11912
J. Phys. Chem. B 2000, 104, 11912-11922
A Combined Temperature-Calorimetric Study of Ion Adsorption at the Hematite-Electrolyte Interface: I. Model of a Homogeneous Oxide Surface W. Rudzin´ ski,* G. Panas, and R. Charmas Department of Theoretical Chemistry, Maria Curie-Sklodowska UniVersity, M. Curie-Sklodowska Sq.3, Lublin, 20-031 Poland
N. Kallay and T. Preocˇ anin Laboratory of Physical Chemistry, Faculty of Science, UniVersity of Zagreb, P.O. Box 163, 10001 Zagreb, Croatia
W. 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: April 4, 2000; In Final Form: August 29, 2000
The theoretical quantitative analysis of the temperature dependence and enthalpic effects of ion adsorption, developed in our earlier publications, was applied here to study the features of hematite/electrolyte interfaces. This is the first time that our set of experimental data could be used to carry out a simultaneous analysis of both the temperature dependence of the titration isotherm and directly measured enthalpic effects. To draw possible general conclusions about the features of the hematite/electrolyte interfaces, we considered two sets of experimental data measured in two laboratories, using the hematite samples prepared in different ways. The differences in sample preparations are manifested by substantially different values of the monitored surface charges and related calorimetric effects. The present quantitative analysis in Part I of this publication was carried out by using the model of an energetically homogeneous solid surface, which is still commonly accepted. Certain inconsistencies were found in the parameter values leading to a good fit of titration isotherms and those that are best to fit directly measured enthalpic effects. Thus, the general conclusion was drawn that this popular model is too crude for a quantitative analysis.
Introduction The features of the electrical double layer, formed due to adsorption of ions at the iron oxide/electrolyte interface, are crucial for many processes important for life and technology. Iron oxides are one of the most important components of soils, as well as of the sediments found in the natural water reservoirs. This explains why the features of the iron oxide/electrolyte interface have been so extensively studied by scientists working in the area of soil science or carrying out environmental research. Iron is still the main component of heavy industrial constructions and automobiles, so corrosion in the presence of water solutions is predominantly governed by the features of iron oxide/ electrolyte interfaces. Also, iron oxides have long been used for the preparation of catalysts, which involves ion adsorption from impregnating solutions. Due to its enormous importance for life on our planet and technology, ion adsorption at the iron oxide/electrolyte interface has been the subject of a large number of publications. Depending on water content and preparation, one may have to deal with a variety of iron oxides, and therefore a variety of physical models has been used to represent the physical situation at the iron oxide/electrolyte interface. Like many authors publishing in this field, we limit our interest to a selected part of this broad research area, namely, the features of the hematite/ electrolyte interface. * To whom correspondence should be addressed.
A variety of methods has been used to study the adsorption system, such as IR and others methods.1 However, as the change in ion concentration (adsorption) at this interface takes place, which is essential for life and technological processes, direct measurements of ion adsorption are of great significance. Moreover, although various experimental techniques provide us with very valuable information about the features of this interface, one experimental technique, calorimetry, is of exceptional importance. Calorimetric measurements provide information about the change, in the course of adsorption, of one of the fundamental thermodynamic functionssthe enthalpy. It has been known for a long time that the enthalpic effects of adsorption are much more sensitive to the nature of an adsorption system than other adsorption quantitiessthe adsorption isotherms, for instance.2 This can be easily explained on the grounds of statistical thermodynamics; namely, enthalpy is related to the temperature derivatives of the system partition functions, and temperature appears in the exponential terms of the related theoretical expressions. It should also be emphasized that the information obtained by using various techniques is accompanied by certain assumptions, such as the assumption about the shape of particles in electrokinetic studies.3 Meanwhile, calorimetric measurements directly yield the values that are theoretically calculated in appropriate statisticals thermodynamic considerations.
10.1021/jp001311z CCC: $19.00 © 2000 American Chemical Society Published on Web 11/18/2000
