ski et al.46,49,50 With this idea in mind, we select for our further analysis three data sets obtained in the following experiments: 1. The calorimetric titration of the aluminum oxide sample in a 0.1 mol/dm3 solution of NaCl, reported by Machesky and Jacobs;35,36 2. The calorimetric titration of a TiO2 sample in a 0.01 mol/dm3 solution of NaCl, reported by Mehr et al.;37 3. Casey’s calorimetric titration measurements of a silica sample in a 0.1 mol/dm3 solution of NaCl.38 In all of these cases, we have at our disposal surface charge isotherms δ0(pH) and heats of proton adsorption Qpr. However, to analyze to which extent comparison with experiment may favor one or another adsorption model a brief discussion of the experimental data seems to be necessary. Namely, we select only the data set which seems to be obtained under conditions close to equilibrium. So, as the first we consider the system Al2O3/NaCl solution investigated by Machesky and Jacobs35,36 at a temperature of 25 °C over a pH range of 4-10 and at three concentrations of NaCl: 0.001, 0.01, and 0.1 mol/ dm3. For the reasons explained in detail in the paper by Rudzin´ski et al.,49 only the calorimetric and potentiometric titrations corresponding to the 0.1 mol/dm3 salt concentration can be subjected to a further quantitative analysis. The TiO2 sample investigated by Mehr and co-workers consists of anatase (60%) and rutile (40%), and its PZC value was equal to 6.4. Both the surface charge isotherms δ0(pH) and the calorimetric curves Qpr(pH) showed hysteresis. Two curves for both δ0(pH) and Qpr(pH) were obtained: one for the base titration and another one for the acid titration. The observed hysteresis was ascribed to kinetics, that is, to the fact that the system did not achieve equilibrium between the next two titration steps. To account for the nonequilibrium effects, Mehr et al.37 proposed to consider the average value from the acid and base titrations corresponding to the same pH value as the best value of the functions δ0(pH) and Qpr(pH). It was observed that the hysteresis was the smallest for the lowest concentration of the inert electrolyte. For that reason, we will take into consideration the titration and calorimetric data, measured at the lowest concentration of NaCl equal to 0.01 mol/dm3. Then, we will fit by our equations the values of δ0(pH) and Qpr(pH) being appropriate arithmetic averages of the two branches corresponding to acid and base titrations. Other details can be found in the paper by Rudzin´ski and co-workers.46 The silica/NaCl system studied by Casey38 was the Aerosil-380 sample whose PZC was equal to 3.8.50 Calorimetric titrations were carried out for the three KCl and NaCl concentrations: 0.01, 0.1, and 1.0 mol/dm3. Nevertheless, only for NaCl solutions the surface charge isotherms δ0(pH) were measured. The calorimetric data exhibited a large hysteresis for the NaCl solution at the concentration of 1.0 mol/dm3, whereas the charge titration isotherms showed a large one for the 0.01 mol/dm3 NaCl solution. Thus, only for the NaCl concentration equal to 0.1 mol/dm3 did both the surface charge isotherm and the measured enthalpic data show no hysteresis. For that reason, only these data are selected for our present quantitative analysis. While investigating these adsorption systems by adopting the 1pK or the 2pK model, the following procedure was applied. In the first step, the surface charge isotherm is calculated in a rough way. To fit the δ0(pH), the following
Ion Adsorption at the Oxide/Electrolyte Interface
Langmuir, Vol. 18, No. 12, 2002 4815
Figure 3. The comparison of the experimental potentiometric data (b) for the Al2O3/NaCl solution system with the theoretical δ0(pH) curves calculated by means of the 1pK model (solid line) and the 2pK model (dashed line). The parameters used in the calculations were collected in Table 1.
