An Isotherm-Based Thermodynamic Model of Multicomponent

Article pubs.acs.org/JPCA

An Isotherm-Based Thermodynamic Model of Multicomponent Aqueous Solutions, Applicable Over the Entire Concentration Range Cari S. Dutcher,*,† Xinlei Ge,† Anthony S. Wexler,†,‡ and Simon L. Clegg*,†,§ †

Air Quality Research Center and ‡Departments of Mechanical and Aerospace Engineering, Civil and Environmental Engineering, and Land, Air and Water Resources, University of California at Davis, Davis, California 95616, United States § School of Environmental Sciences, University of East Anglia, Norwich NR4 7TJ, United Kingdom ABSTRACT: In previous studies (Dutcher et al. J. Phys. Chem. C 2011, 115, 16474−16487; 2012, 116, 1850−1864), we derived equations for the Gibbs energy, solvent and solute activities, and solute concentrations in multicomponent liquid mixtures, based upon expressions for adsorption isotherms that include arbitrary numbers of hydration layers on each solute. In this work, the long-range electrostatic interactions that dominate in dilute solutions are added to the Gibbs energy expression, thus extending the range of concentrations for which the model can be used from pure liquid solute(s) to infinite dilution in the solvent, water. An equation for the conversion of the reference state for solute activity coefficients to infinite dilution in water has been derived. A number of simplifications are identified, notably the equivalence of the sorption site parameters r and the stoichiometric coefficients of the solutes, resulting in a reduction in the number of model parameters. Solute concentrations in mixtures conform to a modified Zdanovskii− Stokes−Robinson mixing rule, and solute activity coefficients to a modified McKay−Perring relation, when the effects of the long-range (Debye−Hückel) term in the equations are taken into account. Practical applications of the equations to osmotic and activity coefficients of pure aqueous electrolyte solutions and mixtures show both satisfactory accuracy from low to high concentrations, together with a thermodynamically reasonable extrapolation (beyond the range of measurements) to extreme concentration and to the pure liquid solute(s).

1. INTRODUCTION Sorption isotherms, such as those of Brunauer−Emmett−Teller (BET)1 and, later, Guggenheim−Anderson−de Boer (GAB)2−4 are able successfully to represent the relationship between water activities (aw) and solute concentrations in aqueous solutions for water activities over the approximate ranges 0 ≤ aw < 0.4 and 0 ≤ aw < 0.75, respectively. The application of sorption isotherms to solutions, first suggested by Stokes and Robinson,5 implies that solute molecules are surrounded by a monolayer of sorbed water molecules. For the BET and GAB isotherms, the water molecules outside the monolayer are assumed to exist in the bulk, or the multilayer, where they may (GAB) or may not (BET) have an association with the solute.6 The ability to model the thermodynamic properties of very concentrated solutions makes these isotherms ideal for applications involving low water content, such as drying processes in food science.7−10 Recently, using statistical mechanics, we developed a sorption isotherm that includes an arbitrary number of sorbed monolayers,6 each with its own energy of adsorption. The resulting expressions, with these multiple layers, are able to represent solvent and solute activities, and solute concentrations, in single solute solutions and aqueous mixtures over the extended range of 0 ≤ aw < 0.95.6,11 The widely used Zdanovskii− Stokes−Robinson12 (ZSR) empirical mixing equation was also derived, for the first time, using a statistical mechanical formalism. © 2013 American Chemical Society

However, a limitation of these isotherm-based approaches is that as the solutions become dilute (i.e., as aw nears unity), their accuracy declines. Ion-interaction models, such as those of Pitzer13 and Pitzer− Simonson−Clegg,14−16 have proven successful at representing the thermodynamic properties of dilute solutions, and solutions at concentrations up to saturation with respect to many common salts. In these and similar models the effects of long-range Coulombic electrostatic forces that arise between charged ions in a dielectric medium are represented explicitly (cf. refs 17−20) using formulations of the Debye−Hückel21 limiting law. Equations in common use, such as those of Pitzer,13 allow extension to more concentrated solutions by incorporating additional terms, and/or adjustable coefficients. However, current ion-interaction models− especially those that use molality as the concentration variable− do not extrapolate well to extremely high concentrations. In this work, we combine the sorption isotherm−based equations of Dutcher et al.6,11 with the mole fraction Debye−Hückel expression of Pitzer13−15 to produce a model, shown schematically in Figure 1, that can be applied over the entire range of solute concentrations. We present equations for any combination Received: November 2, 2012 Revised: January 11, 2013 Published: April 8, 2013 3198

dx.doi.org/10.1021/jp310860p | J. Phys. Chem. A 2013, 117, 3198−3213

The Journal of Physical Chemistry A

Article

the quantities are those of single solute (binary) aqueous solutions. For nj = 2, eq 2 simplifies to the Brunauer−Emmett− Teller (BET)1 sorption isotherm when Kj is equal to unity, and to the Guggenheim−Anderson−deBoer (GAB)2−4 sorption isotherm when Kj ≠ 1.0. Dutcher et al.11 expanded the sorption model to multicomponent aqueous mixtures containing an arbitrary number of solutes, A, B, ..., Z, and derived the following expressions for the activity, aj, and molality, mj, of each solute j in the mixture:11 ⎛ ⎞r j 1 − a w Kj ⎜ ⎟ * aj = xj ⎜ nj − 1 i−1 ⎟ i ⎝ 1 − ∑i = 1 ((a w Kj) (1 − Cj , i) ∏k = 1 Cj , k) ⎠

Figure 1. Schematic of a solution containing two solutes A and B (lettered circles) with sorbed monolayers (rings) of water molecules and a multilayer containing only water (small, unlabeled circles). The arrow represents the long-range Debye−Hückel interactions between solute ions that have been added to the model in this study. The multilayers (the nth adsorption layer) of all solutes do not differ in terms of their energies and are therefore not distinguished in this figure.

∑ j

=1 (4)

⎛ ⎞ ∑j Xj , n ⎟ G /kT = Nw ln⎜⎜ ⎟ ∑ ( ) K X X + j j , n 1 j , n − ⎝ j ⎠

2. THEORETICAL DEVELOPMENT Consider an aqueous solution containing a single solute j, on which there are rj sorption sites per molecule. Water molecules directly sorbed to one of these sites constitute the first monolayer, characterized by an energy of sorption parameter Cj,1. Further, weaker sorption can occur to form higher order monolayers, 2 to (nj − 1), with associated energy of sorption parameters Cj,2 to Cj,nj−1, respectively. Water molecules not involved in monolayer sorption are located in the multilayer or bulk layer, nj, which has energy parameter Kj. Such a system is illustrated in Figures 1d and 2b of Dutcher et al.6 In that work the following expressions for the solute activity, aoj , and solute molality, moj , were derived using statistical mechanics:6 ⎛ ⎞r j 1 − a wo Kj ⎟ = ⎜⎜ nj − 1 i−1 ⎟ o i ⎝ 1 − ∑i = 1 ((a w Kj) (1 − Cj , i) ∏k = 1 Cj , k) ⎠

mjo

where the summation in eq 4 is over all solutes, and xj* in eq 3 is the dry mole fraction of solute j which is calculated from the amounts of all solutes present but excluding the amount of water solvent from the denominator (e.g., see text following eq 20 of Dutcher et al.11). Equation 4, in which the molalities in pure aqueous solutions (moj ) and the mixture (mj) are for the same water activity, is equivalent to the Zdanovskii−Stokes−Robinson (ZSR) mixing rule.5 The expression for the Gibbs energy of the solution from which eqs 1−4 were derived, for an indefinite number of solutes each with nj sorption layers consisting of (nj − 1) monolayers and a bulk layer, is given by eq A17 of Dutcher et al., repeated here for convenience:11

of charged (“electrolyte”) and neutral (“nonelectrolyte”) solutes. The treatment of nonelectrolytes in the equations is addressed specifically, when it is different from that of the electrolytes. Equations for solute molalities, activities, and activity coefficients are derived, including a function for the conversion of activity coefficients to a reference state of infinite dilution with respect to the solvent. In developing the model, our initial interest was the accurate simulation of highly concentrated aqueous solutions in atmospheric aerosol particles relevant to climate and air quality. However, multicomponent, electrolyte-containing aqueous solutions are important in a wide range of applications from grain drying (food science) to the properties of brines (geochemistry, soil science, and chemical engineering). The application of the model to predict the solvent and solute activity coefficients of these highly complex solutions should lead to improved treatments of condensed phase physicochemical processes.

