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J. Phys. Chem. B 2008, 112, 1212-1217
Osmotic Coefficients of Electrolyte Solutions Elsa Moggia* Biosafety Engineering, Via M. Pippo 9, 17044, Stella (SV), Italy, and Department of Biophysical and Electronic Engineering (DIBE), UniVersity of Genoa, Via Opera 11A, 16145, Genoa, Italy ReceiVed: June 15, 2007; In Final Form: NoVember 1, 2007
In this paper, the osmotic coefficient, φ, of electrolyte solutions is considered. According to the GibbsDuhem equation, the calculation of φ follows from that of the mean activity coefficient, γ, based on a pseudolattice approach recently proposed. For any given electrolyte, the whole range of concentrations providing γ e 1 is considered. The major feature of the pseudolattice approach is given by the fact that γ can be calculated without using adjustable parameters where the (upper) concentration, clim, exists at which the electrolyte solution exhibits γ ) 1. In the remaining cases, a unique parameter is required, that is, the value of clim that should ideally give γ )1 for the electrolyte. Known values of clim from 1 up to 9 M (about) are available for several aqueous electrolytes at 25 °C. 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 illustrates an application of the pseudolattice model1 to the calculation of the osmotic coefficients, φ, of aqueous electrolyte solutions at 25 °C. Starting from the Gibbs-Duhem equation, one has
φ)1+
1 m
∫0m md(ln γ) ) 1 + ln(γ) - 1m ∫0m ln γ dm
(1)
In eq 1 m indicates the molal concentration. By means of eq 1, the calculation of φ follows from that of γ for molalities from 0 to m. In so doing, one has to afford the well-known problem of the theoretical prediction of γ.2-4 This problem has been greatly investigated for many decades,2-23 from the earliest Debye-Hu¨ckel (DH) theory5-6 to the Pitzer theory14 or more recent mean spherical approximation/nonrandom two liquid (MSA-NRTL) models.22 On the one hand, models that do not require adjustable parameters tend to fail at high concentrations, e.g., in the DH Theory, or in some approaches15-17 where electrolyte parameters are set to a constant value to limit the use of fitting procedures. On the other hand, several adjustable parameters are needed in most theories providing the highest agreement with experimental data.12-14,19,21-23 These parameters can be determined only by means of best-fitting numerical techniques from experimental data, even in the case of extremely dilute solutions.24,25 In this general context, the pseudolattice approach provides a one-parameter model, the main advantage being that the unique parameter, that is, a suitable concentration referred to as clim, is sometimes experimentally known. The Pseudolattice Approach In summary, the pseudolattice approach1 starts from a disordered lattice of free point-charge ions moving quite randomly within the continuous solvent, at extremely dilute solutions, and then comes to a lattice of local arrangements of both solute ions and solvent molecules, at higher concentrations. * To whom correspondence should be addressed. Phone: +39 010 3532899. E-mail:
[email protected];
[email protected].
