A Simple G-Excess Model for

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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A Simple G‑Excess Model for Concentrated Aqueous Electrolyte Solutions Ricardo Macías-Salinas*

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ESIQIE-SEPI, Departamento de Ingeniería Quimica, Instituto Politécnico Nacional, Ciudad de México 07738, Mexico ABSTRACT: A simple yet accurate thermodynamic model was developed here to represent the nonideal behavior of single electrolytes in water at very high molalities and within a wide temperature range. The present model was obtained from an analytical expression of the excess Gibbs free energy (G-excess) which comprises three major contributions; in this context, a chemical term in the model handles the most predominant short-range ion− solvent interactions by means of a chemical equilibrium approach based on a stepwise ion solvation, whereas another term in the model of physical nature also contributes in describing the aforementioned interactions by incorporating a simple Margules equation, and last, a continuum-solvent term given by the explicit mean-spherical-approximation (MSA) expression serves to account for long-range ion−ion forces. The resulting G-excess model was applied to the representation of experimental mean ionic activity coefficients and osmotic coefficients of various representative aqueous electrolyte solutions: AgNO3, CaCl2, HCl, HClO4, HF, HNO3, KF, KOH, LiBr, LiCl, LiNO3, NaCNS, NaOH, NH4NO3, ZnBr2, and ZnCl2 salts in water at 25 °C (and from 0 up to 300 °C only in the case of KOH) and at high concentrations (up to 83.263 M). The results indicated a very good agreement between the experimental data and those calculated using the present G-excess model for the majority of the electrolyte solutions considered here.

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INTRODUCTION A precise understanding and quantification of the nonidealities occurring in aqueous electrolyte solutions is paramount for the optimum design and operation of many geophysical and chemical processes containing this type of solutions. As a matter of fact, a large amount of experimental studies have so far appeared in the literature dealing with the measurement of thermodynamic properties of aqueous electrolyte solutions over diverse molality and temperature conditions. Such studies have effectively demonstrated the highly nonideal behavior that many aqueous electrolyte systems exhibit, particularly at high salt concentrations. Many theoretical and semiempirical models have been developed in an attempt to describe the nonideal behavior of electrolyte solutions. There are several excellent reviews on electrolyte solution models available in the literature.1−5 In general, models for electrolyte solutions fall into two main categories: (1) local-composition models based on the excess Gibbs energy,6−15 and (2) equations of state derived from analytical expressions of the Helmholtz free energy.16−26 Electrolyte models based on the Helmholtz energy, unlike those based on the local-composition concept, apart from using a larger number of model parameters, are usually more elaborate thus making them somewhat cumbersome to use. In the case of electrolyte models based on the excess Gibbs energy (G-excess), the majority of them treat the electrolyte as completely dissociated (or fully ionized).7,8,10,11,13,15 On the other hand, few G-excess models have been so far reported in the literature that assume partial dissociation or ionization of the salt in solution.6,9,12,14 Although the application of aforementioned models often delivers good results, they do © XXXX American Chemical Society

not describe well the nonideality of aqueous electrolytes at high concentrations; namely, they are usually restricted to salt concentrations of about 6 M. One possible remedy to improve the performance of a G-excess model at very high concentrations (up to 20 M or more) is to consider a suitable solution chemistry such as ion solvation (or hydration) to better represent the short-range ion−solvent interactions that take place over the entire molality range. To the best of my knowledge, Chen et al.12 and Simonin et al.14 are the only authors proposing so far the inclusion of not only ion hydration but also ion association into their so-called electrolyte-NRTL models. Although the application of these two models yielded improved correlations of the mean ionic activity coefficients of various aqueous electrolyte solutions at room temperature, both authors limited their correlating efforts up to a maximum concentration of 6 M; however, as demonstrated by Chen et al.,12 model predictions beyond 6 M were reasonable but not necessarily accurate for some concentrated aqueous electrolyte systems. The purpose of the present work is therefore to contribute a simple G-excess model that can accurately represent the thermodynamic properties of fully ionized electrolytes in water at very high concentrations and within a wide range of temperature. Recognizing the fact that the major contribution to nonideality due to short-range ion−solvent interactions Special Issue: Latin America Received: December 7, 2018 Accepted: April 3, 2019

A

DOI: 10.1021/acs.jced.8b01176 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

comes from ion solvation, the present approach incorporates a stepwise ion-association chemistry thus making it possible to estimate the extent of hydration of the dissolved ions at diverse temperatures and concentrations. Physical interactions between ion and solvent molecules are also important but they constitute only a minor contribution in the present model. In fact, the physical contribution is presently described by a simple Margules equation rather than using elaborate multiparameter expressions based on the local-composition concept as those proposed by virtually all the authors of previous G-excess models cited here.6−15 It is worth mentioning that the work of Zerres and Prausnitz27 is perhaps the only one so far published that uses a similar modeling approach to that proposed here. The authors utilize practically the same aforementioned chemical and physical contributions in order to represent the net short-range interactions; however, the present work differs from that of Zerres and Prausnitz27 in the way the long-range or electrostatic forces were modeled. Accordingly, the meanspherical-approximation (MSA) approach was preferred here over the traditional Debye−Hückel type expression used by the authors since it yields much better predictions of ionic properties at higher molalities4,14,24−26 and it also confers a discrete nature to the dissolved electrolyte by assuming that both cations and anions are electrically charged spheres of finite diameters.

with a cation that has NH binding sites, whereas k0 is regarded as an adjustable equilibrium-constant parameter. In regards to NH, its value was presently set equal to 5 for monovalent cations. Furthermore, solvation numbers for multivalent cations should be higher;30 accordingly, a value of NH = 15 was set here for bivalent cations which is somewhat higher than the one chosen by Zerres and Prausnitz27 (NH = 12) but ensures a better representation of the experimental data. Returning to eq 2, the chemical equilibrium constant is expressed in terms of activities: aCi Ki = aC0 ·(a wf )i (4)

MODEL DESCRIPTION The present thermodynamic model is based on a semiempirical expression for the molar excess Gibbs energy: GEX(T, P, x) where T is the temperature, P is the pressure, and x is the mole fraction of each species in solution (cations, anions, and solvent). The excess Gibbs energy is the sum of three contributions: two arising from short-range forces (solvation and physical) and a third from long-range electrostatic forces (represented by the MSA approach) as follows,

while the total number of bound water molecules is obtained from

assuming that the activity a is equal to the “true” mole fractions ξ for all species under solvation conditions:

