An extended solvation theory for electrolyte osmotic and activity coefficients. I. Application at 298.15 K

Fluid Phase Equilibria 499 (2019) 112243

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An extended solvation theory for electrolyte osmotic and activity coefficients. I. Application at 298.15 K Javier Temoltzi-Avila a, Gustavo A. Iglesias-Silva a, *, Mariana Ramos-Estrada b, Kenneth R. Hall c, **, 1
gico de Celaya, Celaya, Guanajuato, CP 38010, Mexico Departamento de Ingeniería Química, Instituto Tecnolo
s de Hidalgo, Morelia, Mich, CP 58030, Mexico Facultad de Ingeniería Química, Universidad Michoacana de San Nicola c Bryan Research & Engineering, Bryan, TX, USA a

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 25 February 2019 Received in revised form 11 July 2019 Accepted 12 July 2019 Available online 13 July 2019

This paper presents a new model to correlate osmotic and activity coefficients of electrolyte solutions. The model contains a long-range interactions term based upon Debye-Hückel theory (Debye- Hückel, DH or Pitzer-Debye-Hückel, PDH) and a short-range interactions term based upon extended solvation theory (ST). Testing the correlative capability of the new model used 56 electrolyte aqueous solutions: 1:1, 1:2, 1:3, 1:4, 2:1, 2:2, 3:1, 3:2 and 4:1 at 298.15 K. The average percentage deviation, standard deviation and its bias from the experimental data are (0.48, 0.25 and
0.001) and (0.69, 0.26 and 0.005) for DH-ST and PDH-ST, respectively. The results demonstrate that the new equation correlates experimental data effectively. © 2019 Elsevier B.V. All rights reserved.

Keywords: Mean ionic activity coefficient Osmotic coefficient Solvation Electrolyte solutions Aqueous solutions

1. Introduction Recently, interest in calculation of thermodynamic properties and phase equilibria of electrolyte solutions have increased significantly because of their use in processes such as separation, extraction, absorption, biological processes, oil recovery, sea water desalination [1,2]. This effort requires models for osmotic and activity coefficients of electrolyte solutions to account for their nonideality. One of the first models to correlate the activity coefficient of these solutions is the Debye and Hückel [3,4] limiting law. This model utilizes the theory of interaction-attraction effects, and it applies successfully to dilute solutions. This theory has been the basis for developing activity coefficient models that consider interactions among ions and solvent molecules. Examples of these equations are models developed by Pitzer et al. [5e7], Archer (modifications of the Pitzer model) [8e15], Hamer et al. [16], and Bromley [17]. Other models have considered different approaches.

* Corresponding author. ** Corresponding author. E-mail address: [email protected] (G.A. Iglesias-Silva). 1 K. R. Hall is also Professor Emeritus at Texas A&M University, College Station, TX, USA. https://doi.org/10.1016/j.fluid.2019.112243 0378-3812/© 2019 Elsevier B.V. All rights reserved.

Cruz and Renon [18] and Chen et al. [19] use NRTL and the PitzerDebye-Hückel equation to describe short-range and long-range activity coefficients,. Triolo et al. [20], Planche and Renon [21], Bell et al. [22] Taghikhani [23] and Lotfikian and Modarress [24] provide models for activity coefficients based upon the mean spherical approximation. Stokes and Robinson [25] introduced the solvation concept to describe the activity coefficient of electrolyte solutions. They postulate that at high solute concentrations in an electrolyte solution an adsorbent (salt)-adsorbate (water) model approximates the system, so their model resembles a Brunauer-Emmett-Teller (BET) adsorption isotherm [25]. Later, they extended the model using a stepwise hydration-equilibrium model for osmotic coefficients of strong, highly-soluble electrolytes [26,27]. In 1993, Lin et al. [28,29] proposed a model for the activity and osmotic coefficients. They used a numerical solution of the PoissonBoltzmann equation to describe the long-range ion-ion interactions and a short-range term that accounts for the ionmolecule interactions using the solvation concept. Specifically, for the short-range interactions, they use an electric potential between ion-dipole using Coulomb's law. Their original model had four characteristic parameters, but later they reduced it to three. Lin et al. [30,31] modified the model by replacing the term proposed by

2

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Pitzer [5] for the long range forces. Then, Pazuki [32,33] combined the Debye-Hückel equation with the solvation term to develop an equation for the activity and osmotic coefficients. This work extends the solvation theory (ST) and proposes a model for the osmotic and mean activity coefficients that contains the Debye-Hückel or the Pitzer-Debye-Hückel equation for long range interactions. The new equation correlates the experimental activity and osmotic coefficients of different aqueous solutions at 298.15 K. These equations apply to systems that other models cannot correlate.

2. Theory and osmotic and activity coefficient models In 1993, Lin et al. [28,29] developed a model to correlate the activity and osmotic coefficients of strong electrolytes in binary solutions. Their equation has a term that describes the long-range ion-ion interactions based upon the Poisson-Boltzmann equation, and a term that describes the short-range interactions based upon solvation theory. The short-range interaction term uses Coulomb's law for the electrical potential between ion, i, and the solvent molecule, in addition to their expression for the activity coefficient

lngSR MX ¼

S I2n ðnþ þ n
Þ

(1)

in which S is a solvation parameter, I is the ionic strength and n is a characteristic parameter obtained from experimental data. They fixed n at 0.645. Later, Lin et al. [30,31] combined the solvation term for the activity coefficient with the Pitzer-Debye-Hückel expression for the long-range interaction activity coefficient. Recently, Ge et al. [34,35] use the Lin et al. original model, but they consider bMX , S and n as variables The dielectric constant is solvent-specific and S is independent of temperature in this work because the paper only deals with the single temperature 298.15 K. Under these assumptions, the mean activity coefficient becomes

" lngMX ¼
jzM zX jAf

  I 1=2 2 þ ln 1 þ bMX I 1=2 1=2 bMX 1 þ bMX I

# (2)

S I 2n þ ðnþ þ n
ÞT

 e2 m  2 2 h z n þ h z n M X MS XS M X 2B2 k

(3)

in which m is the water dipole moment, his is an interaction parameter between ion and solvent molecule; k is the Boltzmann constant (1.3805  10
23 J K
1); e is the unit charge (1.6021  10
19 C); and vi is the stoichiometric number of ion i. In 2005, Pazuki et al. [32] combined the Debye-Hückel [3,4] equation with the solvation equation of Lin et al. to obtain the activity coefficient of the molecule