Ion Adsorption at a Homogeneous Oxide Surface The enthalpic effects of adsorption can either be deduced from the temperature dependence of isotherms of adsorption or directly measured in the appropriate calorimetric experiments. In the case of ion adsorption at the oxide/electrolyte interfaces, the first temperature studies of these enthalpic effects were reported by Berube and Bruyn as long as 30 years ago.4 They studied the temperature dependence of PZC to obtain information about the enthalpic effects of proton adsorption on the negatively charged surface oxygens. Since that time, several papers have reported on this kind of experiment.4-26 At the same time, the temperature dependence of the whole titration curve has rarely been reported.17,21,22,27 Studies of temperature dependence are a relatively simple experiment, but the theoretical interpretation is difficult. This is because changing temperature induces several surface complexation reactions to occur simultaneously. It was only recently that Rudzinski et al.28-30 published a quantitative analysis of the temperature dependence of ion adsorption. As for the direct calorimetric measurement of the enthalpic effects accompanying ion adsorption at the oxide/electrolyte interface, the first experiment of that kind was reported by Griffiths and Fuerstenau31 as early as in 1981. However, it has only been during the past decades that a substantial body of such data has been reported that is suitable for an advanced theoretical analysis.32-41 Although semiquantitative analyses by Machesky and co-workers,20,23,34-36 Kallay and coworkers,2,24,39-41 and Kosmulski19 have provided very valuable information, only recently has a fully quantitative analysis been attempted by De Kaizer et al.,33 and Rudzinski and co-workers. Such a theoretical quantitative analysis was presented in our series of recent publications.42-51 Here for the first time, both the direct calorimetric measurements of these enthalpic effects (titration calorimetry data) and the monitored temperature dependence of PZC are subjected to a simultaneous quantitative analysis within the frames of a certain theoretical model. Then, a number of such models and the related theoretical approaches are used for comparison. Such combined experimentalstheoretical studies bring us to a much better level of understanding the nature of these adsorption systems. Although in our previous publications we have dealt mainly with developing still more accurate equations corresponding to more refined adsorption models, this time we are going to show the applicability of our new equations to a very important example of the oxide/electrolyte interfaces. Experimental Section In this paper, the theoretical analysis of experimental results obtained with hematite (Alfa Johnson Mattey GmbH, Karlsrue, Germany) are reported. Hematite, with a specific surface area of 8.8 m2/g, was extensively purified. To avoid CO2 contamination, the experiments were performed under an argon atmosphere. The data, reported previously,52 and were subjected to a simplified analysis that neglected the counterion association. The calorimentric titration of hematite suspension was described in detail elsewhere.52 To avoid coagulation and to minimize the heat of neutralization, the slightly acidic suspension was titrated with acid (HNO3). Also, the slightly basic suspension was titrated by base (KOH). The data are reported in Tables 1 and 2. Surface charge was determined by the simultaneous potentiometric titration of the hematite suspension. The point of zero charge was measured by the mass titration method. According to this method, the pH of a concentrated suspension corresponds to the point of zero charge.53,54
J. Phys. Chem. B, Vol. 104, No. 50, 2000 11913 TABLE 1: Results of Calorimetric and Potentiometric Titrations with Base, Obtained in Three Sets of Experimentsa δ0(pH1) [C/m2]
δ0(pH2) [C/m2]
pH1
pH2
6.66 7.42 8.08 8.83 9.58 10.02 10.24 10.42
7.42 8.08 8.83 9.58 10.02 10.24 10.42 10.58
Experiment No. 1 -0.013095 -0.038205 -0.038205 -0.063265 -0.063265 -0.096255 -0.096255 -0.132408 -0.132408 -0.153171 -0.153171 -0.162788 -0.162788 -0.170027 -0.170027 -0.175836
6.76 7.32 8.14 8.98 9.65 10.02 10.21 10.38
7.32 8.14 8.98 9.65 10.02 10.21 10.38 10.51
Experiment No. 2 -0.016384 -0.034580 -0.034580 -0.065896 -0.065896 -0.103589 -0.103589 -0.153306 -0.153306 -0.161564 -0.161564 -0.168438 -0.168438 -0.173412 -0.173412 -0.176782
9.74 9.95 10.08 10.18 10.27 10.35 10.41 10.46 10.51
9.95 10.08 10.18 10.27 10.35 10.41 10.46 10.51 10.56
Experiment No. 3 -0.139987 -0.150073 -0.150073 -0.155967 -0.155967 -0.160184 -0.160184 -0.163971 -0.163971 -0.167137 -0.167137 -0.169462 -0.169462 -0.171455 -0.171455 -0.173442 -0.173442 -0.175001
Qs [J/sample]
Q ∆pr [kJ/mol of adsorbed protons]
0.174924 0.192252 0.276297 0.294351 0.162328 0.068572 0.048892
42.5 46.6 51.2 49.9 47.7 42.8 40.7
0.103624 0.231051 0.300347 0.260846 0.130817 0.051442 0.043225 0.028684
34.6 45.1 48.4 49.2 45.1 39.6 36.0 35.8
0.078598 0.037339 0.022939 0.020700 0.017652 0.011295 0.009540
31.0 23.6 20.7 21.8 22.3 17.8 20.1
a pH1 and pH2 are the initial and final pH values, whereas δ0(pH1) and δ0(pH2) are the corresponding surface charges. Further Qs is the heat evolved during the titration step from pH1 to pH2. The last column collects the calculated Q ∆pr values.
TABLE 2: Results of the Calorimetric and Potentiometric Titrations with Acid, Obtained in the Three Sets of Experimentsa Q ∆pr [kJ/mol of adsorbed protons]
pH1
pH2
δ0(pH1) [C/m2]
5.34 4.44 3.88 3.53 3.31 3.16 3.04 2.94
4.44 3.88 3.53 3.31 3.16 3.04 2.94 2.86
0.027849 0.060932 0.084850 0.101022 0.111332 0.118931 0.124725 0.129291
Experiment No. 4 0.060932 -0.039841 0.084850 -0.032091 0.101022 -0.028036 0.111332 -0.018653 0.118931 -0.015672 0.124725 -0.009601 0.129291 -0.011663 0.133395
5.42 3.74 3.43 3.24 3.10 2.99
3.74 3.43 3.24 3.10 2.99 2.90
0.025328 0.091325 0.105753 0.114868 0.121855 0.126929
Experiment No. 5 0.091325 -0.045618 0.105753 -0.035049 0.114868 -0.030363 0.121855 -0.025505 0.126929 -0.021972 0.131298 -0.022262
4.2 14.8 20.2 22.4 26.2 31.4
5.44 4.54 3.98 3.63 3.41 3.26 3.14
4.54 3.98 3.63 3.41 3.26 3.14 3.04
0.024556 0.056945 0.080382 0.096311 0.106517 0.114072 0.119853
Experiment No. 6 0.056945 -0.038822 0.080382 -0.028449 0.096311 -0.022083 0.106517 -0.024013 0.114072 -0.018699 0.119853 -0.015244 0.124423 -0.014791
7.3 7.4 8.4 14.3 15.1 16.0 20.0
a
δ0(pH2) [C/m2]
Qs [J/sample]
The meaning of the symbols is the same as in Table 1.
7.3 8.2 10.5 11.0 12.6 10.1 15.6
11914 J. Phys. Chem. B, Vol. 104, No. 50, 2000
Rudzin´ski et al.