ajo

mj

+

⎛

⎛

⎝

⎝

∑ Nj⎜⎜ln(x*j ) + rj ln⎜⎜1 − j

Xj ,1 ⎞⎞ ⎟⎟ rjNj ⎟⎠⎟⎠

(5)

where Nw is the number of water molecules, Nj is the number of molecules of solute j, Xj,i is the number of sorbed water molecules in layer i of solute j, k is Boltzmann’s constant, and T is temperature. Equations 1−5 accurately express solution properties up to water activities near 95% but become less accurate at higher water contents because no long-range electrostatic interactions are included. 2.1. Long Range Interactions. To more accurately predict solvent and solute activities in dilute solutions, the mole fraction based Debye−Hückel excess Gibbs energy expression of Pitzer13−15,21 has been included in the Gibbs energy given by eq 5 and used to obtain modified equations for solute molalities, and solvent and solute activities. The additional term accounts for long-range interactions due to electrostatic screening of the ions in solution. The equation for the Gibbs energy, including the additional term, is

(1)

and

⎛ ⎞ ∑j Xj , n ⎟ G /kT = Nw ln⎜⎜ ⎟ ⎝ ∑j Kj(Xj , n − 1 + Xj , n) ⎠

⎛ 1 − a wo Kj ⎞ n −1 i−1 i o ⎜ ⎟(1 − ∑i =j 1 ((a w Kj) (1 − Cj , i) ∏k = 1 Cj , k)) M w rja wo Kj ⎠ ⎝ mjo = ⎛ (1 − a o K )2 ∑nj − 2 (p(a o K ) p − 1 ∏ p C ) + ⎞ w j w j k=1 j,k p=1 ⎜ ⎟ ⎜((n − 1) − (n − 2)a o K )(a o K )nj − 2 ∏nj − 1 C ⎟ j w j w j ⎝ j k=1 j,k ⎠

+

⎛

⎛

⎝

⎝

∑ Nj⎜⎜ln(x*j ) + rj ln⎜⎜1 − j

(2)

aow

(3)

−1

− 2Ax (Nw +

where is the water activity of the system, Mw (kg mol ) is the molecular weight of water, and superscript “o” indicates that

∑ νjNj) j

3199

Xj ,1 ⎞⎞ ⎟⎟ rjNj ⎟⎠⎟⎠

∑j Nj|zj −|zj +νjYj ∑j νjNj + Nw

(6)



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“DH”, is given by

where Ax is the Debye−Hückel coefficient on a mole fraction basis (equal to 2.917 at 298.15 K),22 zi is the charge on ion i (cation j+, or anion j−, of solute j), and νj is the stoichiometric coefficient of solute j, which is the number of moles of ions formed from 1 mol of electrolyte j. Equation 6 can also be obtained by including the long-range electrostatic expression in the constraining statistical mechanical energy expression (eq 10 in Dutcher et al.11) for each species in the solution. For nonelectrolytes νj is equal to unity. The function Yj is given by13−15,21 ⎛ 1 + ρ I 1/2 ⎞ j x ⎟ Yj = (1/ρj ) ln⎜⎜ ref 1/2 ⎟ + 1 ( ) ρ I ⎝ ⎠ j x ,j

K wDH

1 ∑j Nj|zj −|zj +νj 1 ∑j mj|zj −|zj +νj = 2 ∑j νjNj + Nw 2 ∑j νjmj + M w −1

(

)) ⎞⎟⎟

(

⎟ ⎠

(11)

When the values of ρj are the same for all solutes j, the 3/2 1/2 expression for KDH w reduces to exp[2AxIx /(1 + ρIx )] . Because ∑k Xk,n/∑k Kk(Xk,n−1 + Xk,n) is equal to Xj,n/Kj(Xj,n−1 + Xj,n) for any solute j,6 where the summations are over all solutes k, eq 10 can be rewritten as a K DH a w = w̅ w Kj

(7)

where Ix is mole fraction ionic strength of the solution, equal to (1/2)∑i xizi2, xi is the mole fraction of ion i; and Iref x,j is the ionic strength of a pure solution of j at the chosen reference state (either the pure liquid solute or infinite dilution in water). Pitzer and Simonson13 showed that the value of ρj in eq 7 can be related to the hard core collision diameters of the solute ions, as shown in their eq 34, but it is usually treated empirically and in a number of previous studies14,15,23 has been set to a constant value for all solutes. In this work, the model equations do not include an extended Debye− Hückel term (with additional parameter BMX), such as used by Clegg and co-workers,14,15 and so ρj is treated as an adjustable parameter. The term multiplying each Yj in eq 6 represents the fractional contribution to the ionic strength of each electrolyte j. This expression is equivalent to eq 42 of Clegg and Pitzer,14 as long as the amounts of electrolytes present are calculated using the “equivalent fraction” approach of Clegg and Simonson24 (see their eq 4), which was originally proposed by Reilly and Wood.25 The ionic strength, in terms of the moles of solutes present, and alternatively their molalities, is given by Ix =

⎛ 1/2 1/2 ⎜ Ax Ix ∑j Nj|zj −|zj +νj/ 1 + ρj Ix = exp⎜ ∑j νjNj + Nw ⎜ ⎝

(12)

where the sorption contribution, aw̅ , is given by Xj , n aw̅ = X +X j,n−1

j,n

(13)

Likewise, the activity of solute j, aj, is found by differentiating eq 6 with respect to total solute content, Nj, yielding aj = aj̅ KjDH

(14)

where the sorption contribution, aj̅ , is given by

(

aj̅ = x*j 1 − Xj ,1/rjNj

rj

)

(15)

and the Debye−Hückel contribution for electrolyte solutes j, KDH j , is given by KjDH

⎛ ⎡ ⎧ 1/2 ⎞ ⎪ 2 ⎛⎜ 1 + ρj Ix ⎜ ⎢ ⎟ ⎨ ln = ⎜exp⎢ − zj +|zj −|Ax ⎜ ⎢ ⎪ ρj ⎜⎝ 1 + ρj (Ixref, j )1/2 ⎟⎠ ⎩ ⎝ ⎣

) ⎞⎟ ∑

⎛ 1 − 2I / z | z | x j+ j− +⎜ ⎜ ∑ (νkNk) + Nw ⎝ k

(

(8)

⎟ ⎠

k

⎫⎤ ⎞ j ⎪⎥ ⎟ ⎬⎥ ⎟ 1/2 1/2 2Ix (1 + ρk Ix ) ⎪⎥⎟ ⎭⎦ ⎠ v

Nk|zk −|zk +νk

(16)

where j+ and j− refer to the cation and anion of electrolyte j, respectively, and the summations are over all electrolytes and neutral (uncharged) solutes. As noted above, the reference state used for Iref x,j can be either = 0) or the pure liquid solute the limit of infinite dilution (Iref x,j or fused salt (e.g., Iref = 1/2 for a 1:1 electrolyte). In section 3, x,j and in Appendix A, we describe the conversion from one reference state to another. Here, we assume the pure fused salt reference state for the activities of the solutes, for which Iref x,j is given by Ixref, j = lim Ix , j = Nw → 0

where the summations are over all solutes k in the solution. The following equation for the activity of electrolyte solute j is obtained for the case where all ρj are the same:

(

aj = x*j 1 − Xj ,1/rjNj

⎛ ⎡ ⎧ 1 − 2Ix /zj +|zj −| ⎪ ⎜ × ⎜exp⎢ −zj +|zj −|Ax ⎨Ix1/2 ⎢ ⎪ ⎜ 1 + ρIx1/2 ⎩ ⎝ ⎣ v ⎫⎤⎞ j ⎛ 1/2 ⎞⎪ 2 ⎜ 1 + ρIx ⎟⎬⎥⎟ + ln⎜ ⎥⎟⎟ ρ ⎝ 1 + ρ(Ixref, j )1/2 ⎟⎠⎪ ⎭⎦⎠

|zj −|zj + 2

(9)

(17)

For nonelectrolyte solute j, in a solution containing electrolytes, the Debye−Hückel term is

The expression for the water activity, aw, is found by differentiating eq 6 with respect to total water content, Nw, yielding ⎛ ⎞ ∑j Xj , n ⎟ a w = K wDH⎜⎜ ⎟ ⎝ ∑j Kj(Xj , n − 1 + Xj , n) ⎠

rj

)

⎛ A I 1/2∑ (N |z |z ν / 1 + ρ I 1/2 ⎞ )) ⎟ x x k k− k+ k ( k x k KjDH = exp⎜⎜ ⎟ ∑k νkNk + Nw ⎝ ⎠

(10)

(18)

where the Debye−Hü ckel contribution KwDH , which is distinguished from the energy parameters Kj by the superscript

This is the same as the expression for the contribution to the water activity, given by eq 11 (see section 3 for further discussion). For solutions containing only nonelectrolytes, the 3200



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In addition, for a water activity equal to unity, the osmotic coefficient, ϕ, must also equal unity. The osmotic coefficient is related to the molality of solute j in a pure aqueous solution, and the water activity, by

Debye−Hückel expressions within the exponents in eqs 11 and 18 are zero, and only the sorption terms remain. 2.2. Pure Solutions. For practical applications of the model eqs 10−18, the amounts of solutes (Nj) and solvent (Nw), and the activities (aj and aw) can be related to Xj,i and the model parameters Cj,i in the same way as before.6 For an aqueous solution containing a single solute j with an arbitrary number of sorption layers, n, the molality of the solute (moj ) is given by

