Langmuir 1998, 14, 6179-6191
6179
Calorimetric Effects of Simple Ion Adsorption at the Metal Oxide/Electrolyte Interfaces: An Analysis Based on the Four Layer Complexation Model Robert Charmas†,‡ Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie Skłodowska University, Pl. Marii Curie Skłodowskiej 3, 20-031 Lublin, Poland Received January 5, 1998. In Final Form: June 5, 1998 A comparative theoretical study of the heat accompanying ion adsorption at the oxide/electrolyte interface, predicted by both the triple layer model (TLM) and the four layer model (FLM) of oxide/electrolyte interface, is presented. The four layer model predicts that even small differences reported between PZC and IEP may affect strongly by the heat effects below PZC. The developed expressions are successfully applied to describe quantitatively heat effects accompanying ion adsorption at the TiO2/NaCl solution interface. An assumption that the inner capacitances are temperature dependent is essential for that successful quantitative description.
Introduction Models of surface complexation are the most frequently applied ones to describe ion adsorption at the metal oxide/ electrolyte interface. Of all the models published so far, the most frequently used is the so called “triple layer model” (TLM),1 a schematic picture of which is shown in Figure 1. Despite being very successful, TLM requires some modifications. The modifications of TLM went into three directions. One group of scientists has challenged the charging mechanism of the surface oxygens. They invoke Pauling’s principle2 of local neutralization of charge to show that the protonated surface oxygens will, in general, have fractional values of charge. That hypothesis is commonly called the 1-pK charging mechanism.3 Moreover, they argue that various surface oxygens may have different 1-pK charging mechanisms, depending on their local coordination to cations. That model is known in the literature as the ”multisite complexation model” (MUSIC) of oxide electrolyte interface.4-7 Another group of scientists has accepted the classical 2-pK charging mechanism, as shown in Figure 1, but emphasized the importance of a small degree of crystallographic organization of the surface oxygens, compared to the situation in the interior of an oxide crystal. That smaller degree of surface organization should lead to a †
Fax: (48)(81) 5375685. E-mail: RCharmas@ hermes.umcs.lublin.pl. ‡ Temporary address: INPL, ENSG, Laboratoire Environnemental et Mineralurgie, CNRS UMR 7569, BP 40, Rue du Doyen Marcel Roubault, F-54501 Vandoeuvre les Nancy, France. Fax: (33) 383 575404. E-mail:
[email protected]. (1) Davis, J. A.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 480. (2) Pauling, L. In The Nature of the Electrostatic Bond, 3rd ed.; Cornell Univ. Press: Ithaca, NY, 1967. (3) Bolt, G. H.; Van Riemsdijk, W. H. In Soil Chemistry, B. Physicochemical Models, 2nd ed. Bolt, G. H., Ed.; Elsevier: Amsterdam, 1982; p 459. (4) Hiemstra, T.; Van Riemsdijk, W. H.; Bolt, G. H. J. Colloids Interface Sci. 1989, 133, 91. (5) Hiemstra, T.; De Wit, J. C. W.; Van Riemsdijk, W. H. J. Colloids Interface Sci. 1989, 133, 105. (6) Hiemstra, T.; Van Riemsdijk, W. H. Colloids Surf. 1991, 59, 7. (7) Hiemstra, T.; Van Riemsdijk, W. H. J. Colloid Interface Sci. 1996, 179, 488.
Figure 1. Diagrammatic presentation of the triple layer model of the metal oxide/electrolyte interface: ψ0, δ0, the surface potential and the surface charge density in the 0-plane; ψβ and δβ, the potential and the charge coming from the specifically adsorbed ions (cations C+ and anions A-) of the basic electrolyte in one plane; ψd, δd, the diffuse layer potential and its charge; c1, c2, the electrical capacitances, constant in the regions between planes.
quite different status of the surface oxygens, resulting in what is called surface energetic heterogeneity”.8-15 Some basic properties of these adsorption systems still remain unexplained. Among fascinating problems there (8) Kinniburgh, D. G.; Barkes, J. A.; Whitfield, M. J. J. Colloid Interface Sci. 1983, 95, 370. (9) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, J. J. Colloid Interface Sci. 1986, 109, 219. (10) Van Riemsdijk, W. H.; De Witt, J. C. M.; Koopal, L. K.; Bold, G. H. J. Colloid Interface Sci. 1987, 116, 511. (11) Rudzin´ski, W.; Charmas, R.; Partyka, S. Langmuir 1991 7, 354. (12) Rudzin´ski, W.; Charmas, R.; Partyka, S.; Thomas, F.; Bottero, J. Y. Langmuir 1992, 8, 1154. (13) Rudzin´ski, W.; Charmas, R.; Partyka, S.; Bottero, J. Y. Langmuir 1993, 9, 2641. (14) Rudzin´ski, W.; Charmas, R. Adsorption 1996, 2, 245. (15) Rudzin´ski, W.; Charmas, R.; Borowiecki, T., In Adsorption on New and Modified Inorganic Sorbents; Dabrowski, A., Tertykh, V. A., Eds.; Elsevier: Amsterdam, 1996; p 357.
S0743-7463(98)00027-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/19/1998
6180 Langmuir, Vol. 14, No. 21, 1998
Figure 2. Diagrammatic presentation of the four layer model of the metal oxide/electrolyte interface: ψ0, δ0, the surface potential and the surface charge density in the 0-plane; ψC, ψA and δC, δA, the potentials and the charges coming from the specifically adsorbed ions (cations C+ and anions A-) of the basic electrolyte in two different planes; ψd, δd, the diffuse layer potential and its charge; c+, c-, cx, c2, the electrical capacitances, constant in the regions between planes.
is more or less manifested divergence between the values of PZC (point of zero charge) and IEP (isoelectric point). The first papers referred to the cases in which this divergence was significant. With the progress of accuracy in PZC and IEP measurements it was believed that inequality of PZC and IEP is a fundamental feature of these adsorption systems, being more or less drastic, depending on the adsorption system. This feature as well as some others led to modification of TLM, called the four layer model (FLM). The idea of the four layer model was introduced to the literature by Bowden et al.16,17 as well as in the papers by Barrow.18,19 A new layer (the fourth one as the name indicates, but situated as the second, next to the surface layer “0” where protons are adsorbed) was reserved first for the bivalent metal ions or anions of multiproton oxy acids. Cations and anions of basic electrolyte were placed still in the same layer as in the triple layer model. Next Bousse et al.20 presented in their paper a diagram of a four layer model, in which there are only ions of the basic electrolyte and the potential determining ions H+. They argued that anions and cations of the basic electrolyte are not located in the same layer (as in TLM) but in two separate ones. A schematic picture of such FLM is shown in Figure 2. A first rigorous thermodynamic description based on that physical model, and providing theoretical expressions for all the experimentally measured physicochemical quantities, has been published recently by the author and his colleagues.21,22 The developed expressions predict that (16) Bowden, J. W.; Nagarajah, S.; Barrow, N. J.; Posner, A. M.; Quirk, J. P. Aust. J. Soil Res. 1980, 185, 49. (17) Bowden, J. W.; Posner, A. M.; Quirk, J. P. In Soil with Variable Charge; Theng, B. K. G., Ed.; New Zealand Society of Soil Science: Lower Hutt, New Zealand, 1980; p 147. (18) Barrow, N. J. Adv. Agronomy 1985, 38, 183. (19) Barrow, N. J.; Bowden, J. W. J. Colloid Interface Sci. 1987, 119, 236. (20) Bousse, L.; De Rooij, N. F.; Bergveld, P. Surface Sci. 1991, 135, 479. (21) Charmas, R.; Piasecki, W.; Rudzin˜ski, W. Langmuir 1995, 11, 3199. (22) Charmas, R; Piasecki, W. Langmuir 1996, 12, 5458.