Theory Principles of the Adsorption Model. The equations used in our present publication have already been developed in our earlier publications.43,44,47 We believe, however, that repeating certain principles and final expressions will be appreciated by readers for the purpose of convenience. We consider the adsorption of ions as a result of the following surface reactions suggested by Davis, Leckie, and other authors:55-59
SOH+ 2
K int a1
798 SOH + H 0
+
(1a)
K int a2
SOH0 798 SO- + H+
(1b)
*K int A
0 + SOH+ 2 A 798 SOH + H + A
(1c)
*K int C
SOH0 + C+ 798 SO-C+ + H+
(1d)
where SO- denotes the outermost surface oxygens, H+ is the proton, A- and C+ denote the anion and cation, respectively. int int int Then K int a1 , K a2 , *K A , and K C are the equilibrium constants for the reactions (1a-d). Introducing the notation
In eq 6 c1 is the first integral capacitance, δ0 is the monitored surface charge, defined as
δ0 ) Bs[θ+ + θA - θ- - θC], Bs ) eNs
and Ns is the surface density (sites/m2). Taking into account eq 6, the nonlinear equation system 3 can be transformed into the following nonlinear equation with respect to δ0:
K + f+ + K A fA - K C fC - 1
δ0 ) B
1+
θ- ) [SO-]/N ˜s ) 1 -
∑i θi (i ) 0, +, A, C)
2.303(PZC-pH) ) where β is given by
β)
Ki fi 1+
, i ) 0, +, A, C
(3)
∑i Ki fi
( )
2e2Ns K int a2 cDLkT K int a1
1/2
(10)
where r is the relative permittivity of solvent, 0 is the permittivity of free space, and I is the ionic strength of solution (ions/m3). The value of cStern is assumed to be 0.2 F/m2. At the point of zero charge (PZC), from eq 8 we have int 1 + (*K int 1 C /K a2 )aC 1 int int (12) PZC ) (pK a1 + pK a2 ) - log 2 2 1 + (K int/*K int)a a1
K+ )
1
KC )
int K int a1 K a2
*K int C int K a2
KA )
1 int K int a2 *K A
(4)
{
f0 ) exp -
}
eψ0 - 2.3pH , f+ ) f 20 kT
{
fC ) aC exp -
{
(11)
r 0
where
1 K0 ) int K a2
(9)
1 2kT/e 1 ) + cDL (8 kTI)1/2 cStern
[SO-C+] + [SOH+ 2 A ] (2)
θi )
( )
eψ0 eψ0 + sinh-1 kT βkT
In eq 10, cDL is the linearized double-layer capacitance, which we will theoretically calculate as described in Bousse’s work:61
N ˜ s ) [SO-] + [SOH0] + [SOH+ 2] +
one arrives at the following set of Langmuir-like equations:
∑i
i ) 0, +, A, C (8)
K i fi
which can be easily solved by means of an iteration method to give the value of δ0 for each pH value. Having calculated these values, one can easily evaluate the individual adsorption isotherms θi from eq 3. As in our previous works, to express the ψ0(pH) dependence, we accept the relation used by Yates et al.60 and by Bousse et al.:61,62
˜ s, θ+ ) [SOH+ ˜ s, θA ) [SOH+ ˜s θ0 ) [SOH0]/N 2 ]/N 2 A ]/N
θC ) [SO-C+]/N ˜ s,
(7)
}
eδ0 eψ0 + kT kTc1
eδ0 eψ0 - 4.6pH fA ) aA exp kT kTc1
}
where ψ0 means the surface potential whereas ψβ is the mean potential at the plane of the specifically adsorbed counterions
ψ β ) ψ0 -
δ0 c1
int
δ0(pH)PZC) )
*K C a H2 H2a + int int - 1 ) 0, int int K a1 K a2 K a2 *K A K int a2 H ) 10-PZC (13a)
or
K+H2 + KAH2a - KCa - 1 ) 0 (5d)
(6)
A
For pH ) PZC, δ0 ) 0, and ψ0 ) 0, and eq 8 can be transformed then into the following form:
(5a,b)
(5c)
A
(13b)
Looking into the reported experimental data, one can see that in the majority of the investigated systems, the value of the point of zero charge does not practically depend on the salt concentration in the equilibrium bulk solution, so all the surface charge (titration) curves have a common intersection point (CIP) at δ0 ) 0. This is also the case in the data reported here. In a series of previous publications,42-51 we drew readers’ attention to the fact that drawing formal mathematical conse-
Ion Adsorption at a Homogeneous Oxide Surface
J. Phys. Chem. B, Vol. 104, No. 50, 2000 11915
quences of the existence of a CIP leads one to establish relations between the equilibrium constants. Thus, the independence of PZC of the salt concentration can be formally expressed as follows:
( ) ∂δ0 ∂a
(pH)PZC)
)0
(14)
Solving the set of eqs (13,14) we obtain
K int a2 )
H2 K int a1
*K int A )
H2 *K int C
(15)
int int Relations (15) between the parameters K int a1 , K a2 , *K A , and int *K C reduce from four to two the number of parameterssthe intrinsic equilibrium constants, to be found by fitting the experimental data. This is also true in the systems where PZC and CIP do not coincide. Only the form of the two interrelations is different from those in eq 15.63,64 While accepting relations (15), we arrive at the following expressions:
1 int PZC ) (pK int a1 + pK a2 ) 2
(16a)
1 int PZC ) (p*K int C + p*K A ) 2
(16b)
However, for the particular TLM model considered here, the relation (16a) can be obtained in another way. Here, the prerequisite is that the PZC and IEP (isoelectric point) values are the same or very close to each other. Let us remark that the Langmuir-like isotherm can be rewritten to the following form: int 4.6pH ) -2.3 log(K int a1 K a2 ) -
θ+ 2eψ0 - ln kT θ-
(17)
At pH ) PZC, ψ0 ) 0, and θ+ ) θ-, so, from eq 17, we arrive again at eq 16a. From eqs 16a,b we have
∂PZC Qa1 + Qa2 ) 2.3 2k ∂(1/T) ∂PZC QaA - QaC ) 2.3 2k ∂(1/T)
+
- +
SO + C T SO C -
+
-
SO + 2H + A T Qai ) -k
SOH+ 2A
d ln K aiint , i ) 1,2 d(1/T)
QaC ) -k QaA ) -k
int d ln(K int a2 /*K C )
d(1/T) int d ln(K int a2 ‚*K A )
d(1/T)
reaction type
equilib constant
heat of reaction
+ H T SOH+ 2 SO- + H+ T SOH0
-pK int a1 -pK int a2 -p*K int A -p*K int C int -pK int a1 - pK a2 int -pK int p*K a2 A int int p*K C - pK a2 int pK int a1 - p*K A
Qa1 Qa2 QaA - Qa2 Qa2 - QaC Qa1 + Qa2 QaA QaC QaA - Qa1 - Qa2
SOH0
+
SOH0 + H+ + A- T SOH+ 2A SO-C+ + H+ T SOH0 + C+
(18a)
-
+ SOH+ 2 + A T SOH2 A
QaA - (Qa1 + Qa2) ) QaC
QaA - (Qa1 + Qa2) ) 0, and QaC ) 0
(20b) (20c)
For the reader’s convenience, Table 3 collects the considered surface reactions, their equilibrium constants, and the related nonconfigurational heats of adsorption.