ϕ=

(19)

aow̅

where is the sorption contribution term, defined in eq 13, for the pure aqueous solution. The term aow̅ is equal to (Kjaow/ DH,o KDH,o is given by w ), where Kw

(20)

Iox

where is the ionic strength of the single-solute solution. Equation 19 for solute molality is the same as eq 2, except that water activity, aow, is replaced by the sorption contribution, aow̅ . The activity of the solute (aoj ) in a pure aqueous solution of j is given by

ajo = a ̅joKjDH,o

(21)

where a ̅jo

⎛ ⎞r j 1 − a w̅ o ⎜ ⎟ =⎜ nj − 1 i−1 ⎟ o i ⎝ 1 − ∑i = 1 ((a w̅ ) (1 − Cj , i) ∏k = 1 Cj , k) ⎠

(22)

and KjDH,o

⎛ ⎡ ⎧ 1 − 2Ixo/zj +|zj −| ⎪ ⎜ ⎢ = ⎜exp⎢ −zj +|zj −|Ax ⎨Ix1/2 ⎜ ⎢ ⎪ 1 + ρj (Ixo)1/2 ⎩ ⎝ ⎣ o 1/2 ⎞⎫⎤⎞ ⎛ 2 ⎜ 1 + ρj (Ix ) ⎟⎪⎥⎟ ⎬ + ln⎜ ρj ⎝ 1 + ρj (Ixref, j )1/2 ⎟⎠⎪⎥⎥⎟⎟ ⎭⎦⎠

(24)

Substituting eq 19 into eq 24, and using L-Hôpital’s rule to evaluate the limit, yields rj equal to νj at aw equal to 1.0. Assuming that the intensive parameter rj, which is the number of sorption sites on solute molecule j, is independent of concentration then rj is equal to νj for all concentrations and solution compositions. Parameter rj was also treated as adjustable by Dutcher et al.6,11 and the fitted values (2.003 for NaCl, 1.638 for NH4NO3, 2.160 for NaOH, and 1.005 for glycerol) are similar to νj. The elimination of Kj, and the setting of rj to fixed values, reduces the number of adjustable parameters in the model by two. Final expressions for the solute molality (eqs 14 and 15) and solute activity (eqs 16 and 17) with Kj = 1.0, rj = νj, and for nj = 1 to 5 are given in Tables 1 and 2, respectively. Note that the expressions for solute molality are implicit, due to the presence of the ionic strength in the Debye−Hückel term, and should be solved numerically. In Appendix B we examine the sorption and Debye−Hückel contributions to solvent and solute activity coefficients and show that, as expected, the Debye−Hückel term accounts for the observed limiting law behavior in dilute solutions. This leaves a very simple sorption contribution, which is nearly linear with respect to molality for both the osmotic coefficient and solute activity coefficient in the example chosen. 2.4. Solution Mixtures. Generalizing the equations in the same way as described in section 3 and Appendix A.2 of Dutcher et al.,11 it is found that for an arbitrary mixture of j electrolyte and nonelectrolyte solutes: mj ∑ o =1 m̅ j j (25)

⎛ 1 − a ̅wo ⎞ ⎟(1 − ∑in=j −11 ((a w̅ o )i (1 − Cj , i) ∏ik−=11 Cj , k)) ⎜ M wrja ̅wo ⎠ ⎝ mjo = ⎛ (1 − a o )2 ∑nj − 2 (p(a o ) p − 1 ∏ p C ) + ⎞ w w ̅ ̅ k=1 j,k p=1 ⎜ ⎟ o o n − ⎜((n − 1) − (n − 2)a )(a ) j 2 ∏nj − 1 C ⎟ j w w ̅ ̅ ⎝ j k=1 j,k ⎠

⎛ ⎞ 2Ax (Ixo)3/2 ⎟ K wDH,o = exp⎜⎜ o 1/2 ⎟ ⎝ 1 + ρj (Ix ) ⎠

−(ln a wo ) M w mjoνj

and aj = x*j KjDHa ̅jo

rj

(26)

where m̅ oj is the molality of solute j in a pure aqueous solution at the aw̅ of the mixture (which is given by eq 12 with Kj = 1.0). Quantity aoj̅ is the sorption contribution to the activity of j in a pure aqueous solution, given by eq 22, at the aw̅ of the mixture. Thus,

(23)

KDH,o j

⎛ 1 − a ̅w ⎞ ⎜ ⎟(1 − ∑in=j −11 ((a w̅ )i (1 − Cj , i) ∏ik−=11 Cj , k)) M a ν ⎝ w j ̅w ⎠ m̅ jo = ⎛ (1 − a )2 ∑nj − 2 (p(a ) p − 1 ∏ p C ) + ⎞ w w ̅ ̅ k=1 j,k p=1 ⎜ ⎟ ⎜((n − 1) − (n − 2)a )(a )nj − 2 ∏nj − 1 C ⎟ j w w ̅ ̅ ⎝ j k=1 j,k ⎠

For nonelectrolyte solutes, is equal to unity. Equation 21 for the solute activity aoj in a pure aqueous solution is similar to eq 1 except for the factor KDH,o , which contains the Debye− j Hückel contribution to electrolyte activity, and the use of aow̅ rather than water activity (aow) itself in the calculation of the sorption term aoj̅ . 2.3. Dilute Solutions. To use eqs 19−23 to the limit of infinite dilution with respect to solute j, we first recognize that as aow approaches unity, both moj and aoj must approach zero. These limits, applied to eq 19 and eq 21, require that Kj equals 1.0 (defined in eq 12) when aow is equal to unity. Therefore, the constant Kj in the equations above must be equal to unity. This finding is consistent with our earlier work,6,11 wherein the Debye− Hückel term was not included and Kj was treated as an adjustable parameter whose fitted value was found to be near unity for several solutes (e.g., 0.985 for NaCl, 1.0015 for NH4NO3, 0.9864 for NaOH, and 0.99995 for glycerol; see Table 5 of Dutcher et al.6).

(27)

⎛ ⎞ 1 − a w̅ ⎟ a ̅jo = ⎜⎜ nj − 1 i−1 ⎟ i ⎝ 1 − ∑i = 1 ((a w̅ ) (1 − Cj , i) ∏k = 1 Cj , k) ⎠

νj

(28)

DH The values of KDH w and Kj , which are needed to relate aw ̅ and aoj̅ to water activity and solute activity, are given by eq 11 and eq 16, respectively. For cases where all ρj are equal, KDH w and KDH reduce to the same form as eqs 20 and 23. Although j the mixing relationship expressed by eq 25 has the same

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Table 1. Expressions for Single Solute Molality (moj ) as a Function of the Sorption Contribution to Water Activity (aow̅ ) for the Numbers of Sorbed Layers nj = 1−5a nj

mjo

1

⎛1 − ao ⎞ w ̅ ⎟ ⎜⎜ o⎟ ν M a ⎝ w j w̅ ⎠

2

⎛ 1 − ao ⎞ w ̅ ⎟(1 − a o (1 − C )) ⎜⎜ w j ,1 ̅ o⎟ M ν C ⎝ w j j ,1a w̅ ⎠

3

⎛ 1 − a o ⎞⎛ 1 − a wo (1 − Cj ,1) − (a wo )2 Cj ,1(1 − Cj ,2) ⎞ ̅ ̅ w ̅ ⎟⎜ ⎟ ⎜⎜ o ⎟⎜ ⎟ o 2 (1 − a w̅ ) + (2 − a w̅ o )a w̅ o Cj ,2 ⎝ M wνjCj ,1a w̅ ⎠⎝ ⎠

4

⎛ 1 − a o ⎞⎛ 1 − a wo (1 − Cj ,1) − (a wo )2 Cj ,1(1 − Cj ,2) − (a wo )3 Cj ,1Cj ,2(1 − Cj ,3) ⎞ ̅ ̅ ̅ w ̅ ⎟⎜ ⎟ ⎜⎜ o ⎟⎜ ⎟ o 2 o o o 2 ν M C a − + + − (1 a ) (1 2 a C ) (3 2 a )( ⎝ w j j ,1 w̅ ⎠⎝ ⎠ w w w ̅ ̅ j ,2 ̅ a w̅ ) Cj ,2Cj ,3

5

⎛ 1 − a o ⎞⎛ 1 − a wo (1 − Cj ,1) − (a wo )2 Cj ,1(1 − Cj ,2) − (a wo )3 Cj ,1Cj ,2(1 − Cj ,3) − (a wo )4 Cj ,1Cj ,2Cj ,3(1 − Cj ,4) ⎞ ̅ ̅ ̅ ̅ w ̅ ⎟⎜ ⎟ ⎜⎜ o ⎟⎜ ⎟ o 2 o o 2 o o 3 ν M C a (1 a ) (1 2 a C 3( a ) C C ) (4 3 a )( a ) − + + + − ⎝ w j j ,1 w̅ ⎠⎝ ⎠ w w w j ,2 j ,3 w w ̅ ̅ j ,2 ̅ ̅ ̅ Cj ,2Cj ,3Cj ,4

aow̅ KDH,o w a

a wo K wDH,o ⎛ ⎞ 2Ax (Ixo)3/2 ⎟ exp⎜⎜ o 1/2 ⎟ ⎝ 1 + ρj (Ix ) ⎠

The expressions in this table were obtained using eqs 19 and 20, with Kj = 1.0 and rj = νj.