Charmas
the difference between the PZC and IEP is a fundamental feature of these adsorption systems. So far, the new theoretical description of ion adsorption at the oxide/ electrolyte interface, based on FLM, focused solely on the adsorption isotherms of ions and accompanying electrokinetic effects. That new theoretical description, however, left out of analysis one very important class of experiments. These are measurements of the calorimetric effects accompanying ion adsorption at the oxide/electrolyte interface. It is more and more realized that the knowledge of the enthalpy changes accompanying ion adsorption may provide an answer to many fundamental questions concerning the mechanism of ion adsorption at the oxide/ electrolyte interfaces. The first attempts to estimate the enthalpy changes were made 25 years ago, but it was only the last decade when the data to constrain the energetics of the surface acid-base reactions were accumulated rapidly. These data were elucidated first from the temperature dependence of ion adsorption isotherms,23-26 but now they are more and more frequently measured directly by using appropriate experimental calorimetric sets.27-33 In their review published in 1994, Kallay and Zalac34 emphasized that the measured heat effects carry, however, very complicated information which needs an advanced theoretical analysis. Interpretation of the experimental data was carried out mostly on a qualitative level. The first attempts to carry out a quantitative analysis were reported by De Keizer et al.31 in 1990. In 1991, we published our first theoretical papers concerning that problem.11,35 The present paper is aimed at providing the first in the literature theoretical analysis of these calorimetric effects, based on FLM. The present analysis will assume the traditional 2-pK charging mechanism and will be extended in the author’s future publications by considering other charging mechanisms. Theory 1. Comparison of the Triple Layer and Four Layer Models. Isotherms Equations. The potential determining ions H+, the cations C+, and anions A- of the basic electrolyte form according to 2-pK complexation models (TLM and FLM) the following surface complexes: SOH0, SOH2+, SO-C+, and SOH2+A-, where S is the surface metal atom. The concentrations of these complexes on the surface are denoted by [SOH0], [SOH2+], [SO-C+], and [SOH2+A-], respectively. [SO-] is the surface concentration of the free sites, “unoccupied” surface oxygens which (23) Berube, Y. G.; De Bruyn, P. L. J. Colloid Interface Sci. 1968, 27, 305. (24) Tewari, P. H.; Campbell, A. B. J. Colloid Interface Sci. 1976, 55, 531. (25) Blesa, M. A.; Figliolia, N. M.; Maroto, A. J. G.; Regazzoni, A. E. J. Colloid Interface Sci. 1984, 101, 410. (26) Fokkink, L. G. J.; De Keizer, A.; Lyklema, J. J. Colloid Interface Sci. 1989, 127, 116. (27) Griffiths, D. A.; Fuerstenau, D. W. J. Colloid Interface Sci. 1981, 80, 271. (28) Foissy, A. Ph.D. Thesis, Universite´ de Franche-Comte´, Bescanc¸ on, France, 1985. (29) Machesky, M. L.; Anderson, M. A. Langmuir 1986, 2, 582. (30) Mehr, S. R.; Eatough, D. J.; Hansen, L. D.; Lewis, E. A.; Davis, J. A. Thermochim. Acta 1989, 154, 129. (31) De Keizer, A.; Fokkink, L. G. J.; Lyklema, J. Colloids Surf. 1990, 49, 149. (32) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 297. (33) Machesky, M. L.; Jacobs, P. F. Colloids Surf. 1991, 53, 315. (34) Kallay, N.; Zˇ alac, S. Croat. Chem. Acta 1994, 67, 467. (35) Rudzin˜ski, W.; Charmas, R.; Partyka, S.; Foissy, A. New J. Chem. 1991, 15, 327.
Calorimetric Effects of Simple Ion Adsorption
Langmuir, Vol. 14, No. 21, 1998 6181
Table 1. Surface Reactions and Equilibrium Equations surface reaction
equilibrium equation
int
K a1
SOH2+ 798 SOH0 + H+
K int a1 )
int
K a2
SOH0 798 SO- + H+
K int a2 )
(aH)[SOH0] [SOH2+] (aH)[SO-]
int
*K C
SOH0 + C+ 798 SO-C+ + H+
*K int C )
*K int C ) int
*K A
SOH2+A- 798 SOH0 + H+ + A-
*K int A )
*K int A )
are likely engaged into adsorption of nondissociated water molecules. Except for protons, which are located in the potential layer ψ0, according to FLM the anions are situated within the layer of potential ψA, whereas the cations are within the layer of potential ψC. According to TLM both anions and cations are situated within the same layer of potential ψβ. For the reactions occurring at the oxide/electrolyte interface, the mass action equations and the related intrinsic equilibrium constants for both TLM and FLM are collected in Table 1. There, aC, aA, and aH are the activities of cations, anions, and protons, respectively. One can see in Table 1 that the equations describing the desorption of protons (reactions 1 and 2) are the same for both complexation models. According to Figures 1 and 2, the surface charge density, δ0, must be proportional (through the number of adsorption sites in appropriate units of charge, B) to the sum of the concentrations of the following surface complexes:
δ0 ) B([SOH2+] + [SOH2+A-] - [SO-] - [SO-C+]) (1) But the charges of the specifically adsorbed ions of the basic electrolyte in their planes are, of course, different in TLM and FLM. They are respectively given by
δβ ) B([SO-C+] - [SOH2+A-]) δC ) B[SO-C+]
δA ) -B[SOH2+A-]
(2) (3)
For the whole compact layer, there must be fulfilled the electroneutrality condition (i.e. the sum of all charges must be equal to zero), so for both models the diffuse layer charge, δd, is given by the same expression:
δd ) B([SO-] - [SOH2+])
(4)
The total number of the sites capable of forming the surface complexes on the surface, Ns, is given by
Ns ) ([SO-] + [SOH0] + [SOH2+] + [SOH2+A-] + [SO-C+]) (5) For both complexation models, the relationships among
0
[SOH ]
{
exp -
{
exp -
}
eψ0 kT
}
eψ0 kT
{
[SO-C+](aH)
}
e(ψ0 - ψβ) (TLM) kT
exp -
0
[SOH ](aC)
[SO-C+](aH) 0
[SOH ](aC)
[SOH0](aH)(aA) [SOH2+A-] [SOH0](aH)(aA) [SOH2+A-]
}
{
e(ψ0 - ψC) (FLM) kT
exp -
{
e(ψ0 - ψβ) (TLM) kT
{
e(ψ0 - ψA) (FLM) kT
exp -
exp -
}
}
the capacitances, the potentials and the charges within the individual electric layers were given in our previous papers.21,22 Introducing next the surface coverages θi’s by the individual surface complexes (i ) 0, +, C, A) and free sites (i ) -),
[SOH2+] [SOH0] [SO-C+] θ0 ) θ+ ) θC ) Ns Ns Ns θA )
[SOH2+A-] [SO-] θ- ) Ns Ns
(6)
one may consider an equivalent description of the reactions leading to the adsorption of ions onto the free sites [SO-], presented in Table 2. The set of the nonlinear equations in Table 2 can be transformed into an equivalent one, having the form of multicomponent Langmuir-like adsorption isotherms of ions θi’s (i ) 0, +, A, C):
θi )
K if i 1+
i ) 0, +, A, C
∑i Kifi
(7)
Here fi’s (i ) 0, +, A, C) are the functions of activity of protons and salt ions in the equilibrium bulk electrolyte.21,22 The functions f0 and f+ are the same for both complexation models and have the following form
{
f0 ) exp -
{
f+ ) exp -
eψ0 - 2.3pH kT