(22)
This hypothesis was used by Rudzin´ski and co-workers in their quantitative interpretation of calorimetric effects of ion adsorption, measured directly in various types of experiments. Direct Calorimetric Measurements and Their Interpretation. In 1986, Machesky and Anderson34 and then other authors33,35-41 measured the heat evolved while changing the pH from pH1 to pH2 and ascribed it solely to proton adsorption (desorption) accompanying that change of pH. Therefore, as the final output of their experiment, they considered the socalled “heat of proton adsorption” Q ∆pr
( ) [( ) ( ) ( )] ∫pHpH ∑Qi 2
Q ∆pr
) 2
∂θ+
∂pH
i
+2
2
T
∂θi
dpH
∂pH
∂θA
∂pH
T
+
T
∂θ0
dpH ∂pH T i ) 0, +, C, A (23)
For the model of a homogeneous solid (oxide) surface considered here in the first part of this publication, we have43
Q0 ) Qa2 - eψ0 -
(19a)
(20a)
(21)
which is obtained from eqs 18a,b. This relation suggests that the heat effect related to the cation adsorption on an “empty” oxygen atom, QaC, is the same as the heat of the attachment of anion to the already existing complex SOH+ 2 . As it is true for so many oxide/electrolyte systems, Rudzin´ski et al.50,51 have launched the hypothesis that they are the same because they are simply nonexistent:
1
(19b)
-
Recently, Rudzin´ski and co-workers have drawn attention to the meaning of the relation
∫pHpH (18b)
-
SO- + 2H+ T SOH+ 2 SO- + 2H+ + A- T SOH+ 2A SO- + C+ T SO-C+
1
where Qa1, Qa2, QaC, QaA are the nonconfigurational heats accompanying the surface reactions (1ab) and the following reactions: -
TABLE 3: Surface Reactions, Equilibrium Constants, and Heats of Reactions
( ) ( )
e ∂ψ0 T ∂(1/T)
Q+ ) Qa1 + Qa2 - 2eψ0 -
( ) ( ) ( ) ( )
(24a)
{θi},pH
2e ∂ψ0 T ∂(1/T)
{θi},pH
e ∂ψ0 + T ∂(1/T) {θi},pH δ0 eδ0T ∂c1 ∂ ln aC e + +k 2 c1 (c ) ∂T {θi},pH ∂(1/T) 1
pH
e ∂ψ0 T ∂(1/T) {θi},pH δ0 eδ0T ∂c1 ∂ ln aA e + k c1 (c )2 ∂T {θi},pH ∂(1/T) 1
pH
QC ) QaC - eψ0 -
QA ) QaA - eψ0 -
( )
( )
(24b)
(24c)
(24d)
11916 J. Phys. Chem. B, Vol. 104, No. 50, 2000
Rudzin´ski et al.
Following Blesa’s recommendation,22 we treated c1 as the linear function 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 for R R,0 c1 around T ) T0, so that cL1 (T0) ) cL,0 1 and c1 (T0) ) c1 . 44 We have shown in our previous publication that the derivatives (∂ln ai/∂(1/T))θi i ) C, A occurring in eqs 24 can be expressed as follows:
∂ ln ai ∂(1/T)
) -2.0172 × 10-8 T 4ln γi
i ) A, C (26)
The derivatives (∂ψ0/∂(1/T))pH occurring in eqs 24 are evaluated from eqs 9:
( )
[
]
t Qa2 - Qa1 ) -ψ0T + + β + t β + t 2kT ∂(1/T) pH kT β ∂PZC 2,3 (27) e β + t ∂(1/T) ∂ψ0
β
where
t)
[( ) ] eψ0 2 +1 βkT
-1/2
(27a)
Finally, the derivatives (∂θi/∂pH) in eq 23 are evaluated from the equation system 3. Their explicit form can be found in our earlier publications.49 From eq 23, it follows that Qpr must be dependent on pH, and it was observed in some of the reported experiments. Meanwhile, De Kaizer et al.,33 who measured the cumulative heat of adsorption Qcum(δ0)
Qcum(δ0) )
∫δδ
0,2
0,1
(Qpr(δ0))T dδ0
(28)
reported that Qcum(δ0) is a linear function of δ0 for both rutile and hematite. This would mean that Qpr does not depend on δ0. In contrast to the results reported by De Kaizer et al.33 Casey38 reported recently that in the case of silica, the “heat of proton adsorption” Qpr varies with the varying charge of the surface δ0. In addition to the “batch” calorimetric experiment first applied by Griffiths and Fuerstenau31 and the titration calorimetry developed by Machesky and Anderson,34 still another technique has been recently applied by Rudzinski et al.42 This is the flow calorimetry technique. Results and Discussion Experimental Results. The results of our calorimetric and accompanying potentiometric titrations are presented in Tables 1 and 2. The initial and final pH, together with the associated exothermic heat, is given for each addition of titrant. Six experimental runs are presented: three runs were performed by the addition of acid to the slightly acidic suspension, and in the other three, the base was added to a slightly basic suspension. In doing so, the extent of neutralization was minimized, so that no significant error due to subtraction of neutralization heat was introduced. For the temperature 25 °C, at which the titration experiments were carried out, i.e.p. ) PZC ) 6.2.
Figure 1. The pH dependence of our experimental Qs, (( ( () and Qpr, (f f f) values. Every Qs and corresponding Q ∆pr value have been assigned to pH ) (pH1 + pH2)/2.
Before we start our quantitative analysis of the experimental data in the next section, certain comments will help the readers to understand their interpretation and make future use of our experimental data possible. So, let us consider that our directly measured data Qs are related to the numerator on the right hand side of eq 23:
∫pH ∑Qi i pH2 1
( )
NA × 10-3
∂θi
∂pH
T
dpH ) Qs mSNs × 1018
(29a)
where Qs is expressed in J/sample, NA is Avogadro’s number, S is the surface area of our sample, equal to 8.8m2/g, m ) 1.8 g is the mass of oxide sample used in our experiments, and Ns is the number of surface oxygens per square nanometer. While expressed in kJ/mol of adsorbed protons, Qpr in eq 23, is given by
Qpr )
FQs × 10-3 mS∆δ0
(29b)
where F is Faraday’s constant, equal to 96 500 C/mol, and ∆δ0 is the change (δ0(pH2) - δ0(pH1)) expressed in C/m2. Our experimental data are presented in Figures 1 and 2. Our study of the calorimetric effects accompanying ion adsorption at the hematite/ electrolyte (KNO3) interface is the second extensive study reported so far in the literature. Calorimetric effects and the temperature dependence of ion adsorption at the hematite/KNO3 interface were a decade ago the subject of Fokkink’s Ph.D. thesis in Prof. Lyklema’s laboratory. These results were next reported and discussed in two papers published by Fokkink et al.21 and De Kaizer et al.33 In addition to temperature dependence of the surface-charge isotherms δ0(pH), some limited data were also published by De Kaizer et al., obtained using the calorimetry titration technique. Our present experimental study was oriented toward more extensive studies of the enthalpic effects measured in that way. Despite the limited calorimetric data reported by Lyklema and co-workers, it should be interesting to compare their and our results to see to what extent sample preparation may affect their features.
Ion Adsorption at a Homogeneous Oxide Surface
J. Phys. Chem. B, Vol. 104, No. 50, 2000 11917
Figure 2. The data for the charge titration isotherm δ0(pH) obtained in the six experiments. The black circles (b b b) are the data obtained in experiments no. 1 and no. 4, the black squares (9 9 9) are the data obtained in experiments no. 2 and no. 5, and the black triangles (2 2 2) are the data from experiments no. 3 and no. 6.