Table 2. Expressions for Single Solute Activity (aoj ) as a Function of the Sorption Contributions to the Solute Activity (aoj̅ ) and Water Activity (aow̅ ) for the Number of Sorbed Layers nj = 1−5a nj 1 2

aoj̅

(1 − a w̅ o )νj ⎛ ⎞νj 1 − a w̅ o ⎜⎜ ⎟⎟ o ⎝ 1 − a w̅ (1 − Cj ,1) ⎠

3

⎛ ⎞νj 1 − a w̅ o ⎜ ⎟ ⎜ 1 − a o (1 − C ) − (a o )2 C (1 − C ) ⎟ ⎝ w j ,1 w j ,1 j ,2 ⎠ ̅ ̅

4

⎛ ⎞νj 1 − a w̅ o ⎜ ⎟ ⎜ 1 − a o (1 − C ) − (a o )2 C (1 − C ) − (a o )3 C C (1 − C ) ⎟ ⎝ w j ,1 w j ,1 j ,2 w j ,1 j ,2 j ,3 ⎠ ̅ ̅ ̅

5

⎛ ⎞νj 1 − a w̅ o ⎜ ⎟ ⎜ 1 − a o (1 − C ) − (a o )2 C (1 − C ) − (a o )3 C C (1 − C ) − (a o )4 C C C (1 − C ) ⎟ ⎝ w j ,1 w j ,1 j ,2 w j ,1 j ,2 j ,3 w j ,1 j ,2 j ,3 j ,4 ⎠ ̅ ̅ ̅ ̅

aoj KDH,o j

aoj̅ KDH,o j v ⎛ ⎡ ⎤⎞ j o ⎧ ⎛ 1 + ρ (I o)1/2 ⎞⎫ 1 2 / − | | I z z ⎪ ⎪ x x j + j − ⎜ ⎢ 2 j ⎟⎬⎥⎟ + ln⎜⎜ (Ixo)1/2 ⎜⎜exp⎢ −zj +|zj −|Ax ⎨ o 1/2 ref 1/2 ⎟ ⎥⎟ ρ ⎟ ⎪ 1 ( ) + (1 ( ) ) + I I ρ ρ j ⎝ ⎠⎪ j x j x ,j ⎭⎥⎦⎠ ⎩ ⎝ ⎢⎣

(

)

The expressions in this table were obtained using eqs 21−23, with Kj = 1 and rj = νj. The expressions for aow̅ and KDH,o for this system are the same w as those given in Table 1.

a

3. REFERENCE STATES FOR SOLUTE ACTIVITY The incorporation of a Debye−Hückel term into the model extends its range of application to include very dilute solutions. In the equations presented in section 2, for the sorption contribution to solute activity, the reference state

form as that in eq 4 (i.e., Zdanovskii−Stokes−Robinson), eq 25 is equivalent to this rule only for constant aw̅ , and not the water activity (aw) itself. Table 3 contains the equations for the two-solute, three-sorption layer case, as an example. 3202



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Table 3. Molalities (mA, mB) and Solute Activities (aA, aB) in a System Containing Two Solutes, with Three Sorption Layersa quantity

expression

mA 2

+

mB (1 − a ̅w )(1 − a ̅w (1 − C B,1) − (a ̅w )2 C B,1(1 − C B,2))

=1

mA, mB

(1 − a ̅w )(1 − a ̅w (1 − CA,1) − (a ̅w ) CA,1(1 − CA,2))

aAb

xA*KADH⎜⎜

aw̅

aw/KwDH

KwDH

⎛ A I 1/2 m |z |z v / 1 + ρ I 1/2 + m |z |z v / 1 + ρ I 1/2 ⎞ ( A A− A+ A ( ) )) ⎟ x x B B− B+ B ( A x B x exp⎜⎜ ⎟ νAmA + νBmB + M w −1 ⎝ ⎠

b KDH A

v ⎛ ⎡ ⎧ ⎫⎤⎞ A 1/2 ⎞ ⎛ ⎛ (1 − 2I /z |z |) m |z |z v / 1 + ρ I 1/2 + m |z |z v / 1 + ρ I 1/2 ⎞⎪ 2 ⎜ 1 + ρA Ix ( ( ) ( )) x + − − + x − + x A A A A A A B B B B ⎟ ⎜exp⎢ −z |z |A ⎪ A B ⎥ ⎟ ⎜ ⎟⎟⎬ ⎨ ⎜ ⎢ A + A − x⎪ ρ ln⎜ 1 + ρ (I ref )1/2 ⎟ + ⎜ ⎪ ⎟ (2Ix1/2)(νAmA + νBmB + M w −1) ⎠⎭⎥⎦⎠ ⎠ ⎝ ⎩ A ⎝ A x ,A ⎝ ⎣

M w νACA,1a ̅w ((1 − a ̅w )2 + (2 − a ̅w )a ̅w CA,2)

M w νBC B,1a ̅w ((1 − a ̅w )2 + (2 − a ̅w )a ̅w C B,2)

⎛

⎞νA 1 − a w̅ ⎟⎟ 2 ⎝ 1 − a w̅ (1 − CA,1) − (a w̅ ) CA,1(1 − CA,2) ⎠

a DH The expressions were evaluated from eqs 25 and 26. bExpressions for aB and KDH B are the same as for aA and KA , except with all the subscripts A and B switched.

basis and for the infinite dilution reference state can therefore be written as

for solute activity coefficients is the pure liquid solute or pure fused salt. Existing BET-type models do not extend to dilute solutions,6,24,26 and for these models the reference state for solute activity coefficients must be some fixed aw or concentration (if it is not for the pure liquid). For practical applications, and to aid comparisons of calculated solute activity coefficients with available measurements, we have derived a general expression for a reference state conversion to infinite dilution in water, for all solutes j. This is described in Appendix A. The conversion is constant over the entire concentration range and therefore needs to be derived only for the infinite dilution limit (pure water, aw = 1.0), although it does depend on the relative amounts of the solutes present. The derivation yields the following expression for f *j± ̅ , the adsorption contribution to the mean activity coefficient of electrolyte j, relative to a reference state of infinite dilution in water: ⎛ ∑ ν m + M −1 ⎞ k k w ⎟⎟(a ̅jo)1/ νj f j̅ *± = ⎜⎜ k ∑ ν m ⎝ ⎠ k k k

⎛ ∏nj − 1 C ⎞ m = 1 j , m ⎟ o DH 1/ νj γ±, j = ⎜⎜ ⎟( a ̅ j K j ) M ⎝ w ∑k νkmk ⎠

nj−1 and for a nonelectrolyte: γj = aoj̅ KDH j ∏ m=1 Cj,m/Mw∑k ν kmk. An additional adsorption term involving parameters Tj,j′,1 was derived by Dutcher et al.11 for the interaction of pairs of solutes (see their section 3.1.1), and can be applied without change within the model developed in this work. However, a reference state conversion is also required. In brief, the right-hand side of the activity coefficient 1/ν j expressions in this section should be multiplied by (T mix j ) mix,∞ 1/νj mix (for fj±̅ ) and (T mix /T ) (for f ̅ * ) where T is j j j± j given by

⎛ ⎛ ⎞ 1 ⎟ T jmix = exp⎜ −⎜⎜ Z ⎜ ∑ N⎟ ⎝ ⎝ i=A i ⎠ Z ⎡ ⎛ Xi ,1Tj , i ,1 ⎞ × ⎢(1 − x*j ) ∑ ⎜ ⎟− ⎢⎣ 2νi ⎠ i=A ⎝

nj − 1

∏ Cj ,m m=1

(30)

(29)

Correspondingly, for a nonelectrolyte, the activity coefficient relative to the same reference state, fj̅ *, is equal to ((∑k νkmk j−1 + Mw−1)/∑k νkmk)(aoj̅ )∏nm=1 Cj,m. The mean activity coefficient of electrolyte j, including the Debye−Hückel term, is given by f j± * = f j± *̅ KDH j , where the reference state for KDH in eq 16 is infinite dilution in the solvent j (thus Iref x,j = 0). For a nonelectrolyte, the activity coefficient f j* is given by f j* = f j*̅ for the case in which the moles of solute j are not counted in the calculation of Ix (i.e., they are not included in the total moles of all species in the denominator), or f *j = f *j̅ KDH otherwise, where j KDH is given by eq 18. We note that this is an approximaj tion, which does not distinguish between the characteristics of the solvent and dissolved neutral solutes j in the calculation of the Debye−Hü ckel term. This is most likely to be reasonable when Nw ≫ Nj. Using the same definition for KDH j , the mean activity coefficient of solute j on a molality