}
(8a)
}
2eψ0 - 4.6pH kT
(8b)
The functions fC and fA are different. For the TLM they are given by
{
fC ) aC exp -
{
fA ) aA exp -
}
eδ0 eψ0 + kT kTc1
eδ0 eψ0 - 4.6pH kT kTc1
(9a)
}
(9b)
6182 Langmuir, Vol. 14, No. 21, 1998
Charmas
Table 2. Alternative Description of the Surface Reactions Leading to the Adsorption of Ions onto Free Sites and Their Equilibrium Constants surface reaction
equilibrium constant
equilibrium equation
{ } { } { }
SO- + H+ T SOH0
K0 )
1 K int a2
K int a2 exp
SO- + 2H+ T SOH2+
K+ )
1 int K int a1 K a2
int K int a1 K a2 exp
SO- + C+ T SO-C+
KC )
*K int C
(aH)θeψ0 ) kT θ0 2eψ0 (aH)2θ) kT θ+
int K int a2 /*K C exp
K int a2
{ } { } { }
int K int a2 /*K C exp
KA )
SO- + 2H+ + A- T SOH2+A-
1
int K int a2 *K A exp
int K int a2 K A
int K int a2 *K A exp
{
{
fA ) ae exp -
}
eψC (aC)θ) (FLM) kT θC
(aH)2(aA)θe(2ψ0 - ψβ) ) kT θA
(TLM)
e(2ψ0 - ψΑ) (aH)2(aA)θ) (FLM) kT θA
1 2kT/e 1 ) + cDL (8 kTc)1/2 cStern
whereas for the FLM they take the following form
fC ) aC exp -
(aC)θeψβ ) (TLM) kT θC
(14)
r 0
eψ0 eδ0 + kT kTc+
(10a)
}
eδ0 eψ0 eδ* eδ* + - 4.6pH kT kTc+ kTc+ kTcwhere δ* ) δA + δd (10b)
The way of solving the set of eqs 7-9 for TLM and the set of eqs 7, 8, and 10 for FLM, to obtain the individual adsorption isotherms θi’s and surface charge δ0 as a function of pH, was shown in our previous publications.11,21,22,35 The surface potential function ψ0(pH) appearing in these equations can be calculated from the equation used by Bousse and co-workers,36
( )
eψ0 eψ0 + sinh-1 2.303(PZC - pH) ) kT βkT
Here r is the relative permittivity of solvent, 0 is the permittivity of free space, and c is the concentration of the electrolyte (ions/m3). The value of the cStern is assumed to be 0.2 F/m2.37 The Rudzin´ski-Charmas criterion12,21,35 for the common intersection point (CIP) can next be applied to study relations between the intrinsic equilibrium constants of the reactions listed in Table 1. If we introduce the notation PZC ) pHδ0)0,ψ0)0 ) -log H, where H is the activity of protons in the bulk solution int int int ) -log Kai (i ) 1, 2), and p*Kint at PZC, pKai i ) -log*Ki (i ) C, A), the relations reducing the number of the independent equilibrium constants have, for the FLM, the following form:21,22 int PZC ) 1/2(pK int a1 + pK a2 - log Y*)
(15a)
int PZC ) 1/2(p*K int C + p*K A + log X*)
(15b)
(11)
the simplified linear form of which is given by
β 2.3kT (PZC - pH) ψ0 ) β+1 e
Here
(12)
showing the difference between the ψ0(pH) dependence and the Nerstian one. Only for relatively large values β, β/β + 1 = 1, the potential change corresponding to one pH unit in this equation becomes that one predicted by the Nernst equation i.e. 2.3kT/e volts for a pH unit. The value of β is calculated from the following equation:13
( )
2e2Ns K int a2 β) cDLkT K int a1
1/2
(13)
∂P* X* ) P* + a ∂a 2 H ∂P* +1 Y* ) a2 ∂a K int*K int a2 A
(16a) (16b)
and
{
P* ) exp -
(
)} ( )
*K int eB C a 1 1 kT 2K int + H + 2*K inta c- c+
(
a2
C
2Ka2int + H
dln
(17a)
)
1 1 c - c+ da (17b)
In eq 13 cDL is the linearized double-layer capacitance. The value of cDL can be theoretically calculated (depending on the salt concentration in the solution) from another Bousse’s relation:37
1 ∂P ) P* ln P* + ∂a a 2K int + H + 2*K inta a2 C
(36) Van den Vlekkert, H.; Bousse, L.; de Rooij, N. F. J. Colloid Interface Sci. 1988, 122, 336.
(37) Bousse, L.; De Rooij, N. F.; Bergveld, P. IEEE Trans. Electron Devices 1983, 30, 8.
Calorimetric Effects of Simple Ion Adsorption
Langmuir, Vol. 14, No. 21, 1998 6183
where a is the activity of the cations and anions (a ) aC ) aA) of the 1:1 inert electrolyte considered here. The same application of Rudzin´ski-Charmas criterion for TLM gives simple equations,
/2(pK int a1
1
PZC ) PZC )
1
/2(p*K int C
+
pK int a2 )
(18a)
+
p*K int A )
(18b)
which have been known in the literature for a long time. The procedure of determining the equilibrium constants int pK int a1 and p*K A from eqs 15, for a given fixed pair of p int int K a2 and p*K C values, was discussed in the papers.21,22 In both cases, the application of the Rudzin´ski-Charmas criterion reduces the number of the freely chosen (bestfit) equilibrium constants from 4 to 2. Contrary to TLM, the FLM eqs 15-17 show that relations between the parameters are not independent of the concentrations of the electrolyte ions. This, at a first view, might be considered as a nonphysical result. However, our numerical exercises showed22 that the concentrations of ions and cations affect only slightly the calculated values of two parameters, when the two other ones are fixed. The explanation for that is following: Strictly speaking, CIP does not exist. The experimentally observed CIP is, as a matter of fact, a set of closely lying points which, at the present standard accuracy of titration experiments, are treated as one CIP. Thus, relations (15)-(17) should be treated as a way of estimating the interrelations of intrinsic equilibrium constants to a degree of accuracy which seems to be satisfactory at the present accuracy of experiment. Using these relations is very essential for more exact elucidation of the values of these parameters by fitting numerically the theoretical expressions to experimental data. Scientists involved in such computer exercises know how quickly their reliability decreases with the growing number of best-fit parameters. 2. Titration Calorimetry. The first experiments aimed at studying formation of the surface complexes SOH0, SOH2+, SOH2+A-, and SO-C+ were reported by Griffiths and Fuerstenau27 and by Foissy.28 They measured the heat of immersion of an outgassed solid sample into a solution of certain pH, Qim(pH). The theoretical interpretation of this “batch adsorption calorimetry” is rather difficult as we discussed in our previous publication.38 The heat effects measured in the experiment called “titration calorimetry”, reported first by Machesky and Anderson29 and Mehr et al.30 and next by De Kaizer et al.31 and Machesky and Jacobs,32,33 are much easier to interpret. Here, after introduction of an outgassed solid sample into a solution, the pH of that solution and sometimes also the concentration of other ions in the equilibrium bulk electrolyte are measured. Then, a titration step is carried out (from the base or acid side), and the evolved heat is recorded. Depending on the reported experiments, the consumption (adsorption) of protons and the ions of the inert electrolyte is monitored, accompanying the change of pH by ∆ pH. The recorded heat effect, which may be defined as the “molar differential heat of proton adsorption”, Qpr, is given by the equation38,39 (38) Rudzin´ski, W.; Charmas, R.; Cases, J. M.; Franc¸ ois, M.; Villieras, F.; Michot, L. J. Langmuir 1997, 13, 483. (39) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Cases, J. M.; Franc¸ ois, M.; Villieras, F.; Michot, L. J. Colloids Surf. 1998, 137, 57.