Figure 4. The cumulative heats of adsorption Qcum reported by De Kaizer et al. (b b b), their spline approximation (s), and the Qpr values (f f f) calculated by differentiation of that spline function. The (b b b) values were found by digitizing the data in Figure 3b in the paper by De Kaizer et al.33
Figure 3. (A) The temperature dependence of PZC observed in the data reported by Fokkink et al. (+++), and in the data obtained by us (4 4 4). The solid line is the same spline approximation for both sets of the data. (B) The Qa1 + Qa2 values calculated according to eq 18a from the derivative ∂PZC/∂(1/T) of the spline function approximating the data of Fokikink et al. (+++), and the PZC(T) data measured by us (4 4 4). Fokkink’s data have been digitized by us from Figure 6 in the paper by Fokkink et al.21
Figure 5. The fit by eq 29a of our experimental Qs data, found in the six calorimetric titrations, using the parameters collected in Table 4. The black rhombuses (( ( () are the experimental data, whereas white rhombuses () ) )) are the theoretically calculated points.
Analysis of Experimental Data Based on the Model of an Energetically Homogeneous Solid Surface. We begin our discussion by drawing the reader’s attention to the fact that, despite differences in sample preparation, the temperature dependences of PZC reported by Fokkink et al.21 and found in our experiments seem to have certain features in common. This is shown in Figure 3. Because the two sets of the PZC data in Figure 3 can be approximated well by the same spline function, we deduced that their temperature derivatives must be similar. Thus, Part B of Figure 3 shows that indeed, the temperature dependence of (Qa1 + Qa2) of both Fokkink’s and our data is similar. Fokkink’s data (Qa1 + Qa2) are greater only by a factor of 2.5. Now, let us focus our attention on the data obtained by De Kaizer et al. using titration calorimetry. Of the two sets of calorimetric data reported by De Kaizer et al. for KNO3 concentrations, 0.2 M and 0.02 M, only the data set corresponding to 0.02 M can, eventually, be subjected to a more detailed analysis. This is because these are 10 points, exhibiting a subtle, but clearly seen, regular nonlinear shape. They are shown in Figure 4.
De Kaizer et al. approximated their Qcum data by a straight line, arriving in that way to the constant Qpr value equal to 36 kJ/mol. Our closer inspection into the subtle nonlinear behavior of Qcum yields Qpr values ranging from 31 to 44 kJ/mol. It should be emphasized that, despite a certain experimental scatter seen in the De Kaizer’s data, a regular nonlinear subtle shape can be easily observed. Now let us turn our attention to Figure 1. From the point of our forthcoming quantitative analysis of the Qs and Qpr data, a certain feature of the Qpr data has to be taken into consideration. This is large scatter in the Qpr data calculated from the data corresponding to acidic titrations, that is, to pH < PZC. Although similar in trend, the Qpr data in parts A, B, and C of Figure 1 show quite different values. The difference between Qpr values in Parts B and C is by the factor two. This can easily be understood by looking into eq 29a. The possible experimental errors in Qs and ∆δ0, can be multiplied determining Qpr. Figures 5 and 6 show the computer best-fit of the measured Qs and Q ∆pr values. Table 4 collects the values of the parameters found in that way. Our numerical fits of Qs values were not very sensitive to the choice of Ns. Taking one or another value of Ns suggested in the literature for this adsorption system will lead to a very similar fit and to the estimated values of parameters.
11918 J. Phys. Chem. B, Vol. 104, No. 50, 2000
Rudzin´ski et al.
Figure 6. The fit by eq 29b of the experimental Qpr data, deduced from the six calorimetric titrations, using the same sets of parameters as in Figure 5. The black squares (9 9 9) are the experimental data, the white squares (0 0 0) are the discrete theoretical Q ∆pr values, and the solid line is the Qpr function (30).
TABLE 4: Parameters Used to Fit the Experimental Qs and Q ∆pr Data in Figures 5 and 6: The Ns Value Was Taken as Equal to 8 sites/nm2, and PZC ) 6.23 Best-Fit Parameters pK int a1
p*K int C
Qa1
Qa2
RL1
RR1
cL1
cR1
4.00
5.30
-2.0
57.3
-0.008
-0.006
1.0
1.0
Calculated Parameters pK int a2
p*K int A
QaA
8.46
7.16
55.3
So far, only a few papers have been published reporting experimental Qpr values. These were the papers by Machesky and Jacobs35,36 investigating the alumina/NaCl interface, the paper by Mehr on the TiO2/NaCl interface,37 and the paper by Casey38 reporting on Qpr values for the silica/KCl and silica/ NaCl interfaces. These data have been subjected by us to a quantitative analysis, presented in our previous publications.30,49,51 There, the experimental Qpr values representing molar differential heats of proton adsorption, averaged between pH1 and pH2, were fitted by the function Qpr:
( ) ( ) ( ) ( ) ∑i
Qpr )
∂θ+
+2
2
∂pH
∂θi
Qi
T
∂pH
∂θA
∂pH
T
+
T
(30)
∂θ0
∂pH
T
Of course, Q ∆pr f Qpr as (pH2 - pH1) f 0. To see the difference between Qpr and Q ∆pr due to real experimental conditions, that is, to finite intervals (pH1 - pH2), we have drawn in Figure 6 both Q ∆pr (discrete) values and the corresponding Qpr values. One can see that the functions Qpr and Q ∆pr are practically the same. So, in our forthcoming analysis of the experimental Qpr values, we will fit them by the continuous Qpr function, as in our previous publications.30,49,51 The values of parameters collected in Table 4 also yield an excellent fit of the titration isotherm data shown in Figure 2. We do not show this best-fit for the reason already discussed
in our earlier publications,49 namely, that it is well-known that the experimental titration isotherms δ0(pH) can be fitted almost equally well by a variety of equations corresponding to various adsorption models. Even within the frame of the same theoretical approach, one can find many different sets of parameters leading to an equally good fit of titration isotherm. So, titration isotherms alone can hardly be used to determine in a reliable way the values of the adsorption parameters occurring in the applied theoretical approach. The situation changes when one has titration isotherms measured at various temperatures. The description of the temperature dependence of δ0(pH) involves considering the same functions and quantities that appear in the theoretical description of enthalpic effects.28-30 However, titration isotherms measured even at only one temperature can still be applied as a useful control of the reliability of the parameters giving fit to the experimental Qs and Qpr values. It appears that not all the sets of parameters giving good fits of Qs and Qpr also yield good fits of titration isotherms. So, a simultaneous fit of Qpr and δ0(pH) data leads to a strong elimination of false sets of parameters. Of course, every best-fit is accompanied by a certain margin of uncertainty as to the exact values of the determined parameters, and the only question is how large or narrow that margin is. To check it, we carried out numerous model calculations. Thus, we changed values of one of the best-fit parameters, while keeping the others constant, to see how its change affects the fit of experimental data. Such numerous computer exercises showed that the margin of uncertainty in our case was fairly narrow. Therefore, both a success and a failure of the obtained best fits has to be ascribed either to the goodness (accuracy) of experimental data or to the features of the applied theoretical approach or to both. Looking at Figures 5 and 6, one can see an important advantage of fitting Qs and Qpr data simultaneously. Figure 5 shows a good agreement for all the base titrations (Experiments 