2986
Langmuir 2004, 20, 2986-2988
Standard States and Activity Coefficients of Interfacial Species Nikola Kallay,*,† Tajana Preocˇanin,† and Suzana Zˇ alac‡ Laboratory of Physical Chemistry, Department of Chemistry, Faculty of Science, University of Zagreb, Marulic´ ev trg 19, P.O. Box 163, 10001 Zagreb, Croatia, and Research & Development PLIVA d.d., Ul. baruna Filipovic´ a 25, 10000 Zagreb, Croatia
is expressed by concentration or molality (rB ) cB/cQ; cQ ) 1 mol dm-3; rB ) bB/bQ; bQ ) 1 mol kg-1). The concept of the standard state involves two aspects, composition and ideality. The state of the component with respect to the ideality is quantitatively described by the activity coefficient being γ ) 1 in the arbitrarily chosen ideal condition. To analyze the physical meaning of the activity coefficient, we shall introduce the relative content of species B in eq 1:
Received November 21, 2003. In Final Form: January 9, 2004
µB ) µXB + RT ln rB
The standard state of species at the solid-liquid interface, such as the metal oxide-aqueous electrolyte solution interfacial layer, is a rather poorly defined concept. The main reason for this situation is the fact that in most of the cases of interfacial reactions the standard state composition cancels so that the choice of the physical quantity describing the standard composition and its value do not influence the result, that is, the value of the equilibrium constant. In a recent article, Sverjensky1 has summarized the literature on standard states of interfacial species and proposed an original approach, which is useful for comparison of the interfacial reactions with the reactions in the bulk of the liquid medium. We shall follow the IUPAC recommendations2 and define the standard states of interfacial species in a somehow different manner that is consistent with the definitions for a liquid or solid mixture, gaseous mixture, and solution (solute, solvent)2. At the interface, the electrostatic effect is commonly taken into account by introducing the Boltzmann correction term into the definition of the interfacial equilibrium constant.3 Another, more informative route, is to consider the species involved in the reaction and their activity coefficients.3 The chemical potential µ (partial molar Gibbs energy) for species B is generally2 given by
Consequently, the quantity µXB is not the standard chemical potential of species B. It is rather the chemical potential at the standard composition (rB ) 1) in the real state (e.g., chemical potential of solute B at the concentration of 1 mol dm-3). The quantity µXB may be considered as the “semi-standard” chemical potential because it is “standardized” only with respect to the composition but the ideality is not taken into account. The comparison of eqs 1-3 leads to the thermodynamic definition of the activity coefficient as
µB ) µQB + RT ln aB
The question that remains is over which surface species i the summation of their amounts n should be performed. One possibility is to take into account all surface species. However, this approach is not applicable if all of the surface groups cannot be properly defined. Another, more reasonable, possibility is to select the species in such a way that the summation includes only those species that might be (by surface reaction) converted into interfacial species B. For example, within the “2-pK” model of surface complexation at a metal oxide interface3, the fraction of surface sites MOH in the presence of cations C+ and anions Ais
(1)
The relative activity (a) depends on the arbitrary choice of the standard state. The activity of species B is
a B ) γ B rB
(2)
where γB is the activity coefficient of species B and rB we shall call the “relative content” with respect to species B. Several quantities may be used to describe the composition of a system. The common practice, as recommended by IUPAC,2 is as follows. For solid and liquid mixtures, the relative content is described by the amount fraction (rB ) xB; xQ ) 1), for gaseous mixtures in terms of its partial pressure (rB ) pB/p.Q; pQ ) 105 Pa), and for solutions different definitions apply for the solvent and solutes. For the solvent, the relative content is defined as for liquid mixtures (rB ) xB). For the solutes, the relative content * To whom the correspondence should be addressed. E-mail:
[email protected]. URL: http://www.chem.pmf.hr/∼nkallay/. † University of Zagreb. ‡ Research & Development PLIVA d.d. (1) Sverjensky, D. A. Geochim. Cosmochim. Acta 2003, 67, 17. (2) Mills, I.; Cvitasˇ, T.; Homann, K.; Kallay, N.; Kuchitsu, K. Quantities, Units and Symbols in Physical Chemistry, 1st ed.; Blackwell Scientific Publications: Oxford, 1988. (3) Kosmulski, M.; Sprycha, R.; Szczypa, J. In Interfacial Dynamics; Kallay, N., Ed.; Marcel Dekker: New York, 2000; Chapter 4.
