J. Phys. Chem. B 2007, 111, 3183-3191
3183
Mean Activity Coefficient of Electrolyte Solutions Elsa Moggia*,† and Bruno Bianco†,‡ Department of Biophysical and Electronic Engineering (DIBE), UniVersity of Genoa, Via Opera 11A, 16145, Genoa, Italy, and InteruniVersity Center for Interactions between Electromagnetic Fields and Biosystems (ICEmB), at DIBE, UniVersity of Genoa, Via Opera Pia 11A, 16145, Genoa, Italy ReceiVed: October 30, 2006; In Final Form: January 18, 2007
In this paper, we deal with the mean activity coefficient, γ, of electrolyte solutions. The case γ e 1 is investigated. As is generally recognized, the most accepted models (specific ion interaction/Pitzer theory) have the disadvantage of the dependence on semiempirical parameters. These are not directly accessible from experimental measurements, but can only be estimated by means of best-fitting numerical techniques from experimental data. In the general context of research devoted to the achievement of some reduction of complexity, we propose a model of electrolyte solution that allows us to calculate γ without using fitting parameters where the (upper) concentration exists at which the electrolyte solution exhibits γ ) 1 (molality scale). In the remaining cases, we show that a unique parameter is required, that is, the concentration that should ideally give γ ) 1 for the electrolyte. Compared to other models that do not require adjustable parameters, the present one is generally applicable over a wider range of concentrations; moreover, it does not impose any restriction on the ion-size variations. Our model follows a pseudolattice approach, starting from the primitive idea of a disordered lattice of solute ions within a continuous solvent at extremely dilute solutions and coming to a disordered lattice of local arrangements of both solute ions and solvent dipoles at higher concentrations. Compared to other theories based on lattice models, this work stresses the role of statistical deviations from any time-averaged (lattice) configuration. All formulas in this paper are applied for 1:1, 2:2, 1:2, and 2:1 aqueous electrolytes at 25 °C.
Introduction This work is concerned with the properties of electrolyte solutions and, more precisely, with their mean activity coefficient, a topic widely treated in the literature.1-4 The Debye-Hu¨ckel theory5,6 represents a step of paramount importance within the general frame of electrolyte solutions, mainly because it provides a simple formula to calculate the mean activity coefficient in good agreement with experiments, at least when (very) dilute solutions are considered. As is wellknown, the DH limiting law (DHLL), despite some theoretical inconsistencies,1,2,7 does not contain arbitrary fitting parameters. Subsequent theories8-15 have thoroughly investigated the role of basic phenomena, such as ion-pairing or solvation effects;4,8-21 however, several adjustable parameters are needed in most theories providing the highest agreement with experimental data, such as in Pitzer theory.15 These parameters can be determined only by means of best-fitting numerical techniques from experimental data, even when extremely dilute solutions are considered.22,23 In some theories, electrolyte parameters are set to a constant value to avoid any fitting procedure;24,25 however, these theories tend to fail at high concentrations or for large variations in the ion size. To provide more manageable formulas for applicative purposes, in recent years, several models have been proposed.16,17 Unfortunately, their applicability is generally salt-dependent or relatively limited with respect to the range of allowable * Corresponding author. Tel +39 010 3532899. E-mails: elsa@ biosafetyengineering.it;
[email protected]. † DIBE. ‡ ICEmB.
Figure 1. Plot (continuous line) of mean activity coefficient γ of NaCl versus molality. Experimental clim ) 5.4 M. Circles refer to experimental data. To make a comparison, the dotted-dashed line refers to γ calculated using the DHLL.
concentrations. Moreover, effective ion diameters resulting from numerical evaluations are often difficult to use due to lack of consistent data from different authors.2,11,16,17 The general problem of both unambiguously defining and actually measuring the solvated-ion parameters is well-known in the literature.11,26,27 In the general context of research devoted to the achievement of some reduction of complexity, we propose a model based on a pseudolattice approach. In comparison with other quasilattice models,28 we focus on statistical deviations from any time-averaged (pseudolattice) configuration. We start from the primitive idea of a disordered lattice29 of point charges fluctuating quite randomly within the continuous solvent at extremely dilute solutions, and we come to a lattice of local arrangements of both solute ions and solvent dipoles at higher concentrations. In so doing, we obtain a one-parameter model for the calculation of γ over a wide range of concentrations, the advantage of this
10.1021/jp067133c CCC: $37.00 © 2007 American Chemical Society Published on Web 03/08/2007
3184 J. Phys. Chem. B, Vol. 111, No. 12, 2007
Figure 2. Plot of γ of NaOH versus molality (calcd γ, continuous line; exptl γ, circles; exptl clim ) 4.44 M). Plot of γ of LiNO3 versus molality (calcd γ, dashed line; exptl γ, diamonds; exptl clim ) 3 M). Plot of γ of HI versus molality (calcd γ, dotted-dashed line; exptl γ, × symbols; exptl clim ) 1.15 M).