Z−1

−

Z

⎛ x *X T i i ′ ,1 i , i ′ ,1

∑′ ∑ ′ ⎜⎜ i = A i ′= i + 1 ⎝

2νi ′

+

Z

⎛ x *X T ⎞ i j ,1 j , i ,1 ⎟ ⎟ 2νj ⎠ ⎝

∑ ⎜⎜ i=A

⎞ xi*′ Xi ,1Ti , i ′ ,1 ⎞⎤⎟ ⎟⎟⎥ 2νi ⎠⎥⎦⎟⎠

(31)

and ⎛ ⎡ Z ⎛ * xi Tj , i ,1 ⎞ ⎟ T jmix, ∞ = exp⎜ −⎢(1 − 2x*j ) ∑ ⎜⎜ ⎟ ⎜ ⎢ 2 ⎠ ⎝ i = A ⎣ ⎝ ⎤⎞ * * − ∑′ ∑ ′ xi xi ′ Ti , i ′ ,1⎥⎟ ⎥⎦⎟ i = A i ′= i + 1 ⎠ Z−1

3203

Z

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Table 4. Fitted Parameters for Aqueous Solutions at 298.15 Ka model parameters solute NaCl

range fitted (m: mol kg−1)

no. of layers (n)

m: 0.001−16.68 aw: ∼1.0−0.39

4

m: 0.001−111.01 aw: ∼1.0−0.27 m: 0.001−29.00 aw: ∼1.0−0.06

NH4NO3 NaOH

HNO3

(NH4)2SO4c

CaCl2

glycerole

5

4

m: 0.001−56.45 aw: ∼1.0−0.07

8

m: 0.129−29.60 aw: 0.995−0.37

9

m: 0.001−10.00 aw: ∼1.0−0.18

6

m: 0.180−961.1 aw: 0.997−0.026

5

parameter

value

uncertainty

RMSEb

C1 P ρ P ρ C1 P ρ C1 P ρ C1 P ρ C1 P ρ P

2.5452 −1.6332 13.9086 0.1304 13.4745 48.7097 −2.1340 15.341 8.5978 −0.2520 19.1567 1.5413 −0.2778 13.516 33.7063 −1.5887 26.7131 −0.2309

0.0777 0.0242 0.4252 0.0064 0.6132 0.4785 0.0080 0.4262 0.0422 0.0016 0.3102 0.0037 0.0012 0.1423 2.5323 0.0080 0.9612 0.0021

1.27%

1.04% 1.03%

1.99%

0.78%d

1.74%

1.18%

a

The numbers of sorption sites per solute molecule (r) is equal to 2 for solutions of electrolytes NaCl, NH4NO3, NaOH and HNO3, 3 for (NH4)2SO4, Na2SO4 and CaCl2, and 1 for glycerol. The energy parameter Kj of the outmost multilayer is 1.0 for all the solutions. The power law coefficient P is used to calculate energy parameter C for the second to (n − 1)th layers, except for NH4NO3 and glycerol, for which P is used for the first layer as well (hence Ci = (i/n)P, where i is the layer number). bRMSE is the root-mean-square percent error, equal to np meas 2 (ϕfit ) /np)]1/2 × 100%, where np is the number of nonzero weighted data points in the fit. cThe parameters for (NH4)2SO4 are defined [(∑i=1 i − ϕi by a different power law relationship: Ci = C1iP. dThe RMSE was calculated using data from the following sources only: Wishaw and Stokes,29 Tang and Munkelwitz,30 Filippov et al.,31 and Clegg et al.32 eFor solutions containing only a nonelectrolyte, such as glycerol−water, there is no Debye−Hückel term.

where the reference state in the expression for KDH,o is j infinite dilution with respect to the solvent (thus Iref = 0). x,j For a nonelectrolyte, the corresponding equation is γoj = −1 n j aoj̅ ∏m=1 Cj,m/Mwm̅ oj . Dividing eq 30 by eq 34 yields

where the summations are over all solutes A to Z, and the primes on the summation indicate that the index cannot be solute “j”. The variables in the expressions for T mix and j T mix,∞ were defined previously.11 j

4. ACTIVITY COEFFICIENTS AND THE MCKAY−PERRING EQUATION For a system that obeys the Zdanovskii−Stokes−Robinson relationship, the activity coefficient of any solute can be expressed by eq 13 of Stokes and Robinson.5,26 For an electrolyte solute this is equivalent to ln(γ±, j) =

ln(γ±o, j)

⎛ νjmjo ⎞ ⎟⎟ + ln⎜⎜ ⎝ ∑k νkmk ⎠

⎛ 1 ⎞ ⎛ KjDH ⎞ ⎛ νjm̅ jo ⎞ ⎟⎟ ln(γj ±) = ln(γjo±) + ⎜⎜ ⎟⎟ ln⎜⎜ DH,o ⎟⎟ + ln⎜⎜ ⎝ ∑k νkmk ⎠ ⎝ νj ⎠ ⎝ Kj ⎠ (35)

which is the same as the McKay−Perring expression in eq 33 above, except for the addition of the term that contains the Debye−Hückel contributions and for the fact that it applies at constant aw̅ rather than aw. This result is consistent with the fact that the ZSR relationship applies for the present model at constant aw̅ generally, and at constant aw for solutions containing only neutral solutes (for which the Debye−Hückel terms in the above equation are unity and cancel). For a nonelectrolyte j in a solution containing ionic species, the mixing rule is ln(γj) = o ln(γj̅ o) + ln(KDH j ) + ln(m̅ j /∑k νkmk). Using the parameter Tj,j′,1 11 derived by Dutcher et al. for the interaction of pairs of solutes (see their section 3.1.1), as discussed after their eq 41, the mixing DH,o rule is ln(γj±) = ln(γoj̅ ±) + (1/νj) ln(KDH ) + ln(νjm̅ oj / j /Kj mix mix,∞ ∑k νkmk) + (1/νj) ln(T j /T j ).

(33)

where γ±,j is the mean activity coefficient in the mixture and γo±,j is the mean activity coefficient of j in a pure aqueous solution at the water activity of the mixture. The summation in eq 33 is over the molalities of all solutes k, and moj is the molality of j in the pure aqueous solution at the water activity of the mixture. This relation was first derived by McKay and Perring5,27,28 for a ternary system. o (For nonelectrolytes νj is unity, and γ±,j and γ±,j should be replaced by γj and γoj , respectively.) When the equations in section 3 are used to obtain the mean molal activity coefficient of electrolyte j in a pure aqueous solution at the same value of aw̅ as the mixture, the following expression is obtained: γ±o, j

⎛ ∏nj − 1 C ⎞ m = 1 j , m ⎟ o DH,o 1/ νj ) = ⎜⎜ o ⎟( a ̅ j K j ⎝ M wνjm̅ j ⎠

5. MODEL APPLICATIONS In this section, we test the model on single electrolyte and nonelectrolyte solutions, on mixtures of electrolytes, and on an electrolyte/nonelectrolyte mixture. It was found necessary to include more sorption layers than in our previous studies to accurately fit osmotic and activity coefficients in dilute solutions.

(34) 3204



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Figure 2. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous NaCl at 298.15 K. Lines: solid, the current model; dashed, three-layer model with four parameters (Dutcher et al.6); dotted, the current model without the contribution of the Debye−Hückel term (part c only). Symbols: square, Archer;34 circle, Tang et al.;35 triangle, Cohen et al.;36 diamond, Chan et al.37 (a) Water activities, plotted against the total mole fraction of ions, xI (equal to 2mNaCl/(2mNaCl + 1/Mw), where mNaCl is the molality of NaCl and Mw (0.0180152 kg) is molar mass of water). (b) Osmotic coefficients, plotted against the total fraction of ions. (c) Osmotic coefficients, plotted against the square root of NaCl molality (m). The vertical dotted lines in (a) and (b) mark the lower limit (3.0 mol kg−1) of data fitted in the model of Dutcher et al.6 The vertical arrows mark the saturation concentration of NaCl (6.144 mol kg−1) at 298.15 K.

Figure 3. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous NH4NO3 at 298.15 K. Lines: solid, the current model; dashed, three-layer model with four parameters (Dutcher et al.6); dotted, the current model without the contribution of the Debye−Hückel term (part c only). Symbols: square, Wishaw and Stokes;38 circle, Chan et al.;39 triangle, Kirgintsev and Lukyanov.40 (a) Water activities, plotted against the total mole fraction of ions, xI (equal to 2mNH4NO3/ (2mNH4NO3 + 1/Mw), where mNH4NO3 is the molality of NH4NO3 and Mw (0.0180152 kg) is the molar mass of water). (b) Osmotic coefficients, plotted against the total mole fraction of ions. The inset shows detail at very low concentration. (c) Osmotic coefficients, plotted against the square root of NH4NO3 molality (m). The vertical dotted lines in (a) and (b) mark the lower limit (0.5 mol kg−1) of data fitted in the model of Dutcher et al.6 The vertical arrows mark the saturation concentration of NH4NO3 (26.33 mol kg−1) at 298.15 K.