∆µB ) µXB - µQB ) RT ln γB
(3)
(4)
The previous relationship, connecting the activity coefficient and the difference in chemical potentials between the real and the ideal states, will be applied to interfacial species. The quantity, which should be considered for the definition of the standard composition at the interface, is the surface coverage θ (“amount fraction” or “mole fraction”), which for interfacial species B is equal to
θB ) nB/
∑i ni
(5)
θ(MOH) ) n(MOH) n(MOH) + n(MOH2+) + n(MO-) + n(MOH2+‚A-) + n(MO-‚C+) (6)
By applying the surface coverage concept, it is natural to adopt the standard composition at a surface coverage θQB ) 1. Consequently, the relative content of interfacial species B and their activities becomes rB ) θB and aB ) γBθB. The second reasonable solution is to use the concept of the surface concentration Γ, defined as the amount of surface species n divided by surface area A. For species B,
10.1021/la036185f CCC: $27.50 © 2004 American Chemical Society Published on Web 02/10/2004
Notes
Langmuir, Vol. 20, No. 7, 2004 2987
ΓB ) nB/A
(7)
The standard value of the surface concentration, because there is no other accepted practice, is ΓQ ) 1 mol m - 2, so that the relative content and activity of the interfacial species are rB ) ΓB/ΓQ and aB ) γBΓB/ΓQ. The shortcoming of this approach is that in some cases the concept of the surface area is not clear, for example, due to the surface roughness. On the other hand, this concept is consistent with the application of the Gouy-Chapman theory, which relates the surface charge density with the corresponding surface potentials. The common practice in the interpretation of interfacial equilibrium is to take into account the effect of the electrostatic potential on charged interfacial species.3 According to this practice, as the ideal state we shall define the state at zero (overall) electrostatic potential. In reality, interfacial species B of the charge number zB are exposed to the electrostatic potential φB. This potential is the difference between the electrostatic potential affecting the state of interfacial species B and the one in the bulk of the solution. The difference between the chemical potential of interfacial species B in the real and that in the ideal states is, therefore, given by
∆µB ) zBFφB
(8)
Accordingly to eqs 4 and 8, in the case of simple (ionic) species B at the interface, the activity coefficient is
γB ) exp(zBFφB/RT)
(9)
For ionic associates (ion pairs) in the interfacial layer, one should take into account that they act as oriented dipoles in such a way that two charged (ionic) end groups are exposed to different electrostatic potentials. Association of the charged surface site S of charge number zS with the counterion G of charge number zG results in the surface ion pair S‚G. This interfacial ion pair is oriented and acts as a dipole so that the charged side S is exposed to the electrostatic potential φS while side G is oriented toward the liquid phase and exposed to the electrostatic potential φG. To evaluate the activity coefficients of these interfacial ion pairs, one should calculate the change in the chemical potential for the process in which these dipoles are transferred from the ideal condition (zero electrostatic potential) to the interface where two ends of the dipoles are exposed to different electrostatic potentials
∆µS‚G ) RT ln γS‚G ) zSFφS + zGFφG
tMOH + H+ a tMOH2+; exp(φ0F/RT)Γ(MOH2+) Γ(MOH)aH+
(11)
tMOH a tMO- + H+; exp(-φ0F/RT)Γ(MO-)aH+ (12) KQd ) Γ(MOH) +
The potential affecting the tMOH2 surface species is denoted by φ0, and the charge numbers are +1, 0, and -1 (4) Sposito, G. J. Colloid Interface Sci. 1983, 91, 329.
tMOH2+ + A- a tMOH2+‚A-; KQA
(13)
tMO- + C+ a tMO-‚C+; KQC
(14)
By introducing appropriate activity coefficients, according to eq 10, and taking into account that the two ends of the interfacial ion pair are exposed to different potentials, that is, MOH2+ (of charge number +1) is exposed to φ0 while A- (of charge number -1) is exposed to φβ, one obtains for the association of anions A- with positive surface groups
KQA )
exp[(φ0 - φβ)F/RT]Γ(MOH2+‚A-) exp(φ0F/RT)Γ(MOH2+)aA-
)
exp(-φβF/RT)Γ(MOH2+aA-) Γ(MOH2+)aA-
(15)
Analogously, for the association of cations C+ with negative surface groups one obtains
KQC )
exp[(φβ - φ0)F/RT]Γ(MO-‚C +) exp(-φ0F/RT)Γ(MO-)‚aC+
)
exp(φβF/RT)Γ(MO-‚C+) Γ(MO-)aC+
(16)
(10)
The approach just described will be demonstrated by the “2-pK” model,4 which assumes protonation and deprotonation of amphotheric surface tMOH groups
KQp )