At the mesoscopic scale of observation, these local arrangements can be suitably assimilated to free carriers of charge, said effective carriers of charge, or, equivalently, arrangements of effective charge. Rigorously, one should calculate the “extent” of a carrier as the size of the arrangement in which, at the microscopic scale, some suitable density of charge provides a nonzero contribution. However, the idea of effective carriers allows dealing only with carrier-carrier interactions, regardless of their microscopic nature, and thus without any direct calculation of microscopic densities. That said, the latticelike distribution of carriers has to be intended as their time-average distribution over a suitable time interval. The standard deviations of carriers from their mean positions are particularly of interest; in fact, they can be used to account both for the free-carrier size effects (actually caused by microscopic phenomena, as above said) and for the stochastic motion of carriers within the solution under Brownian-like forces (which are per se mesoscopic26). More precisely, the mean value (with respect to the ensemble of carriers), U, of these standard deviations provides the linear standard deviation of the so-called effective (or mesoscopic) density charge of each carrier, which is Gaussianlike and centered on the lattice position of the carrier. From the effective densities, one obtains (by means of a Poisson equation) the mean electrostatic potential that a carrier experiences by effect of the others, and then the corresponding mean electrostatic energy, E. Calculations provide an expression of E as a function of U. Finally, the calculation of γ follows from the Robinson “charging process”,3,27 that is, from ln(γ) ) (1/2)(E/kBT), where kB is the Boltzmann constant and T the temperature (K), provided that the chemical potential change is interpreted as due only to carrier-carrier electrostatic interactions. At very dilute solutions, the value of U depends only on the stochastic motion of carriers, since they behave actually as zero-extent, point-charge ions. For increasing concentrations, the number of carriers with a nonzero extent increases, this process being maximum at a concentration, namely, clim, at which, by definition, each carrier “extends” up to ∞ (a limit situation that corresponds to the highest uncertainty in attributing any effective charge within any limited arrange-
10.1021/jp074648a CCC: $40.75 © 2008 American Chemical Society Published on Web 01/08/2008
Osmotic Coefficients of Electrolyte Solutions
J. Phys. Chem. B, Vol. 112, No. 4, 2008 1213
Figure 1. Plot of φ of NaClO4 vs m (calcd φ, continuous line; exptl φ, × symbols). Plot of γ of NaClO4 vs m (calcd γ, dashed line; exptl γ, circles). Experimental clim ) 8.63 M (mlim ) 14.66 m).
ment), so U f ∞. The dependence of γ on clim is expressed through the dependence of U on clim. One finds analytically that γ ) 1 at c ) clim, which also provides a practical definition of clim. If saturation occurs at csat < clim, then clim can be seen as the unique adjustable parameter in the model. Equations below summarize the main formulas obtained from the pseudolattice model for symmetric electrolytes. Formulas for the asymmetric case are not reported here for brevity (see ref,1 eqs 10-18) but have been used, when required, in the following. The number N of solute molecules per 1 m3 of solution is obtained from the molar concentration c by means of N ) NAc × 103, where NA is Avogadro’s number.28 For symmetric electrolytes, the mean volume of solution per ion, R3, is equal to 1/(2N). Then, one can consider an ensemble of 2N effective carriers, N of them having an effective charge equal to the nominal charge, QA, of one cation and the remaining ones that of one anion, that is, QA ) (Q (at very dilute solutions this partitioning clearly tends to 2N solute ions). The face-centered cubic (fcc) lattice29 is assumed to be the time-average distribution of the 2N carriers, which are disposed so as to have +Q and -Q charges, alternately, at the fcc vertices. One first introduces the following normalized concentration
s≡
c ,0ese1 clim
(2)
Figure 2. Plot of φ of NaCl vs m (calcd φ, continuous line; exptl φ, circles, experimental clim ) 5.4 M (mlim ) 6.1 m)). Plot of φ of HNO3 vs m (calcd φ, dashed line; exptl φ, × symbols, experimental clim ) 3.85 M (mlim ) 4.4 m)).
Figure 3. Plot of φ of KF vs m (calcd φ, continuous line; exptl φ, circles, experimental clim ) 5.57 M (mlim ) 6.08 m)). Plot of φ of NaOH vs m (calcd φ, dashed line; exptl φ, × symbols, experimental clim ) 4.44 M (mlim ) 4.6 m)).
Finally, after calculating the mean electrostatic potential experienced by one carrier from the others, the calculation of γ follows giving
ln(γ) ≡ 1
Q2
(x2π)3
2kBTU
du )
Calculations give
U≈ log10.5
1 1 1 1/10.5 κ s s
()
(3)
In eq 3 the DH inverse screening length κ ) x2NQ2/kBT has been evidenced ( is the solvent permittivity). Rigorously, for s e e-10.5 one should put U ) 1/xRπ/2 1/κ, R ≈ 1.7476 being the Madelung constant of fcc ionic crystals; however, eq 3 is good in practice also for all the realistic strong dilutions. As a note, the DH screening parameter 1/κ appears only as a factor within eq 3, and only at very dilute solutions (that is, when ion-ion interactions only are significant) do U and 1/κ become proportional. With concern for the analytical form of eq 3, it must be added that it has been strictly derived1 for s , 1/2 and for 1/2 e s e 1. For the intermediate values of s between these ranges, eq 3 can be used only in reason of a mere analytical continuity. Moreover, for s , 1/2, eq 3 is rigorously valid for free ions only; however, using existing data on dissociation constants,30,31 ion association effects can be evaluated in case they are significant at dilute solutions (some examples will be shown in the next paragraph).