" lngMX ¼
3jzM zX jAf

I 1=2 1 þ bMX I 1=2

#

jz z j þ M X SIn T

ri ¼

in which S ¼ ½e2 m=ð2B2 kÞ½ðhMS nX þ hXS nM Þ=ðnþ þ n
Þ. In the solvation theory developed by Lin et al. [28,29], the electrical potential between ion an i and a solvent molecule (water) is

(5)

Bi In

(6)

in which Bi is a proportionality parameter (different for cations and ions) and n is a parameter obtained from experimental data. As the concentration increases, the interactions in an electrolyte solution become more complex because of the presence of ions in the solution, random movement caused by thermic energy, ion-solvent interactions, solvent-solvent interactions, etc. The first neighbor model [36] can describe ion hydration. In this model, the structure function of a solution depends upon the structure functions of ion aggregates, “free” water and their respective compositions. Therefore, the ion dipole distance should depend upon a function of the ionic strength that accounts for all the above interactions

r2 ¼

B2i gðIÞ

(7)

Assuming that the ion-dipole distance of a solvated ion (p± 0 ) is inversely proportional to the ionic strength, I n0 can change in the presence of a particle, p± 1 , in the same direction because of the concentration and nature of the solute. This particle could be an ion or a contra-ion with a different tendency for solvation, an ionic pair or a dissociated electrolyte particle (this particle would not modify substantially the distance because of its neutral charge). This effect is similar to the contribution of the structure functions in the first neighbor model. Here, the ion-dipole distance modification appears in the ionic function as a term a± I n1 , in which a± is the intensity of the ionic strength caused by p± 1 . The distance can change in the presence of another aggregate by an amount b± I n2 . Therefore, the square of the distance ion-dipole is

I n0

þ

aI n1

B2i þ bI n2 þ /

(8)

Substituting Eq (8) into Eq (5) provides the electrical potential

Gis ¼

 his zi em  n0 I þ aI n1 þ bI n2 þ / Bi

(9)

The dimensionless electrical potential for a ion and its surrounding solvent molecules is

Fis ¼

eGis kT

(10)

in which Gis is electrical potential between an ion and a solvent molecule. The dimensionless electrical potential defined for a ion and a surrounding ion is

Fir ¼ (4)

his zi em r 2i

in which ri is the distance between the ion and the solvent molecule. They assume that this distance decreases as the concentration increases, so it can be inversely proportional to the ionic strength

r 2i ¼

in which Af is the Debye-Hückel constant; bMX is the parameter describing the closest approach distance of ions; T is the absolute temperature; nþ and n
are stoichiometric coefficients of the cation or anion; zi is the absolute charge number of ion i; and S is the solvation parameter

S¼

Gis ¼

eJir kT

(11)

in which Jir is the electrical potential of an ion within an electric field. Applying the charge process theorem, the work necessary to increase the charge of an ion from 0 to z,e equals the electrostatic potential of the surrounding ions, kT lngi . Then, the activity coefficient of an ion is

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

ðz ðz 1 ðJir þ Gis Þedz ¼ ðFir þ Fis Þdz kT

lngi ¼

0

(12)

ðz his 0

 zi e2 m  n0 I þ aI n1 þ bI n2 þ / dz Bi kT

 z2i e2 m  n0 I þ aI n1 þ bI n2 þ / 2Bi kT

" (13)

 z2i  S I n0 þ S1;i In1 þ S2;i I n2 þ / T 0;i

(14)

(15)

with

lngMX ¼
3jzM zX jAf

I 1=2 1 þ bMX I 1=2

# (25)

3  2 2ln 1 þ bMX I 1=2 3jzM zX jAf 4 bMX I 1=2 þ 2 5
f
1 ¼
bMX I 1=2 bMX þ b2MX I 1=2 b2MX I 1=2  n n þ k1 jz z j S In þ þ M X S I nþk1 1þn 0 nT 1 þ n þ k1 1  n þ k2 þ S2 I nþk2 þ / 1 þ n þ k2 (26)

S0;i ¼

his e2 m 2Bi k

(16)

S1;i ¼

ahis e2 m 2Bi k

(17)

S2;i ¼

bhis e2 m 2Bi k

(18)

The mean ionic activity coefficient results from lngMX ¼ ðnM lngM þ nX lngX Þ=ðnM þ nX Þ and using electronegativity nM zM ¼ jnX zX j,

lngSR MX ¼

(24)

 jz z j  þ M X S0 I n þ S1 I nþk1 þ S2 I nþk2 þ / nT

or

lngi ¼

lngMX dm

Considering f ðIÞ to be the Debye-Hückel equation, the activity and osmotic coefficients become

which becomes upon integration

lngi ¼ his

m ð

0

Neglecting the ion term and considering only the solvation term, the short range activity coefficient is

lngi ¼

f ¼ 1 þ lngMX

0

1
m

3

 jzM zX j  n0 S0 I þ S1 I n1 þ S2 I n2 þ / nT

If f ðIÞ is the Pitzer-Debye-Hückel for the long-range interactions term, the osmotic and activity coefficients become

"

#   I 1=2 2 1=2 þ ln 1 þ b I MX 1 þ bMX I 1=2 bMX  jz z j  þ M X S0 I n þ S1 I nþk1 þ S2 I nþk2 þ / nT

lngMX ¼
jzM zX jAf

(27) " f
1 ¼
jzM zX jAf

(19)

þ

#  I 1=2 jzM zX j n S In þ 1þn 0 nT 1 þ bMX I 1=2

n þ k1 n þ k2 S I nþk1 þ S I nþk2 þ / 1 þ n þ k1 1 1 þ n þ k2 2

 (28)

with

  e2 m hMS nX hXS nM S0 ¼ þ 2k BM BX 

with the slope of the Debye-Hückel law

(20) Af ¼



S1 ¼ a

e2 m hMS nX hXS nM þ 2k BM BX

S2 ¼ b

  e2 m hMS nX hXS nM þ 2k BM BX

(21)

(22)

in which S0 , S1 , S2 , /, are temperature dependent. However, this work considers only a single temperature, 298.15 K, thus they are constants. Temperature dependence of one of these parameters has appeared previously [37]. A future work can develop the temperature dependence for all these parameters. Considering n ¼ n0 ¼ n1
k1 ¼ n2
k2 ¼ / and k1 sk2 s /, the activity coefficient becomes

lngMX ¼ f ðIÞ þ

 jzM zX j  n S0 I þ S1 I nþk1 þ S2 I nþk2 þ / nT

(23)

in which n ¼ ðnM þ nX Þ and the k
is are characteristic parameters. The osmotic activity coefficient comes from