It was also found that, beyond the first layer, the Ci parameters often obey a simple power law relationship: Ci = (i/n)P, where 2 ≤ i < n, Ci is the energy parameter of the ith layer, n is the total number of layers including the outmost or bulk layer, and P is a power coefficient. For each solute, the sorption model is generally fitted with only two parameters: C1 and P.

For electrolytes, the Debye−Hückel parameter ρj was also fitted, and typically ranged in value from 10 to 30. The model parameters for some single solute solutions are presented in Table 4 and were determined by directly fitting to molality-based osmotic coefficients at 25 °C taken from the literature. These include, for a number of solutes, concentrations that are highly supersaturated with respect to the solid 3205



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Figure 4. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous NaOH at 298.15 K. Lines: solid, the current model; dashed, three-layer model with four parameters (Dutcher et al.6); dotted, the current model without the contribution of the Debye−Hückel term (part c only). Symbols: circle, calculated values using the equation of Hamer and Wu.41 (a) Water activities, plotted against the total mole fraction of ions, xI (equal to 2mNaOH/ (2mNaOH + 1/Mw), where mNaOH is the molality of NaOH and Mw (0.0180152 kg) is the molar mass of water). (b) Osmotic coefficients, plotted against the total mole fraction of ions. (c) Osmotic coefficients, plotted against the square root of NaOH molality (m). The vertical dotted lines in (a) and (b) mark the lower limit (1.5 mol kg−1) of data fitted in the model of Dutcher et al.6

Figure 5. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous HNO3 at 298.15 K. Lines: solid, the current model; dashed, model of Clegg and Brimblecombe;33,42 dotted, the current model without the contribution of the Debye−Hückel term (part c only). Symbol: circle, data from sources shown in Figure 1 of Clegg and Brimblecombe.33 (a) Water activities, plotted against the total ions, xI (equal to 2mHNO3/(2mHNO3 + 1/Mw), where mHNO3 is the molality of HNO3 and Mw (0.0180152 kg) is the molar mass of water). (b) Osmotic coefficients, plotted against the total mole fraction of ions. (c) Osmotic coefficients, plotted against the square root of HNO3 molality (m).

model and the ZSR relationship are compared to measurements in Figure 12. Measured and calculated mean activity coefficients of NaCl in aqueous NaCl−Na2SO4 solutions at different ionic strengths are compared in Figure 13. 5.1. Pure Aqueous Solutions. Parts a and b of Figures 2−8 show calculated water activities and osmotic coefficients over the entire concentration range, together with literature data. Parts c show model predictions made using the same set of parameters but both with and without the contribution of the Debye− Hückel term. The new model incorporating the Debye−Hückel term, and using only a few fitted parameters, successfully

solutes. Measured and fitted osmotic coefficients and water activities are compared for single electrolyte solutions (containing NaCl, NH4NO3, NaOH, HNO3, (NH4)2SO4, and CaCl2) and a nonelectrolyte solution (glycerol) in Figures 2−8, together with values from our previous model.6 Predictions for some aqueous mixtures (NaCl−NaNO3, NaCl−glycerol, and NH4NO3−(NH4)2SO4) are compared to measurements in Figures 9−11. For the four-ion system of NaCl−KCl−Na2SO4−K2SO4 both model-calculated osmotic coefficients and predictions using the 3206



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Figure 6. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous (NH4)2SO4 at 298.15 K. Lines: solid, the current model; dashed, three-layer model of Dutcher et al.6 (with parameters r = 1.872, C1 = 0.0812, C2 = 34.77, and K = 0.9876 fitted in this study); dotted line, the current model without the contribution of Debye−Hückel term (part c only). Symbols: square, Wishaw and Stokes;29 circle, Tang and Munkelwitz;30 triangle, Filippov et al.;31 diamond, Clegg et al.;32 inverted triangle, Clegg et al.43 (a) Water activities, plotted against the mole fraction of the total mole fraction of ions, xI (equal to 3m(NH4)2SO4/(3m(NH4)2SO4 + 1/Mw), where m(NH4)2SO4 is the molality of (NH4)2SO4 and Mw (0.0180152 kg) is the molar mass of water). (b) Osmotic coefficients, plotted against the total mole fraction of ions. (c) Osmotic coefficients, plotted against the square root of (NH4)2SO4 molality (m). The vertical dotted line in (a) and (b) marks the lower limit of data fitted (3 mol kg−1) using the model of Dutcher et al.6 The vertical arrow marks the saturation concentration of (NH4)2SO4 (5.813 mol kg−1) at 298.15 K.

Figure 7. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous CaCl2 at 298.15 K. Lines: solid, the current model; dotted, the current model without the contribution of the Debye−Hückel term (part c only). Symbols: circle, from the review of Goldberg and Nuttall.44 (a) Water activities, plotted against the total mole fraction of ions, xI (equal to 3mCaCl2/(3mCaCl2 + 1/Mw), where mCaCl2 is the molality of CaCl2 and Mw (0.0180152 kg) is the molar mass of water). (b) Osmotic coefficients, plotted against the total mole fraction of ions. (c) Osmotic coefficients, plotted against the square root of CaCl2 molality (m).

Figures 2−4 and 6) is not valid at low concentrations due to the absence of a Debye−Hückel term. Figure 5 shows that the model performs well for HNO3 compared to the model of Clegg and Brimblecombe33 which employs more fitted parameters (four) than the present model (up to three; see Table 4). Another example, aqueous CaCl2, is shown in Figure 7, again indicating the good model performance, in this case for an unsymmetrical electrolyte. The effect of the inclusion of the Debye−Hückel term is illustrated in parts c of Figures 2−7. When this term is included, the sorption contribution to the osmotic coefficient varies

represents the observed changes in solution properties over the entire concentration range, including the dilute subrange, and satisfies the limits. In contrast, our earlier three-layer model6 for NaCl, NH4NO3, NaOH, and (NH4)2SO4 (see parts a and b of 3207



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Figure 8. Measured and calculated osmotic coefficients (ϕ) of aqueous glycerol at 298.15 K. Lines: solid, the current model; dashed, three-layer model with four parameters (Dutcher et al.6); Symbols: square, Ninni et al.;45 circle, Rudakov and Sergievskii;46 triangle, Scatchard et al.47 (a) Osmotic coefficients, plotted against the mole fraction (x) of glycerol (calculated as m/(m + 1/Mw), where m is the molality of glycerol and Mw (0.0180152 kg) is the molar mass of water). (b) Osmotic coefficients of dilute solutions, plotted against the mole fraction of glycerol (x). Figure 9. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous NaCl-NaNO3 mixtures at ∼298.15 K, plotted against the square root of the total molality of all solute species (mT, equal to mNa+ + mCl− + mNO3−). Lines: solid, the current model (fitted) for pure aqueous NaNO3 and NaCl (labeled); dotted, dashed, and dash-dot lines, predictions of the current model for mixtures containing 1:3, 1:1, and 3:1 mole ratios of NaCl to NaNO3, respectively. The model parameters used for NaCl are given in Table 4. The following model parameters for NaNO3 were obtained by fitting to data from sources listed by Clegg et al.:48 C1 = 1.2174, C2 = 0.2874, C3 = 13.9837, C4 = 0.0778, C5 = 0.9463, C6 = 2.73654, C7 = 1.4400, C8 = 1.0203, and ρ = 11.6426. Symbols: circle, solid dot, and triangle, data of Chan et al.37 for mixtures of 1:3, 1:1, and 3:1 mole ratios of NaCl to NaNO3, respectively (a) Water activities. (b) The same data, plotted as osmotic coefficients.

very simply with concentration in dilute solutions. For nonelectrolytes, we demonstrated that the number of sorption sites must equal the stoichiometric coefficient, which is unity (vj = rj = 1.0) and that the multilayer energy parameter must be unity (Kj = 1.0). The results for glycerol in Figure 8 show that these constraints greatly improve the fit in the dilute concentration range. At very high concentrations, even extrapolated to the pure liquid solutes, the model performs well and yields monotonically decreasing water activities with increasing solute mole fraction. 5.2. Mixtures. Results for NaCl−NaNO3, NH4NO3− (NH4)2SO4, and NaCl−glycerol aqueous solutions are presented in Figures 9−11. Note that the original measurements are water activities from an electrodynamic balance and/or water activity meter, and the uncertainty in the data leads to significant scatter when plotted as osmotic coefficients (for example, the NaCl−glycerol system in Figure 11b). However, the model generally reproduces the measurements well. In Figure 12 measured and calculated osmotic coefficients for dilute NaCl−KCl−Na2SO4−K2SO4 mixtures are shown, together with osmotic coefficients calculated using the ZSR relationship. It is clear that the present model is more accurate, with errors that are similar in magnitude to the scatter in the data (Figure 12c). In contrast, predictions using the ZSR relationship are too low by about 0.005. Finally, measured and calculated mean activity coefficients of NaCl in NaCl−Na2SO4 mixtures are compared in Figure 13.