( ) [( ) ( ) ( )] ∂θi
∫pHpH+∆pH∑Qi ∂pH T dpH i
Qpr )
∂θ+
∂θA
∂θ0
∫pHpH+∆pH 2 ∂pH T + 2 ∂pH T +
∂pH
dpH
T
(19)
i ) 0, +, A, C where Qi is the molar heat of formation of ith complex. The molar differential heats of adsorption Qi’s are “configurational”, as they must depend on the concentrations of the surface complexes θi’s. Their detailed explicit form can be developed by applying the appropriate thermodynamic relations,38,39
[
]
b ∂ µSOH - µH Q0 ) -k kT ∂(1/T)
)k
{θi}
( ) ∂F0
∂(1/T)
{θi}
] ( )
[
b ∂ µSOH2+ - 2µH Q+ ) -k kT ∂(1/T)
[
{θi}
]
b ∂ µSO-C+ - µC QC ) -k kT ∂(1/T)
{θi}
)k
)k
[
∂(1/T)
{θi}
( )
]
b b ∂ µSOH2+A- - µA - 2µH QA ) -k kT ∂(1/T)
∂F+
∂FC
∂(1/T)
(20b)
(20c)
)
{θi}
k
{θi}
(20a)
( ) ∂FA
∂(1/T)
{θi}
(20d)
where µSO-C+, µSOH2+A-, µSOH, and µSOH2+ are the chemical potentials of these surface complexes whereas µbj (j ) H, C, A) are those in the bulk solution. For the TLM and FLM models, the functions of Fi (i ) 0, +, A, C) are different and they will be discussed in next sections. Three years ago, Kallay and co-workers34,40,41 started a theoretical-experimental study aimed at determining the experimental conditions under which the obtained experimental data could be free of the coulombic contributions to the measured heats of adsorption and refer only to the nonconfigurational heat values. This would eliminate the necessity of carrying out a complicated theoretical-numerical analysis. In one of their papers the authors postulated that the “average” heat of proton adsorption should correspond to the heat of titration from (PZC - ∆pH) to (PZC + ∆pH). The consumption of protons from the bulk solution was monitored in Kallay’s experiment, so the “average” molar heat of proton adsorption, QAv is given by38,42
( ) [( ) ( ) ( )] ∂θi
PZC ∆pH Q dpH ∫PZC ∆pH ∑ i ∂pH T +
-
QAv )
i
∂θ+
∂θA
PZC-∆pH 2 +2 + ∫PZC+∆pH ∂pH T ∂pH T
i ) 0, +, A, C
∂θ0
∂pH
T
dpH (21)
(40) Kallay, N.; ZÄ alac, S.; SÄ tefanic, G. Langmuir 1993, 9, 3457. (41) ZÄ alac, S.; Kallay, N. Croat. Chem. Acta 1996, 69, 119. (42) Rudzin´ski, W.; Charmas, R.; Piasecki, W.; Kallay, N.; Cases, J. M.; Franc¸ ois, M.; Villieras, F.; Michot, L. J. Adsorption, in press.
6184 Langmuir, Vol. 14, No. 21, 1998
Charmas
Equations 19 and 21 show the behavior of the heat obtained by titration calorimetry as a function of pH, for two types of calorimetric experiments discussed above. The functions Fi and the derivatives (∂θi/∂pH)T from eqs 19 and 21 are different for the two investigated models of surface complexation. For TLM, these derivatives were developed earlier,35 but for the reader’s convenience they are given in Appendix A. Also, the (∂θi/∂pH)T equations corresponding to FLM are given in Appendix A. 3. Explicit Expressions for Enthalpic Effects in the Triple Layer Model. For the triple layer model and a homogeneous solid surface, the functions Fi (a difference between the chemical potentials of components in different phases) occurring in eqs 20 can be obtained from the appropriate relations collected in Table 2. They have the following forms:
int F+ ) -ln(K int a1 K a2 ) -
int FC ) -ln(K int a2 /*K C ) -
eδ0 eψ0 θC + + ln aC - ln ) 0 kT kTc1 θ(22c)
eδ0 eψ0 + ln aA - 4.6pH kT kTc1 θA ln ) 0 (22d) θ-
Q0 ) Qa2 - eψ0 -
( )
e ∂ψ0 T ∂(1/T)
Q+ ) Qa2 + Qa1 - 2eψ0 -
Table 3):
dln K int ai
Qai ) -k
( )
( ) ( )
{θi},pH
( ) ( )
e ∂ψ0 T ∂(1/T) δ0T ∂c1 e 2 (c ) ∂T
(23c)
pH
-e
( )
pH
Here the nonconfigurational heats Qa1, Qa2, QaC, and QaA have the following definitions (heats of the reactions in
d(1/T) int dln(K int a2 *K A )
QaA ) -k
d(1/T)
(24a)
(24b)
(24c)
While calculating the derivative (∂c1/∂T){θi},pH in TLM, we assumed43,44 that there are two different values of c1 parameters, depending on the pH value. One for the acidic region (pH < PZC) and another one for the basic region (pH > PZC) on the oxide surface. Then, following Blesa’s recommendation,25 we treat c1 as the linear functions of temperature, L c1 ) cL1 ) cL,0 1 + R1 ∆T pH < PZC
(25a)
R c1 ) cR1 ) cR,0 1 + R1 ∆T pH > PZC
(25b)
which may be considered as the formal Taylor expansions R R,0 around T ) T0, such, that cL1 (T0) ) cL,0 1 and c1 (T0) ) c1 . While evaluating the bulk activity of cations and anions, we assumed that the activity coefficients γi’s are given by the equation proposed by Davies:45
[
]
xI - 0.3I 1 + xI
i ) A, C
(26)
Here z is the valency of ion “i”, I (mol/dm3) is the ionic strength of the suspension, and A is given by
1.825 × 106 (rT)3/2
(27)
In eq 27 r is the relative permittivity of the solvent. We have shown in our previous publication35 that the derivatives (∂ai/∂(1/T)){θi},pH, i ) C and A, occurring in eqs 23 can be approximated well by the equation
∂ai (23d)
i ) 1, 2
int dln(K int a2 /*K C )
A)
( )
δ0 c1 {θi},pH ∂ln aA +k ∂(1/T) {θi},pH
(23b)
d(1/T)
QaC ) -k
logγi ) -Az2
δ0 e ∂ψ0 +e + T ∂(1/T) {θi},pH c1 δ0T ∂c1 ∂ln aC +k e 2 ∂T ∂(1/T) {θi},pH (c )
1
heat of reacn Qa1 Qa2 QaA - Qa2 Qa2 - QaC Qa1 + Qa2 QaA QaC QaA - Qa1 - Qa2
(23a)
{θi},pH
2e ∂ψ0 T ∂(1/T)
1
QA ) QaA - eψ0 -
equil constants -pK int a1 -pK int a2 -p*K int A -p*K int C int -pK int a1 - pK a2 int * -pK a2 - p K int A int p*K int C - pK a2 * int pK int a1 - p K A
θ+ 2eψ0 - 4.6pH ) 0 (22b) kT θ-
Neglecting the temperature dependence of pH36 and taking into account that the constant set of θi’s also means a constant value of δ0, from eqs 22 we arrive at the following expressions for configurational heats accompanying the formation of surface complexes, Qi:38
QC ) QaC - eψ0 -
reacn type SOH0 + H+ T SOH2+ SO- + H+ T SOH0 SOH0 + H+ + A- T SOH2+A+ SO-C+ + H+ T SOH0 + C+ SO- + 2H+ T SOH2+ SOH- + 2H+ + A- T SOH2+ASO- + C+ T SO-C+ SOH0 + H+ T SOH2+
θ0 eψ0 - 2.3pH - ln ) 0 (22a) kT θ-
F0 ) -ln K int a2 -
int FA ) -ln(K int a2 *K A ) -
Table 3. Surface Reactions, Equilibrium Constants, and Heats of Adsorption
∂(1/T)
) -2.0172 × 10-8T4ai ln γi i ) A, C
(28)
(43) Sprycha, R. J. Colloid Interface Sci. 1984, 102, 173. (44) Blesa, M. A.; Kallay, N. Adv. Colloid Interface Sci. 1988, 28, 111. (45) Davies, C. W. Ion Association; Butterworths: London, 1962. (46) Casay, W. H. J. Colloid Interface Sci. 1994, 163, 407.