G (T , P , x) = G

EX,Solv

+G

EX,Phys

+G

EX,MSA

NH

(6)

i=0

NH

n wb = nC0∑ iK i(ξwf )i

(7)

i=0

dividing eq 7 by eq 6 one obtains the average solvation number (number of bounds water molecules per cation): N

h=

(1)

∑i =H0 iK i(ξwf )i N

∑i =H0 K i(ξwf )i

(8)

This number along with the “apparent” mole fractions based on complete dissociation (xw, xC. and xA) allows the determination of the value of ξwf as follows: ξwf =

x w − hxC x w − hxc + xC + xA

(9)

since h depends on ξwf, it is necessary to employ procedure to find ξwf. Once ξwf is known, the mole

an iterative fractions of

the unsolvated ions can be readily obtained from ξC0 =

(2)

xC/Σ x w − hxc + xC + xA

(10)

xA x w − hxc + xC + xA

(11)

and

where C0 is the unsolvated cation, Ci is the solvated cation with i water molecules. Also, according to Schönert,29 Ki is defined as follows: i NH y ln K i = i ln k 0 + lnjjjj zzzz k i {

(5)

nC = nC0 ·∑ K i·(ξwf )i

Solvation Contribution. As stated by Zerres and Prausnitz,27 short-range ion−solvent interactions are often described by the formation of stoichiometric complexes, namely solvated ions; in the case of strong electrolytes dissolved in water, ions become hydrated (water molecules bound to the ions due to the strong electrostatic attraction between the charged ion and the polar water). Experimental evidence indicates that in many aqueous electrolyte systems, cations are more extensively hydrated (or solvated) than anions.27−30 Following the work of Stokes and Robinson28 by assuming that only the cation is solvated, the following stepwise solvation chemistry applies with a characteristic equilibrium constant Ki for each solvation step: C0 + i(water) ⇔ Ci

ξC0(ξwf )i

here ξwf is the “true” mole fraction of free water (nonbound water molecules). As shown by Zerres and Prausnitz,27 the total number of solvated cations is given by

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EX

ξCi

Ki =

ξA 0 =

where Σ is the summation appearing in eq 6. From the condition of chemical equilibrium, that is, the equality of chemical potentials (μw = μfw, μC = μC0 and μA = μA0), Zerres and Prausnitz27 derived the following expressions for the activity coefficients of each species based on free water and unsolvated ions:

(3)

where the binomial coefficient NH represents the number of i distinguishable ways that i identical water molecules can bind

( )

B


Journal of Chemical & Engineering Data ij ξ f yz ln γwSolv = ln jjjj w zzzz k xw {

ij NH yz ji ξC zy ln γCSolv = lnjjj 0 zzz + lnjjjj∑ K i zzzz j xC z j i=0 z k { k { ln γASolv

ij ξA yz = lnjjj 0 zzz j xA z k {

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approximation and it does not adversely affect the prediction capabilities of the MSA approach, particularly within the dilute region where the electrostatic forces dominate. This oversimplification also leads to the fact that VEX → 0, hence

(12)

2Γ 3Vw GEX,MSA AEX,MSA (1 + 1.5σ Γ) ≈ =− RT RT 3πNant

(13)

here Γ is the screening parameter defined by (14)

Γ=

As a matter of fact, the activity coefficients of the ionic species (eqs 13 and 14) are unsymmetrically normalized to the standard state for the ions: γi = 1 as xi → 0. Lastly, the G-excess expression due to ion solvation is written in terms of γSolv as i follows: GEX,Solv = x w ln γwSolv + xC ln γCSolv + xA ln γASolv RT

GEX,Phys = αs,wxsx w − xs ln γs∞ = αs,wxs(x w − 1) RT

κ=

(20)

e 2Na2nt ∑ xiZi2 εw ε0RTVw ions

(21)

εw in the above equation is the dielectric constant of pure water, Zi is the ionic charge, e is the unit electronic charge, and ε0 is the permittivity in a vacuum. It is noteworthy that the dielectric constant of water and its derivatives were obtained from the equation of Uematsu and Franck,34 which gives εw as a function of temperature and density over a wide range of temperatures and pressures. As a matter of fact, the MSA model approaches the Debye−Hückel expression in the limit σ → 0. Lastly, it is important to reconcile the inconsistency between the different thermodynamic frameworks used here to derive the three G-excess contributions; for example, both solvation and physical contributions were devised under the Lewis−Randall framework, for which the independent variables are T, P, and the number of moles of each species ni, whereas the MSA contribution was obtained within the McMillan−Meyer framework having T, V, number of moles of the ionic species, and the chemical potential of water as independent variables. Fortunately, according to Cardoso and O’Connell,35 this theoretical discrepancy is of little importance for single-solvent systems such as the ones considered in the present modeling effort.

(15)

(16)

which has been already normalized to meet the previously defined standard-state criteria for the ions. In the above equation, αs,w is a specific interaction parameter between the electrolyte (s) and water, whereas xs = xC + xA. MSA Contribution. Ion−ion interactions are presently described by the MSA approach32 which is based on the socalled primitive model (ions are charged hard spheres in a medium of uniform dielectric constant); its solution is analytical and it has been used to predict ionic properties in aqueous electrolyte systems. Unlike the Debye−Hü ckel approach, the MSA model gives reasonable results for activity coefficients up to moderate ion concentrations (m ≈ 0.1 M) provided that suitable values of the ionic diameters σi are used. The original form of the MSA model, however, involves an implicit solution of a complicated equation for the so-called screening factor Γ that complicates to some extent differentiation of the resulting expressions with respect to volume and number of moles. A Γ-explicit approximation of the MSA by Harvey et al.33 was therefore used here; it is mathematically simple and provides similar results to those obtained from the full MSA. This explicit approximation assumes that all ions in the mixture have the same effective diameter σ given by

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DERIVED IONIC PROPERTIES Expressing the present G-excess model (eq 1) in terms of T, P, and n, the activity coefficient of species i can be readily obtained from the following derivative: ij ∂n GEX /RT yz zz ln γi = jjj t zz j ∂ n i k {T , P , nj(i)

(22)

