Generalized Quasi-Random Lattice model for electrolyte solutions: Mean activity and osmotic coefficients, apparent and partial molal volumes and enthalpies

Accepted Manuscript Generalized Quasi-Random Lattice model for electrolyte solutions: Mean activity and osmotic coefficients, apparent and partial molal volumes and enthalpies Elsa Moggia PII:

S0378-3812(18)30388-1

DOI:

10.1016/j.fluid.2018.09.008

Reference:

FLUID 11942

To appear in:

Fluid Phase Equilibria

Received Date: 5 July 2018 Revised Date:

2 September 2018

Accepted Date: 15 September 2018

Please cite this article as: E. Moggia, Generalized Quasi-Random Lattice model for electrolyte solutions: Mean activity and osmotic coefficients, apparent and partial molal volumes and enthalpies, Fluid Phase Equilibria (2018), doi: https://doi.org/10.1016/j.fluid.2018.09.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Generalized Quasi-Random Lattice model for electrolyte

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solutions: mean activity and osmotic coefficients,

Elsa Moggia a *

Applied Electromagnetics Group (AEM) at Department of Naval, Electric, Electronic and

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a

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apparent and partial molal volumes and enthalpies

Telecommunications Engineering (DITEN), University of Genoa, Via Opera Pia 11A, 16145 Genoa, Italy * Corresponding author. Tel + 39 3402536867; [email protected]; [email protected]

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ABSTRACT

Electrolytes are the subject of a vast number of theoretical and experimental investigations concerned with a variety of modern applications. The modeling of thermodynamic properties of ionic solutions is

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thus a fundamental research topic that has been supported in many ways for many decades. There is

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however still a lack of models that are truly predictive over wide ranges of concentration, temperature and pressure conditions.

In this article, the Quasi-Random Lattice (QRL) model is presented in a generalized form that allows for evaluating relevant thermodynamic properties of binary electrolyte solutions. The semi-predictive character of the model yields a powerful and competitive representation of electrolyte data over well defined concentration ranges. The additional experimental information to support the generalized version of QRL is very modest compared to the number of data points typically used in regression techniques for best-fit purposes. The thermodynamic consistency of the improved QRL model is demonstrated by the

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level of agreement with experimental data concerning mean activity and osmotic coefficients, apparent and partial molal volumes and enthalpies, for a variety of aqueous 1:1, 2:1, and 3:1 electrolytes.

1.

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KEYWORDS: Pseudolattice Theory; Mean activity coefficient model; Thermodynamic data prediction.

INTRODUCTION

Electrolyte solutions are the subject of an impressive number of theoretical, computational and

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experimental investigations, concerned with a variety of modern applications in industrial, technological, chemical, biological and environmental areas [1-3]. The modeling of thermodynamic properties of ionic

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solutions is thus a fundamental research topic that has been afforded in many ways, for many decades [4]. However, despite notable advances in the field, mainly promoted by innovation or improvement of experimental techniques and computer-based facilities, there is still a lack of models able to be truly predictive over wide ranges of concentration, temperature and pressure conditions. Classical theories run the risk of being simplified such that limit their actual applicability (e.g., Primitive-Model based theories

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[4, 5]) or, conversely, over expanded to multi-parameter equations [1] that are better intended and used for data correlation rather than data prediction.

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In this article, the Quasi-Random Lattice approach (QRL) [6-10] is presented in a generalized form that allows for evaluating, consistent with experimental thermodynamic data, relevant properties of binary

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electrolyte solutions, that is, their mean activity and osmotic coefficients and their apparent and partial molal volumes and enthalpies. The generalization of the model yields a powerful and competitive representation of electrolyte data over well defined concentration ranges that extend to medium-high molalities. This article shows that the additional parameterization introduced in the model requires a very limited number of measured data. In addition, and perhaps more importantly, it illustrates the semipredictive character of QRL in that, to determine the concentration parameter clim – this parameter defines the range of applicability of the model, [0, clim] on molar scale or [0, mlim] on molal scale - an

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experimental procedure can be followed starting from an arbitrary concentration until convergence to clim occurs within a few steps. In these introductory remarks main ideas of the QRL approach [6-10] are briefly summarized then followed by a comparison with recent literature models. QRL approach