{(
Q2
1 (x2π)
∫
1/ x 2 0
3
n)+∞
∑ (-1)ne-R (n u /2U ) n)-∞ 2
2 2
2
) } 3
-1 ×
∫1/x2 {ϑ43(0,e-(R u /2U )) - 1} du 2k TU 0 2 2
2
(4)
B
In eq 4, the fourth Jacobi θ function,32 θ4(ξ,z), appears
θ4(ξ,z) ≡ n)+∞
(-1)n e2jnξ zn , with ξ ) 0 and z ) e-(R u /2U ) ∑ n)-∞ 2
2 2
2
(5)
From the mathematical point of view, the elliptic function θ4(ξ,z) has been introduced to provide an analytical insight into eq 4. It is obtained putting N ) ∞, since the number N of solute molecules is sufficiently large even at very dilute solutions. However, the calculation of eq 5 gives an excellent accuracy by summing up a number of terms of the order of 10/|z|. As a note, the presence of θ functions arises from the manipulation of triple alternating series, which are per se slowly converging.33 Application to Single Aqueous Electrolytes at 25 °C The results of the application of the model to some representative aqueous electrolytes at 25 °C are shown in Figures 1-9, where calculated and measured osmotic coefficients are reported for each electrolyte over the whole range of concentra-
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Figure 4. Plot of φ of SrCl2 vs m (calcd φ, continuous line; exptl φ, circles, experimental clim ) 2.65 M (mlim ) 2.85 m)). Plot of φ of SrBr2 vs m (calcd φ, dashed line; exptl φ, × symbols, experimental clim )1.62 M (mlim ) 2.12 m)).
Figure 8. Plot of φ of H2SO4 vs m (calcd φ, continuous line; exptl φ, circles). Experimental clim ) 8.78 M (mlim ) 14.5 m)). Plot of γobs of H2SO4 vs m (calcd γobs, dashed line; exptl γobs, × symbols).
Figure 5. Plot of φ of Cu(NO3)2 vs m (calcd φ, continuous line; exptl φ, circles, experimental clim ) 2.78 M (mlim ) 3.15 m)). Plot of φ of Zn(ClO4)2 vs m (calcd φ, dashed line; exptl φ, × symbols, experimental clim ) 1.02 M (mlim ) 1.1 m)).
Figure 9. Plot of φ of ZnSO4 vs m (calcd φ, continuous line; exptl φ, upward triangles). Plot of γobs of ZnSO4 vs m (calcd γobs, dashed line; exptl γobs, × symbols, theoretical clim ) 7.4 M).
Figure 6. Plot of φ of ZnCl2 vs m (calcd φ, continuous line; exptl φ, circles, experimental clim ) 7.4 M (mlim ) 10.75 m)).
Figure 7. Plot of φ of CsCl vs m (calcd φ, continuous line; exptl φ, upward triangles, theoretical clim ) 18 M). Plot of φ of K2CrO4 vs m (calcd φ, dashed line; exptl φ, downward triangles, theoretical clim ) 9.5 M).