The resulting activity coefficients serve in turn to derive important ionic properties such as the mean ionic activity coefficient:

∑ions xiσi ∑ions xi

1 [ 1 + 2σκ − 1] 2σ

whereas κ is the reciprocal Debye screening length

Physical Contribution. Physical interactions between ions and water are properly accounted for by the two-suffix Margules equation.31 This equation was chosen here because of its simplicity and ease of use. Its corresponding G-excess expression is given by

σ=

(19)

γ± = (γCνCγAνA)1/ ν

(17) 33

(23)

and the osmotic coefficient (water activity): xw φ=− ln(x wγw ) xC + xA

According to Harvey et al., the above mixing rule has been successfully tested against experimental data for ion-size ratios characteristic of most electrolytes. Their excess thermodynamic expression is given in terms of the Helmholtz energy:

(24)

(18)

where νC is the number of cations, νA is the number of anions and ν = νC + νA. Furthermore, the molality-based mean ionic activity coefficient γ±,m is obtained from

where V is the solution volume. To circumvent the tedious calculation of V, it seems reasonable to assume that V = Vw (pure water volume); this equality constitutes a good

(25)

AEX,MSA 2Γ 3V =− (1 + 1.5σ Γ) RT 3πNant

ij 1000 yz zz γ±, m = γ±jjj j 1 + M w mν zz k {

C



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where γ± is the mole-fraction-based mean ionic activity coefficient, m is the molality, and Mw is the molar mass of the solvent (water).

diameters: the most widely used ionic diameters by existing electrolyte models are those due to Pauling40 who employed quantum mechanical results to determine the distances of inorganic ions in ionic crystals. (2) Goldschmidt diameters: these crystal diameters were empirically obtained by Goldschmidt41 from X-ray diffraction measurements and geometrical considerations. (3) Cavity diameters: these diameters (greater than the crystal ones) were determined by Rashin and Honig42 by assuming that ions are charged hard spheres centered in a cavity inside the solvent; following the authors, cations form larger cavities than anions in aqueous solutions. For each set, Table 2 lists the values of the diameters for the

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RESULTS AND DISCUSSION The present modeling approach was first applied to the correlation of experimental mean ionic activity coefficients (γ±,m) at 25 °C and 1 atm of some representative concentrated aqueous electrolyte solutions. The following electrolytes were chosen for this purpose because of their available experimental ionic properties at high molalities and at various temperatures: 13 1−1 electrolytes (AgNO3, HCl, HClO4, HF, HNO3, KF, KOH, LiBr, LiCl, LiNO3, NaCNS, NaOH, and NH4NO3) and 3 2−1 electrolytes (CaCl2, ZnBr2, and ZnCl2). Table 1 gives

Table 2. Sets of Ionic Diameters Used in the Present Modelingsa

Table 1. Aqueous Electrolyte Solutions Considered in This Study T, °C

electrolyte 1−1: AgNO3 2−1: CaCl2 1−1: HCl 1−1: HClO4 1−1: HF 1−1: HNO3 1−1: KF 1−1: KOH 1−1: LiBr 1−1: LiCl 1−1: LiNO3 1−1: NaCNS 1−1: NaOH 1−1: NH4NO3 2−1: ZnBr2 2−1: ZnCl2

P, bar

mmax, mol/kg

Sources for (γ±,m)exp and φexp

25

1.013

15.0

Hamer and Wu36

25

1.013

10.0

25 25

1.013 1.013

16.0 16.0

Staples and Nutall37 Hamer and Wu36 Hamer and Wu36

25 25

1.013 1.013

20.0 28.0

Hamer and Wu36 Hamer and Wu36

25 0, 25, 50, 100, 150, 200, 250, 300 25 25 25

1.013 1.013− 85.879 1.013 1.013 1.013

17.0 83.263 20.0 19.219 20.0

Hamer and Wu36 Hamer and Wu36, Li and Pitzer38 Hamer and Wu36 Hamer and Wu36 Hamer and Wu36

25

1.013

18.0

Hamer and Wu36

25

1.013

29.0

Hamer and Wu36

25

1.013

25.0

Hamer and Wu36

25

1.013

20.1

Goldberg39

25

1.013

23.193

Goldberg39

ion

σiP, Å

σiG, Å

σiC, Å

Ag+ Ca2+ H+ K+ Li+ Na+ NH4+ Zn2+ Br− Cl− ClO4− CNS− F− NO3− OH−

2.52 1.98 2.4b 2.66 1.2 1.9 2.96 1.48 3.9 3.62 4.8c 4.26c 2.72 3.58c 2.66c

2.26 2.12 2.4b 2.66 1.56 1.96 2.9 1.66 3.92 3.62 4.8c 4.26c 2.66 3.58c 2.66c

2.868 3.724 2.4b 4.344 2.632 3.36 4.26 2.676 4.174 3.874 4.8c 4.26c 2.846 3.58c 2.996

a

Notation: P, crystal Pauling diameters;40 G, empirical Goldschmidt diameters;41 C, cavity diameters taken from Rashin and Honig.42 bvan der Waals diameter.45 cEstimated diameters by Jenkins and Thakur.43

eight cations and seven anions considered here for modeling purposes; in the cases of the ClO4−, CNS−, NO3−, and OH− anions, their diameters were taken from Jenkins and Thakur43 who estimated them using the Kapustinskii equation;44 last, in the particular case of the H+ cation, its diameter was fixed equal to that of van der Waals.45 A least-squares fit based on the Levenberg-Marquadt method was then performed to obtain the model parameters (k0 and αs,w) for each one of the three aforementioned sets of ionic diameters. The minimization of the following objective function served for this purpose:

the 16 electrolytes considered in this work along with their temperatures, pressures, and maximum molalities at which experimental data were measured, as well as the sources of the experimental γ±,m and osmotic coefficients (φ). As a matter of fact, experimental data for all the 1−1 electrolytes were obtained from Hamer and Wu.36 Only in the case of KOH, were the experimental data of Li and Pitzer38 also considered to test the modeling capabilities of the present approach over a wide temperature range (from 0 to 300 °C) and at very high molality (up to 83.263 M). A quick inspection of eqs 3 and 16 indicates that the present G-excess model contains two characteristic parameters, namely k0 and αs,w, that should be determined from experimental γ±,m data. Prior to the application of the electrolyte model proposed here, suitable values of the ionic diameters σi must be provided to estimate the effective ionic diameter (eq 17) needed in the MSA contribution. Accordingly, three different sets of ionic diameters were presently considered. (1) Crystal Pauling