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1.1

The first idea in QRL is the adoption of an “ionic-lattice frame” in place of the classical “central-ion frame” [4, 11] for describing the statistical behavior of ions and molecules in an electrolytic solution. The

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second idea is that of considering ions and molecules as if they formed groupings - called “carriers of charge” - that are continuously changing size and composition. On the whole, these groupings form a

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crystal-like structure that appears widely distorted and continuously modified. The average effect of the carrier population of interest in a given lattice cell is that of reproducing a sort of density of charge that can spread out, even widely, from that cell. Such a density of charge is called “effective carrier of charge” (assumed as Gaussian-distributed with the centre on the reference lattice-point of the cell). In actual fact, effective carriers rather than mere carriers, are used to derive the main equations in QRL. The third idea

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in QRL is that of calculating the mean activity coefficient γ± (molal scale, Lewis-Randall frame [1, 4]) starting from calculating the mean energy of an effective carrier related to its interactions with the others.

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Calculations, reported in details in Refs.[6-10], yielded main equations that will be summarized in the Theory section. It is here referred to as the fundamental result that, at c=clim, it is γ±= 1. This states a

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rigorous definition of clim (which depends, in general, on pressure P, temperature T, and specific salt considered) as the (upper) concentration at which the mean molal activity coefficient is equal to 1. At present the limiting condition of applicability of the model to the range [0, clim] is due to the mesoscopic approach [10, 12] driven by concepts like those of carrier and effective carrier. 1.2

Comparison with literature

The QRL approach, though belonging to the class of pseudo-lattice theories [3, 4, 13-16], shows the fundamental advantage of being consistent with the Debye-Hückel Limiting Law (DHLL) [4,11], which is valid at the infinite-dilution limit [5, 17]. The QRL convergence to DHLL was derived in Ref. [10].

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This important theoretical result is not available from other pseudo-lattice models due to their dependence on the so called “cube-root law” [3, 4,16] which is inconsistent with DHLL. In this article, comparison of QRL with literature models is mainly concerned with recent approaches -most of them proposed as developments or simplified forms of classical theories [4] - that have been applied to a wide

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variety of binary electrolytic solutions, and results forthrightly compared with experimental data. Therefore, despite their theoretical importance, general formalisms like Integral Equations [4-5] or Fluctuation Solution Theory [17] will not be considered. Similarly, the Mean Spherical Approximation

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(MSA) [18] will not be discussed since it has already been debated elsewhere resulting in controversial opinions [19], thereby suggesting that DH extended models would be comparable or even preferable.

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Table 1 shows a schematic overview arising from comparison of QRL with literature models. Further

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comparisons can be found in previous articles [6-10].

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Table 1. Mean activity coefficient models for electrolyte solutions

Hückeltype Equation c

Pitzer Theory d

Isothermbased e

REUNIQUAC g

MSANRTL f

a

Assumes DHLL a priori; extends DH and adds terms (powers of molal concentration). Assumes DHLL a priori; extends DH and adds a virial-type equation. Often combined with localcomposition models to account for short-range interactions. Developed for adsorption processes; includes activity models from literature (NRTL f, Pitzer Theory d). Assumes DHLL a priori; extends DH with inclusion of effective ionic radii of solute species; includes UNIQUAC f to account for 'short-range' interactions. Derived from Statistical Mechanics; ions are modeled as hard spheres (MSA f); converges to DHLL at infinite dilution. Includes NRTL f to model short-range interactions.

Requires best-fit procedure (3 adjustable parameters).

Semi-empirically set, from best-fit results: up to medium or high molalities.

h

Not available

Requires best-fit procedure (2 to 3 adjustable parameters).

Requires best-fit procedure (2 to 3 adjustable parameters for each apparent property).

Requires best-fit procedure (3 to 4 adjustable parameters).

Requires best-fit procedure (3 to 4 adjustable parameters for each apparent property).

Requires best-fit procedure (no less than 4 adjustable parameters).

main issues

For electrolytes that show ion association even at high dilution, the association constant is also required. At present the model does not apply to concentrations > clim.