tions for which γ e 1. Other examples are given in the Supporting Information (see below). To obtain molarities c from molalities m, the following formula28 has been used
c)
103md 103 + mWs
(6)
In eq 6, d is the absolute density of the solution (g/cm3), and Ws is the molecular weight of the solute. Experimental data concerning osmotic coefficients, activity coefficients, and solution densities are collected in ref 34. Major literature referring to experimental work is explicitly reported in refs 27 and 3537. Densities of aqueous NaCl solutions at 25 °C are reported in ref 38. Osmotic and activity coefficients concerning ZnSO4 are reported in ref 25. In some cases, several sources of data have been used to cover the overall range of concentrations for which γ e 1. For example, in the case of NaClO4 (Figure 1), osmotic and activity data are from ref 37 up to 6m and from ref 39 for higher concentrations; density data are from ref 40. Hereafter, mlim refers to the molality corresponding to clim. The values of clim and mlim have been given in Figure captions. Osmotic coefficients have been obtained from eq 1 via the numerical integration of ln(γ). Molalities range from 0 to mlim. Calculations have been made putting ) 78.30 ( ) water permittivity; 0 )vacuum permittivity) and T ) 298.15 K. As a note, very fast converging formulas exist,33 which could be used to improve the numerical computation of eq 4; however, results are sufficiently accurate even from the present form of eq 4 (for the asymmetric case, see ref 1, eq 17). For a better understanding of results in Figures 1-9, a comparative examination of the pseudolattice model in the context of recent literature can be useful, especially with respect to models that show common features with this one. Accordingly, one can first say that the pseudolattice approach belongs to the class of the “effective-charge” theories, such as the dressed ion theory.41,42 These theories implicitly involve ion-ion interactions (long-range effects) and solvation/many-body cor-
Osmotic Coefficients of Electrolyte Solutions relations (short-range effects) by means of suitable effective densities and effective potentials. In the pseudolattice model, the lsd, U, characterizes the effective charge density “generated” by the stochastic motion of a carrier within the solution. Ionion interactions are predominant at dilute solutions (while U is not largely different from its extreme-dilution expression), whereas local effects, which are included through the “extent” of each carrier, are the most contributing term in U at the highest concentrations (because of the increasing extent of carriers).1 However, in general U derives from the not trivial composition of all the contributions and their separate evaluation could be a difficult task. The explicit modeling of each contribution is a typical procedure in theories such as Pitzer theory14 and MSANRTL,22 but these theories unfortunately require a large number of fitting parameters to this aim. The NRTL offers also an example of a model in which lattice concepts are adopted, and the electrostatic-exclusion effects, allowing only charge with opposite sign to be immediately neighboring, suggest a similarity with the ionic fcc scheme summarized in the previous paragraph. However there is an important difference, since in the pseudolattice approach the fcc scheme (or hexagonal close-packed (hcp) for 1:2/2:1 systems1) does not model the local composition of an arrangement of ions and dipoles (that is, of a carrier); instead, it models the time-average distribution of the ensemble of arrangements within the solution. Figures 1-3 refer to some representative 1:1 electrolytes for which clim values are experimentally known, thus neither numerical fitting nor unknown quantities must be accounted for in all these cases. For all these systems except for NaClO4 (Figure 1) and KF (Figure 3), activity coefficients have been reported in ref 1. In Figure 1 the mean activity coefficients of NaClO4 are also reported to illustrate an example of the curve γ(m) obtained by eq 4. On the whole, results appear satisfactory for all the electrolytes; however the agreement between calculated and measured osmotic coefficients depends on that of the activity coefficients, by means of their contribution into eq 1. Larger deviations from measured γ are evident at intermediate concentrations with respect to the range [0,clim] for any given electrolyte.1 However, this is expected since the analytical form of U of eq 3 is not rigorously valid for c in the neighbors of clim/2 (s )1/2) and less, as summarized in the previous paragraph; thus calculated γ values at these concentrations are expected to be less accurate than elsewhere. Indeed, for NaCl the maximum