N

min f =

]2 ∑ [lnγ±exp,m,i − γ±calc ,m,i

(26)

i=1

γexp ±,m

where N is the number of experimental points, and γcalc ±,m are the measured and calculated mean molal ionic activity coefficients, respectively. Table 3 summarizes for each electrolyte system the correlating results in terms of the absolute average deviations (AAD) between calculated and experimental γ±,m values using the three sets of ionic diameters. Excluding the cases of HF, ZnBr2, and ZnCl2, which will be further discussed, it can be seen from Table 3 that the ability of the present approach in representing the experimental data is quite good with overall AAD values of 2.75% using Pauling diameters, 2.68% using Goldschmidt diameters, and 2.24% using cavity diameters based on a total of 584 experimental D



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salts show ion-pair formation (to ZnBr− or ZnCl− ions) and/or association to ZnBr2 or ZnCl2.28 Moreover, concentrated solutions of either ZnBr2 or ZnCl2 are strongly complexed thus forming complex ions such as ZnBr3− or ZnCl3− and even ZnBr4−2 or ZnCl4−2 at much higher concentrations; de facto, ZnCl2 forms complex ions more easily than ZnBr2.28 The correlating results so far obtained here demonstrate that the predictive capabilities of the MSA approach (eqs 17 to 21) depend on the use of a suitable set of ionic diameters. This gives good grounds to treat the MSA effective ionic diameter σ as a third adjustable parameter (besides k0 and αs,w) in an attempt to see how much further the MSA contribution can improve the correlating performance of the present approach. Table 4 gives the optimized σ values and the resulting AAD

Table 3. Resulting Model Deviations at 25 °C Using Different Sets of Ionic Diameters %AAD γ±,m electrolyte

N

σiP

AgNO3 CaCl2 HCl HClO4 HF HNO3 KF KOHa KOHb LiBr LiCl LiNO3 NaCNS NaOHa NaOHb NH4NO3 ZnBr2 ZnCl2 overallc

37 65 39 39 45 51 40 43 43 43 43 43 41 52 52 48 105 117 584

0.69 5.53 1.56 3.71 321 3.09 2.12 1.06 3.64 4.37 2.54 2.85 3.28 2.67 6.64 0.73 20.5 30.2 2.75

σiG

σiC

0.52 5.38 1.56 3.71 320 3.09 2.17 1.06 3.64 4.12 2.38 2.65 3.25 2.65 6.61 0.73 20.6 30.2 2.68

0.94 3.15 1.44 3.71 322 3.09 0.54 2.02 2.53 3.61 1.98 1.94 2.52 2.23 5.74 1.17 21.6 32.5 2.24

Table 4. Resulting Model Deviations at 25 °C Using Regressed Effective Ionic Diameters σ

a

NH = 4. bNH = 5. cExcluded AAD values: HF, ZnBr2, ZnCl2 as well as KOH and NaOH with NH = 5.

points. On one hand, the use of Pauling and Goldschmidt diameters yielded comparable results in the majority of the cases with similar AAD values; on the other hand, the correlating performance of the present model was better using cavity diameters, particularly for CaCl2, KF, LiBr, LiCl, LiNO3, and NaCNS with lower AAD values. As also revealed by Table 3, the present approach yielded significant improved results for the two electrolyte solutions containing the OH− anion (KOH and NaOH) by setting their values of NH equal to 4 rather than NH = 5; this choice makes sense since the decrease of bound water molecules in these two alkali metal cations (K+ and Na+) is presumably ascribed to the solvation of the OH− anion to the point that competes with the hydration of the cation. Even though all anions are assumed here to be anhydrous, the present modeling results are highly satisfactory for the aforementioned electrolytes with NH = 4. The Particular Cases of HF, ZnBr2, and ZnCl2. Table 3 also shows that the present G-excess model delivered very poor correlating results for the electrolyte systems containing HF, ZnBr2, and ZnCl2 with large AAD values varying from 21% up to 322%. These particular electrolytes exhibit strong tendencies to associate with the possibility of forming ion pairs and/or complex ions under certain temperature and molality conditions; unfortunately, the present modeling approach is not equipped with a partial dissociation chemistry to properly account for this fact. For example, HF, unlike other halide acids, is a weak acid with a very low ionization constant (6.7 × 10−4 at 25 °C)28 and with a strong tendency to associate and form HF2− even at low concentrations, thus leading to low values of the mean ionic activity coefficients within the dilute region; it is precisely within this region that the present model largely fails to represent the experimental data of HF. In regards to ZnBr2 and ZnCl2, both of them are extremely soluble and complex-forming salts. Dilute solutions of either ZnBr2 or ZnCl2 are strong electrolytes below 1 M (full dissociation occurs); however, above this concentration, these

electrolyte

N

σ, Å

%AAD γ±,m


37 65 39 39 45 51 40 43 43 43 43 43 41 52 52 48 105 117 584

2.874 4.562 3.635 3.034 0.361 5.189 3.910 2.789 3.640 3.301 3.183 4.416 4.336 2.859 3.066 3.274 3.011 1.338

0.46 1.35 1.20 3.63 285 0.84 0.22 0.99 2.51 3.63 2.01 0.56 2.16 2.29 5.85 0.73 20.3 22.1 1.54

a

NH = 4 bNH = 5 cExcluded AAD values: HF, ZnBr2, ZnCl2 as well as KOH and NaOH with NH = 5.

values yielded by the present model in its three-parameter version during the correlation of the experimental γ±,m data of the 16 electrolytes at 25 °C. As expected, the inclusion of σ as a model parameter yielded better correlating results than those previously obtained using any of the three sets of ionic diameters (Table 3), particularly for the AgNO3, CaCl2, HNO3, KF, KOH, LiNO3, and ZnCl2 electrolytes. Although these results are indeed encouraging, it is preferable to use ionspecific parameters (σi from known sources) rather than electrolyte-specific parameters (those regressed from a model) thus preserving the versatility of the present approach and keeping the number of adjustable parameters to a minimum. Accordingly, taking into account the previous correlating results given in Tables 3 and 4, a set of recommended ionic diameters (taken from Table 2) was devised in an attempt to obtain the best possible performance of the present twoparameter model. Table 5 lists the values of the recommended cation and anion diameters for the present G-excess model; de facto, the majority of the recommended diameters are of the cavity type. A fit of the experimental γ±,m data at 25 °C was again performed using the recommended ionic diameters. The E