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Does not require best-fit procedures. Requires following experimental data: apparent and partial molal volume / enthalpy at clim; slope of partial molal volume / enthalpy at clim.

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b

Semi-empirically set from best-fit results: up to low-medium molalities. Non-electrostatic interactions should be negligible.

additional parameterization (apparent molal volume / enthalpy)

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SiS

Assumes DHLL a priori; extends DH with inclusion of various defined "closestapproach" distances between ions.

Does not require best-fit procedures. Requires following experimental data: clim; osmotic coefficient at clim; slope of either mean activity or osmotic coefficient at clim.

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a

main parameterization (mean activity coefficient)

Theoretically set: mean molal activity coefficient must be ≤1, that is, the molar concentration must be ≤clim. From experimental values of clim, the model is available up to medium-high molalities.

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QRL

Converges to DHLL at infinite dilution, starting from an independent approach. The DH "central-ion" frame is replaced by an "ioniclattice" frame. DH solute ions are replaced by “carriers of charge” that include both solute ions and solvent molecules. Effective carriers describe charge densities arising from the statistical treatment of the carrier population.

concentration range (at given temperature and pressure)

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Model

relationship with DHLL (Debye-Hückel Limiting Law)

h

Not available

Requires best-fit procedure (no less than 3 adjustable parameters for each apparent property).

This article and Refs. [6-10]. b Refs. [20, 21]; c Refs. [4, 22-27]; d Refs. [4, 22, 26-30]; [33]. h To the knowledge of this author.

e

The number of experimental data points required by regression techniques depends on the considered salt, the target property and the concentration range. This number typically ranges from 10-20 to 150-200. Values of parameters, that are obtained from numerical best-fitting, strongly depend on the concentration range, therefore models are not predictive. Parameters can sometimes become meaningless, and related properties either unphysical or inconsistent with measurements.

Refs. [31, 32]; f Ref. [1];

g

Ref.

Table 1 outlines main aspects related to parameterization that models require in order to provide accurate results. In general, compared to QRL, a relevant disadvantage of other models is caused by their need for best-fit numerical procedures that usually ask for a large set of measurements in order to be reliable. In addition, number and values of adjustable parameters strongly depend on the optimized concentration

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ranges which are, as a rule of thumb, only qualitatively, or semi-empirically established. In the IsothermBased Model [31-32] some of the parameters can either be calculated from suitable equations or numerically found from best-fit procedures, however, differences appear relevant and do not allow for any immediate interpretation. For example, in the case of aqueous CaCl2 (at 25°C), the interspatial

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distance between cation and anion is 4.06 Å if the parameter that represents the dipole moment of the solute is calculated, while the same distance is 1.93 Å if the best-fit value of the same parameter is used [32] (the number of hydration layers also changes from 4 to 6 and it was obtained by a preliminary best-

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fit [31] over the optimized concentration range). Too often concepts that should substantially refer to similar properties conversely yield a spread of results that become difficult to judge. For example, with

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aqueous dilute solutions of LaCl3 (at 25 °C), where ion association is likely to be absent [34], the Singleion-Shell (SiS) Theory [20] states that the average closest-approach distance between Cl- ions is about 2.87 Å whereas the first peak position in the Cl--Cl- radial distribution function is about 4.5 Å from Molecular Dynamics simulations performed at 0.89 M [35]; analogous differences can be seen with respect to other salts [20, 36]. These differences cannot easily be ascribed to “model simplification” tout-

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court, nor does any generic categorization such as electrostatic/non-electrostatic potentials/effects [2021]. This remains an issue [2] that is still under theoretical and experimental investigation [37]. In recent

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years great effort has been spent to develop optimization procedures by manipulation of huge amounts of data [38], however, the significance of parameters tends to become obscure and related properties

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difficult to understand. The main issue emerging from all these considerations is that theories are far from predictive, even despite their consolidated use and importance in data correlation procedures, as in the case of Pitzer Theory [4, 22, 26-30] and Hückel Equation [4, 22-27]. Theoretical aspects of QRL, computational methods, and their application to representative aqueous 1:1, 2:1, 3:1 electrolytes (at 25°C) are presented in the following sections.