There is good agreement and, for the low mole fractions of NaCl for which the influence of Na2SO4 on γNaCl is greatest, there is a significant improvement relative to predictions made using the ZSR model and the McKay−Perring relation. These results, and all those described in this section, were obtained without mixture-specific parameters and are based entirely on the properties of the pure aqueous solutions.

6. SUMMARY In earlier work, we derived expressions for molality and solvent and solute activities in multicomponent solutions, valid for concentrations ranging from the pure liquid solute(s) down to 3208



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Figure 11. Measured and predicted water activities (aw) and osmotic coefficients (ϕ) of aqueous NaCl−glycerol mixtures at ∼298.15 K, plotted against the square root of the total molality of all solute species (mT, equal to mNa+ + mCl− + mGlycerol). Lines: solid, the current model (fitted) for pure aqueous NaCl and glycerol (labeled); dashed, predictions of the current model for mixtures of 1:1 mole ratios of NaCl to glycerol. The model parameters used for NaCl and glycerol are given in Table 4. Symbol: solid dot, Choi and Chan49 for mixtures of 1:1 mole ratios of NaCl to glycerol. (a) Water activities. (b) Osmotic coefficients. Note that measurements at low molalities, obtained using a water activity meter, have quite large uncertainties (±0.003 in aw). Measurements at high molalities, obtained using a scanning electrodynamic balance, have uncertainties of about ± 0.9% aw.

Figure 10. Measured and calculated water activities (aw) and osmotic coefficients (ϕ) of aqueous NH4NO3−(NH4)2SO4 mixtures at ∼298.15 K, plotted against the square root of the total molality of all solute species (mT, equal to mNH4+ + mNO3− + mSO42−). Lines: solid, the current model (fitted) for pure aqueous (NH4)2SO4 and NH4NO3 (labeled); dotted, dashed, and dash-dot, predictions of the current model for mixtures of 1:2, 1:1, and 4:1 mole ratios of NH4NO3 to (NH4)2SO4, respectively. The model parameters used for NH4NO3 and (NH4)2SO4 are given in Table 4. Symbols: circle, solid dot, and triangle, data of Chan et al.39 for mixtures containing 1:2, 1:1, and 4:1 mole ratios of NH4NO3 to (NH4)2SO4, respectively. (a) Water activities. (b) Osmotic coefficients.

small but finite solute molalities.6,11 This study improves upon that work in a number of significant ways. (1) A Debye−Hückel term has been added to the model equations, thus extending their range of application to infinite dilution of the solutes in the solvent. Analytical expressions for converting activity coefficients between reference states (i.e., from the fused-salt reference state of the adsorption isotherms to an infinite dilution one more commonly used in solution chemistry) have been derived. (2) The number of sorption sites per solute molecule has been shown to be equivalent to the stoichiometric coefficient (υj) of the solute for the case where the equations are extended to the infinitely dilute solution, eliminating one unknown. (3) The energy parameters of the multilayer (Kj for each solute) must be equal to 1.0 in order that the activity of the solute and its molality approach zero, and the osmotic coefficient of the solution approaches unity, as water activity approaches 1.0 (an infinitely

dilute solution). This removes further unknowns. (4) For many solutes the remaining energy parameters can be related to each other (by a simple power law in the cases presented here), resulting in a further reduction in the number of parameters needed to represent solvent and solute activities even when many monolayers are used. (5) The relationship between solvent activity and solute concentration in multisolute systems has been shown to conform to a modified Zdanovskii−Stokes−Robinson equation, and activity coefficients to a modified McKay−Perring relation. The mixing rules relate the solute molalities and solute activity coefficients not to the actual solvent activity, as is usually the case, but rather to the sorption contribution to the solvent activity. 3209



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Figure 12. Measured and calculated osmotic coefficients (ϕ) of aqueous NaCl−K2SO4−KCl−Na2SO4 mixtures at 298.15 K, plotted against the square root of ionic strength (I, in mol kg−1). Lines: solid line, the current model; dotted, calculated using the ZSR relationship. To better compare the accuracy of the two mixing models, the pure electrolyte solutions were modeled using extra adjustable parameters (NaCl: C1 = 4.7829, C2 = 0.0714, C3 = 61.5592, C4 = 0.5826, and ρ = 17.3562, obtained by fitting the data shown in Figure 2. K2SO4: C1 = 4.7829, C2 = 0.0714, C3 = 61.5592, C4 = 0.5826, and ρ = 17.3562, obtained by fitting osmotic coefficients from Hamer and Wu41 and Ninkovic et al.;50 KCl: C1 = 4.7829, C2 = 0.0714, C3 = 61.5592, C4 = 0.5826, and ρ = 17.3562, obtained by fitting osmotic coefficients from Hamer and Wu.41 Na2SO4: C1 = 0.0147, C2 = 2.4829, C3 = 32.3837, C4 = 6.96, C5 = 0.0054, C6 = 0.0858, C7 = 276.2929, and ρ = 17.2362, obtained by fitting osmotic coefficients from sources listed by Clegg et al.48). All data for the mixtures are from Robinson et al.51 Symbols in plot a correspond to the following molar ratios NaCl:K2SO4:KCl:Na2SO4: square, 0.236:0.118:0.600:0.046; inverted triangle, 0.445:0.102:0.302:0.151; left-pointing triangle, 0.336:0.158:0.355:0.151; diamond, 0.340:0.170:0.293:0.197. In plot b the molar ratios are: circle, 0.583:0.051:0.244:0.122; triangle, 0.267:0.134:0.532:0.067; right-pointing triangle, 0.331:0.166:0.343:0.160; narrow diamond, 0.310:0.184:0.338:0.168. (c) Differences between measured and calculated osmotic coefficients (Δϕ, measured − fitted). Symbols: solid, the current model; open, the ZSR relationship (noting that the symbol types in (c) correspond to the molar ratios of solutes in (a) and (b)).

In the model without the Debye−Hückel term,6,11 the solvent activity was found to be equal to the ratio of the total amount of solvent in the multilayer to the total amount of solvent in both the multilayer and outermost monolayer of each solute, multiplied by a “bulk energy parameter”, Kj, which was treated as an adjustable parameter. In the present model the bulk energy parameter is replaced by the solvent Debye− Hü ckel expression. Similarly, the solute activity in single solute solutions was found to be the ratio of the number of sorption sites occupied in the first monolayer to the total number of available sites.6 In mixtures, that ratio was multiplied by the dry mole fraction of the solute.11 Here, it is found that the same expression must also be multiplied by a solute Debye−Hückel term. In general, the inclusion of the long-range Coulombic electrostatic forces between ions in a dielectric solvent acts to modify the relative equilibrium concentrations of “free” and “sorbed” solvent molecules, thus altering the solvent and solute activities. We have shown that the changes and additions to the model yield improved accuracy in both the representation of solvent and solute activities for pure aqueous solutions (of glycerol, NaCl, NaOH, NH4NO3, HNO3, CaCl2, and (NH4)2SO4), and for mixtures (e.g., NaCl−glycerol, NaNO 3−NaCl, NH4NO3−(NH4)2SO4, NaCl−Na2SO4, and NaCl−KCl− Na2SO4−K2SO4). The model is able to represent osmotic and activity coefficients to very high concentrations, and extrapolates satisfactorily to the pure liquid state. The fact that good results were obtained for the mixtures without the use of any mixture-specific parameters is an encouraging result.

of the pure liquid solute to one of infinite dilution in the solvent. For electrolytes, the adsorption contribution to activity, aj̅ , can be written as aj̅ = (fj±̅ )νj xνj+j+(j)xνj−j−(j),17 where xj+ and xj− are the ionic mole fractions of cation j+ and anion j− in the mixture and νj+(j) and νj−(j) are the stoichiometric coefficients of the two ions in electrolyte solute j. Using eq 14, and rearranging for the mean activity coefficient, fj±̅ , we obtain fj̅ ±

⎛ ⎞1/ νj xj*a ̅jo = ⎜⎜ νj+(j) νj−(j) ⎟⎟ xj − ⎠ ⎝ xj +

(A1)

For nonelectrolytes, the activity can simply be written as aj̅ = fj̅ xj and activity coefficient fj̅ = (xj*aoj̅ /xj). Equation A1 can be rewritten in several useful ways, including ⎛ ⎞1/ νj xj*a ̅jo ⎜ ⎟ )⎜ νj + (j) νj − (j) ⎟ ⎝ (mj +) (mj −) ⎠

(A2)

⎞1/ νj ⎛ ∑ ν m + M − 1 ⎞⎛ xj*a ̅jo k k w k ⎟ ⎟⎟⎜⎜ = ⎜⎜ ⎟ ν ν ∑k νkmk ⎝ ⎠⎝ (x*j +) j+(j) (x*j −) j−(j) ⎠

(A3)

fj̅ ± = (∑ νkmk + M w k

−1

and fj̅ ±

where mj+ and mj− are the molalities of ions j+ and j− in the mixture and x*j+ and x*j− are the dry ionic mole fractions defined as x*j+, − =

■

Nj+, − ∑k νkNk

=

mj+, − ∑k νkmk

(A4)

where subscript j+,− indicates that the quantity can be for either cation j+ or anion j−. For a nonelectrolyte, eqs A2 and A3 can be rewritten as fj̅ = (∑k νkmk + Mw−1)(xj*aoj̅ /mj) and fj̅ = ((∑k νkmk + Mw−1)/∑k νkmk)(∑k νkxk*)aoj̅ .