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Table 5. Recommended Ionic Diameters for the Present Modela σi, Å

cation

G

+

2.26 3.724C 2.4b 2.66P 2.632C 3.36C 2.9G 1.48P

Ag Ca2+ H+ K+ Li+ Na+ NH4+ Zn2+

anion

σi, Å

−

4.174C 3.874C 4.8c 4.26c 2.846C 3.58c 2.66c

Br Cl− ClO4− CNS− F− NO3− OH−

a

Notation: P, crystal Pauling diameters;40 G, empirical Goldschmidt diameters;41 C, cavity diameters taken from Rashin and Honig.42 bvan der Waals diameter.45 cEstimated diameters by Jenkins and Thakur.43

resulting model deviations for the 16 electrolytes are summarized in Table 6, which shows the AAD values, the Table 6. Performance of the Present Two-Parameter Model at 25 °C Using Recommended Ionic Diameters correlated γ±,m electrolyte

N

% AAD


37 65 39 39 45 51 40 43 43 43 43 43 41 52 52 48 105 117 584

0.52 3.15 1.44 3.71 322 3.09 2.00 1.06 3.64 3.61 1.98 1.94 2.52 2.25 5.90 0.73 20.8 30.7 2.20

Figure 1. Relative deviations between correlated and experimental mean ionic activity coefficients for 13 aqueous electrolyte systems at 25 °C.

predicted φ

%Bias

% AMD

% AAD

%Bias

% AMD

−0.073 1.493 0.631 −0.442 −303 2.028 1.035 0.306 0.827 −0.035 0.121 0.828 0.420 0.062 0.188 −0.048 −1.137 −11.70 0.543

3.79 8.42 4.72 18.8 966 7.02 3.93 1.88 13.8 17.0 5.60 3.72 14.7 6.19 17.5 1.85 50.0 86.6 7.51

0.83 1.46 0.69 1.56 94.1 1.54 1.06 0.51 1.24 1.25 0.96 0.98 1.47 1.16 2.42 1.02 12.0 16.9 1.14

−0.387 −0.219 −0.205 −0.274 67.1 −0.610 −0.236 −0.115 −0.011 −0.096 −0.196 −0.148 −0.276 −0.213 0.190 0.134 −1.713 2.266 -0.219

5.53 3.98 1.83 4.70 251 5.05 2.37 1.38 5.04 4.65 2.80 2.46 8.75 3.33 7.53 3.48 33.5 40.7 3.87

obtained for HClO4 at 16 M. Not shown in Figure 1 are the relative deviations of HF, ZnBr2, and ZnCl2, as well as KOH and NaOH with NH = 5 because of the reasons given earlier. Figure 2 now graphically shows the excellent agreement between calculated and experimental γ±,m values for seven of

a

NH = 4. bNH = 5. cExcluded statistical values: HF, ZnBr2, and ZnCl2 as well as KOH and NaOH with NH = 5.

statistical bias (Bias) and the absolute maximum deviation (AMD) between correlated and experimental γ±,m values; also shown in this table are the same type of model deviations between predicted and experimental osmotic coefficients (φ). Overall, as revealed by Table 6, the correlative abilities of the present two-parameter model (with an overall AAD value of 2.2%) were slightly better using the recommended ionic diameters than those using the three sets of ionic diameters (2.75, 2.68, and 2.24% from Table 3), and somewhat inferior to those using regressed ionic diameters (1.54% from Table 4). Furthermore, the low overall Bias value of 0.543% yielded by the present approach indicates a very good distribution of all model deviations around zero mean. Accordingly, Figure 1 confirms this result where the majority of the relative deviations in correlated γ±,m values versus molality fall within ±4%; nevertheless, the largest relative deviation (18.8%) was

Figure 2. Correlated mean ionic activity coefficients for seven concentrated aqueous electrolyte solutions at 25 °C.

the most representative concentrated aqueous electrolyte systems at 25 °C within a wide molality range (from 0.001 up to 28 M); however, in the particular case of LiBr, the model tends to somewhat underestimate the experimental data as its highest molality (20 M) is approached. Interestingly, the two electrolytes having the NO3− anion (AgNO3 and NH4NO3) exhibit γ±,m values that always remain below unity over their F



Article

values with molality for 13 electrolytes, whereas Figure 4 shows the comparison between predicted and experimental φ values for seven electrolytes. Qualitatively speaking, Figures 3 and 4 are very similar to Figures 1 and 2, respectively, but quantitative differences are notable. For example, the largest relative deviation plotted in Figure 3 is −8.75% corresponding to NaCNS at 18 M versus 18.8% obtained for HClO4 at 16 M (see Figure 1). Furthermore, the φ values depicted in Figure 4 vary over a very narrow range (from 0.35 up to 4.2) in comparison with the γ±,m values that vary with almost 4 orders of magnitude (from 0.085 up to 486) as shown in Figure 2. It is worth mentioning that, as revealed by Figure 4, the present model predicts reasonably well the maximum exhibited by HNO3 and NaOH at high molalities; in the case of LiBr, however, the model wrongly predicts a maximum and underestimates the experimental data as the saturation point of LiBr is approached. The present approach also gives a satisfactory representation of the experimental data within the dilute region (from 0.001 up to 0.1 M) as demonstrated by Figure 5a,b that show correlated γ±,m and predicted φ values of seven electrolytes, respectively; both panels use a logarithmic x-axis for a better visualization of the dilute region. As evidenced by Figure 5, beyond 0.1 M, the present model tends to underestimate the experimental data of some electrolytes (particularly HNO3 and LiNO3); nevertheless, it does a pretty good job in representing the minimum exhibited by the following electrolyte solutions: HNO3, LiBr, LiCl, LiNO3, and NaOH. Moreover, the model captures the correct molality at which the experimental minimum occurs for each of the aforementioned electrolyte systems. Figure 6 graphically shows how each term appearing in the present G-excess model (eq 1) distinctively contributes to the total value of either γ±,m (Figure 6a) or φ (Figure 6b) in the case of NaOH in water at 25 °C. This value is denoted by the line passing through the experimental points in Figure 6 and is obtained from the different contributions as follows:

entire molality ranges presumably due to their very low levels of hydration; this issue will be further discussed in more detail. Returning to Table 6, it can be seen that the abilities of the present model in predicting the experimental φ values were remarkably good yielding both individual and overall model deviations that are even smaller than those obtained from correlated γ±,m values by almost 50% for most electrolytes. This result is likely attributed to the fact that ionic species exhibit a wider variation in their γ±,m values (which is harder to model) than those exhibited by the solvent in terms of φ values. Figure 3 depicts the variation of the relative deviations in predicted φ

Figure 3. Relative deviations between predicted and experimental osmotic coefficients for 13 aqueous electrolyte systems at 25 °C.