2. THEORY 2.1

Reference lattice frame

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We hereinafter suppose that in a volume V there are N solute molecules giving rise to ν+N carriers with charge Q+, and ν-N carriers with charge Q- . We also put ν = ν++ ν- and ρ = N/V. The mean inter-ionic distance R is: R = 1/(νρ)1/3

(1)

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In terms of the molar concentration c [mol/dm3], N = 103NAvc (V=1 m3), where NAv is the Avogadro Number. Concentration scales are converted from each other by means of standard formulae. c=

md 1 + mM 2

(2)

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In Eq. (2), m is the molal concentration of solute [mol/kg], d is the absolute solution density [g/cm3], and

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M2 is the molecular weight of solute [kg/mol]. Hereinafter, Q+=z+e and Q- =z-e, where z+ and z- are the ion valences, and e = 1.66x10-19 C. The electro-neutrality condition reads: ν+Q++ ν-Q- = 0. For a given reference ionic-lattice frame, the set of spatial points is indicated with {RA}, and the set of net charges with{QA}, where either QA= Q+ or QA= Q- . For the sake of simplicity, and without loss of generality [10], electrolytes of the same order will hereinafter refer to the same space group and all space groups will

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belong to the cubic system [39]. Table 2 summarizes main properties of lattice geometries, and {RA} points of reference space-frames for the electrolytes considered in this article, including Madelung

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constants of the corresponding ionic crystals [40-41]. Regarding 3:1/1:3 electrolytes, lattice-geometry details (Table 2) and pertinent formulae can be obtained by the same mathematical methods previously

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used [6-10], so their derivation is omitted here for brevity.

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Table 2. QRL model. Ionic-crystal reference space-frame space group (cubic system)

vector position RA of lattice point A, with respect to mean inter-ionic distance R

Madelung Constant

a

b

a, c, d

N. 225 ( Fm3m ) (NaCl prototype)

RA/R=(n, m, p). Either a positive or a negative ion according to the sign of (-1)n+m+p Divalent ions: RA/R=

symmetric

e

(2n, 2m, 2p); asymmetric (1:2 ; 2:1)

N. 225 ( Fm3m ) (CaF2 prototype)

1.74756 (1.74756)

(2n+1, 2m, 2p);

(2n, 2m+1, 2p);

(2n, 2m, 2p+1)

Univalent ions: RA/R=

3 1 1 1 n + ,m + ,p + 2 2 2 2

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electrolyte

2.5414 (2.5194 )

Trivalent ions: RA/R= (2n, 2m, 2p) Univalent ions: RA/R= 3.760 N. 221 ( Pm3m ) (2n+1, 2m, 2p); (2n, 2m+1, 2p); (2.985) (2n, 2m, 2p+1) a Refs. [39,40]; b R is given in Eq. (1); c Refs. [40, 41]; d Madelung constants here refer to R , values between parentheses refer to the minimum distance between ions of opposite sign; e (n, m, p) indicates any arbitrary triplet of integers.

Mean activity coefficient

The main QRL formula [6,9-10] is:

ln γ ± =

QAQB 1 1 R  erf  AB  ∑ 4πεk BT 2νN A,B RAB  2U 

(3)

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A≠ B

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2.2

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asymmetric (1:3 ; 3:1)

In Eq. (3), RAB=|RA-RB| for any pair of RA and RB reference space points determined according to the

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reference space group (Table 2); kB is the Boltzmann Constant, T the absolute temperature; the symbol ε indicates the dielectric permittivity of solvent; the symbol erf indicates the Error Function. In Eq. (3) the

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½ factor avoids a same pair contribution to be counted twice in the double sum. Details on numerical implementation of Eq. (3) will be given in the Computational Methods section. Eq. (3) can be used provided that an expression for U is given. Concerning U, the following equations represent a generalization of the unified expression in Ref. [10]. U=

 U e 3k c 3  2  e 2k c 2  2 2  UD  ek c  2 2 2   6k 2 + 1 − 6k 2  − 2 2  3k + 1 − 3k 2  + D 2 k + 1 − 2 k 1 + 2  2 3   k c α 2 k c α π  2k cL  α π L  L    

0≤

c ≤ exp( − k ) cL

(

)