Figure 4. The comparison of the experimental Qpr data (b) for the Al2O3/NaCl solution system with the theoretical Qpr functions calculated by means of the 1pK model (solid line) and the 2pK model (dashed line). The parameters used in the calculations were collected in Table 1.
parameters can be changed: surface reaction equilibrium int int int constants (pKint a1 , pKa2 , p*KC , and p*KA for the 2pK 1 int model, whereas for the 1pK model p1Kint H , p KC , and int p1KA ) and electrical capacitances of the double layer (cL,0 1 and cR,0 1 for both models). Furthermore, the relations 8a,b and 20a,b were taken into account to decrease the number of unknown parameters. The values of surface site densities Ns were the same as those in earlier papers.46,49,50 In the next step, the calorimetric data were fitted. The parameter values obtained in the first step were precisely adjusted, and the values of additional parameters were found at this stage. The additionally heat parameters were Qa1, Qa2, QaC, and QaA for the 2pK model; Q1aH, Q1aC, and Q1aA for the 1pK model; and RL1 and RR1 for both models. Relations 14 and 22 were used to reduce the number of unknown parameters. In the course of these calculations, the assumptions Q1aC ) 0 and QaC ) 0 were taken into consideration. That means that both the nonconfigurational heat of cation attachment to the free surface site SOH1/2- (or SO- ) and the nonconfigurational heat of anion attachment to the SOH21/2+ (or SOH2+) group are assumed equal to zero. This conclusion is a consequence of the assumption that the interactions of electrolyte ions with the surface are purely electrostatic. As follows from Figure 3, both the 1pK and the 2pK models fit equally well potentiometric titration data for the system Al2O3/NaCl solution. However, in the case of heat effects presented in Figure 4 these two models provide different theoretical Qpr(pH) curves. In this case, the 2pK approach is more flexible than the 1pK one. It is easy to model the shape of the Qpr(pH) curve in the former model, and it is difficult to obtain the desired shape in the latter. It is especially clear in the pH range of 4-7. The bend of the theoretical curve Qpr(pH) is due to different values of RL1 and RR1 . As we can see in Table 1, in the case of the 2pK model there are four main parameters: two equilibrium conint stants of proton dissociation, pKint a1 and pKa2 , which are int connected by the relationship (1/2)(pKa1 + pKint a2 ) ) PZC (see eq 20a), and two heats, Qa1 and Qa2, corresponding to them. In the 1pK model, we have only two main parameters: the proton dissociation constant p1Kint H and the heat of proton adsorption Q1aH. One should remember
that the p1Kint H value is fixed and equals PZC as follows from eq 8b. So, we have two parameters fewer in the 1pK model. In the case of the TiO2/NaCl system, both models reproduce the potentiometric titration data well enough (Figure 5). As previously, the heat of proton adsorption is better mimicked by the 2pK model although the results obtained by means of the 1pK approach are also correct (Figure 6). As follows from Figure 7 for the SiO2/NaCl system, the 1pK model incorrectly describes the potentiometric titration data. It was impossible to curve the line, which represents the δ0(pH) function. The shape of the charging curve of silica is different from those found for other metal (hydr)oxides. It is due to another attribution of charge to the surface groups in the case of SiO2. The charging process of silica in the analyzed pH range should be described by one protonation step.10 Namely, the reactive neutral surface group SiOH releases a proton evolving into the negative group SiO- . Many other metal (hydr)oxides have positively and negatively charged surface species, for example, SOH21/2+ and SOH1/2- . For these two different cases, the behavior of the potentiometric curve was clearly demonstrated in Figure 11 in ref 10. So, the 1pK model is not a proper approach to describe charging of silica. In this case, the 2pK or the MUSIC model is a better choice. At the same time, both models satisfactorily describe the heat effects (Figure 8). It should be stressed, however, that we have at our disposal calorimetric data for a narrow pH range from 7 to 9 and the experimental points are arranged along the straight line. Recently, Sverjensky and Sahai have proposed a new method of theoretical prediction of single-site enthalpies of surface protonation for oxides and silicates in water.56 They applied for that purpose their method developed earlier to estimate PZC and surface protonation equilibrium constants, which is based on the combination of crystal chemical and Born solvation theory.57,58 Sverjensky and Sahai analyzed available calorimetric data thoroughly and proposed simple equations which enable calculation (56) Sverjensky, D. A.; Sahai, N. Geochim. Cosmochim. Acta 1998, 62, 3703. (57) Sverjensky, D. A. Geochim. Cosmochim. Acta 1994, 58, 3123. (58) Sverjensky, D. A.; Sahai, N. Geochim. Cosmochim. Acta 1996, 60, 3773.