Calorimetric Effects of Simple Ion Adsorption
Langmuir, Vol. 14, No. 21, 1998 6185
Finally, the derivatives occurring in eqs 23 (∂ψ0/∂(1/ T))pH are evaluated from eq 11 and take the form
( ) ∂ψ0
∂(1/T)
pH
[
) -ψ0T
]
t Qa2 - Qa1 β + + β+t β+t 2kT kT β ∂PZC 2.3 (29) e β + t ∂(1/T)
where
t)
eδ0 eψ0 eδ* eδ* + + kT kTc+ kTc+ kTcθA ln aA - 4.6pH - ln ) 0 (33b) θ-
int FA ) -ln(K int a2 *K A ) -
1
x( ) eψ0 βkT
2
(30)
Transformations similar to those outlined in the case of TLM lead to the following expressions for configurational heats of formation of complexes SO-C+ and SOH2+A-:
QC ) QaC - eψ0 -
+1
In eq 29 the derivative (∂PZC/∂(1/T)) is replaced by the term (Qa1 + Qa2)/4.6k, according to the result of the formal differentiations of PZC in eq 18a and the definition of Qa1 and Qa2 in eq 24a,
∂PZC Qa1 + Qa2 ) 2.3 2k ∂(1/T)
(31a)
( ) ( ) ( )
e ∂ψ0 T ∂(1/T) eδ0T ∂c+ (c )2 ∂T +
QA ) QaA - eψ0 -
( )
e(δ0 + δ*)T ∂c+ ∂T (c )2 +
e ∂ψ0 T ∂(1/T)
δ0 +e + c {θi},pH + ∂ln aC +k ∂(1/T) {θi},pH
{θi},pH
( )
(31b)
in view of the definitions of QaC and QaA in eqs 24b,c. From eqs 31a,b we have
QaA - (Qa1 +Qa2) ) QaC
(32)
This means that the heat effect related to cation adsorption on an empty oxygen atom (QaC) is the same as the heat related to the attachment of anion to the already existing complex SOH2+. Thus, in our previous publications38,39 we have launched the hypothesis that they are always the same, because they are simply nonexistent. Such a conclusion was, in fact, drawn some three years ago by Kosmulski,47 who used radiometric methods to study the individual adsorption isotherms of ions of the inert electrolyte. Although it was not said expressis verbis, such a point of view was also expressed by Machesky et al.,32,33 who called Qpr defined in eq 19 the heat of proton adsorption. 4. Explicit Expressions for Enthalpic Effects in the Four Layer Model. Comparing to TLM, the four layer model leads to different expressions for FC and FA, whereas F0 and F+ remain unchanged. According to the appropriate equations in Table 2, FC and FA take the following form: int FC ) -ln(K int a2 /*K C ) -
eδ0 eψ0 + + ln aC kT kTc+ θC ln ) 0 (33a) θ-
(47) Kosmulski, M. Colloids Surf. A: Physicochem. Eng. Aspects 1994, 83, 237. (48) Smit, W.; Holten, C. L. M. J. Colloid Intertace Sci. 1980, 78, 1. (49) Kallay, N.; Babic´, D.; Matijevic´, E. Colloids Surf. 1986, 19, 375. (50) Thomas, F.; Bottero, J. Y.; Cases, J. M. Colloids Surf. 1989, 37, 281. (51) Wood, R.; Fornasiero, D.; Ralston, J. Colloids Surf. 1990, 51, 389.
( ) ( )
δ* eδ*T ∂c+ c- (c )2 ∂T -
k
Similarly to eq 18a, the same differentiation with respect to (1/T) in eq 18b leads to the result
∂PZC QaA + QaC ) 2.3 2k ∂(1/T)
(δ0 + δ*) c+
-e
+e
{θi},pH
(34a)
pH
+
{θi},pH
∂ln aA ∂(1/T)
(34b)
pH
Here the nonconfigurational heats Qa1, Qa2, QC, and QA are still given by eqs 24. Assuming that c+ and c- are the linear functions of temperature, we have two new parameters RC and RA,
c+ ) c0+ + R+∆T
(35a)
c- ) c0- + R-∆T
(35b)
These linear functions may be considered as the formal Taylor expansions around T ) T0, such that c+(T0) ) c0+ and c-(T0) ) c0-. The derivatives ψ0 and ai with respect to (1/T) are the same as in TLM but not ∂PZC/∂(1/T), because PZC is expressed now through eqs 15. Now, we take the advantage of having two expressions for PZC(T) to develop interrelations of some temperature derivatives of interest. Thus we calculate ∂PZC/∂(1/T) from two equation systems 15a, 16, 17 and 15b, 16, 17
(∂F(1) PZC/∂(1/T))PZC ∂PZC )∂(1/T) (∂F(1) PZC/∂PZC)1/T
(36a)
(∂F(2) PZC/∂(1/T))PZC ∂PZC )∂(1/T) (∂F(2) PZC/∂PZC)1/T
(36b)
(2) where F (1) PZC and F PZC are given by eqs 15a,b, respectively:
1 1 1 int int F(1) PZC ) /2pK a1 + /2pK a2 - /2 log Y* - PZC ) 0 (37a) 1 1 1 int int F(2) PZC ) /2pK C + /2p*K A + /2 log X* - PZC ) 0 (37b)
After certain rearangements, eqs 36 can be written as follows:
6186 Langmuir, Vol. 14, No. 21, 1998
Charmas
Qa1 + Qa2 1 ∂ log Y* 4.6k 2 ∂(1/T) ∂PZC ) 1 ∂ log Y* ∂(1/T) 1+ 2 ∂PZC
(38a)
QaA + QaC 1 ∂ log X* + 4.6k 2 ∂(1/T) ∂PZC ) 1 ∂ log X* ∂(1/T) 12 ∂PZC
(38b)
Table 4. Values of Best-Fit Parameters and Calculated Values Obtained for the Adsorption System TiO2/NaCl (0.01 mol/dm3) by Numerical Calculations for TLM params
calcd values
pK int a2 ) 8.80 p*K int C ) 7.00 cL1 ) 0.80 F/m2 cR1 ) 1.05 F/m2
a pK int a1 ) 4.00 a p*Kint ) 5.80 A
Qa1 ) 31 kJ/mol Qa2 ) 43 kJ/mol RL1 ) 0.003 F/(m2‚deg) RR1 ) -0.003 F/(m2‚deg)
Here the derivatives (∂ log Y*/∂PZC)1/T, (∂ log Y*/∂(1/T))PZC, ∂ log X*/∂PZC)1/T, (∂ log X*/∂(1/T))PZC have pretty complicated explicit forms, given in Appendix B. Finally, from eqs 38ab one can arrive at the following interrelation of the nonconfigurational heats of adsorption (similar to eq 32 for TLM):
a
QaA ) 74 kJ/molb QaC ) 0 kJ/molc
Using eqs 18. b Using eq 32. c Assumed.