(

)

(

)

(4a)

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U=

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UD

exp(− k ) ≤

1/ k

 c  c  log k  L    c  cL 

c ≤1 cL

(4b)

In Eqs. (4a, 4b), the DH Screening Length [4, 11] is evidenced and indicated with UD. UD =(εkBT)1/2/(νρ|Q+Q- |)1/2

(5)

πα k = e ln(k ) 2 1

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Furthermore, in Eqs. (4a, 4b), the QRL constant k is obtained by solving [6, 8]: 1
(6)

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In Eqs. (4a, 6), the symbol α indicates the Madelung Constant of the reference ionic lattice (Table 2). For symmetric electrolytes, α = 1.747, so k = 10.65. For asymmetric electrolytes, Eq. (6) yields k = 14.54 for

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2:1/1:2 electrolytes (α=2.541), and k = 19.69 for 3:1/1:3 electrolytes (α = 3.760). Concerning clim, it will from now on be embedded into a generalized concentration-parameter indicated with cL that will be a function of c, clim, P and T, still considering that clim = clim(P,T). Once cL is determined, Eq. (3) will no longer depend on unknown parameters but only on experimental information, that is, the solution density

2.3

QRL parameterization

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and the solvent dielectric permittivity at given concentration, temperature and pressure.

The generalized concentration-parameter cL is introduced in order to improve the agreement between

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measured and predicted values of lnγ±. Although QRL was found satisfactory on its whole [6-10],

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approximations included in the model yielded less accurate results relative to the intermediate concentrations with respect to the range [0, clim] (clim typically ranges from 1 M to 10 M [42]). The problem of accuracy is also important when considering the fact that a relative inaccuracy in the prediction of lnγ± might become unacceptable when calculating partial derivatives of lnγ±. That said, cL should not deviate greatly from clim and must fulfill the requirement that cL=clim if c=clim. This requirement implies that U→+∞ in the limit c→clim (from Eq. (4b)). When U→+∞ one also has that lnγ±→0, as it is seen by evaluating Eq. (3) in the limit U→+∞.

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c →clim

→

Q A Q B 1 RAB 1 ∑ 8πεk BTνN A,B RAB π U A≠B

∑Q In Eq. (7) the relation

A

QB

A,B A ≠B

=

νN

= U →+∞

− ∑ Q 2A A

νN

Q+Q−

→0

3

8 π εk BTU

(7)

U → +∞

= Q + Q − is used [10], and the Error Function is approximated

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R  R by the first-order expansion erf  AB  ≈ AB , which is applicable since U→+∞. Eq. (7) is consistent πU  2U 

with the definition of clim as the (upper) concentration corresponding to γ±=1.

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If the experimental value of γ± is known at given c, T and P, then the corresponding cL can be searched for by imposing Eq. (3) to yield that measured γ± (within a suitable tolerance). It is important to state that

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this procedure does not require knowledge of clim. Figure 1 shows cL values obtained by using the procedure in the depicted cases of aqueous NaCl, CaCl2 and LaCl3 (25°C, 0.1 MPa). In general, cL moderately deviates from clim provided that c is not too small. In addition, cL trends appear almost the same for any electrolyte. These considerations suggest a general approximation for cL (at given T and P),

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corresponding to a second-degree polynomial centered on c=clim. cL ≈ clim + cL,1 (c − clim ) + cL,2 (c − clim ) / 2 2

(8)

The values of clim, cL,1 and cL,2 can be determined by means of an experimental procedure that will be

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presented in details in the Results section. Here some key steps are shortly anticipated. To determine clim

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the procedure takes advantage of the semi-predictive character of QRL in that an estimate of clim can be performed at any c < clim. Such an estimate is progressively more accurate as c approaches clim (examples of clim estimates for NaCl, CaCl2 and LaCl3 are represented by large symbols in Figure 1).

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Figure 1. Trends of cL values. Aqueous electrolytes (25 °C, 0.1 MPa) [42]. Red symbols, LaCl3; blue symbols, CaCl2; green symbols, NaCl. Dashed lines: cL from Eq.(8), values of clim, cL,1 and cL,2 are shown in Table III (last column). Large symbols refer to cL values used to estimate clim (Table III, fourth column) according to the iterative procedure (described in the Results and Discussion section) that correlates clim, cL,1 and cL,2 to experimental data.