for tMOH2+, tMOH, and tMO-, respectively. The standard values of the surface concentration, in eqs 11 and 12, cancel. Equations 11 and 12 are identical to those derived by the “intrinsic concept”, which assumes the twostep process:3 the first step in protonation is the transfer of H+ ions from the bulk of the solution to the “intrinsic state” at the interface characterized by the surface potential φ0. This equilibrium is described by the Boltzmann distribution.3 The second step is assumed as the binding of “intrinsic” ions H+ to surface tMOH groups and is described by the “intrinsic equilibrium constant”, Kint. The procedure suggested in this article clearly shows that these “intrinsic surface equilibrium constants” are nothing but thermodynamic equilibrium constants defined in the same way as equilibrium constants for any other reaction system. The association of anions A- and cations C+ is commonly described by the following reaction equations
Equations 15 and 16 are identical to those derived on the basis of the “intrinsic concept”, and the corresponding thermodynamic equilibrium constants are identical to the commonly used “intrinsic equilibrium constant”. The special case is association of multivalent ions.4 Here, we shall consider the simple case of binding of divalent cations to negative surface sites
tMO- + C2+ a tMO-‚C2+; exp(2φβF/RT)Γ(MO-‚C2+) Q (17) K1C ) Γ(MO-)aC2+ Another possible mechanism is the formation of surface groups
2988
Langmuir, Vol. 20, No. 7, 2004
Notes
2 tMO- + C2+ a (tMO-)2‚C2+; KQ2C )
exp(2φβF/RT)Γ((MO-)2‚C2+)‚ΓQ Γ(MO-)2aC2+
(18)
In this case, the standard values of the surface concentrations are only partly canceled so that the value of the thermodynamic equilibrium constant depends on the choice of standard state composition. Experimentally, one cannot distinguish between mechanisms 17 and 18. It might be assumed that both reactions take place at the same time but with different extents. The problem of formation of the (tMO-)2‚C2+ species may be approached in the more correct way by taking into account that the surface is a condensed system and that surface sites are not distributed in the space but rather closely packed, so that each negative surface site is already surrounded by neutral sites (negative sites in the vicinity are not probable because of electrostatic repulsion). Accordingly, we may write equations for the two-step process
tMOH(tMO-) + C2+ a tMOH(tM-)‚C2+; exp(2φβF/RT)‚Γ[MOH(MO-)‚C2+] Q (19) KC1 ) Γ[MOH(MO-)]aC2+ tMOH(tMO-)‚C2+ a (tMO-)2‚C2+ + H +; KQC2
)
exp(-φ0F/RT)Γ[(MO-)2‚C2+]aH+ Γ[MOH(MO-)‚C2+]
(20)
The summary reaction is
tMOH(tMO-) + C2+ a (tMO-)2‚C2+ + H +; KQC1,2 ) KQC1KQC2 ) exp[(2φβ - φ0)F/RT]Γ[(MO-)2‚C2+]aH+ Γ[MOH(MO-)]aC2+
(21)
Equation 21 probably better represents the formation of (tMO-)2‚C2+ sites. An analysis may show that the equilibrium constant KQC2 corresponds to the deprotonation (eq 12) of neutral sites KQd . To describe the equilibrium state at the solid-liquid interface, it is necessary to define the mechanism of the
reactions taking place at the interface. The corresponding definition of the thermodynamic equilibrium constants is based on relative activities so that one should define the standard states for interfacial species. For that purpose, it is proposed to introduce either surface coverage or surface concentration. The recommended values of these quantities are θQ ) 1 and ΓQ ) 1 mol m-2. The activity coefficients are defined by considering the change in chemical potentials for the transfer of ionic species from the ideal to the real state. In this article, only simple electrostatic contributions were considered. However, it may be concluded that this effect plays the major role. For example, the surface potential φ0 affecting the states of the charged surface species formed as a result of interactions with potential-determining ions is for metal oxides approximately a linear function of pH with the slope somehow lower in magnitude than the Nernstian.5 The value of φ0 may be6 about 200 mV, at pH ) pHpzc - 4. This potential leads to γ(MOH2+) > 2000. On other side, at pH ) pHpzc + 4, γ(MOH2+) < 5 × 10-4. It means that, in the common experiment, in which the pH was changed by eight units, the activity coefficient will be changed more than 106 times. Apart from the major effect of the surface potential, a more refined treatment would consider some other effects. For example, the charge is not homogeneous at the interface so that the local electric field in the vicinity of charged surface site would directly influence the reactivity of neighboring sites. It is desirable to define the “ideal state” in such a way that it could be achieved experimentally. This requirement was accomplished here because at the “point of zero potential” φ0 ) 0 and at the isoelectric point both the electrokinetic and the φβ potentials are equal to zero (ζ ) φβ ) 0). The procedure just mentioned was described on the example of the “2pK” model of surface complexation. However, it is a general approach and could be simply applied to reactions written in a different way and also to “1-pK” or any other mechanism or stoicheiometry describing the interfacial reactions.3 Acknowledgment. We are grateful to Professor Vladimir Simeon and Dr. Davor Kovacˇevic´ for helpful discussions. LA036185F (5) Blesa, M. A.; Kallay, N. Adv. Colloid Interface Sci. 1988, 28, 111. (6) Avena, M. J.; Camara, O. R.; De Pauli, C. P. Colloids Surf. 1993, 69, 217.