 3=2 ð2pNA rÞ1=2 e2 3 4pε0 εr kT

(29)

in which NA is Avogadro's number; εr and ε0 are the dielectric constants of the solvent and (at vacuum) respectively; r is the solvent density; k is the Boltzman constant; and e is the absolute electron charge. Values of εr ¼ 78:38 , ε0 ¼ 8.8541878176  10
12, and r ¼ 997:05 Kg=m3 come from the literature [38]. In Eqs (25)e(28) the adjusting parameters are the closest approach parameter, bMX , the solvation parameters, S0 , S1 , S2 , /; and the characteristic parameter, n. In this paper, ki ¼ i
12 for i  1 are fixed parameters. It is possible to calculate single ion activity coefficients from Eq (15) using the long range contribution, fi ðIÞ, given by the DebyeHückel or the Pitzer-Debye-Hückel equation:

lngi ¼ fi ðIÞ þ

 z2i  S0;i I n þ S1;i I nþk1 þ S2;i I nþk2 þ / T

(30)

and the mean ionic activity coefficient comes from its definition using

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J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

P

" fi ðIÞ þ

z2i T



S0;i

In

þ S1;i

lngMX ¼

I nþk1

þ S2;i

I nþk2

#  þ /  ni

bias ¼

nM þ nX

N 1 X Dyi N i¼1

(36)

(31) This work provides the single ion activity coefficient of different cation-anion pairs using Eq. (30). Then the mean ionic activity coefficient comes from Eq. (31). Data for the single ion activity coefficient come from the literature [39,40]. Several authors [16,41,42] report the mean ionic activity coefficients for these aqueous solutions.

3. Methodology This work uses a least squares method to obtain the fitting parameters in Eqs (25)e(28). Experimental activity and osmotic coefficients at 298.15 K come from the literature. The objective function to optimize the activity and osmotic coefficient equations is

J¼

N1  X

exp

lngi


lngcalc i

2

i¼1

þ

N2  X

exp

fi


fcalc i

2

(32)

i¼1

in which N1 and N2 are the number of points of activity and osmotic coefficients, respectively. This work uses a least squares method developed by Levenberg-Marquardt [43e45] embedded in the Wolfram Mathematica® software. The established confidence level for parameters is 95% in all systems. Fitting and fixed parameters in the activity and osmotic coefficient models appear in Table 1. The percentage error is

yexp
ycalc i i

Dyi ¼

!

yexp i

,100

(33)

in which yi can be the mean activity coefficient, g, or the osmotic coefficient, f. In this work, the average absolute percentage error is

AAPD ¼

N 1 X jDyi j N i¼1

(34)

in which N is the number of data points. The standard deviation is

2

 2 31=2 exp calc N y
y X i i 6 7 SD ¼ 4 5 N
Np i¼1

(35)

in which Np is the number of adjustable parameters; and the bias is

Table 1 Adjusted parameters in the new models. Parameter

DH-MS

bMX S0 S1 S2 « k1 k2 « n

Fit to experimental Fit to experimental Fit to experimental Fit to experimental « Fixed parameter Fixed parameter « Fit to experimental

4. Results and discussion A parametric study indicates that k1 ¼ 1=2 when the short range interactions term ends at the second solvation term. In most cases, a second term is sufficient to correlate the activity and osmotic coefficients. Table 2 contains average absolute percentage deviations, the standard deviations and biases for the electrolyte solutions considered in this work. Also, the value of the parameters together with the asymptotic standard error of the parameter appear in Table 3. Figs.1 and 2 demonstrate the correlative capability of the two models for activity and osmotic coefficients respectively. Figs. 3 and 4 show the correlation of the models for the activity and osmotic coefficient at concentrations as high as 36 molal. The models developed by Lin et al. [28,29] and Ge et al. [34,35] can correlate these systems at concentrations as high as 6 molal. Some systems exist in which truncation at the second solvation term is not sufficient. Therefore, the equation must include more terms in the short-range interaction term. We have compared correlative capability of the new solvation equations with the extended Pitzer model developed by Archer [8,10]. Previously, reference [46] indicates that the model developed by Archer performed better than the Pitzer model [5,47] and some of their modifications [48,49]. Figs. 5 and 6 show that the new solvation models correlate the osmotic and activity coefficients slightly better than the Archer model, see for example the aqueous systems Y(NO3)3 and Th(NO3)4. Fig. 7 illustrates that for the activity and osmotic coefficient of the CCl3COONa and (C3H7)4NBr aqueous systems, the new models should include more terms in the solvation term. Table 4 illustrates the correlative capability (in terms of AAPD) of the extended (two-terms) solvation equation DH-MS and PDH-MS for single ions of different cation-anion pairs in aqueous solutions. The Table also contains calculations of the mean ionic activity coefficient using the solvation model for the single ions. The new equations agree with the experimental data as shown in Fig. 8. Parameters for the models, Eq. (15), with the Debye-Hückel or the Pitzer-Debye-Hückel equation appear in Table 5. Consideration of the equivalence between the solvation models DH-MS and PDH-MS and the Pitzer model. The Pitzer [5,47] model is an improvement of the model proposed by Guggenheim [50,51]. The Pitzer model contains two terms: one for long-range electrostatic interactions and a second that considers short-range interactions expressed by molality terms. Pitzer [5,47] calls these terms virial coefficients recalling the virial equation of state. The equation for the mean activity coefficient truncated at the third virial coefficient for a single electrolyte is

lngMX ¼ jzM zX jf g þ

2nM nX

n

g

mMX BMX þ

2ðnM nX Þ3=2

n

g

m2MX C MX

(37)

and the respective equation for the osmotic coefficient is PDH-MS data data data data

data

Fit to experimental Fit to experimental Fit to experimental Fit to experimental « Fixed parameter Fixed parameter « Fit to experimental

data data data data

f
1 ¼ jzM zX jf f þ

n

mMX BfMX þ

2ðnM nX Þ3=2

n

m2MX C fMX

(38)

In this work, the solvation models for the activity and osmotic coefficients truncated at two solvation terms are

lngMX ¼ jzM zX jf g þ data

2nM nX

and

 jzM zX j  n S0 I þ S1 I nþk1 nT

(39)

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

5

Table 2 Comparison between DH-MS and PDH-MS models truncated models in the second solvation term for the correlation of the activity and osmotic coefficients of aqueous electrolyte solutions at 298.15 K Electrolyte