At each iteration, given a trial concentration c the corresponding cL value is searched for by imposing equality between measured and calculated lnγ±. The trial concentration at the ith iteration generally corresponds to cL obtained at the (i-1)th iteration. The loop is repeated until cL converges to clim, that is,

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until lnγ± converges to 0. Concerning cL,1, it will be shown that it can also be obtained as a by-product of the above procedure, whilst cL,2 can be determined by means of the experimental osmotic coefficient at

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m=mlim. Figure 1 also shows that Eq. (8) adequately approximates cL for concentrations from clim down to almost high dilution (0.05 M about), where QRL becomes insensitive to parameterization values.

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The relationship between γ± and the osmotic coefficient Φ is given by the Gibbs-Duhem Equation [4]. Φ =1 + ln γ ± −

m

1 ln γ ± dm m ∫0

(9)

According to Pitzer formalism and definitions [1, 4], Eq. (9) also gives 1 ∂ G − G id = ln γ ± νRGTn 1M 1 ∂m

(

)

(10)

In Eq. (10) RG indicates the Gas Constant, n1 indicates the number of solvent moles, M1 is the molecular weight of the solvent, while G − Gid represents the total excess Gibbs energy, where G and Gid refer to Gibbs energy of real and ideal solution [1, 4]. Eq. (10) is obtained considering the set of independent

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variables (P, T, n1, m). Eq. (3) demonstrates that c≤clim implies lnγ± ≤0, the case lnγ± = 0 occurring at c=0 or at c=clim. From Eq. (10), the partial derivative of G − Gid with respect to m is negative for 0clim. So, c=clim corresponds to a minimum in the Excess Gibbs Energy with respect to m, provided that the other independent variables (P, T and n1) are kept fixed.

Volumetric and thermal properties

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2.4

QRL is based on the modeling of lnγ±, so the most natural route for calculating molal enthalpies and

relationships below apply [43]:

= m, P

− A2 νRGT 2

∂ ln γ ± ∂P

= m ,T

V 2 − V 20 νRG T

(11)

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∂ ln γ ± ∂T

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volumes passes through pertinent partial derivatives of lnγ± with respect to T and P. To this aim the

Eq. (11) involves the partial molal volume of solute V2 , the corresponding Standard State value V20 , and the partial molal enthalpy of solute A2 [43]. Regarding the apparent molal volume of solute V2, its

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 d −d M2   , where d0 indicates determination is usually based on density measurements, since V2 =  0 + d   d 0 dm

the solvent density [4, 43]. In so doing the determination of V2 implies that of

∂d . To this aim, the ∂m

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density d should be measured with high precision in order to obtain reliable results. Problems of experimental accuracy can become relevant in the strong-dilution range as far as V2 approaches the

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infinite-dilution value [44]. Using partial derivatives of lnγ± to calculate apparent properties allows for evaluation of the internal consistency of QRL from a thermodynamic point of view after comparing with measured data in the mPT frame [4]. For readability, we hereinafter put: s≡

c cL

 ek s  2  e2 k s 2  2  e3 k s 3  2 2   f(s) ≡ 1 + 2  (6k 2 + 1) − 6k 2  − 2  (3k 2 + 1) − 3k 2  + ( 2k + 1) − 2k 2  2   2k  α   k  α  2k  α

(12)

(13)

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Eqs. (14-16) are straightforwardly obtained by combining Eqs. (1, 2, 4-6, 12, 13).