deviations occur between 1 and 2.5 M (about), and clim is 5.4 M; for NaOH, the range is 0.8-2 M (about), and clim is 4.4 M and so on (see Figures in ref 1). It is interesting to note that goodness of results does not depend on the value of clim; indeed NaClO4, with clim ) 8.63 M, shows an accuracy somewhat better than other cases corresponding to lower clim. Moreover, the deviation from experimental data generally has the same order of magnitude for all the 1:1 electrolytes (for both activity and osmotic coefficients). The accuracy of results from the pseudolattice model can be compared with that from other treatments (see also the Supporting Information) and first from other one-parameter models, which are generally based on the MSA, and always require, it must be recalled, a fitting procedure because of the presence of the unknown ion-diameter parameter. In ref 18, ranges of concentrations for alkali metal halides are shorter than those used with the pseudolattice model, for example, clim is 5.57 M for KF, whereas the highest concentration is 1.4 M in ref 18. In ref 20, the corrected DH theory provides osmotic coefficients within an error comparable with that from the pseudolattice model; however, available ranges of concentrations are not
J. Phys. Chem. B, Vol. 112, No. 4, 2008 1215 exceeding 2 M. The limited applicability of these MSA-based models can be related to their difficulty in accounting for shortrange effects (such as ion association) at elevated concentrations by means of ion-size-based parameters only. Among the multiparameter theories, present results can be compared with those from MSA-NRTL,22 where the ranges of concentrations are equal or larger (even much larger) than those used here. Moreover, the deviations from experimental data (for both activity and osmotic coefficients) are generally very little in ref 22. However, besides the obvious need of strong numerical fittings, some questions concern the unclear physical meaning of parameters used in the NRTL part of the model to account for the “nearest neighboring” probability between particles.22 Moreover, the inclusion of hydration effects43,22 seems to cause new limitations (e.g., shorter ranges of allowable concentrations) unless introducing further terms (and parameters) in the model itself. The problem of a clearly understandable use of multiparameter models has been often discussed in literature.2,3,10 Figures 4-6 refer to some representative asymmetric electrolytes for which clim values are experimentally known. For all cases except for SrBr2 (Figure 4), activity coefficients have been reported in ref 1. Previous comments concerning the general trend of results still apply in the asymmetric case, and also the comparison with other mono-20 and multiparameter22 models allows similar conclusions. However, the deviations of activity (and thus of osmotic) coefficients from measured data are more heterogeneous for 2:1/1:2 than for 1:1 electrolytes. As a general trend (but with some exceptions), systems involving I-, Br-, Cl-, and (ClO4)- show deviations comparable to those of 1:1 electrolytes, whereas deviations are more significant for electrolytes involving (NO3)-. On the one hand, this more electrolyte-specific dependence can be explained with a stronger approximation in using U (cf. ref 1, eq 16) to calculate γ in the intermediate range of concentrations. On the other hand, this could also depend on the “tout-court” use of the hcp scheme for all the 1:2/2:1 electrolytes,1 though there are many other representative prototypal schemes29(conversely, the fcc lattice is the most representative for the symmetric case29). Besides these considerations, there are also wider differences among asymmetric electrolytes due to ion association effects occurring at low concentrations. For example, Figure 6 shows the case of ZnCl2, which experimentally exhibits an anomalous plateau effect44 in the range 0.1-0.4 m (about), probably depending on the formation of (ZnCl)+ complexes in that range of molalities.44 Since clim ) 7.4 M, one has s ) 1/2 at c ) 3.7 M, so the range above belongs to s , 1/2 where the pseudolattice model is strictly available for free ions only. Although (neutral) ion pairs at low concentrations can be suitably considered in the model (see examples below); nevertheless including charged complexes at these concentrations requires changes even at the reference-lattice level. However, these specific situations will be considered in future works, since the overall overview of the pseudolattice model and of its general potentialities are the main concerns at present. Figure 7 shows some representative cases, that is, CsCl and K2CrO4, for which the saturation concentration, csat, is less than clim, so clim is an unknown parameter that has to be theoretically estimated. Such an estimate derives from an elementary fitting of calculated activity coefficients to the experimental ones.1 For both systems, the accuracy of results is comparable to that given by many-parameter models.14,21 As yet mentioned in ref 1, the case of CsCl exemplifies also a situation where, clim being equal to 18 M, the mean distance between ions should be 3.56 Å at c ) clim, that is, shorter than the crystallographic one. However