4, 5, and 6); therefore, we are confident of the Qs values measured at pH > PZC. At the same time, such a good agreement in this pH region between the experimental and theoretical Qpr values can only be observed in Parts A and B of Figure 6. This must raise our doubts as to the accuracy (reliability) of the basic part of the titration data δ0(pH) found in experiment no. 6. For the same reason, we have doubts as to the reliability (accuracy) of the δ0(pH) data, found in experiment no. 2. The experimental Q ∆pr values in Figure 6B, corresponding to the acidic titrations, show quite different behavior from those in Parts A and C of Figure 6. Therefore, while investigating the behavior of the Qpr function, we will consider only the experimental data, obtained in experiments No. 1, 3, 4, and 5. Meanwhile, let us consider the values of the parameters collected in Table 4. Perhaps the most striking data in Table 4 int are the values of Qa1 and Qa2. While pKint a1 and pKa2 values still fulfill eq 16a, the obtained Qa1 and Qa2 values do not fit the corresponding eq 18a. The analysis of the temperature dependence of PZC treated as a linear function of 1/T suggests that (Qa1 + Qa2) is 42.5 kJ/mol52 or is equal to 40-41 kJ/mol at a temperature 25 °C, as suggested by Figure 3. Meanwhile, our numerical exercises clearly showed that we could never arrive at the fit of the experimental Qs and Qpr values using (Qa1 + Qa2) < 55 kJ/mol. However, we would like to mention that the value QaA was still calculated from eq 21, following the assumption that QaC ) 0. That means we assumed that the Coulombic adsorption of an anion on the complex SOH+ 2 and of a cation on SO- are not accompanied by enthalpic effects, so these adsorption processes are purely entropy driven.
Ion Adsorption at a Homogeneous Oxide Surface
J. Phys. Chem. B, Vol. 104, No. 50, 2000 11919
[
]
(32a)
[
]
(32b)
PZC(T2) ) PZC(T1) +
Qa1 + Qa2 1 1 4.6k T2 T1
PZC(T2) ) PZC(T1) +
QaA - QaC 1 1 4.6k T2 T1
or
Figure 7. The surface charge isotherms δ0(pH), reported by Fokkink et al.21 for the four KNO3 concentrations and two temperatures. (The data was digitalized from Figure 5 of the paper by Fokkink et al.).
At the same time, still intriguing is the difference between (Qa1 + Qa2) values determined from the temperature dependence of PZC and from fitting the directly measured heats of ion adsorption Qs and Qpr. So, to get more information concerning that problem, we decided to subject Fokkink’s data, as well, to a quantitative analysis. Figure 7A shows the titration isotherms measured by Fokkink et al.21 at the two temperatures: 5 °C and 60 °C. When analyzing their δ0(pH) isotherms measured at four temperatures, Fokkink et al. launched a “temperature congruence” hypothesis. It says that changing temperature mainly shifts δ0(pH) isotherms on the pH scale, without changing essentially the shape of the δ0(pH) functions. They further argued that studying the shift of pH(δ0) with T for δ0 * 0 may lead to a more accurate estimation of the temperature dependence of PZC than the traditionally studied shift of actual PZC with T. They called this procedure the “∆T titration”. While using such a procedure, they arrived at the value (Qa1 + Qa2) ) 72.6 kJ/mol, instead of the value 98 kJ/mol found by us from the analysis of the temperature dependence of their true PZC data. (See Figure 3). Trying to extract more information for comparison, we subjected the temperature dependence of Fokkink’s titration isotherms to quantitative analysis. To calculate δ0(pH) at different temperatures, one must have the following parameters calculated for different temperatures:
[ [
] ]
int pKint a1 (T2) ) pKa1 (T1) +
Qa1 1 1 2.3k T2 T1
(31a)
int pKint a2 (T2) ) pKa2 (T1) +
Qa2 1 1 2.3k T2 T1
(31b)
[ [
] ]
int p*K int C (T2) ) p*K C (T1) +
Qa2 - QaC 1 1 (31c) 2.3k T2 T1
int p*K int A (T2) ) p*K A (T1) +
QaA - Qa2 1 1 (31d) 2.3k T2 T1
While analyzing Blesa’s data in one of our previous publications,28 we also accepted the following relations:
Blesa et al.22 were the first to analyze quantitatively the temperature dependence of the whole titration isotherms, measured at various electrolyte concentrations. They studied the magnetite/KNO3 interface. Blesa’s investigation strategy was int int int to determine the values of K int a1 , K a2 , *K C , and *K A for every temperature by using a popular graphical extrapolation method. They also assumed that the capacitance parameters cL1 , cR1 , RL1 , and RR1 do not depend on the inert electrolyte concentration. int int int The determined parameters K int a1 , K a2 , *K C , and *K A satisfied the relation 16a,b, and the theoretical δ0(pH) curves fitted the experimental titration data measured at the highest concentration of the inert KNO3 electrolyte, 10-1 M. The least successful was the fit in the case of the smallest concentration of the inert electrolyte, 10-3 M. When commenting on the discrepancies between theory and experiment, Blesa et al. ascribed them to the existence of sites of different reactivities (energetic heterogeneity of surface oxygens). Meanwhile, our earlier theoretical works suggested that, in addition to the surface heterogeneity effects, there is still another physical factor that may influence the behavior of these adsorption systems, namely, that the electric capacitances may by affected by the concentration of the inert electrolyte.43,44 Although this effect does not result conceptually from the hitherto theoretical treatments of the triple-layer model, we decided to include it into our analysis by assuming that the (bestfit) parameters cL1 , cR1 , RL1 , and RR1 may be different at various electrolyte concentrations. Next, we decided to check how considering this effect only may improve the agreement between theory and experiment, without taking into account the energetic heterogeneity of surface oxygens. The result of that study has been published in one of our recent papers.28 To better fit Blesa’s experimental titration curves, we were forced to assume that the parameters cL1 , cR1 , RL1 , and RR1 are also functions of the concentration of the inert electrolyte KNO3. We expect to face a similar situation in the case of the hematite/KNO3 system and in the related titration data reported by Fokkink et al. To make our analysis more transparent, we decided to analyze the temperature dependence of only one set of Fokkink’s titration data, measured at the electrolyte concentration 0.02 M. The choice of that concentration was caused by the fact that only for this particular electrolyte concentration did Fokkink et al. report the 10 data points obtained in their calorimetric titration, shown in Figure 4 and analyzed by us. (Only five points were reported for the other concentration, 0.2 M). So, we will subject to a simultaneous analysis the following three types of Fokkink’s experimental data: (1) the temperature dependence of PZC, shown and analyzed in Figure 3, (2) the calorimetric titration data at 0.02 M, shown and analyzed in Figure 4, (3) the three titration isotherms, measured at one electrolyte concentration, 0.02 M, and at the three temperatures, 5 °C, 20 °C, and 60 °C. Figure 8 shows our best-fits of Fokkink’s titration isotherms, measured at three temperatures but at the same electrolyte concentration, 0.02 M. The strategy of our best-fit calculations
11920 J. Phys. Chem. B, Vol. 104, No. 50, 2000
Rudzin´ski et al. The relations 31 and 32 should also be expressed as appropriate integrals. For instance,
Figure 8. Theoretical best-fit of the Fokkink’s titration isotherms measured at three temperatures: 5 °C (( ( (), 20 °C (b b b), and 60 °C (2 2 2), and at the KNO3 concentration 0.02 M. The solid lines (s), representing the theoretical best-fit, were calculated using the parameters collected in Table 5.