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Table 1. List of Parameters Used in Calculations
Figure 5. The comparison of the experimental potentiometric data (b) for the TiO2/NaCl solution system with the theoretical δ0(pH) curves calculated by means of the 1pK model (solid line) and the 2pK model (dashed line). The parameters used in the calculations were collected in Table 1.
of enthalpies of proton adsorption for any oxides or silicates. These equations include coefficients derived from the statistical analysis of experimental data. As follows from eqs 12a and 21a, for the same oxide the relation (1/2)(Qa1 + Qa2) ) Q1aH should be held. The comparison of the values Q1aH and (1/2)(Qa1 + Qa2) with the heats extracted from the paper by Sverjensky and Sahai is presented in Table 2. For the same system, the values of heats are generally in good agreement with one another. In the case of titanium dioxide, the difference between the value of (1/2)(Qa1 +
Figure 6. The comparison of the experimental Qpr data (b) for the TiO2/NaCl solution system with the theoretical Qpr functions calculated by means of the 1pK model (solid line) and the 2pK model (dashed line). The parameters used in the calculations were collected in Table 1.
Qa2) found by Sverjensky and Sahai and our results is probably due to a different composition of the investigated TiO2 samples. Namely, Sverjensky and Sahai analyzed the data for the rutile sample whereas the data by Mehr et al. were measured for the mixture of rutile and anatase. In the case of alumina and silica, the values of Q1aH (which is a parameter in the 1pK model) are in excellent agreement with Sverjensky’s and Sahai’s results but quality of fit is better for the 2pK model (Figures 3, 4, 7, and 8).
Ion Adsorption at the Oxide/Electrolyte Interface
Langmuir, Vol. 18, No. 12, 2002 4817 Table 3. Values of Free Energy ∆G1aH, Enthalpy ∆H1aH, and Entropy T∆S1aH of Proton Adsorption Determined by Applying the 1pK BSM
Figure 7. The comparison of the experimental potentiometric data (b) for the SiO2/NaCl solution system with the theoretical δ0(pH) curves calculated by means of the 1pK model (solid line) and the 2pK model (dashed line). The parameters used in the calculations were collected in Table 1.
Figure 8. The comparison of the experimental Qpr data (b) for the SiO2/NaCl solution system with the theoretical Qpr functions calculated by means of the 1pK model (solid line) and the 2pK model (dashed line). The parameters used in the calculations were collected in Table 1. Table 2. Comparison of the Values Q1AH and (1/2)(Qa1 + Qa2) Obtained by Means of Both Models for Different Adsorption Systems with the Heats Estimated by Sverjensky and Sahaia
Q1aH 1pK BSM (1/2)(Qa1 + Qa2) 2pK TLM (1/2)(Qa1 + Qa2) Sverjensky and Sahaia a
Al2O3/NaCl [kJ/mol]
TiO2/NaCl [kJ/mol]
SiO2/NaCl [kJ/mol]
46 41 45
33 34 25
17 12 17
Reference 56.
Having at our disposal equilibrium constants of reactions and their enthalpies, we can calculate entropy changes applying the basic thermodynamic equation
T∆S1aH ) ∆H1aH - ∆G1aH ) ∆H1aH - 2.303RT p1Kint aH (24) In this equation, ∆H1aH ) -Q1aH and p1Kint aH ) PZC. Using the 1pK approach, we are able to determine the value of T∆S1aH in an easy and reliable manner. In Table 3, values of free energy ∆G1aH, enthalpy ∆H1aH, and entropy T∆S1aH of proton adsorption are presented.