1:1 Electrolytes (C3H7)4NBr (CH3)4NCl (CH3)4NNO3 CCl3COONa HClO4 HNO3 KF LiBr LiNO2 RbNO2 1:2 Electrolytes (CN3H6)2CO3 (NH4)2B10H10 Cs2S2O8 K2HAsO4 Li2SO4 Na2CO3 Na2CrO4 Na2HASO4 Rb2S2O8 Rb2SO4 1:3 Electrolytes K3AsO4 K3Co(CN)6 K3Fe(CN)6 1:4 Electrolytes K4ATP K4Fe(CN)6 K4Mo(CN)8 2:1 Electrolytes C8H22N2I2 Ca(ClO4)2 Ca(NO3)2 CaBr2 Co(ClO4)2 Co(NO3)2 MgCl2 MnBr2 Pb(ClO4)2 UO2(NO3)2 2:2 Electrolytes BeSO4 CaSO4 CdSO4 CoSO4 CuSO4 3:1 Electrolytes AlCl3 CeCl3 Er(ClO4)3 GdCl3 Ho(C2H5SO4)3 Lu(NO3)3 LuCl3 Nd(C2H5SO4)3 PrCl3 ScCl3 3:2 Electrolytes Al2(SO4)3 Cr2(SO4)3 Lu2(SO4)3 4:1 Electrolytes Pt(en)3Cl4 Th(NO3)4 Total

mmin

mmax

Obs.

DH-MS

PDH-MS

Ref

AAPD

SD

bias

AAPD

SD

bias

0.1 0.1 0.1 0.1 0.001 0.001 0.001 0.001 0.001 0.001

9 19 12 9 16 28 17.5 20 19.9 62.3

26 36 29 26 39 51 41 43 47 90

1.2 0.6 0.1 1.2 0.5 0.3 0.2 1.7 0.1 0.6

1  10
2 6  10
3 1  10
3 1  10
2 0.7 1  10
2 3  10
3 2.4 5  10
3 4  10
3

1  10
3 7  10
4 2  10
4
5  10
4 5  10
2
1  10
5
7  10
5 0.1 5  10
4 2  10
4

1.2 0.6 0.1 1.3 0.7 0.3 0.4 2.0 0.1 0.9

1  10
2 6  10
3 1  10
3 2  10
2 1.0 8  10
3 6  10
3 2.8 5  10
3 7  10
3

2  10
3 7  10
4 3  10
4
6  10
4 0.1
1  10
5
9  10
5 0.1 7  10
4 7  10
4

[54] [54] [55] [56] [16] [16] [16] [16] [57] [57]

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

2.613 3.806 0.109 0.886 3.165 3.1115 4.363 1.029 0.075 1.707

35 40 20 27 38 38 42 29 17 31

0.2 2  10
2 4  10
3 6  10
3 8  10
3 0.1 0.1 6  10
3 4  10
3 9  10
3

2  10
3 2  10
4 3  10
5 6  10
5 7  10
5 1  10
3 4  10
3 6  10
5 4  10
5 8  10
5

2  10
5 3  10
5
1  10
5 1  10
5 2  10
5 2  10
4
3  10
4 0  100 0  100 1  10
5

0.3 0.3 5  10-3 0.1 0.5 0.2 0.2 0.1 4  10-3 0.1

2  10
3 2  10
3 4  10
5 1  10
3 4  10
3 2  10
3 4  10
3 6  10
4 4  10
5 1  10
3

5  10
4 5  10
4
1  10
5 3  10
4 7  10
4 5  10
4
1  10
5 1  10
4 0  100 5  10
5

[42] [42] [42] [42] [42] [42] [42] [42] [42] [42]

0.1 0.01 0.1

0.7 1.311 1.4

7 15 12

0.1 0.7 0.1

1  10
3 2  10
2 1  10
3


3  10
4
3  10
3
3  10
4

0.1 1.3 0.2

4  10
4 2  10
2 2  10
3

2  10
5
4  10
3
3  10
4

[41] [58] [41]

0.1 0.1 0.1

2.4 0.9 1.4

24 9 12

0.6 0.4 0.4

4  10
3 4  10
3 1  10
3

0  100
2  10
3
4  10
4

0.5 0.4 0.5

4  10
3 3  10
3 2  10
3

5  10
5
1  10
4
3  10
4

[59] [41] [41]

0.001 0.1 0.005 0.001 0.1 0.001 0.1 0.001 0.001 0.001

4 6 20 9.21 3.5 5.79 5.9188 5.64 10.25 5.511

40 23 24 62 18 49 36 47 65 47

0.7 0.5 0.3 2.3 0.3 0.3 0.4 0.2 0.4 0.9

5  10
3 0.1 4  10
3 4.7 3  10
2 7  10
3 0.1 1  10
2 0.1 2  10
2

1  10
3 4  10
4 3  10
4 0.1
3  10
4 5  10
4
2  10
3 2  10
4
7  10
5 2  10
4

0.6 0.7 0.6 3.1 0.4 0.5 0.5 0.4 0.5 1.0

4  10
3 4  10
2 6  10
3 6.4 3  10
2 1  10
2 0.1 9  10
3 0.1 2  10
2

8  10
4 3  10
3 5  10
4 0.1
4  10
4 9  10
4
8  10
4 8  10
4 8  10
4 1  10
3

[60] [41] [61] [62] [63] [64] [65] [66] [66] [66]

0.1 0.0001 0.1 0.0001 0.1

4 0.02 3.5 0.1 1.4

19 5 18 7 12

0.3 1.0 0.4 0.1 0.5

2  10
3 2  10
2 4  10
3 1  10
3 4  10
3


9  10
4
6  10
3 2  10
3
3  10
5 1  10
3

0.6 1.0 0.3 0.2 0.2

6  10
3 2  10
2 2  10
3 2  10
3 2  10
3


3  10
3
6  10
3
2  10
4
6  10
5 8  10
5

[41] [67] [41] [67] [41]

0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.1

1.8 2 4.6221 3.5898 1.1 7.1806 4.1239 1.1 2 1.8

14 15 29 23 11 41 26 11 15 14

0.4 0.4 0.8 0.2 0.2 0.8 0.3 0.2 0.3 0.3

6  10
3 6  10
3 5.6 3  10
3 3  10
3 2  10
2 3  10
2 3  10
3 4  10
3 3  10
3


2  10
3
2  10
3
0.3
3  10
4
9  10
4 3  10
4
3  10
4
8  10
4
1  10
3
7  10
4

0.8 0.8 1.1 0.7 0.3 0.6 0.7 0.3 0.6 0.5

1  10
2 1  10
2 3.6 1  10
2 4  10
3 1  10
2 2  10
2 4  10
2 8  10
3 7  10
3


4  10
3
4  10
3 2  10
2
1  10
3
1  10
3
6  10
4 1  10
5
1  10
3
3  10
3
3  10
3

[41] [41] [68] [69] [70] [71] [69] [70] [41] [41]