Figure 3. Plot of mean activity coefficient, γobs, of ZnSO4 versus molality. The dashed line refers to experimental γobs; × symbols refer to calculated γobs; circles refer to calculated γobs by numerical cubic interpolation. Theoretical clim ) 8 M.
Figure 4. Plot (continuous line) of mean activity coefficient γ of CaCl2 versus molality. Circles refer to experimental data. To make a comparison, the dotted-dashed line refers to γ calculated using the DHLL. Experimental clim ) 2.25 M.
model being that the unique parameter (indicated with clim) is sometimes experimentally known. The agreement with experimental data is illustrated in Figures 1-10 (see also the Supporting Information), where common 1:1, 2:2, 1:2, and 2:1 aqueous electrolytes at 25 °C are considered over the whole range of molalities where γ e 1. Figures 8-10 show a comparison with the mean spherical approximation (MSA),13,24 the corrected Debye-Hu¨ckel theory (CDH),17 the Pitzer theory,15,23 and with the simulation results from the hypernetted chain equation (HNC)24,30 and the complete Poisson-Boltzmann equation (IPBE).23 Symmetric Electrolytes Below, we present our general modeling and results concerning symmetric electrolytes. For more readability, derivations of eqs 3, 6, 8 are given in Appendix A. Asymmetric electrolytes are considered in the next paragraph. Application to our formulas of both symmetric and asymmetric electrolytes is presented in the last paragraph.
Moggia and Bianco
Figure 5. Plot of γ of Zn(ClO4)2 versus molality (calcd γ, continuous line; exptl γ, circles; exptl clim ) 1 M). Plot of γ of Cu(NO3)2 versus molality (calcd γ, dashed line; exptl γ, asterisks. exptl clim ) 2.78 M). Plot of γ of ZnCl2 versus molality (calcd γ, dotted-dashed line; exptl γ, × symbols; exptl clim ) 7.4 M).
Figure 6. Plot of γobs of H2SO4 versus molality. Experimental clim ) 8.78 M. The dashed line refers to experimental γobs; × symbols refer to calculated γobs; circles refer to calculated γobs by numerical (cubic) interpolation.
Figure 7. Plot of γ of CsCl versus molality (calcd γ, continuous line; exptl γ, circle symbols; theoretical clim ) 18 M). Plot of γ of NaNO3 versus molality (calcd γ, dashed line; exptl γ, asterisks; theoretical clim ) 45 M). Plot of γ of K2CrO4 versus molality (calcd γ, dotted-dashed line; exptl γ, × symbols; theoretical clim ) 9.5 M).
Let N be the number of solute molecules per 1 m3 of solution, which is obtained from the molar concentration, c, by means of the standard conversion formula.31
N ) NAvc × 103
(1)
In eq 1, NAv is the Avogadro number. Let R3 be the mean volume of solution per ion, then
R)
1 (2N)1/3
(2)
We start from extremely dilute solutions, in which only ionion interactions are relevant,26,27 and the continuous model for the solvent is very satisfactory. At this extreme, we assume that an equilibrium-like distribution is reached from the ions, at least on time average over a suitable time interval, τ. More precisely, we assume the FCC (face-centered cubic) lattice as the timeaveraged distribution for symmetric electrolytes. We then evaluate the linear standard deviation, UI-I (lsd), of an ion from
Mean Activity Coefficient of Electrolyte Solutions
J. Phys. Chem. B, Vol. 111, No. 12, 2007 3185
Figure 8. Plot of γ of HI versus c. Continuous line, our model (exptl clim ) 1.15 M); points, MSA; squares, HNC; diamonds, experimental data.
Figure 9. Plot of γobs of MnSO4 versus molality. Continuous line, our model (thoretical clim ) 7.5 M); dashed line, IPBE model; squares, Pitzer theory; points, experimental data.