Table 5. Values of Best-Fit Parameters and Calculated Values Obtained for the Adsorption System TiO2/NaCl (0.01 mol/dm3) by Numerical Calculations for FLM params
1 ∂ log X* 12 ∂PZC + QaC QaA ) (Qa1 + Qa2) 1 ∂ log Y* 1+ 2 ∂PZC 1 ∂ log X* 12 ∂PZC ∂ log Y* ∂ log X* - 2.3k (39) 2.3k 1 ∂ log Y* ∂(1/T) ∂(1/T) 1+ 2 ∂PZC
pK int a2 ) 8.80 p*K int C ) 7.00
c- ) 0.80 F/m2 c+ ) 1.05 F/m2 Qa1 ) 31 kJ/mol Qa2 ) 43 kJ/mol R- ) 0.003 F/(m2‚deg) R+ ) -0.003 F/(m2‚deg) c
calcd values a pK int a1 ) 4.00 a p*Kint ) 5.82 A
dln(1/c- - 1/c+)/da ) -120 dm3/molb
QaA ) 74.6 kJ/molc QaC ) 0 kJ/mold
a Using eqs 15-17. b Using eq 40 in order to obtain IEP ) PZC. Using eq 39. d Assumed.
One should remember that (∂ log Y*/∂(1/T))PZC is the function of QaA and the above equation can be solved as an implicit one using the iteration method. Discussion So far only few experiments were reported where in the course of titration also the accompanying heat effects were measured upon the addition of an acid or base. The first experiments of that kind were performed by Machesky and Anderson29 in 1986, Mehr et al.30 in 1989, De Keizer et al.31 in 1990, Machesky and Jacobs32,33 in 1991, Kallay et al.40 in 1993, and Casey46 in 1994. While selecting the experimental data for our analysis, we looked for those which could be easily read either from tables or from good quality figures. Then, there should be given some additional information, like the titration data δ0(pH) for instance. Finally, the value of PZC should not be far from 7, to have at the disposal well-represented features of the system at both negative and positive values of the surface charge δ0. All these requirements have led us to choose the data published by Mehr et al.30 They represent the “titration calorimetry” in the system TiO2/ NaCl. The TiO2 sample was anatase in 60% and rutile in 40%, and its PZC value was equal to 6.4. Other experimental details are available in the original paper by Mehr et al.30 For the reasons described in our previous publication39 we took into consideration the titration and calorimetric data, measured at the lowest concentration of NaCl equal to 0.01 mol/dm3 (the smallest hysteresis between acid and base titration). Then, we fitted by our equations the values of δ0(pH) and Qpr(pH) being appropriate arithmetic averages of the two values obtained in the course of acid and base titrations. The best-fit parameters found by computer are collected in Tables 4 and 5 for TLM and FLM, respectively. Figure 3 shows the agreement between the experimental and theoretical values of the surface charge δ0(pH). The * int pK int a2 and p K C values were treated as the free (best-fit) * int parameters, whereas the other two pK int a1 and p K A
Figure 3. Plots of δ0 vs pH for TiO2 in 0.01 mol/dm3 NaCl. The black circles (b) are arithmetic averages of the data recorded by Mehr et al.30 in the course of acid and base titrations. The broken line is the best fit obtained by using the expressions corresponding to TLM and the parameters collected in Table 4. The solid line is the best fit obtained with FLM and the parameters values collected in Table 5.
values were calculated every time from the relations (15)(17) for FLM and (18) for TLM. Then, also the values of cL1 , cR1 , and c-, c+ had to be found by computer. The main difference in the predictions of TLM and FLM discussed in our previous publications, concerned electrokinetic features of adsorption systems.21,22 The Rudzin´ski-Charmas criterion applied to TLM led to the prediction that PZC and IEP must have identical values. In the case of FLM, the Rudzin´ski-Charmas criterion led to the prediction of a difference between PZC and IEP, given by the equation22
IEP - PZC ) 1/2(β + 1) log Y*
(40)
where Y* is defined in eq 16b. An analysis of eqs 16 and 17 showed that one of the parameters responsible for the
Calorimetric Effects of Simple Ion Adsorption
Langmuir, Vol. 14, No. 21, 1998 6187
Figure 6. Contribution of Qpr to formation of the four kinds of surface complexes. The broken lines are the contributions C A Q0pr, Q+ pr, Qpr, and Qpr, calculated by accepting TLM. Their sum gives the values represented by the broken line in Figure 5. The solid lines denote the contributions calculated by accepting FLM. Their sum is the value of Qpr represented by the solid line in Figure 5.
Figure 4. Individual isotherms of the surface complexes θ0, θ+, θC, and θA along with θ-, calculated by using TLM (- - -) and the related values of parameters collected in Table 4 and calculated by FLM (s) by using the corresponding values of parameters collected in Table 5.
int L R parameters pK int a2 , p*K C , and c1 , c1 or c-, c+, found by fitting the titration curve shown in Figure 3. In addition to the above parameters, the following had to be known to calculate the theoretical Qpr(pH) curve: Qa1, Qa2, QaC, QaA. According to the assumption that the enthalpic effects of the purely coulombic adsorption of big alkali metal cations on the SO- sites, and of the anions on the complexes SOH2+, are negligible, we put QaC ) 0. Thus, while fitting the experimental data for Qpr(pH), we had the following four free (best-fit) parameters: Qa1, Qa2, RL1 , and RR1 for TLM and Qa1, Qa2, R-, and R+ for FLM. Their values were found by computer in the course of fitting best the experimental Qpr(pH) values. Figure 5 shows a reasonably good agreement between the experimental and theoretical heats of adsorption. The next Figure 6 shows the composite values of Qpr, due to the formation of the four kinds of surface complexes, A SOH, SOH2+, SOH2+A-, and SO-C+, Q0pr, Q+ pr, Qpr, and C Qpr, respectively,
∫pHpH+∆pHQi(∂pHi )T dpH ∂θ
Qipr ) Figure 5. Comparison of experimental and theoretical heats of proton adsorption. The black circles (b) are the arithmetic averages of the experimental values of Qpr recorded by Mehr et al.30 in the course of acid and base titrations. The broken line is the best fit obtained by accepting TLM and the corresponding parameter values collected in Table 4. The solid line is the best fit obtained with FLM and the parameters values collected in Table 5.
difference between IEP and PZC is the derivative,
dln
(
)
1 1 c- c+ da
(41)
For the adsorption system TiO2/NaCl investigated here and the assumption IEP ) PZC, the derivative value is equal to -120 dm3/mol. Figure 4 shows the individual adsorption isotherms of the surface complexes. Figure 5 shows the agreement between the experimental and theoretical “heats of proton adsorption” Qpr, calculated by using eq 19, and the
[
]
∫pHpH+∆pH 2(∂pH+ )T + 2(∂pHA )T + (∂pH0 )T ∂θ
∂θ
∂θ
dpH (42)
i ) 0, +, A, C As can be seen in Figure 6 the heat effect due to formation of the surface complexes SOH2+ is almost negligible. The substantial contributions Q0pr and QApr can be easily understood by considering the fact that they contain effects of one or two protons adsorption, characterized by high nonconfigurational values. At the same time somewhat striking is the large contribution from QCpr, in spite of the zero value characterizing the nonconfigurational heat of cation adsorption. Our computer exercises showed that there is not much room for maneuvering while choosing the Qa1, Qa2, RL1 , RR1 , or R-, R+ parameters. A slightly different choice of any of them will destroy substantially the agreement between theory and experiment. First we show that although the estimated values of the parameters RL1 , RR1 , or R-, R+ are small, they are
6188 Langmuir, Vol. 14, No. 21, 1998
Charmas Table 6. Examples of the Difference between PZC and IEP Values Taken from the Literature system R -alumina/NaBr anatase/NaNO3 alumina Ma/KCl boehmite/KNO3 a
value of PZC value of IEP 4.5 6.5 8.5 8.5
3.1-3.5a 4.5 8.4 9.1
ref Smit et al. (48) Kallay et al. (49) Thomas et al. (50) Wood et al. (51)
Depending on the salt concentration.
Figure 7. Changes in the behavior of the theoretical curves Qpr presented in Figure 5 after putting the parameters RL1 , RR1 or R-, R+ equal to zero. The two families of the broken (TLM) and solid (FLM) lines have been calculated for one fixed value of Qa1 ) 26kJ/mol and two different values of Qa2, marked in the figure. Other parameters are the same as in Figure 5.