0.1 0.1 0.05

1 1.2 0.89415

10 11 19

1.3 0.3 1.7

8  10
3 3  10
3 8  10
3

7  10
4
5  10
4 7  10
4

3.0 1.7 2.0

2  10
2 1  10
2 9  10
3


6  10
3
5  10
3 1  10
3

[41] [41] [72]

0.01 0.1

0.33 5

9 21

0.7 0.9 0.4795

6  10
3 8  10
3 0.2504

2  10
3 1  10
3 ¡0.0015

1.1 1.2 0.6856

7  10
3 1  10
2 0.2565

1  10
3 3  10
3 0.0051

[73] [41]

6

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

Table 3 Parameters for the truncated in the second solvation term DH-MS and PDH-MS models for the correlation of the activity and osmotic coefficients of aqueous electrolytes at 298.15 K. (Asymptotic standard error of the parameters is given in parenthesis). Electrolyte 1:1 Electrolytes (C3H7)4NBr (CH3)4NCl (CH3)4NNO3 CCl3COONa HClO4 HNO3 KF LiBr LiNO2 RbNO2 1:2 Electrolytes (CN3H6)2CO3 (NH4)2B10H10 Cs2S2O8 K2HAsO4 Li2SO4 Na2CO3 Na2CrO4 Na2HASO4 Rb2S2O8 Rb2SO4 1:3 Electrolytes K3AsO4 K3Co(CN)6 K3Fe(CN)6 1:4 Electrolytes K4ATP K4Fe(CN)6 K4Mo(CN)8 2:1 Electrolytes (C8H22N2)I2 Ca(ClO4)2 Ca(NO3)2 Co(ClO4)2 Co(NO3)2 MgCl2 MnBr2 Pb(ClO4)2

DH-MH bMX

S0

S1

n

PDH-MH bMX

S0

S1

n

5.5481  10
5 (2.8971  10
5) 0.1599 (0.0543) 0.0506 (0.6938) 1.2308 (1.2398) 2.4246 (0.0317) 1.4859 (0.0234) 1.4636 (0.0052) 2.7048 (0.1044) 1.4998 (0.0288) 0.7870 (0.0043)

206.5507 (1.8858) 260.0737 (31.6232) 243.9069 (457.6111) 222.1811 (206.1566) 102.0336 (1.1349) 106.7533 (2.3602) 27.6736 (0.3155) 77.3872 (2.3399) 124.3241 (3.0940)
4.1390 (0.31425)


43.8951 (0.4181)
24.1638 (2.8416)
23.2672 (17.2968)
45.0190 (32.4685)
12.7955 (0.0660)
14.1422 (0.2600)
4.6250 (0.0425)
12.2689 (0.3001)
18.1966 (0.3491) 0.3356 (0.0213)

1.3922 (0.0107) 0.8812 (0.0035) 0.8847 (0.1723) 0.8091 (0.2547) 1.5722 (0.0072) 1.0412 (0.0076) 1.7392 (0.0056) 1.7244 (0.0144) 1.1097 (0.0102) 1.403 (0.0221)

3.8852  10
14 (7.8620  10
14) 0.2038 (0.0868) 0.8675 (0.0061) 1.6988 (2.6616) 4.5348 (0.1028) 2.3062 (0.0461) 2.6044 (0.0248) 5.3882 (0.2850) 2.3070 (0.0640) 1.4298 (0.0089)

206.7610 (1.8801) 276.0850 (33.1010) 20.0294 (1.0134) 275.0740 (280.7984) 115.5024 (1.8405) 141.5287 (2.8353) 39.2884 (0.8642) 85.8580 (2.9966) 160.2305 (4.1866)
0.4468 (0.0896)


43.9050 (0.4236)
25.0007 (2.7909)
4.5665 (0.1410)
52.863 (39.8415)
13.7079 (0.1068)
17.9423 (0.2900)
6.2817 (0.1064)
13.4376 (0.3770)
22.4844 (0.4398) 0.0423 (0.0078)

1.3913 (0.0107) 0.8858 (0.0039) 1.6904 (0.0383) 0.8018 (0.2484) 1.5169 (0.0102) 0.9809 (0.0063) 1.6155 (0.0107) 1.6867 (0.0163) 1.0463 (0.0099) 1.8977 (0.0573)

1.1549 (0.0416) 1.3000 (0.0011) 0.5692 (0.0194) 1.7673 (0.0010) 1.2772 (0.0002) 1.3511 (0.0103) 1.5134 (0.016) 1.6618 (0.0010) 0.8819 (0.0410) 1.1079 (0.0006)


172.0511 (10.6905) 5.7564 (0.2488)
1.9760 (0.5589)
70.3257 (0.1662)
4.4585 (0.0694)
62.048 (2.4305)
63.8883 (3.2815)
95.6649 (0.1506)
80.7000 (27.7985)
30.4414 (0.1855)

29.4557 (1.4481)
1.4054 (0.0533) 39.8611 (7.8016) 27.6052 (0.0430) 3.0994 (0.0338) 15.4012 (0.4952) 19.2391 (0.8744) 28.3458 (0.0376) 83.1407 (18.8941) 8.7025 (0.0351)

0.8195 (0.0173) 1.1808 (0.0180) 0.5323 (0.0812) 1.2170 (0.0031) 1.3961 (0.0034) 1.0662 (0.0170) 1.0522 (0.0161) 1.1827 (0.0022) 0.8944 (0.0103) 1.2418 (0.0043)

1.3343 (0.0320) 1.8637 (0.1261) 0.8411 (0.0902) 2.8600 (0.0051) 2.1434 (0.7862) 2.1055 (0.0083) 2.3919 (0.0121) 2.5907 (0.0047) 1.1308 (0.0828) 1.9751 (0.0459)


46.8381 (6.6347) 75.9390 (17.2967)
3.1756 (2.4317)
2.8490 (1.0816) 17.3943 (61.3586)
7.5725 (1.0502)
7.1662 (1.3836)
31.2613 (0.5786)
6.0014 (8.4966)
25.5496 (6.1947)