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this causes no physical inconsistencies because clim is only a limit value that does not affect ionic parameters at their true concentrations (ecsat). On this subject, it has to be noted that conceptual difficulties are often encountered in literature, especially with respect to rubidium or cesium halides according to the decreasing trend of mean diameters for alkali metal halides with cations in the order Li+, Na+, K+, Rb+, and Cs+.2,10,18,20,22 In Figures 8-9, two electrolytes are considered for which ion-pairing effects seem to be relevant even at very dilute solutions,30,31 that is, H2SO4 and ZnSO4. Let γobs be the observed mean activity coefficient and Rf the dissociation degree,2,3 then one has1
1 1 γobs ≡ Rfγf ≈ Rfγ, s , ; γobs ≡ γ, s g 2 2
(7)
As stressed in eq 7, when s g 1/2 ion association is implicitly accounted for by the pseudolattice approach. This can be better understood considering that, at high concentrations, the concept of “effective carrier” imposes no restriction on the composition of the carrier itself. Indeed, although there should be one ion per carrier on the average (the number of effective carriers is always equal to that of solute ions), nevertheless, has been discussed elsewhere41,42 basing on rigorous statistical mechanics, even the solvent molecules can behave as charged structures when regarded at the local-arrangement scale. Moreover, a carrier is somewhat a “dynamic” structure changing in composition for increasing concentrations, while keeping unaltered the overall charge. The extent of a carrier increases with concentrations allowing the carriers to overlap upon each other because of their large U and thus allowing more than one ion to “belong” to the same carrier (complying with possible ion association), though the role of each included particle has not been so far better pre´cised by the model. It can be noted that even at very dilute solutions one has large U; however, the solvent behaves as an electroneutral medium, so each carrier corresponds exactly to one solute ion (and large U are only due to the stochastic motion of carriers). Thus, ion association must be explicitly included for s , 1/2. This can be done according to refs 2 and 3 by using eq 7 and the association constant, KA
Rf ) 1 - KAγobs2m )
-1 + x1 + 4γ2KAm 1 ,s, (8) 2 2 2γ KAm
KA ) 204 has been used31 for ZnSO4, whereas KA ) 83 has been used30 for H2SO4. As to H2SO4, since KA is small compared to that of metal sulfates, ion pairs have been roughly treated as neutral at strong dilutions, to avoid changes in the hcp scheme. The crucial point is that one has to go from γobs ) Rfγ at very dilute solutions to γobs ) γ at high concentrations. The values of γobs at intermediate concentrations cannot be known at present unless adopting a suitable numerical interpolation of Rfγ and γ so as to satisfy the constraints of eq 7. However, numerical difficulties arise from the use of too a rough interpolation. From the cubic interpolation of γobs proposed in ref 1 and from the rough estimate of the range of its applicability, osmotic coefficients are not satisfying when calculated using eq 1. Indeed, if γobs becomes sufficiently small (as it occurs for H2SO4 and ZnSO4 at their intermediate concentrations) then ln(γobs) becomes so large that even a small numerical sharpness caused by the interpolation becomes significant in eq 1. This can be seen in Figures 8 (H2SO4) and 9 (ZnSO4), where the calculated γobs, deriving from the smoother interpolation of eq 9 below, are very close to the experimental values. Nevertheless,
by use of eq 1, the deviations from the measured osmotic coefficients are more significant than for all the cases so far considered and more remarkably for ZnSO4 (that shows lower γobs compared to H2SO4). That said, the interpolating formula of eq 9 has been applied over the whole range of concentrations from 0 up to mlim (or up to the saturation molality, msat, when msat < mlim)
γobs(m) )
x
( )
m γ, m0 m from 0 to the minimum between (msat, mlim) (9) (1 - tanh(m))Rf2 + tanh