γ±, m = γ±Solv γ Physγ MSA , m ±, m ±, m

(27)

φ = φSolv + φ Phys + φ MSA

(28)

It can be seen from Figure 6 that the MSA contribution meets the correct boundary condition (i.e., limiting Debye−Hückel law) near the dilute region of the electrolyte solution (from 0.001 up to 0.1 M); but beyond 0.1 M, predictions from the MSA contribution deviate from the experimental data. As the concentration of ions in solution increases, short-range intermolecular forces become increasingly important. Such interactions are properly accounted for by the solvation and physical contributions. As shown in Figure 6, both contributions give comparable values below 2 M; from this molality up to the saturation point of the salt (28 M), however, solvation or ion hydration prevails as the most dominant contribution to the total value of either γ±,m or φ. It is actually mainly responsible for qualitatively capturing the correct variation of either γ±,m or φ with molality, from moderate to very high electrolyte concentrations. An additional modeling effort was undertaken in an attempt to obtain improved γ±,m correlations for some of the most “problematic” electrolyte systems already considered here, namely, CaCl2, HClO4, HF, HNO3, LiBr, ZnBr2, and ZnCl2 at 25 °C (the ones with the highest AAD values listed in Table

Figure 4. Predicted osmotic coefficients for seven concentrated aqueous electrolyte solutions at 25 °C. G



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Figure 6. Model contributions to the ionic properties of NaOH in water at 25 °C: (a) mean ionic activity coefficients, (b) osmotic coefficients.

GEX,Phys = xsx w[αs,w + βs,w (xs − x w )] − xs(αs,w − βs,w ) RT (29)

which has been already unsymmetrically normalized to satisfy the standard state for the ions. The main purpose of the above modification is to give a major role to the physical term over the hydration term when modeling the complex intermolecular forces, particularly at high salt concentrations. Equation 29 introduces, though, a third adjustable parameter (βs,w) into the present approach. A final least-squares fit of the experimental γ±,m data at 25 °C was thus carried out to obtain the three model parameters for CaCl2, HClO4, HF, HNO3, LiBr, ZnBr2, and ZnCl2. For each one of these electrolytes, Table 7 lists the

Figure 5. Performance of the present model near the dilute region for seven concentrated aqueous electrolyte systems at 25 °C: (a) correlated mean ionic activity coefficients, (b) predicted osmotic coefficients.

6). In doing so, the physical contribution to GEX, that is, eq 16 was replaced by a more elaborated Margules equation containing two specific interaction parameters (αs,w and βs,w): H



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Table 7. Improved Modeling Results for Some Selected Electrolyte Solutions at 25 °C Using the Three-Parameter Model correlated γ±,m

predicted φ

electrolyte

N

%AAD

%Bias

%AMD

%AAD

%Bias

%AMD

CaCl2 HClO4 HF HNO3 LiBr ZnBr2 ZnCl2

65 39 45 51 43 105 117

2.77 2.87 243 2.47 3.08 12.1 9.38

1.399 0.822 −216 1.126 1.100 4.101 3.870

8.40 16.5 816 5.68 8.40 41.1 30.2

1.36 0.91 104 1.36 0.89 8.43 6.90

−0.199 −0.141 55.52 −0.249 −0.042 −3.833 −2.69

3.83 4.10 252 3.67 2.51 27.5 22.2

Table 8. Regressed Model Parameters at 25 °C two-parameter model

three-parameter model

electrolyte

k0

αs,w

AgNO3 CaCl2 HCl HClO4 HF HNO3 KF KOHa LiBr LiCl LiNO3 NaCNS NaOHa NH4NO3 ZnBr2 ZnCl2

1.54909 1.78316 4.29863 14.5798 19.4592 0.000331 2.74768 8.13960 11.8514 5.37517 0.684627 1.54480 9.50870 0.684250 0.221525 0.333460

4.04545 −4.09510 −4.01735 −3.87134 15.1981 −4.11684 −1.14760 −3.56964 −3.50579 −3.24262 −3.70601 −1.88386 −1.79404 1.64026 −5.70299 −3.95748

k0

αs,w

βs,w

1.91718

−2.81926

0.40397

10.1331 1.95861 0.001624

−9.07357 −20.4352 −3.65067

−1.96958 −14.2967 0.23877

7.10776

−9.40959

−2.17686

0.000976 0.000375

−9.20200 −9.84225

−1.28829 −2.36301

a

NH = 4.

resulting model deviations from correlated γ±,m and predicted φ values, whereas Table 8 gives the corresponding regressed parameters (k0, αs,w, and βs,w). Also included in Table 8 are the k0 and αs,w values of the two-parameter model that were regressed earlier for the 16 electrolyte solutions considered here. By comparing the AAD values given in Tables 6 and 7 for the seven electrolytes in question, it is evident that the threeparameter GEX model performed better than its two-parameter counterpart in all cases, particularly in the cases of ZnBr2 and ZnCl2 (two of the salts with strong tendencies to form ion pairs and complex ions). Despite such an improvement for these two Zn-based salts, their resulting AAD values are not low enough to claim an accurate modeling outcome. This is graphically demonstrated in Figure 7 where the correlative and predictive abilities of the three-parameter in representing the observed γ±,m (Figure 7a) and φ values (Figure 7b) of LiBr, ZnBr2, and ZnCl2 were better than those of the two-parameter model, particularly at high concentrations (above 10 M); only in the cases of ZnBr2 and ZnCl2, below 10 M, the threeparameter model still fails to give a good representation of the experimental data (at these concentrations both electrolytes form ion pairs and molecular salt, the present models are not able to account for this since they lack of an ion association chemistry). It is noteworthy that, as depicted in Figure 7b, the three-parameter model corrects the anomaly yielded by the two-parameter model by not predicting the maximum in the φ value of LiBr at high concentration. Furthermore, a close inspection of all k0 values listed in Table 8 reveals that the magnitude of k0 tells much about the