10.5577 (1.0829)
11.5728 (1.8727) 72.4272 (19.8416) 1.7776 (0.6623) 4.0660 (144.8544) 2.6904 (0.3526) 3.9100 (0.5777) 12.5076 (0.1938) 79.4141 (4.8285) 8.7548 (0.9789)

1.0699 (0.0600) 0.8137 (0.0640) 0.5585 (0.1483) 3.1238 (0.4696) 0.9667 (5.4064) 1.6078 (0.0588) 1.4558 (0.0433) 1.7974 (0.0334) 0.6582 (0.1339) 0.5205 (0.0220)

1.1757 (0.3534) 1.3364 (0.0294) 5.0398 (0.4629)

165.1806 (123.7916)
4.5621 (8.9397)
422.2008 (19.6416)


20.5939 (19.4920) 1.5514 (2.9036) 62.3910 (2.0178)

0.3093 (0.1739) 1.7897 (1.0366) 0.4083 (0.0089)

1.2706 (0.1113) 2.1109 (1.8988) 22.5705 (12.7193)

342.5058 (32.7226) 52.6569 (273.8892)
509.9305 (66.7726)


33.2149 (6.0627)
4.6392 (4.6427) 68.6771 (4.8907)

0.4940 (0.0142) 0.8012 (1.9613) 0.3410 (0.0166)

95.3600 (67.2524) 1.4365 (0.0150) 3.4936 (0.6739)


546.7399 (13.3561)
24.8508 (4.4357)
363.2909 (62.0430)

64.7930 (1.0087) 5.4815 (0.7495) 34.2528 (4.0911)

0.3977 (0.0067) 1.3475 (0.0973) 0.4434 (0.0343)

194.6296 (147.8742) 6.6146 (1.0273) 13.7929 (9.3237)


529.1865 (23.0840)
326.9662 (45.5111)
487.9014 (131.6305)

63.4829 (1.7549) 26.8119 (3.8186) 37.9450 (6.6186)

0.4031 (0.0093) 0.3517 (0.0062) 0.3395 (0.0292)

0.3757 (0.0607) 2.1615 (0.0194) 1.4203 (0.0045) 2.2176 (0.0339) 1.6817 (0.0074) 1.7408 (0.0143) 1.8617 (0.0077) 1.6736 (0.0079)

73.0468 (51.4549) 59.7328 (1.1437) 10.2213 (0.2291) 86.4648 (1.5288) 36.4177 (0.6764) 45.3744 (1.0431) 67.4244 (0.6965) 45.2907 (0.5586)

11.7892 (17.8417)
7.3949 (0.0708)
0.9796 (0.0196)
1.6683 (1.5520)
3.7707 (0.0318)
2.8136 (0.1518)
10.2622 (0.0748)
4.9711 (0.0446)

0.3433 (0.2396) 1.5200 (0.0107) 1.4510 (0.0067) 1.31176 (0.0319) 1.4402 (0.0111) 1.4443 (0.0163) 1.4353 (0.0052) 1.4033 (0.0049)

0.5297 (0.3235) 4.0048 (0.0712) 2.6152 (0.0217) 3.4886 (0.1637) 2.7428 (0.0398) 3.0347 (0.0457) 3.2067 (0.0236) 2.9860 (0.0206)

26.3339 (85.8588) 80.4937 (2.2451) 24.2388 (0.8225) 99.8603 (38.6393) 72.9087 (2.3960) 69.0482 (1.8048) 98.5398 (1.2489) 66.4209 (0.8609)

86.3897 (199.0316)
8.9998 (0.1145)
2.1479 (0.0622) 37.4102 (48.1690)
4.0475 (0.3352)
0.8795 (0.6249)
13.9615 (0.1157)
6.8328 (0.0606)

0.1669 (0.3493) 1.4038 (0.0156) 1.2509 (0.0100) 0.9263 (0.2222) 1.1468 (0.0210) 1.2408 (0.0221) 1.3044 (0.0062) 1.2925 (0.0050)

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

7

Table 3 (continued ) UO2(NO3)2 2:2 Electrolytes BeSO4 CaSO4 CdSO4 CoSO4 CuSO4 3:1 Electrolytes AlCl3 CeCl3 Er(ClO4)3 GdCl3 Ho(C2H5SO4)3 Lu(NO3)3 LuCl3 Nd(C2H5SO4)3 PrCl3 ScCl3 3:2 Electrolytes Al2(SO4)3 Cr2(SO4)3 Lu2(SO4)3 4:1 Electrolytes Pt(en)3Cl4 Th(NO3)4

1.6133 (0.0326)

113.4430 (4.4983)


20.0235 (0.6629)

1.2514 (0.0171)

2.6173 (0.0709)

157.1314 (5.7566)


26.4801 (0.7810)

1.1617 (0.0151)

1.8164 (0.0519) 2.4736 (74.223) 2.1333 (0.1358) 1.6554 (0.0433) 1.8181 (0.2757)


120.8117 (10.2717)
1569.6324 (14271.8550)
178.9112 (21.1336) 31.9398 (2.1714)
122.7784 (54.1780)

20.8989 (1.4265) 2404.3802 (5862.1245) 23.5516 (2.1957)
90.7048 (6.8284) 18.9166 (6.0361)

0.6348 (0.0162) 1.0413 (10.8066) 0.5933 (0.0254) 0.1891 (0.0157) 0.7027 (0.1265)

2.5279 (0.0086) 4.3223 (67.7818) 2.6212 (0.0176) 2.4580 (0.0980) 2.5620 (0.0151)

2.9497 (2.6243)
1391.8263 (6488.4237)
17.9911 (2.0703) 13.8481 (2.8000)
8.8363 (2.0845)

0.7404 (3.7050) 2177.8572 (2919.2487) 4.6214 (0.4360) 7.4859 (10.8186) 2.6280 (0.5402)

1.1987 (0.7404) 0.9799 (4.6665) 0.9070 (0.0233) 0.0844 (0.0374) 1.0988 (0.0759)

1.8144 (0.0795) 1.6323 (0.1200) 2.2834 (0.0196) 1.8470 (0.0027) 2.0259 (0.0200) 1.5950 (0.0122) 1.9000 (0.0058) 1.9747 (0.0145) 4.1284 (1.1172) 4.5413 (1.0840)

24.8042 (62.7981) 20.4157 (63.3019) 31.9628 (0.7486) 14.7379 (0.1515) 43.1635 (2.0674) 32.1484 (1.2466) 15.3409 (0.2791) 32.9852 (1.6228)
355.3532 (75.3947)
383.3916 (63.9303)