system
∆G1aH [kJ/mol]
∆H1aH [kJ/mol]
T∆S1aH [kJ/mol]
Al2O3/NaCl TiO2/NaCl SiO2/NaCl
-48.5 -36.5 -20.0
-46.0 -33.0 -17.0
2.5 3.5 3.0
Decreasing of enthalpy values with decreasing of PZC is obvious because PZC is correlated with the acidity of the surface. Enthalpy is connected with chemical bond formation energy, while net solvation changes are responsible for entropy in aqueous solutions.30 The small and almost constant value of entropy suggests that the processes occurring in the solution have second-order significance for the proton adsorption onto the investigated metal (hydr)oxides and silica. Somewhat larger values of proton adsorption entropies were obtained by Machesky (T∆S1aH ranged from 6 to 12 kJ/mol).30 Even these results clearly suggest that the driving force for proton adsorption is enthalpy. As we can see, the 2pK model is more flexible than the 1pK one. However, in the former we have two parameters more. In the 1pK model, two main parameters can be found directly from experiment. The proton dissociation 1 constant p1Kint aH equals PZC, and heat QaH can be determined from temperature dependence of PZC as follows from eq 12a. Both models give comparable results, but the 1pK approach gives a chance to reduce the uncertainty connected with the presence of many parameters in the related theoretical descriptions of the oxide/electrolyte interface. When instead of the 1pK BSM the 1pK triple-layer model (1pK TLM) was applied in calculations with the same set of parameters, the results obtained hardly changed. Only in the case of silica was a noticeable difference in the calculated heats observed, but a slight change of the parameter values in the 1pK TLM led to the same results as in the 1pK BSM. In this paper, we have neglected the energetic heterogeneity of the real solid surfaces. However, the recent experimental results have shown that metal oxide surfaces in solution reveal considerable energetic heterogeneity relative to proton adsorption.59-61 Application of highresolution H+/OH- adsorption isotherms makes it possible to determine the distribution of different local pK’s.61 Thus, we can expect that in the near future surface heterogeneity will be taken into consideration not as an a priori parameter but as the experimentally determined quantity. Acknowledgment. The author expresses his gratitude to Professor W. Rudzin´ski and Dr. R. Charmas for the fruitful discussions during the writing of this paper and critical and helpful reviews of the manuscript. Appendix The formula for (∂ψ0/∂(1/T)){θi},pH is as follows: (59) Contescu, C.; Jagiello, J.; Schwarz, J. A. Langmuir 1993, 9, 1754. (60) Borkovec, M.; Rusch, U.; Cernik, M.; Koper, G. J. M.; Westall, J. C. Colloids Surf., A 1996, 107, 285. (61) Villieras, F.; Charmas, R.; Zarzycki, P.; Thomas, F.; Prelot, B.; Piasecki, W.; Rudzin´ski, W. J. Phys. Chem. B, submitted for publication.
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( ) ∂ψ0
∂(1/T)
Piasecki
where
)
{θi},pH
[(
)
X(-B - 2δ0 + aHXK1A(B - 2δ0)) kT2 Ln + e HXK1+(B - 2δ0) - aK1C(B + 2δ0) 2δ0)K1C((B
- 2δ0)HX a K1AT - (B + (1/T((B + 2δ0)aXT + X(BaT + 2δ0aT + 2BHa2K1AXT 4δ0Ha2K1AXT)) - X(aB2H2K1+X2K1AT 4aδ0H2K1+X2K1AT(B + δ0) - aB2K1CT - 4Bδ0aK1CT 1 1 - 4δ02HXK+T + 4δ02aK1CT + B2HXK+T 1 1 1 (B - 2δ0)HKAX(BHK+XaT - 2δ0HK+XaT + Ba2K1CT + 1 1 + 2δ0HaXK+T + 2δ0a2K1CT - BHaXK+T 1 1 {BHaK+XT - 2δ0HaK+XT))))/(X(-B - 2δ0 + (B 2δ0)HaK1AX)(-(B + 2δ0)aK1C + (B - 2δ0)HK1+X))] (A1) 2 2
1 K+T )
d(K1+) K1+ Q1aH ) 1 k d T
()
K1AT )
d(K1A) K1A Q1aA ) 1 k d T
()
K1CT )
d(K1C) K1C Q1aC ) 1 k d T
()
XT )
eδ0 dX )X 1 kc1 d T
()
aT )
LA025614R
∂a (A2) ∂(1/T)