extent of solvation that the cation undergoes; for example, the smaller is the value of k0 the weaker are the effects of hydration, de facto, as k0 → 0 the hydration contribution to GEX becomes negligible as in the case of the HNO3 electrolyte with k0 = 0.000331 (from Table 8). To illustrate this, Figure 8 shows the variation of calculated ratios of number of dissociated cations to number of unsolvated cations (xC/ξC0) with molality for eight 1−1 electrolytes (HCl, HClO4, HNO3, KF, LiBr, NaCNS, NaOH, and NH4NO3) and one 2−1 electrolyte (CaCl2) along with their corresponding k0 values obtained from the present two-parameter model. It is worth mentioning that very large values of xC/ξC0 (or k0 > 10, except for CaCl2) imply that the cations are highly hydrated; on the other hand, values of xC/ξC0 close to unity (or k0 → 0) signify that the cations remain anhydrous. Accordingly, as shown in Figure 8, cations reach their highest hydration levels within the dilute region where water is present in excess (except in the case of HNO3); nevertheless, as the electrolyte molality increases, the hydration of cations (in terms of xC/ξC0 values) significantly declines down to the saturation point even with 4 orders of magnitude as in the cases of CaCl2, HClO4, LiBr, and NaOH. In the particular case of CaCl2, even though its k0 value is low, this electrolyte exhibits the largest xC/ξC0 values within the dilute region as depicted in Figure 8. This has to do with the number of binding sites (NH) for the bivalent Ca2+ cation which was presently set equal to 15 (three times larger than that for a monovalent cation). Interestingly, the three I



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Figure 8. Calculated variation of xC/ξC0 with molality for nine concentrated aqueous electrolyte systems.

whereas the H+ in HNO3 was the only cation that remained anhydrous in this modeling study. This result suggests that the nature of the anion ultimately dictates the degree of solvation that a particular cation will undergo. It is now worthwhile to verify the abilities of the present three-parameter model in accurately representing the experimental data of a particular aqueous electrolyte system at temperatures other than 25 °C. The experimental γ±,m and φ data of KOH in water measured by Li and Pitzer38 at 0, 25, 50, 100, 150, 200, 250, and 300 °C were chosen for this purpose. As previously done, the three model parameters (k0, αs,w, and βs,w) were obtained by fitting the experimental γ±,m data at each temperature. Table 9 gives, for each temperature, the resulting model deviations and the regressed model parameters. Figure 9 depicts the correlation results in graphical form revealing that there is a quite acceptable agreement between experimental and correlated γ±,m data with an overall AAD value of 3.02%. On the other hand, Figure 10 shows the corresponding model predictions for osmotic coefficients; it can be seen that these predictions are also in good agreement with the experimental data with an overall AAD value of 1.94%; moreover, the present approach is able to correctly predict the two maximums in the osmotic coefficient that were experimentally obtained at 250 and 300 °C and at 41.63 M. Although the agreement between calculated and experimental data is quite good for the eight isotherms, according to Figures 9 and 10, model estimations start somewhat departing from the experimental data at temperatures larger than 100 °C. This is also reflected in the individual model deviations listed in Table 9, namely, those AAD values obtained at 150 °C and above are much larger than those obtained at 0, 25, 50, and 100 °C; in fact, the value of k0 drastically changes from 8.098 at 100 °C to 0.00065 at 150 °C indicating that ion hydration is not existent at temperatures higher than 100 °C. These modeling difficulties may be attributed to ion association

Figure 7. Performance of the two- and three-parameter models for LiBr, ZnBr2, and ZnCl2 at 25 °C: (a) correlated mean ionic activity coefficients, (b) predicted osmotic coefficients.

electrolytes: HCl, HClO4, and HNO3, despite having the same H+ cation, yielded hydration levels that greatly differ from each other; as shown in Figure 8, their hydration levels lie in the order: HNO3 ≪ HCl ≪ HClO4 over their entire molality ranges. As a matter of fact, the H+ in HClO4 was the highest hydrated cation of all 1−1 electrolytes considered here, J



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Table 9. Performance of the Three-Parameter Model for the KOH−Water Solution at Various Temperatures mmax,

%AAD

model parameters

T, °C

N

mol/kg

γ±,m

φ

k0

αs,w

βs,w

0 25 50 100 150 200 250 300 overall

6 6 6 6 7 8 9 10 58

18.503 18.503 18.503 18.503 27.754 41.631 64.76 83.263

1.13 1.12 1.12 1.15 3.41 3.91 4.60 5.17 3.02

0.97 0.96 1.02 1.10 2.03 2.28 2.52 3.31 1.94

8.17077 8.14812 8.12739 8.09832 0.000648 0.000477 0.000516 0.000801

−6.85243 −4.14734 −2.03874 0.89418 −9.51976 −8.81691 −8.45632 −7.80246

−1.30192 −0.25760 0.51667 1.47842 −1.88166 −1.92563 −2.11931 −2.07179

Figure 9. Correlated mean ionic activity coefficients for the KOH− water system at various temperatures.