 1 ∂d 1 ∂cL  ∂s  = s − ∂T  d ∂T cL ∂T 

 1 ∂d 1 ∂cL  ∂s  = s − ∂P  d ∂P cL ∂P 

(15a)

∂U U  1 ∂ε 1 1 ∂d  U  ln(s )   1 ∂d 1 ∂cL  =  + − − − 1 +   exp(−k ) ≤ s ≤ 1 ∂T 2  ε ∂T T d ∂T  ln(s )  k   d ∂T cL ∂T 

(15b)

∂U U  1 ∂ε 1 ∂d  U D ∂f ∂s =  − + ∂P 2  ε ∂P d ∂P  π ∂s ∂P

0 ≤ s ≤ exp(−k )

(16a)

(16b)

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0 ≤ s ≤ exp(−k )

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∂U U  1 ∂ε 1 1 ∂d  U D ∂f ∂s =  + − + ∂T 2  ε ∂T T d ∂T  π ∂s ∂T

(14)

exp(−k ) ≤ s ≤ 1

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∂U U  1 ∂ε 1 ∂d  U  ln(s )   1 ∂d 1 ∂cL  =  − − − 1 +   ∂P 2  ε ∂P d ∂P  ln(s )  k   d ∂P cL ∂P 

It is useful to remember that k only depends on α by virtue of Eq. (6), so k does not change with concentration, temperature or pressure. Regarding partial derivatives of lnγ±, from Eq. (3) the following equations can be obtained (calculations are straightforward but tedious, and will not be reported for

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brevity sake).

  2  − RAB  1 ∂U 1 ∂d   1  1 1 ∂ε 1 ∂d   1 = − + − +  ln γ ±   ∑B Q A Q B exp 4U 2  (17) 3  T ε ∂T 3d ∂T  m ,P   8 π εk BTνU  U ∂T 3d ∂T   N A,  A ≠B    2 ∂ ln γ ±  − RAB   1 ∂ε 1 ∂d   1 ∂U 1 ∂d   1 1 (18) = − − ln γ +    ± ∑B Q A Q B exp 4U 2  3  ∂P m,T U ∂ P 3 d ∂ P N  ε ∂P 3d ∂P    A,   8 π εk B T ν U  A ≠B  Simplification of double sums in Eqs. (17, 18) will be reported in the Computational Methods section. To

AC C

EP

∂ ln γ ± ∂T

use Eqs. (17, 18) partial derivatives of U (with respect to T or P) must also be known, which in turn, depend on partial derivatives of cL. Similarly to what was previously done (Eq. (8)) an approximation function can be adopted for each partial derivative: once again, at given T and P, a second-degree polynomial centered on c = clim.

∂cL ∂cL ≈ ∂T ∂T ∂cL ∂cL ≈ ∂P ∂P

T T + cL,1 (c − clim ) + cL,2 (c − clim ) / 2

(19)

P P + cL,1 (c − clim ) + cL,2 (c − clim ) / 2

(20)

2

c =clim

2

c =clim

13

∂c The values of L ∂T

ACCEPTED MANUSCRIPT T T , cL,1 and cL,2 can be determined following an experimental procedure that will

c =clim

be presented in the Results section, and analogously for

∂cL ∂P

P P , cL,1 and cL,2 . Use of Eqs. (17, 18) is

c = clim

then possible provided that pertinent partial derivatives of d and ε are also known. In practice, solvent

RI PT

density derivatives (with respect to T or P) have often been used in place of those of d due to lack of experimental data, but results have shown that numerical effects are negligible. Note, with QRL the

COMPUTATIONAL METHODS

3.1

Mean activity coefficient

M AN U

3.

SC

apparent molal volume of solute at infinite dilution must also be known.

Eq. (3) is useful for theoretical investigations, while fast numerical computation of lnγ± was proposed via integral equations containing elliptic functions [6-10, 45- 47]. Note, integral transformation methods based on the Mellin Transform of elliptic (Theta) functions are used in literature to deal with

TE D

conditionally convergent Madelung sums and to define these sums by virtue of the existence and the analyticity of the Mellin Transform, that allows for analytic continuation (in the region Re(s)>0 of the

EP

complex s-plane) to multidimensional Zeta functions [46]. Helpful manipulations of lattice sums were also performed according to the space-group geometry [39]. Below, integral expressions arising from Eq.

lnγ± =

AC C

(3) are summarized for the electrolytes investigated in this article. Q+Q− 3

2 2π εk BTU

1/ 2

∫ 0

3 2 2  n = +∞    n 2 n u      ( − 1 ) exp − R 1 −  ∑ du  2U 2     n = −∞ 

1:1 electrolytes (21)