8.2896 (77.2341) 5.9609 (87.8456)
3.3811 (0.0495)
1.9195 (0.0136)
5.3213 (0.8721)
2.8701 (0.0821)
1.8342 (0.0229)
4.7777 (0.5042) 123.9184 (16.2885) 149.3751 (14.7424)

1.1969 (1.531) 1.1187 (2.3347) 1.6539 (0.0103) 1.7800 (0.0049) 1.3957 (0.0665) 1.2315 (0.0132) 1.7632 (0.0081) 1.4945 (0.0636) 0.4732 (0.0322) 0.4611 (0.0243)

3.2652 (0.9542) 2.7913 (3.4220) 4.6155 (0.0666) 3.5118 (0.0353) 3.0191 (0.886) 2.8292 (0.0277) 3.6936 (0.0312) 2.9765 (0.8346) 2.7931 (1.5767) 2.7287 (4.3843)

47.8665 (170.3619) 53.5535 (173.8866) 44.6662 (1.3365) 31.7631 (1.3495) 112.1388 (37.8593) 65.5598 (1.8526) 29.3692 (0.9567) 99.3346 (35.7456) 51.9349 (102.1080) 73.3070 (125.9576)

13.8463 (252.7971) 11.6468 (529.7882)
4.3564 (0.0720)
3.4599 (0.0748) 8.7351 (45.9125)
5.2200 (0.0995)
3.0382 (0.0536) 7.9414 (44.5869) 11.9408 (266.3685) 14.8529 (594.0396)

1.0244 (2.9086) 0.8974 (6.7672) 1.54101 (0.0131) 1.5031 (0.0207) 0.8255 (0.4861) 1.0631 (0.0091) 1.5388 (0.0149) 0.8276 (0.5279) 0.8970 (3.3764) 0.8729 (5.8573)

0.05105 (0.0104) 1.6014 (0.0112) 2.5964 (0.2230)


323.0721 (10.2375)
40.5234 (3.0746)
386.2418 (33.0956)

1228.3185 (8.9053) 10.0653 (0.6734) 41.7185 (3.2254)

0.0178 (0.0121) 0.9826 (0.0203) 0.3825 (0.0079)

19.2013 (179.1146) 2.8935 (0.0846) 4.9344 (0.8634)


645.1747 (1538.0973) 2.9013 (27.0285)
354.5134 (56.9508)

93.7628 (122.2109) 0.7821 (37.3193) 43.6076 (3.8100)

0.3456 (0.2805) 1.3547 (7.2788) 0.3065 (0.0068)

1.2144 (0.0089) 0.1366 (0.0212)


11.1717 (5.9853) 1115.0326 (28.1709)

5.4115 (2.6403)
48.9749 (0.8967)

2.8389 (0.6841) 0.5342 (0.0124)

1.81721 (1.7470) 0.1807 (0.0672)

98.3499 (445.1895) 1130.4541 (59.9386)


25.3303 (74.9661)
43.3592 (2.0998)

0.8772 (1.5348) 0.5487 (0.0225)

Fig. 1. Comparison of the mean activity coefficient correlative capability for several models of some aqueous electrolytic systems at 298.15 K.

8

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

Fig. 2. Osmotic coefficients for some aqueous electrolytic systems at 298.15 K.

Fig. 3. Mean activity coefficients for some aqueous electrolytic systems with molalities greater than 6.

m¼

  jz z j n n þ k1 S0 I n þ f
1 ¼ jzM zX jf f þ M X S1 I nþk1 1þn nT 1 þ n þ k1

2I

(42)

njzM zX j

Substituting the above equation into the Pitzer equation

(40) respectively. In these equations, f g and f f can be the DebyeHückel or the Pitzer-Debye-Hückel equations. In a electrolyte solution with a single electrolyte the ionic strength is

lngMX ¼ jzM zX jf g þ

I ¼ mnjzM zX j=2

f
1 ¼ jzM zX jf f þ

therefore the molality is

(41)

4nM nX g 4ðnM nX Þ3=2 g 2 B C I I þ MX n2 jzM zX j n2 jzM zX j MX

(43)

and the osmotic coefficient becomes

4nM nX f B I n2 jzM zX j MX

þ

4ðnM nX Þ3=2 f 2 C I n2 jzM zX j MX

(44)

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

9

Fig. 4. Osmotic coefficients for some aqueous electrolytic systems with molalities greater than 6.

Fig. 5. Correlative capability of Archer [8,10], DH-MS and PDH-MS models for the activity coefficient of some aqueous electrolytic systems at 298.15 K.

Rewriting the activity coefficient for a single electrolyte truncated at the second solvation term

lngMX ¼ jzM zX jf g þ

jzM zX j n jzM zX j nþk1 S I þ S I nT 0 nT 1

(45)

and the osmotic coefficient is

f
1 ¼ jzM zX jf f þ

jzM zX j n jz z j n þ k1 S In þ M X S I nþk1 nT 1 þ n 0 nT 1 þ n þ k1 1 (46)

Considering n ¼ 1 and k1 ¼ 1 in Eqs (45) and (46), the activity coefficient from the solvation model and the activity coefficient

based upon the Pitzer model are equivalent if

jzM zX j 4nM nX g S0 ¼ B T njzM zX j MX

(47)

and

jzM zX j 4ðnM nX Þ3=2 g S1 ¼ C T njzM zX j MX For the osmotic coefficient, the equivalency occurs if

(48)

10

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

Fig. 6. Correlative capability of Archer [8,10], DH-MS and PDH-MS models for the osmotic coefficient of some aqueous electrolytic systems at 298.15 K.

Fig. 7. Comparison of predicted osmotic and activity coefficients between DH-MS with 4, 5 and 6 adjusted parameters.