For the sake of brevity, in eq 9, m0 indicates the molality corresponding to s ) 1/2 (c ) clim/2). It is easily seen from eq 9 that m f 0 gives γobs ≈ Rfγ, whereas m g m0 gives γobs ≈ γ, according to eq 7. For H2SO4, the experimental value of clim ) 8.78 M has been used34 corresponding to m0 ) 5.4 m; for ZnSO4, the theoretical value clim ) 7.4 M has been used in place of 8 M as reported in ref 1, according to recently revised experimental data concerning this metal sulfate.25,44 To evaluate clim for ZnSO4, calculated γ have been fitted to experimental ones so as to have the very same trend at the highest available concentrations (near saturation). For ZnSO4, the theoretical m0 is 3.95 m. The formation of ion pairs in metal sulfates at very dilute solutions is a debated point in literature.24,25,30,31,44 An objection is that experimental KA values are too largely depending on the measurement method, e.g., for ZnSO4, KA values range from 111 to 309 about.44 Another objection derives from the fact that, by means of suitable shape-based parameters in the Pitzer theory, it is possible to calculate γ and φ without considering ion association at strong dilutions.24,25,44 However, as outlined in ref 24, calculations of the excess free energies from both association- and non-association-based theories are not conclusive at present; that legitimates also the conclusions of the pseudolattice model. Concluding Remarks In this work, an application of the pseudolattice model1 has been presented, that is, the calculation of the osmotic coefficients for symmetric (1:1, 2:2) and for asymmetric (1:2/2:1) aqueous electrolytes at 25 °C, in the general case of mean activity coefficients e1. The presented results seem to confirm also for osmotic coefficients a research interest about the pseudolattice approach. However, many aspects of the model need to be further investigated. Some of them are related to the “effective carrier” concept and its relationships with other effective-chargebased theories,41,42 especially concerning the modeling of shortrange effects. Other aspects concern the analytical improvement of formulas for the intermediate range of concentrations, where the strongest approximations have been so far carried out. Moreover, though the mathematical frame of the model is available also for nonaqueous solvents and for any temperature, detailed applications under more general conditions are to be investigated as well as the application to higher order electrolytes. Also ion association at very dilute solutions will require a more detailed discussion. Besides these considerations, a major point remains the theoretical prediction of clim, so that the model can be of some practical usefulness. Acknowledgment. This work has been supported by the Italian Ministry for University and Research (MIUR), and by Biosafety Engineering, Italy. The author greatly thanks Prof. Bruno Bianco, University of Genoa, Italy, for his encouraging support.
Osmotic Coefficients of Electrolyte Solutions Supporting Information Available: Figures S1-S5 give the plots of φ vs m for the following aqueous electrolytes at 25 °C: NaBr, HCl, NaI, HBr, LiCl, LiBr, Ca(ClO4)2, Mg(ClO4)2, Ba(ClO4)2, CoI2. Figures S6 and S7 show a comparison of the results from the pseudolattice model with those from Hypernetted Chain and Monte Carlo simulations15-17,20 in the cases concerning, respectively, LiI and KOH. The case of CaCl2 is illustrated in figure S8, where a comparison is shown with the MSA combined with the Born-Peng-Robinson equation.21 This information is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Moggia, E.; Bianco, B. J. Phys. Chem. B 2007, 111, 3183. (2) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1973. (3) Conway, B. E. ComprehensiVe Treatise of Electrochemistry; Plenum Press: New York; 1983, Vol. 5. (4) Grenthe, I.; Plyasunov, A. Pure Appl. Chem. 1997, 69, 951. (5) Debye, P.; Hu¨ckel, E. Z. Phys. 1923, 24, 185. (6) Debye, P.; Hu¨ckel, E. Z. Phys. 1924, 25, 97. (7) Bro¨ensted, J. N. J. Am. Chem. Soc. 1922, 44, 877. (8) Guggenheim, E. A. Philos. Mag. 1935, 19, 588. (9) Scatchard, G. Chem. ReV. 1936, 19, 309. (10) Robinson, R. A.; Stokes, R. H. J. Am. Chem. Soc. 1948, 70, 1870. (11) Outhwaite, C. W. J. Chem. Phys. 1969, 50, 2277. (12) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1972, 56, 3086. (13) Bromley, L. A. AIChE J. 1973, 19, 313. (14) Pitzer, K. S. ActiVity Coefficients in Electrolyte Solutions; CRC Press Inc.: Boca Raton, Florida; 1979; Vol. 1. (15) Sloth, P.; Sørensen, T. S. J. Phys. Chem. 1990, 94, 2116. (16) Molero, M.; Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 1992, 88, 1541. (17) Outhwaite, C. W.; Molero, M.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 1993, 89, 1315. (18) Fawcett, W. R.; Tikanen, A. C. J. Phys. Chem. B 1996, 100, 4251. (19) Thoenen, T.; Hummel, W. J. Conf. Abstracts 2000, 5, 997.
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