APPENDIX A Here we derive an equation for the conversion of the adsorption contribution to solute activity coefficients from a reference state 3210



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Substituting eq A6 into eq A5 yields the following analytic expression for the mean activity coefficient. fj̅ ± = (∑ νkmk + M w −1)(xj*a ̅jo)1/ νj × k 1/ νj −ν ⎛⎛ ⎛ * * ⎞⎞ j+(j) ⎞ ⎜ ⎜∑ ⎜ (xk xj ) νj + (k) ⎟⎟ ⎟ × ⎜ ⎜⎝ k ⎜⎝ ∑l (xl* x*j ) m̅ lo ⎟⎠⎟⎠ ⎟ ⎜ ⎟ −νj − (j) ⎜ ⎛ ⎛ (xk* x*j ) νj − (k) ⎞⎞ ⎟ ⎜⎜ ⎜⎜∑k ⎜⎜ ⎟⎟ ⎟ ⎟ ⎟ ⎟ o * * ⎝ ⎝ ⎝ ∑l (xl xj ) m̅ l ⎠⎠ ⎠

(A7)

Equation A7 can be simplified to ⎛ x* ⎞ fj̅ ± = ⎜⎜∑ ko ⎟⎟(∑ νkmk + M w −1) ⎝ k m̅ k ⎠ k × (x*j a ̅jo(∑ xk*νj + (k))−νj+(j) (∑ xk*νj − (k))−νj−(j) )1/ νj k

k

(A8)

For a nonelectrolyte, the simplified form yields fj̅ = (∑k νkmk + Mw−1)(∑k x*k /m̅ ok )(aoj̅ ). To convert from an activity coefficient relative to a fused-salt reference state, fj±̅ , to an infinitely dilute reference state, f ̅ j*±, we must first evaluate fj±̅ at infinite dilution, denoted as f ̅ ∞ j±. Infinite dilution of the mixture involves fixing the relative concentrations of the solutes, as the amount of solvent increases, such that the total solute concentration goes to zero. Because the (1 − aw̅ ) terms from (aoj̅ )1/νj and mok in eq A8 cancel, the following limit applies:

Figure 13. Measured and calculated mean activity coefficients (γ±) of NaCl, with respect to a reference state of infinite dilution, in aqueous NaCl−Na2SO4 mixtures at 298.15 K, plotted against the square root of the total molality of the solutes (mT, calculated as mNaCl + mNa2SO4). Lines: solid, predictions of the current model (the solid upper curve shows values for pure aqueous NaCl); dashed, the McKay−Perring relationship. To better compare the accuracy of the two mixing models, the pure electrolyte solutions were modeled using extra parameters (c.f. Figure 12 caption). Symbols in plot a: circle, Sarada et al.;52 cross, Galleguillos-Castro et al.53 Symbols in plot b: diamond, I = 6.0 m; narrow diamond, I = 4.0 m; inverted triangle, I = 3.0 m; triangle, I = 2.0 m; square, I = 1.0 m; circle, I = 0.5 m; rightpointing triangle, I = 0.3 m; left-pointing triangle, I = 0.2 m, where I is the molality-based ionic strength.

⎛ lim ⎜⎜M w −1(a ̅jo)1/ νj a w → 1⎝

∞ f j̅ ±

⎛ ⎞1/ νj x*j ⎜ ⎟ = n −1 ⎜ ⎟ ν ν ∏mj = 1 Cj , m ⎝ (x*j +) j+(j) (x*j −) j−(j) ⎠ 1

⎛ ∑ ν m + M −1 ⎞ k k w ⎟⎟(a ̅jo)1/ νj f j̅ *± = ⎜⎜ k ∑ ν m ⎝ ⎠ k k k

k

× (xj*a ̅jo(∑ νj + (k)mk )−νj+(j) (∑ νj − (k)mk )−νj−(j) )1/ νj

(A10)

nj − 1

∏ Cj ,m m=1

(A11)

Correspondingly, for a nonelectrolyte, the activity coefficient relative to the same reference state is simply: f*j̅ = ((∑k νkmk + j−1 Mw−1)/∑k νkmk)aj̅ o∏nm=1 Cj,m.

k

(A5)

where the summations are over all solutes k. For a solute k that does not contain cation j+ or anion j−, the stoichiometric coefficients νj+(k) and νj−(k) are zero. Using dry mole fractions of the solutes, and noting that mk = (xk*/xj*)mj, the mixing relationship from eq 25 for the molality of solute j in a mixture containing k solutes can be rewritten as mj = 1

(A9)

For a nonelectrolyte, the activity coefficient evaluated at infinite j−1 dilution is fj̅ ∞ = ∑k νkx*k /∏nm=1 Cj,m. Dividing eq A3 by eq A10, the adsorption contribution to the mean activity coefficient of electrolyte j relative to a reference state of infinite dilution with respect to the solvent is obtained:

fj̅ ± = (∑ νkmk + M w −1)

⎛ ⎛ x * x * ⎞⎞ ⎜∑ ⎜ k j ⎟⎟ ⎜ ⎜ mko ⎟⎟ ⎝ k ⎝ ̅ ⎠⎠

k

⎞ ⎛ νkxk* ⎟ xk* ⎞ ⎜ ⎟ = ∑ m̅ ko ⎟⎠ ⎜⎝ k ∏npj=−11 Cj , p ⎟⎠

Thus eq A8, evaluated at infinite dilution, yields

Equations A2 and A4 yield the following alternative form for fj±̅ :

k

∑

■

APPENDIX B The individual contributions of the long-range (Debye− Hückel) and sorption terms to the properties of the solvent and solute activity coefficients in a pure aqueous solution are examined here. The logarithm of the water activity is equal to the sum of the Debye−Hü ckel contribution to the logarithm of mole fraction activity coefficient of water (KDH w ) plus the sorption term (aw̅ ) that includes both the mole

(A6) 3211



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(noting that rj = 1.005 resulted in a divergence of ϕ from unity at infinite dilution). Figure B1a therefore suggests that the nonsorption contribution (i.e., the Debye−Hückel term) is solely describing the electrolyte-specific behavior of the osmotic coefficient, as expected. Likewise, the logarithm of the mean mole fraction solute activity coefficient, f ̅ j± * , can be written as the sum of the sorption contribution and the Debye−Hückel contributions:

fraction and the sorption contributions to the activity coefficient. Thus ln a w = ln K wDH + ln a w̅

(B1)

where ln K wDH =

2Ax Ix 3/2 1 + ρIx1/2

(B2)

ln f ±* = ln f ±̅ * + ln f±DH, *

The contribution of the sorption term to the molal osmotic coefficient of the solution is therefore given by ln a w − ln K wDH ϕ̅ = − M w mν

(B4)

where the “*” refers to the infinite dilution reference state and DH 1/νj (for the same reference state). f ̅ DH, ± * is equivalent to (Kj ) Values of f ̅*± for aqueous NaCl were calculated using γ± from the study of Archer,34 and assuming a series of different values of ρ. The results are shown in Figure B1b and indicate that f ̅ *± varies approximately linearly with m for dilute solutions. This kind of variation of the sorption contribution to the solute activity coefficient with concentration resembles that of a nonelectrolyte in the dilute limit, in the same way as ϕ̅ shown in Figure B1a.

(B3)

As an example, we calculate the value of ϕ̅ for aqueous NaCl using water activities from the study of Archer,34 and from electrodynamic balance experiments, and assuming a series of different values of ρ in eq B2. The results are shown in Figure B1a

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AUTHOR INFORMATION

Corresponding Author

*E-mail: C.S.D., [email protected]; S.L.C., [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the National Oceanic and Atmospheric Administration (grant NA07OAR4310191) and the Department of Energy (grant DE-FG02-08ER64530). C.S.D. is supported by a National Science Foundation Atmospheric and Geospace Sciences Postdoctoral Research Fellowship.

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Figure B1. The sorption contributions to (a) the osmotic coefficient (ϕ̅ ), and (b) the mole fraction solute activity coefficient relative to an infinite dilution reference state ( f ̅ *± ), as a function of molality (m) of aqueous NaCl. The lines are for different assumed values of ρNaCl in the Debye−Hückel term (indicated on the plot), and are calculated using osmotic coefficients from Archer.34

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