its time-averaged (FCC) position. We will show (cf. Appendix A) that UI-I is inversely proportional to the Debye-Hu¨ckel inverse screening length, κ )
UI-I ≡
x
2 πR
x2NQ2/kBT:
x
kBT
2NQ
2
)
xπR2 1κ
(3)
In eq 3, R is the Madelung constant, whose value is ∼1.7476 for FCC ionic crystals;32,33 kB is the Boltzmann constant, and T, the temperature (K); is the static dielectric permittivity of the solvent; the N molecules of solute are completely dissociated into N ions with charge QA ) Q and N ions with charge QA ) -Q. As seen in Appendix A, UI-I is actually the lsd of a Gaussian probability density pA, centered on the FCC position of QA, which statistically describes the stochastic motion of QA in the time interval τ. This also allows formally associating QA with the density charge or effectiVe distribution FA ≡ QApA in the extreme dilution limit:
e-(|r-rA| )/2UI-I ; FA(r) ≡ QA pA(r) ) pA(r) ) (x2π)3 UI-I3 2
2
e-(|r-rA| )/2UI-I QA (4) (x2π)3 UI-I3 2
2
In eq 4, r is the position vector of a generic point in the frame space, whereas rA is the FCC position of QA. More precisely, if Q is the ion at the origin O ) (0, 0, 0), then QA ) Q(1)n+b+l is the ion at the vertex rA ) R(n, b, l), where (n, b, l) is a triplet of integers. We then study the statistical changes within the solution as the solute concentration increases. If the solvent were not affected by any kind of alterations, then the solution could be modeled (always on time-average) with a lattice of point charges within the continuous solvent, even at high concentrations,
Figure 10. Plot of γ of MgBr2 versus c. Continuous line, our model (exptl clim ) 1.38 M); asterisks, CDH; points, HNC; diamonds, experimental data.
because of some balancing between short and long-range (Coulombic) ion-ion interactions, similarly to what occurs in ionic crystals.32 But the solvent is affected by alterations, and deformation34 and symmetry disruption2,11,26,27 are experienced by the dipolar solvation shells. As a consequence, local iondipole interactions become more and more relevant, involving an increasing number of particles so as to form local arrangements of both ions and dipoles, such as microclusters, as sometimes experimentally observed.35 Let us suppose that at a mesoscopic scale of observation, a given arrangement can be seen as a suitable free carrier of charge: we define such a charge as the effectiVe charge of that arrangement (also called an effectiVe carrier). This idea of effective carriers allows dealing only with carrier-carrier interactions, regardless of the underlying nature of the particles forming an arrangement. At a microscopic level, the effective charge should be calculated from some distribution function, ψeff.charge, depending on the canonical ensemble {Xj} of coordinates of the particles within the arrangement (j denotes the jth particle). However, we do not follow such a complex procedure, but rather, we a priori consider 2N arrangements, each having an effective charge equal to the nominal charge, QA, of one cation or one anion (this partitioning clearly tends to 2N free solute ions at very dilute solutions). The extent of the carrier QA is then defined as the minimum volume ΩA where ψeff.charge)QA provides ∫ΩA ψQA dΩA ≈ 1. Moreover, following the pseudolattice approach, the mean value ∫ΩA ∑j XjψQA dΩA is set equal to rA. So, the “radial size” of a carrier can be estimated by the standard deviation UI-D,A ≡ ∫ ∑ |X -r | ψ dΩ . These statistical estimators will be xsufficient to our aims without explicitly evaluating ψ . We 2
ΩA
j
j
A
QA
A
QA
want now to obtain the effective distribution function, FA(r), of the carrier QA, which generalizes eq 4 when solutions that are not dilute are concerned. By definition, the effective distribution is a (mesoscopic) density charge accounting for both the extent of QA (depending on ψQA, as above said) and the stochastic motion of QA within the solution, remembering that carriers are always moving under Brownian forces.36
Let UA be the lsd x(∫|r-rA|2FA(r)d3r)/3 in the time interval τ. For simplicity, we assume UA ≈ U ∀ QA, U to determine. In the extreme dilution case, we know that U ) UI-I and UI-D, A ) 0 ∀ QA (this because each QA is exactly located within the zero-volume of the point-charge ion QA itself). For increasing concentrations, we can see U as an increasing distortion from UI-I because of two factors: (a) the number NI-I of carriers that behave as point charges decreases; moreover, only a part of them fluctuate with lsd ) UI-I; and (b) the number NI-D of carriers with UI-D,A * 0 increases, and UI-D,A itself is an increasing function of the concentration. The upper limits are NI-I f 0, NI-D f 2N, and UI-D, A f ∞ ∀ QA (this representing the maximum uncertainty in attributing QA within any limited
3186 J. Phys. Chem. B, Vol. 111, No. 12, 2007
Moggia and Bianco
arrangement). Since UI-D,A f ∞ ∀ QA we have U f ∞. Let clim be the concentration of a given electrolyte in such a limit case, and the normalized concentration, s.