4nM nX f jzM zX j n S ¼ B 1 þ n 0 njzM zX j MX T

(49)

and

functions of the ionic strength, the activity and osmotic coefficients are approximations of the equations given by the solvation theory developed in this work. The Guggenheim [50,51] equation also satisfies the extended solvation theory. Its activity coefficient is

lngMX ¼ jzM zX jf g þ Bm

jzM zX j n þ k1 4ðnM nX Þ Cf S ¼ T 1 þ n þ k1 1 njzM zX j MX

(51)

3=2

(50)

From these results, one can conclude that the Pitzer equations for the activity and osmotic coefficient are a specific case of the solvation models, as long as the virial coefficients are not functions of the ionic strength. For the modification of the Pitzer equation g g given by Archer [8e10] in which BMX , C MX , BfMX and C fMX are

and the equation for the osmotic coefficients is

f
1 ¼ jzM zX jf f þ B

m 2

(52)

in which B is a characteristic parameter related to interaction coefficients, f g is the Debye-Hückel function with a maximum approach parameter of 1. If the solvation model, Eqs (45) and (46),

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243 Table 4 Average absolute percentage deviation of the models DH-MS and PDH-MS when correlating the single ion activity coefficient and comparison of the mean ionic activity coefficient to literature values. Electrolyte

CsCl NaBr NaCl NaNO3 KCl KNO3 LiBr LiCl K2SO4 Na2SO4 Total

DH-MS

PDH-MS

Ref.

Cation

Anion

Mean ionic

Cation

Anion

Mean ionic

1.04 1.57 1.05 0.56 0.84 3.00 0.88 0.25 0.31 0.32 0.98

0.15 0.98 0.81 1.83 0.20 2.39 0.93 0.28 6.40 4.92 1.89

0.23 1.38 1.31 0.99 0.98 0.80 0.33 0.12 1.41 0.50 0.81

1.18 1.60 1.04 0.60 1.14 3.18 0.87 0.27 0.53 0.43 1.08

0.19 0.98 0.86 1.85 0.21 2.37 0.62 0.28 6.27 4.82 1.85

0.20 1.39 1.31 0.93 0.83 0.51 0.35 0.18 1.49 0.35 0.75

[40] [39] [39] [40] [39] [40] [40] [40] [40] [40]

11

contains only the first solvation term with n ¼ 1, the equivalence is

jzM zX j 2B S0 ¼ T jzM zX j

(53)

B jzM zX j n S ¼ 1 þ n 0 jzM zX j T

(54)

This result reflects the Guggenheim equation has its basis in Coulomb's law. It is important to mention some considerations about ion pairing in the model. Long-range nondirectional electrostatic forces play an important role in the association of ions in solution into pairs. However, Marcus and Hefter [52] have shown that the Pitzer expressions [6] for the activity coefficients, which have been developed without considering ion pairing explicitly, take into account the ion pairing effects in the B parameter. They

Fig. 8. Correlative capability of DH-MS and PDH-MS for single ion activity coefficients of Naþ and Br
, and calculated mean activity coefficients using Eq. (31) for NaBr.

Table 5 Parameters for the truncated in the second solvation term DH-MS and PDH-MS models for the correlation of single ion activity coefficients of aqueous electrolytes at 298.15 K. Electrolyte

mmax

CsCl

3

NaBr

5

NaCl

5

NaNO3

3.5

KCl

4

KNO3

3.5

LiBr

3

LiCl

3

K2SO4

0.69

Na2SO4

3

Ion

Csþ Cl
Naþ Br
Naþ Cl
Naþ NO
3 Kþ
Cl Kþ NO
3 Liþ
Br Liþ Cl
Kþ SO
2 4 Naþ SO
2 4

DH-MS

PDH-MS

bi

S 0;i

S 1;i

ni

bi

S 0;i

S 1;i

ni

8.6234 1.3115 0.4436 1.2013 3.3452 0.7046 1.1909 0.3820 4.1952 1.4794 4.5135 0.2559 2.2141 1.3815 1.8158 1.0617 0.9833 0.6627 0.9916 1.1539


229.0736
14.4679 21.7268
12.0130 36.8859 7.2038 86.8190
3.9748
194.4368 24.5272
78.7138
3.0275 88.5629 89.6911 108.4549
6.7436
6.2790 67.1132
4.9286
13.7032

69.1034 2.7437 149.4363 8.1829
5.0055 2.5351
29.7447 2.0744 75.6167
4.4181
61.2488 1.6067
29.5434
25.2974 14.0857 25.6502 4.1033
38.8957 1.7335 2.9213

0.5496 0.4286 0.3328 1.4281 1.3580 1.2509 0.9621 4.3340 0.7314 0.9333 0.3679 4.3127 2.0451 0.9541 0.9516 0.5353 4.8380 1.4619 1.6549 1.2469

1.5200 1.4020 0.6043 2.0014 5.7135 1.0273 1.6454 0.6167 1.5874 2.1021 1.7503 0.4038 3.8521 3.9698 4.7105 1.3021 1.3018 0.0857 1.7173 1.8857


25.7749 44.2888 17.2494 1.8355 48.7036 19.3030 115.9558
2.9668 0.5414 52.0977
26.5191
2.2507 100.8316
26.9052
37.4978 55.9021 40.0979 231.0166 0.0940
2.7458

17.5111
13.1547 166.7929 1.9448
2.3386 2.9491
36.6985 1.5637
0.2445
10.0521
0.4723 1.2066
31.0181 62.0258 142.3919
6.7829
19.5317
81.7954 0.0440 0.7052

1.2170 0.9689 0.3232 1.7024 1.1170 1.0578 0.9415 4.5091 5.2028 0.8784 1.2617 4.5061 1.8957 0.2102 0.5025 0.9665 0.9704 1.0375 2.4526 1.6734

12

J. Temoltzi-Avila et al. / Fluid Phase Equilibria 499 (2019) 112243

show that when the second virial term, B, is a function of the ionic strength, one of the parameters of this function relates to the association constant of the ion pairing equilibrium. This work shows that the solvation models DH-MS and PDH-MS are equivalent to the Pitzer models, therefore the S-terms could contain ion pairing effects. Also, the model does not account for saturation of hydration shells, but the flexibility of DH-MS and PDH-MS could approximate this behavior. This flexibility allows a better correlation of the ion activity coefficients because previous proofs indicate that solvation models correlate adequately the ion activity coefficients [53]. 5. Conclusions This paper presents new expressions for the mean activity and osmotic coefficients. The new equations contain a long-range interaction term based upon either the Debye-Hückel or the Pitzer-Debye-Hückel and a short-range interaction based upon an extended solvation theory. The short-range interaction terms consider different interactions among ion-solvent, solvent-solvent, etc. The new equations correlate the mean activity and osmotic coefficients within an average absolute percentage deviation of 0.479 (DH-MS) and 0.686 (PDH-MS) for electrolyte solutions at 298.15 K. The paper contains parameters for 56 electrolyte solutions. The results demonstrate that the Debye-Hückel equation together with the current solvation terms correlates the activity and osmotic coefficients better than previous work and, within the framework of the solvation model, the use of DH or PDH is practically identical. Acknowledgements
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