TABLE 5: List of Parameters Used to Fit the Three Fokkink’s Titration Isotherms Shown in Figure 8; a Brief Description of Our Best-Fit Calculations is also Given T °Cd
5 20 °Cd,e 60 °C f
a pK int a1
a p*K int C
cL1 a
cR1 a
b pK int a2
b p*K int A
PZCc
7.2 6.42 5.39
9.95 9.15 8.08
0.96 0.81 0.87
1.1 1.12 1.08
11.8 11.0 9.93
9.05 8.27 7.24
9.5 8.71 7.66
a At 5 °C, the best fit calculations yield these values. b Parameters calculated from eqs 16a,b. c From experiment. d To fit simultaneously the titration isotherms at 5 °C and 20 °C, the following best fit parameters had to be accepted: Qa1 ) 81 kJ/mol, Qa2 ) 83 kJ/mol, RL1 ) -0.01, RR1 ) 0.001. It was assumed that QaC ) 0. So QaA ) (Qa1 + Qa2). e Values of parameters given for the temperature 20 °C are calculated from the above listed parameter values using eqs 16a,b, 25a,b, and 31a,b. f While accepting these parameters, the following other bestfit parameters had to be accepted to arrive at a good fit of the titration isotherm at 60 °C: Qa1 ) 48 kJ/mol, Qa2 ) 50 kJ/mol, RL1 ) 0.0015, RR1 ) -0.001. As usual, we assumed that QaC ) 0, so QaA ) (Qa1 + Qa2). In this way, parameters were found for the temperature 60 °C.
int was the following. First we assumed certain pKint a1 and p*K C int values for the temperature 5 °C, and the other ones, pK a2 and p*Kint A , were calculated from the relations 16ab. Also, we assumed certain cL1 and cR1 values to fit the titration isotherm at the lowest temperature, 5 °C. Next, we checked whether we may find a set of Qa1, Qa2, RL1 , and RR1 values that could lead us, using relations 31 and 32 to parameters, yielding a good fit of the titration isotherm measured at the higher temperature 20 int °C. When we failed, we looked for another set of pK int a1 , p*K C , L R c1 , and c1 parameters until we found the set leading to a successful fit of both titration isotherms, measured at 5 °C and 20 °C. Having achieved that purpose, we accepted the parameters found for 20 °C and looked for another set of Qa1, Qa2, RL1 , and RR1 parameters that would allow us to find the other parameters, leading to a good fit of the third titration isotherm measured at 60 °C. The parameters estimated in this way are collected in Table 5. This kind of numerical exercise is a simplified way of handling the problem. The fully rigorous way would be considering the integral
δ0(pH, T2) ) δ0(pH, T1) +
∫TT
2
1
( ) ∂δ0 ∂T
pH
dT
(33)
int pK int a1 (T2) ) pK a1 (T1) +
1 2.3k
∫1/T1/T
int pK int a2 (T2) ) pK a2 (T1) +
1 2.3k
∫1/T1/T
2
1
2
1
( ) () ( ) () ∂Qa1 1 d 1 T ∂ T ∂Qa2 1 d 1 T ∂ T
(34a)
(34b)
Handling the problem in such an accurate way would involve knowledge of the functions Qa1(T) and Qa2(T). Meanwhile, from Figure 3, we can know only the sum (Qa1 + Qa2) as a function of T. The parameter values collected in Table 5 represent certain averages between the lower and the next higher temperature. Despite that, they provide us with very interesting information. Thus, the Qa1 and Qa2 values, representing certain effective (average) values in the temperature region 278-293 K, give the sum (Qa1 + Qa2) ) 164 kJ/mol, which corresponds to values found from the temperature dependence of PZC seen in Figure 3B. As for the temperature range 298-333 K, the (average) sum (Qa1 + Qa2), found by fitting the two titration isotherms, is about 100 kJ/mol, that is, looks somewhat lower than the values (Qa1 + Qa2) seen in Figure 3B. However, one should also take into account some uncertainty in determining the temperature dependence of (Qa1 + Qa2) from only three PZC(T) points in this temperature range. Their linear regression, for instance, yields the value 109 kJ/mol, which is not far from the value 98 kJ/mol found by fitting the two titration isotherms. Thus, the quantitative simultaneous analysis of both the temperature dependence of PZC and the temperature dependence of the corresponding titration isotherms, measured at the same electrolyte concentration, would apparently provide a logical picture of the thermodynamic properties of that system. Meanwhile, using the parameters collected in Table 5, we face serious problems in reproducing Fokkink’s experimental function Qpr, (fff), shown in Figure 4. Let us consider that problem in more detail. When fitting titration isotherms measured at different temperatures, we arrive only at certain values of Qa1, Qa2 that are some averages between two temperatures T1 and T2. Not knowing precise values of Qa1, Qa2, RL1 , and RR1 at T1 and T2, one cannot reproduce well the behavior of Qpr at a fixed temperature T1 or T2. However, it is to be expected that while fitting the function Qpr(pH) measured at 20 °C, one should do it successfully using values of the parameters Qa1, Qa2, RL1 , and RR1 that lie between the values determined for the temperature intervals 5 °C-20 °C and 20 °C-60 °C. This, however, does not happen in our present case, as can be seen in Figure 9. Figure 9 shows the Qpr functions, calculated by using various parameter sets Qa1, Qa2, RL1 , and RR1 . The good fit of Fokkink’s experimental Qpr function (fff) is obtained for the set of parameters lying outside the intervals suggested by the data collected in Table 5. Most striking is that the sum (Qa1 + Qa2) ) 74 kJ/mol obtained for 20 °C is even lower than the lower sum (Qa1 + Qa2) ) 98 kJ/mol found for the temperature range 20 °C-60 °C by fitting the experimental titration isotherms. Also, the value of RR1 ) 0.015 found by fitting the Qpr function determined at 20 °C does not lie in the interval (+0.001, -0.001) suggested by the data collected in Table 5. Furthermore, the result Qa1 ≈ Qa2 is very strange, based on what we know about these systems. Looking at the Qa1, Qa2 data reported in