s≡
N c ) 0ese1 Nlim clim
(5)
In eq 5, Nlim is obtained substituting c ) clim into eq 1. We will show (cf. Appendix A) that starting from the evaluation of U in the limit s f 1 (or c f clim), for 1:1/2:2 aqueous electrolytes at 25 °C we have
U(s) ≈
1 1 1 1/10.5 κ log10.5 s s
(6)
()
(An analogous result could be derived for any given solvent or temperature, but these cases are not considered in this paper.) Equation 6 is valid for s g 1/2 and also for s , 1/2, for which we will see (cf. Appendix A) that eq 6 approaches UI-I. If s ≈ 1 we can say that UI-D, A ≈ U, meaning that the effective charge QA can be found within an arrangement whose volume has radius ≈U. Note that U depends on both concentration and thermal energy. This agrees, at least in principle, with criteria usually adopted to determine the effective charge of an ion (or a colloidal particle) starting from a screened Coulomb pair potential.37 To proceed, for continuity from the extreme dilution case (and also due to the lack of more information), we extend eq 4 to the carrier QA, provided that U in place of UI-I is used.
e(-|r-rA| )/2U FA(r) ) QA (x2π)3 U3 2
2
(7)
We then use eq 7 to calculate the mean (over both time and space) electrostatic potential V experienced by one carrier from the others. We will show (cf. Appendix A) that V can be expressed as
V)
Q
1 (x2π)3
n)+∞
∫1/x2 {( ∑ U 0
(-1)n e-R (n u )/2U )3 - 1} du 2
2 2
2
Q2 ln(γ) ≡ (x2π)3 2kBTU
Asymmetric Electrolytes We generalize our results to the case of 1:2 (2:1) electrolytes. Asymmetric electrolytes of higher order are not considered in this work. For more readability, derivations of eqs 15-17 are given in Appendix B. We now have N charges QA ) 2Q and 2N charges QA ) -Q. Moreover,
R)
R0 )
(8)
∫
1/x2 0
{θ34(0,
e
-(R2u2)/2U2
) - 1} du (9)
From a numerical point of view, eq 9 is fast converging. Moreover, we see that γ ) 1 when c ) clim (since U ) ∞), which defines clim operatively for a given electrolyte. The value of clim is salt-specific and sometimes experimentally known.41 If saturation occurs at a concentration csat < clim, then clim can be seen as the unique unknown parameter in our model;
1 (3N)1/3
(10)
Concerning the (time-average) lattice scheme in the asymmetric case, we adopt a simple 1:2 prototypical structure;33,47 that is, the HCP (hexagonal close packing) lattice. Let 2R0 be the minimum lattice distance between two positive charges or two negative charges; then the HCP Madelung constant R32 is 2.39 (R is ∼2.35 for real crystals33 when R0 is not exactly the same for positive and negative charges). In the HCP scheme, there is one molecule per unit cell, and the cell periodicities along the x, y, and z axes are47 2R0, x3 R0, and x32/3 R0. So from standard calculations,47 we obtain
n)-∞
From an analytical point of view, eq 8 can be rewritten using n)+∞ the Fourth Jacobi theta function38,39 θ4(ξ, z) ≡ ∑n)-∞ (-1)n 2 2u2)/2U2 2jnξ n -(R e z , with ξ ) 0 and the nome z ) e . The mean electrostatic energy of interaction per carrier, namely, E, is then obtained from E ) QV. The calculation of γ (molality scale) then follows from that of V based on the Robinson-Stokes “charging process”,1,2,40 ln(γ) being equal to E/2kBT, provided that γ is intended as a measure of the chemical potential change due to carrier-carrier interactions.
1
however, clim can be predicted from known values of U at lower concentrations. At least in principle, this can be done by solving eq 9 with respect to U, using known values of γ at c < clim. We will see (cf. Appendix A) that eq 6 and, thus, eq 9 are generally valid for s g 1/2 but are rigorously valid for free ions only when s , 1/2. Nevertheless, eq 9 is expected to provide a satisfactory agreement with experimental data when ion pairing is not significant in the dilute/moderate solution range, as occurs in most 1:1 salts.42-44 Conversely, when the degree of dissociation is 1 or when mixed electrolytes are involved, something not yet tackled in this article. Among other developments, the CDH theory17 gives satisfactory results, but one adjustable parameter is needed (the ratio between the cation and the anion effective diameters); moreover, c must not exceed 1.8 M generally (0.8 M in the case of NaCl). Some developments16 allow the permittivity to vary with the solute concentration, but effective ion diameters must be considered as fitting parameters, and results for alkali halides are generally valid within shorter ranges of concentrations, as compared to those presented here (cf. Figures 1, 2, 7). Figure 3 shows ZnSO4 as a representative 2:2 electrolyte, for which csat < clim. For metal sulfates, ion association seems to be a relevant factor, even in the extreme dilution range.45,46 To consider ion association, we first observe that if Rf is the degree of dissociation, the mean distance between free ions becomes R/(Rf1/3) > R after substituting N with RfN into eq 2. However, ion-pairing occurring in the extreme dilution range means that ion-size effects must be there relevant for these ions, but this also means that each ion is actually moving within a mean volume