Figure 9. Theoretical surface charge curves δ0 for FLM, calculated by accepting that the derivative (41) is equal to -120 dm3/mol (s) and by assuming the derivative value to be 320 dm3/mol (- - -). Other parameters are as in Figure 3. The solid line is for IEP ) PZC ) 6.4, whereas the broken line is for PZC ) 6.4 and IEP ) 6.2.
Figure 8. Changes in the behavior of the theoretical curves Qpr presented in Figure 5 after putting the parameters RL1 , RR1 or R-, R+ equal to zero. The two families of the broken (TLM) and solid (FLM) lines have been calculated for one fixed value of Qa2 ) 36kJ/mol and two different values of Qa1, marked in the figure. Other parameters are the same as in Figure 5.
essential for good agreement between the experimental and theoretical values of Qpr to be obtained. Figures 7 and 8 show what happens when these parameters are put equal to zero. Figures 7 and 8 show that good agreement between the experimental and theoretical Qpr values can never be obtained when one neglects the temperature dependence of the capacitances RL1 , RR1 or R-, R+. As discussed in our previous papers,21,22 there is a similarity in the physical values played by the parameters RL1 , R- and RR1 and R+, respectively. Now, we are going to show that exact estimation of PZC and IEP is essential for a reliable estimation of parameters characterizing heats of proton adsorption. Let us assume for that purpose that the true value of IEP in our system is 6.2. The difference PZC - IEP ) 0.2 is smaller than the differences between IEP and PZC reported so far in the literature. They are listed in Table 6. With the parameters collected in Table 5, the IEP value equal to 6.2 is observed when the derivative 41 takes the value 320 dm3/mol. That change in the derivative value has practically no remarkable influence on the theoretical surface charge curve δ0 (pH), as it is shown in Figure 9. On the contrary, that change has a serious effect on the estimated heat effects.
Figure 10. Behavior of Qpr curves, calculated by assuming that the derivative (41) is equal to 320 dm3/mol (IEP ) 6.2), for three different values of Qa1: 31 kJ/mol (- - -), 26 kJ/mol (s), and 20 kJ/mol (‚ ‚ ‚). Other parameters are as in Figure 5.
Figure 10 shows that in order to arrive at good agreement between the theoretical and experimental Qpr values when IEP ) 6.2, one has to assume that Qa1 is equal to 26 kJ/mol, instead of 31 kJ/mol assumed previously when IEP ) PZC ) 6.4. Thus, the about 3% change in IEP value causes the change in the estimated values of Qa1 greater than 10%. Figure 10 also shows that the difference between PZC and IEP should have a small effect on the estimated value of the parameter Qa2. An inspection into Figure 10 and next Figure 11 brings an encouraging observation as to the possibility of finding the reliable values of Qa1 and Qa2 from fitting the experimental Qpr curves. Figure 10 shows that the parameter Qa1 affects strongly the experimental Qpr data below PZC, without affecting much Qpr above PZC. On the contrary, the parameter Qa2 affects strongly the basic part of the Qpr curve, without influencing much its acidic branch.
Calorimetric Effects of Simple Ion Adsorption
Figure 11. Behavior of Qpr curves, calculated by assuming that the derivative (41) is equal to 320 dm3/mol (IEP ) 6.2) and Qa1 ) 26 kJ/mol for three different values of Qa2: 40 kJ/mol (- - -), 43 kJ/mol (s), and 48 kJ/mol (‚ ‚ ‚). Other parameters are as in Figure 5.
Langmuir, Vol. 14, No. 21, 1998 6189
PZC caused by the stepwise change of the values cL1 into cR1 and RL1 into RR1 going from low to high pH values through PZC. On the contrary, the FLM model is characterized by continuity of all thermodynamic functions. Calculations of the values Qpr(pH) for TLM and FLM must be made using a computer, and the lack of difference in calculation time promotes application of FLM equations provided that the necessary experimental information is available. There is a difference between both surface complexation models which becomes more evident in calculation of isotherm derivatives in the interpretation of calorimetric titration experiment. As for real values of nonconfigurational heats of adsorption of ions as well as temperature derivatives of capacitances, it seems that, due to sensitivity of theoretical predictions (a small change of parameters causes a great change of theoretical heat look), the accuracy of the experimental data of measured heat must be taken into account. Obtaining such data is not easy and requires perfect knowledge of equipment and experimental technique as well as taking into account various heat corrections from the direct experiment as very well presented in the paper by Machesky and Jacobs.32 Acknowledgment. The author expresses his gratitude to Professor W. Rudzin´ski for helpful discussions. This research was supported by KBN Research Grant No. 3 T09A 078 11. Also, the author wishes to express his thanks and gratitude to the Foundation for Polish Science for the grant making his one-year stay in the Laboratoire Environnement et Mineralurgie, ENS Geologie, CNRSURA 235, Vandoeuvre les Nancy, France, possible. Appendix A
Figure 12. Contributions to the best-fit Qpr curve (s) in Figure C A 10, due to Q0pr, Q+ pr, Qpr, and Qpr. The present figure corresponding to IEP ) 6.2 is to be compared with Figure 6 corresponding to IEP ) PZC ) 6.4.