Figure 10. Predicted osmotic coefficients for the KOH−water system at various temperatures.

occurring at high temperatures for which the present model is not able to account for; for example, as the temperature increases, the dielectric constant of the solution decreases thus causing higher levels of ion association; moreover, at such extremely high molalities (83.26 M), ions are very prone to associate thus forming ion pairs and/or complex ions. Finally, it is pertinent to compare the correlating capabilities of the present modeling approach against those of existing Gexcess models of similar characteristics. In this context, the Gexcess models proposed by Chen et al.12 and Simonin et al.14 were considered here for comparison purposes; both groups of investigators allowed the inclusion of not only ion solvation but also ion association into their so-called electrolyte−NRTL models; the difference between these two modeling works relies on the way the long-range electrostatic forces were modeled: for example, Chen et al.12 used the traditional Debye−Huckel (DH) equation thus proposing two DHNRTL modeling versions (a two-parameter model with solvation only and a five-parameter model with both solvation and association), whereas Simonin et al.14 employed the MSA approach thus yielding a five-parameter MSA-NRTL model. Table 10 gives the comparison of the present two- and three-

parameter models with the aforementioned models of Chen et al.12 and Simonin et al.14 in terms of AAD values between experimental and correlated γ±,m data at 25 °C for some selected electrolytes (CaCl2, HCl, LiBr, LiCl, ZnBr2 and ZnCl2) along with the corresponding maximum molalities at which each model was validated. As revealed by Table 10, for the same basis of comparison (number of parameters and maximum molality) the present two-parameter model was clearly superior to the two-parameter DH-NRTL model in the cases of HCl and LiCl thus yielding AAD values 5 to 7 times lower than those of the Chen et al.12 formulation; moreover, the present two-parameter model and the five-parameter DHNRTL approach give rather comparable results for the two aforementioned electrolyte systems. In the case of the other electrolytes (CaCl2, LiBr, ZnBr2, and ZnCl2), it is not easy to tell which model prevails between the present three-parameter model and the two Chen et al.12 models since the latter yielded lower AAD values but at a maximum concentration of 6 M. The same reasoning applies to the five-parameter MSA-NRTL model proposed by Simonin et al.14 who de facto obtained the lowest AAD values but at much lower maximum molalities (4 M for CaCl2, 6 M for HCl, LiBr, and LiCl) and using a larger K



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Table 10. Comparison of the Present Model with Other Previously Reported Two- and Five-Parameter G-Excess Models in Terms of AAD Values Obtained from Correlated γ±,m Data present model

2-P DH-NRTL12

5-P DH-NRTL12

salt

%AAD

mmax

%AAD

mmax

CaCl2 HCl LiBr LiCl ZnBr2 ZnCl2

2.77b 1.44a 3.08b 1.98a 12.1b 9.38b

10 16 20 19.219 20.1 23.193

3.06 9.72 0.95 10.61

6 16 6 19.219

7.31

6

%AAD

mmax

1.76

16

1.35 1.27

19.219 6

5-P MSA-NRTL14 %AAD

mmax

0.21 0.14 0.28 0.28

4 6 6 6

a

Two-parameter model. bThree-parameter model. mmax in mol/kg.

number of adjustable parameters (from 2 to 3 more than the present models). As mentioned earlier, the only existing Gexcess model that rivals in simplicity and number of parameters with the present modeling approach is the one by Zerres and Prausnitz;27 unfortunately, the authors regressed their model parameters from experimental vapor pressures of aqueous solutions of CaCl2, KOH, LiCl, and KOH at 25 °C and reported their resulting model deviations in terms of this quantity (not in terms of γ±,m). Consequently, a fair comparison regarding the correlation abilities between the Zerres and Prausnitz27 approach and the present twoparameter model was not possible.

■

■

CONCLUSIONS A simple yet accurate G-excess model was developed here for completely ionized aqueous electrolyte solutions being applicable at high molalities and within a wide range of temperature. The present model was successfully applied to the correlation of mean ionic activity coefficients and prediction of osmotic coefficients of concentrated aqueous solutions containing a single electrolyte. The following conclusions can be drawn from this work: • The present modeling approach (in its two- and threeparameter versions) correlated quite well the experimental mean ionic activity coefficients of various concentrated electrolyte systems (CaCl2, AgNO3, HCl, HClO4, HF, HNO3, KF, KOH, LiBr, LiCl, LiNO3, NaCNS, NaOH, and NH4NO3) at 25 °C using recommended ionic diameters which in turn were selected from well-known sources. • Furthermore, the new G-excess model was even more accurate in predicting the observed osmotic coefficient of the aforementioned electrolyte solutions. • The use of a stepwise solvation chemistry within the present modeling work provides a consistent yet relatively simple thermodynamic framework that was proven here to have good correlative and predictive capabilities in representing well the nonidealities of concentrated aqueous electrolyte solutions. • In the particular cases of HF, ZnBr2, and ZnCl2, the present two- and three-parameter models failed to give a good representation of the experimental data; these electrolytes have a strong tendency to form ion pairs and/or complex ions at high concentrations (even at low concentrations as in the case of HF), this highly nonideal behavior cannot be described by the present models since they lack ion association chemistry. • Model parameters (k0, αs,w and βs,w) in the case of the KOH−water system were found to be temperature

dependent; however, the present model should be further validated with other electrolyte solutions at temperatures other than 25 °C in order to establish suitable functionalities of the model parameters with temperature. • Because of its relative simplicity, the present approach may be well suited for use in engineering calculations for solutions of strong and weak electrolytes where ion solvation plays a significant role in their thermodynamic properties.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Ricardo Macías-Salinas: 0000-0002-0372-8190 Notes

The author declares no competing financial interest.

■ ■

ACKNOWLEDGMENTS The author gratefully acknowledges the Instituto Politécnico Nacional for providing financial support for this work. LIST OF SYMBOLS a = Activity A = Helmholtz free energy Ci = Solvated cation C0 = Unsolvated cation e = Unit electronic charge G = Gibbs free energy h = Average solvation number k0 = Adjustable equilibrium-constant parameter Ki = Equilibrium constant for each solvation step m = Salt molality M = Molar mass n = Number of ions nt = Total number of moles N = Number of data points Na = Avogadro’s number NH = Number of binding sites P = Pressure R = Universal gas constant T = Temperature V = Solution volume x = Vector containing the mole fractions of each species in solution xi = Liquid mole fraction of species i Zi = Charge of ionic species I

L



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Greek Letters

αs,w βs,w γ Γ ε ε0 κ ξ μ ν σ σi φ

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Margules binary interaction parameter Margules binary interaction parameter Activity coefficient Screening factor in the MSA contribution Dielectric constant Permittivity in vacuum Reciprocal Debye screening length True mole fraction Chemical potential Number of dissociated ions Effective ionic diameter Diameter of ionic species i Osmotic coefficient

Subscripts

A C Ci C0 m s w ±

Anion Cation Solvated cation Unsolvated cation Mean molal condition Salt Water Mean ionic condition

Superscripts

calc exp EX f MSA Phys Solv ∞

■

Calculated value Experimental value Excess property Free condition Mean-spherical-approximation contribution Physical contribution Ion solvation contribution Infinite dilution

REFERENCES

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