Ion Adsorption at a Homogeneous Oxide Surface
J. Phys. Chem. B, Vol. 104, No. 50, 2000 11921
Figure 9. The various Qpr functions calculated for the Fokkink’s hematite sample and for the temperature 20 °C, at which Kaizer et al. carried out their direct calorimetric measurements. The strongly broken line (‚ ‚ ‚ ‚ ‚) is for the set Qa1 ) 81 kJ/mol, Qa2 ) 83 kJ/mol, RL1 ) 0.01, RR1 ) 0.001, corresponding to the temperature interval 5 °C-20 °C, the broken line (- - -) is for the set Qa1 ) 48 kJ/mol, Qa2 ) 50 kJ/mol, RL1 ) 0.0015, RR1 ) -0.001, corresponding to the temperature range 20 °C-60 °C, and the solid line (s) represents the best fit to the experimental Qpr function, obtained by assuming Qa1 ) 39 kJ/mol, Qa2 ) 35 kJ/mol, RL1 ) -0.0031, RR1 ) 0.015. The experimental Qpr function (f f f) is redrawn from Figure 4.
the literature for various oxide/electrolyte systems, we can see that usually Qa1 < Qa2. This has been shown in the recent paper by Rudzin´ski et al.,65 which treat some fundamental features of these systems, discovered in the theoretical works by Sverjensky and co-workers66-68 and deduced from some reported calorimetric experiments. Turning again to Figure 9, we can see that the theoretical Qpr function has a sharp minimum, indicating a strong discontinuity of its first derivative (∂Qpr/∂δ0) at the minimum. Also, that theoretical minimum does not coincide with the experimental minimum occurring at δ0 > 0, and any other choice of the parameters Qa1, Qa2, RL1 , and RR1 , does not help. Thus, we can see that the analysis based on the model of an energetically homogeneous hematite surface does not allow us to use the same set of parameters to arrive at a simultaneous good fit of both the temperature dependence of the titration isotherms and the pH dependence of the directly measured calorimetric data for Qpr. De Kaizer et al.33 compared the value (Qa1 + Qa2)/2 ) 36.3 kJ/mol obtained from the “∆T titration” with their value of Qpr ) 36 kJ/mol and suggested that the cumulative plot of the calorimetric effects of titration may provide the same information as the temperature dependence of PZC. Meanwhile, on the basis of the TLM surface complexation model, used in their and our considerations, the temperature dependence of PZC provides us with a number, whereas Qpr is a function of pH and electrolyte concentration. For the purpose of illustration, Figure 10 shows the cumulative heat of adsorption, Qcum
Qcum )
∫δδ
02
01
Qprdδ0
(35)
calculated for our Qpr function shown in Figure 6. Looking at Figure 10, one can see two fairly linear sections in the calculated Qcum values: one linear section for δ0 < 0 and another one for δ0 > 0. This would suggest that there are two enthalpic effects: one accompanying the adsorption of protons at δ0 < 0 and another one at δ0 > 0. From the tangents of the corresponding linear regressions, one can estimate these two heats of adsorption as equal to 75.5 kJ/mol and 16.4 kJ/
Figure 10. The cumulative heat Qcum defined in eq 35 (/ / /) calculated for our hematite sample by taking the Qpr function shown in Figure 6. The two solid lines are the linear regressions for the Qpr data (/ / /) for the two regions: δ0 < 0, and δ0 > 0.
mol, respectively. One can hardly relate these values to Qa1, and Qa2 values in Table 4, used to calculate the Qpr function in eq 35, and next the Qcum data seen in Figure 10. Also, the sum (75.5 + 16.4) kJ/mol ) 91.9 kJ/mol can hardly be compared to the values Qa1 + Qa2 = 40-41 kJ/mol, deduced from Figure 3B. This model investigation shows the potential risks of a simple interpretation of directly measured calorimetric data. Only a full theoretical analysis, and related computer calculations, allow us to extract true information about the enthalpic effects of ion adsorption at the oxide/electrolyte interfaces. Of course, this does not exclude the possibility of finding a smart way to a simpler, though less precise interpretation. Kallay and co-workers,2,39-41 for instance, have proposed that the nonconfigurational value (Qa1 + Qa2) could be deduced from the integral PZC-∆pH ∫PZC+∆pH ∑Qi i
( ) ∂θi
∂pH
d(pH)
(36)
T
Recently, this idea has been investigated in more detail in the paper by Rudzin´ski et al.47 The results discussed in this paper show that there is one physical factor that has not received sufficient attention in the studies hitherto of ion adsorption at the oxide/electrolyte interface. This is the temperature dependence of Qa1 and Qa2, and more generally, the heat capacity of the surface complexes. So far, only numbers were reported in the literature on Qa1 and Qa2 values. Their temperature dependence was discussed only in a few publications, such as that by Machesky et al.23 or by Schoonen.69 This problem will be discussed in more detail in Part II of this publication. However, the popular studies of the temperature dependence of PZC alone does not provide with much information and may even be misleading in some cases. Figure 3, showing a similar temperature dependence of PZC for both our hematite sample and the sample investigated by Fokkink et al., would suggest essential similarities exist between their thermodynamic properties. Meanwhile, direct calorimetric measurements, as well as studies of the temperature dependence of the whole titration isotherms, show differences in the values Qa1 and Qa2. Perhaps most striking is the difference between the Qa1 value of our sample, -2 kJ/mol, and the value of about 40 kJ/mol in the case of the hematite sample studied by Fokkink et al. (See Table 5 and Figure 9). The success, but also the shortcomings, of the popular TLM (triple layer model) are well-known. Recently, that surface
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