Finally, Figure 12 shows how changing IEP affects the contributions to Qpr from the heats accompanying formation of various surface complexes. The changes in the A C compositive curves Q0pr, Q+ pr, Qpr, and Qpr are not dramatic but clearly seen. Conclusions The model studies showed that the FLM allows for more accurate estimation of nonconfigurational adsorption heat of the second proton Qa1 having the experimental values IEP and PZC for the adsorption system under consideration. The value of derivative (41) can also be estimated from the difference between the experimental values of IEP and PZC. However, considering other parameters affecting this difference, it is important for the calorimetric experiment to be preceded not only by the potentiometric titration curve δ0(pH) but also by measurements of the ζ-potential and individual isotherms of electrolyte ions adsorption. As shown in previous papers,12,35 the success in obtaining reliable values of parameters in the model description depends on the number of independent experiments taken into consideration. The figures presenting the calculated heats Qpr vs pH for TLM show distincly discontinuity in the point pH )
The derivatives (∂θi/∂pH)T (i ) 0, +, A, C) in eqs 19 and 21 are to be evaluated from the equation systems 22 or 33 for TLM and FLM, respectively. For instance,
| | | |
∂F0 ∂pH ∂F+ ∂pH ∂FC ∂pH ∂FA ∂θ0 ∂pH ) (-1) ∂F ∂pH T 0 ∂θ0 ∂F+ ∂θ0 ∂FC ∂θ0 ∂FA ∂θ0
( )
∂F0 ∂θ+ ∂F+ ∂θ+ ∂FC ∂θ+ ∂FA ∂θ+ ∂F0 ∂θ+ ∂F+ ∂θ+ ∂FC ∂θ+ ∂FA ∂θ+
∂F0 ∂θC ∂F+ ∂θC ∂FC ∂θC ∂FA ∂θC ∂F0 ∂θC ∂F+ ∂θC ∂FC ∂θC ∂FA ∂θC
∂F0 ∂θA ∂F+ ∂θA ∂FC ∂θA ∂FA ∂θA ∂F0 ∂θA ∂F+ ∂θA ∂FC ∂θA ∂FA ∂θA
(A.1)
While evaluating the partial derivatives (∂Fi/∂pH) occurring in the numerator of eq A.1, one has to consider ψ0 as a function of pH defined in eq 11. Then, for both TLM and FLM,
6190 Langmuir, Vol. 14, No. 21, 1998
Charmas
∂F+ ∂F0 t t ) -2.3 ) -4.6 ∂pH t + β ∂pH t+β ∂FC ∂FA β 2t + β ) 2.3 ) -2.3 ∂pH t + β ∂pH t+β
(A.2)
where t is given by eq 30. Further, while evaluating the derivatives (∂Fi/∂θi) (i, j ) 0, +, A, C) one has to consider the equation
δ0 ) B(2θ+ + 2θA + θ0 - 1)
(A.3)
obtained from eqs 1 and 6 and from the dependence ∑jθj ) 1 (j ) 0, +, A, C, -) between the surface coverages. Then, for TLM,
θ0 + θ- ∂F0 ∂F0 1 ))∂θ0 θ0θ∂θ+ θθ + + θ∂F+ 1 ∂F+ ))∂θ0 θ- ∂θ+ θ+θ∂FC ∂FC 1 1 )+w )+ 2w ∂θ0 θ∂θ+ θ∂FA ∂FA 1 1 )-w )- 2w ∂θ0 θ∂θ+ θ∂F0 1 ∂F0 1 ))∂θC θ- ∂θA θ∂F+ 1 ∂F+ 1 ))∂θC θ- ∂θA θ∂FC θC + θ- ∂FC 1 ))+ 2w ∂θC θCθ∂θA θθ A + θ∂FA 1 ∂FA ))- 2w ∂θC θ- ∂θA θAθ-
(A.9)
For TLM, Xi’s have the following explicit form:
X0 ) t[θ- + θC - θ+ - θA] + 2wt[θ-θA - θ+θC] + β[θC - θA] X+ ) t[θ0 + 2θ- + 2θC] + wt[θ0θC + θAθ0 + 4θAθ- + 4θAθC] + β[θC - θA]
Whereas for FLM we obtain
X0 ) t[θ- + θC - θ+ - θA] + w-tθA[θC + 2θ-] w+tθC[θA + 2θ+] + (w- - w+)θCθA[β + 2w+tθ-] + β[θC - θA] (A.4)
X+ ) t[θ0 + 2θ- + 2θC] + w-tθA[2θC + 4θ- + θ0] + w+tθC[2θA + θ0] + (w- - w+)θCθA[β + w+ t(4θ- + θ0)] + β[θC - θA]
(A.5)
XC ) -t[θ0 + 2θ+ + 2θA] + w+t[4θAθ- + θAθ0 + 4θ-θ+ + θ-θ0 + θ+θ0] - w-tθAθ0 - (w- w+)βθA[2θ- + θ0] - β[θ0 + θ+ + θ- + 2θA]
Then for FLM, we obtain the derivatives ∂F0/∂θi and ∂F+/ ∂θi (i ) 0, +, A, C) having the same form as for TLM, because the form of equations F0 and F+ is identical for both models. The other derivatives of interest take the following explicit form:
where
2.3θi Xi ∂θi )i ) 0, +, A, C ∂pH t+β Y
Y ) 1 + w[θA(2θ- + 2θC + θ0) + θC(2θ+ + 2θA + θ0)] (A.10)
δ* ) B(θ- - θ+ - θA) ) B(1 - 2θ+ - 2θA - θ0 - θC) (A.6)
∂FA 1 )+ w+ - w∂θC θ-
After calculating the determinants, we arrive at the following explicit form of the derivatives ∂θi/∂pH:
XA ) t[θ0 + 2θ- + 2θC] - wt[θ0θ+ + θ-θ0 + 4θ-θ+ + 4θ+θC] + β[θ0 + θ+ + θ- + 2θC]
In the case of FLM, we have additionally to remember that from eq 10b
∂FC 1 )+ w+ ∂θ0 θ∂FA 1 )- w∂θ0 θθ C + θ∂FC )∂θC θCθ-
(A.8)
XC ) -t[θ0 + 2θ+ + 2θA] + wt[θ0θ+ + θ-θ0 + 4θ-θ+ + 4θ-θA] - β[θ0 + θ+ + θ- + 2θA]
where
eB w) kTc1
eB eB w- ) kTc+ kTc-
w+ )
∂FC 1 )+ 2w+ ∂θ+ θ∂FA 1 )- 2w∂θ+ θ∂FC 1 )+ 2w+ ∂θA θθA + θ ∂FA )- 2w∂θA θAθ(A.7)
XA ) t[θ0 + 2θ- + 2θC] - w-t[2θCθ+ + 4θ-θ+ + θ-θ0 + θ+θ0] - 2w+tθCθ+ + (w- - w+)[β(θ- + θ+) w+tθC(4θ-θ+ + θ-θ0 + θ+θ0)] + β[θ0 + θ+ + θ- + 2θC] Y ) 1 + w+θC[3θA + 2θ+ + θ0] + w-θA[θC + 2θ- + θ0] + w+(w- - w+)θCθA[2θ- + θ0] (A.11) When w+ ) w- ) w (FLM reduces to TLM) eqs A.11 reduce to eqs A.10. Appendix B The derivatives (∂ log Y*/∂PZC)1/T, and (∂ log X*/∂PZC)1/T in eqs 38 have the following form:
(
)
∂log Y* ∂PZC
(
(
1/T
∂log X* ∂PZC
)
H2 ∂P* ∂2P* a2 - 4.6 2.3Y* K int*K int ∂a∂PZC ∂a a2 A (B.1)
)
)
1/T
)
(
∂2P* ∂P* 1 +a 2.3X* ∂PZC ∂a∂PZC
)
(B.2)
Calorimetric Effects of Simple Ion Adsorption
Langmuir, Vol. 14, No. 21, 1998 6191
[(
where
2.3HP* ln P* ∂P* ) ∂PZC 2K int + H + 2*K int a a2
(B.3a)
C
(B.3b)
The derivatives (∂log Y*/∂(1/T))PZC and (∂log X*/∂(1/T))PZC in eqs 38 can be written in the following way:
)
)
PZC
(
∂2P* a2 H2 + int int 2.3Y* K *K ∂a∂(1/T) a2 A
(
where
)
PZC
)
(
2K int a2
)
∂a ∂P* ∂P* 1 + + 2.3X* ∂(1/T) ∂(1/T) ∂a ∂2P* (B.5) a ∂a∂(1/T)
)
)
Q Qa2 - QaC ∂a int a2 + 2a*K int -a C K a2 k k ∂(1/T) +H+
2 2*K int C a)
(
TRTR+ 1 1 + c- c+ (c )2 (c )2 + H + 2*K int a C +
[
∂2P* ) P* ln P* ∂a∂(1/T)
(
int 2*K int C (2K a2 + H)
d2 ln
(
)
1 1 c- c+
da d(1/T)
]
)
+H+
+
(B.6a)
-
)
Qa2 - QaC Q ∂a int a2 + 4a*K int -a C K a2 k k ∂(1/T)
a(2K int a2
∂ ln a ∂P* QaA ∂P* + (B.4) 2 k ∂a ∂(1/T) ∂a ∂log X* ∂(1/T)
(
*K int C a
int 2 (2K int a2 + H + 2*K C a)
∂log Y* ∂(1/T)
int *K int C (2K a2 + H)
(2K int a2
ln P* + 1 ∂P* ∂P* ∂2P* ) ∂a∂PZC P* ln P* ∂a ∂PZC 4.6H*K int C P* ln P*
(
)
∂P* 1 1 1 eB × ) - P* k T c- c+ ∂(1/T)
]
2 2*K int C a)
-
2K int a2 + H ln P* + 1 ∂P* ∂P* 1 ∂a + int 2 ∂(1/T) P* ln P* ∂a ∂(1/T) a 2K a2 + H + 2*K int a C (B.6b)
Thus, in the case of FLM there appears one new parameter:
d2 ln
(
)
1 1 c- c+
da d(1/T) However, the numerical exercises showed that this value does not affect the theoretical calculated functions, so this value was set to zero. LA980027K