In Eq. (21), the Fourth Jacobi Theta Function [45-47] θ4(ξ,z) ≡

n = +∞

∑ (−1)

n

e 2 jnξ z n can also be 2

n = −∞

 R 2u 2   −  exp evidenced after putting ξ = 0 and z = 2 . 2 U   6

lnγ±=

Q+Q− 3

6 2π εk BTU

∫ {2(A (u ) − 1) + 6 A(u ) B

1/ 2

3

2

}

(u ) + ( A(u ) + B (u ) ) − 2C 3 (u ) − 1 du 1:2/2:1 electrolytes(22a) 3

0

14

ACCEPTED MANUSCRIPT

For readability, in Eq. (22a) the auxiliary functions A(u), B(u) and C(u) are introduced.  2(3 / 2) 2 / 3 R 2 n 2 u 2 U2 

n = +∞

∑ exp −

n = −∞

C (u ) =

 (3 / 2 ) 2 / 3 R 2 ( n + 1 / 2 ) 2 u 2 2U 2 

n = +∞

∑ exp −

n = −∞

lnγ± =

2/3 2 2 2 n = +∞   B (u ) = ∑ exp − (3 / 2) R (2n + 1) u  2   2U n = −∞   

  

∫ {4(A (u) − 1)− 6 A

(22b)

}

1/ 2

Q+Q−

3

3

8 2π εk BTU

2

(u ) B (u ) + 2 A(u ) B 2 (u ) du

0

In Eq. (23a) the auxiliary functions are: n =+∞

 21 / 3 R 2 n 2u 2  exp ∑  − U 2  n =−∞  

B (u ) =

n =+∞

 2 −2 / 3 R 2 ( 2n − 1) 2 u 2   2U 2  

∑ exp −

n =−∞

(23a)

(23b)

SC

A(u ) =

1:3 / 3:1 electrolytes

RI PT

A(u ) =

In Eqs. (22-23) U is assumed to be the same for positive and negative effective-carriers, according to

3.2

Volumetric and thermal properties

M AN U

considerations done in Refs. [9, 10] and different from what was done in Refs.[6-8].

Double summations in Eqs. (17, 18) can be reduced to single summations after applying symmetry rules [39-40] of the concerned space groups (for brevity derivation of equations below will be omitted,

TE D

however it is analogous to what was previously done [6-10] to yield Eqs. (21-23)). 1:1 Electrolytes:

∂ ln γ ± ∂P

m ,T

EP

m,P

3   − R2n 2   1 ∂U 1 ∂d   n =+∞ n  +  − 1    ∑ (-1) exp  (24a) 4U 2    U ∂T 3d ∂T   n =−∞    3   − R2n 2  1 ∂U 1 ∂d   n =+∞  1 ∂ε 1 ∂d  Q+Q−  n  ∑ (-1) exp  (24b) = − − +  − 1  ln γ ±     3  n =−∞ 4U 2    ε ∂P 3d ∂P   8 π εk BTU  U ∂P 3d ∂P      

 1 1 ∂ε 1 ∂d  Q+Q− = − + −  ln γ ± 3  T ε ∂T 3d ∂T  8 π εk BTU

AC C

∂ ln γ ± ∂T

1:2/2:1 Electrolytes: ∂ ln γ ± ∂T

∂ ln γ ± ∂P

m,P

{

}

Q+Q− 1 ∂d  3  1 ∂U 1 ∂d  3 2 3  1 1 ∂ε + = − + −   2(A0 − 1) + 6 A0 B0 + ( A0 + B0 ) − 2C 0 − 1  ln γ ± 3 U ∂ T 3 d ∂ T    T ε ∂T 3d ∂T  24 π εk BTU

(25a)

 1 ∂U 1 ∂d  1 ∂d  Q+Q−  1 ∂ε 3 + = − −   2(A03 − 1) + 6 A0 B02 + ( A0 + B0 ) − 2C03 − 1  ln γ ± 3 U ∂ P 3 d ∂ P   ε ∂P 3d ∂P  24 π εk BTU 

(25b)

{

m ,T

}

For readability, in Eqs. (25a, 25b) the auxiliary terms A0, B0 and C0 are introduced. A0 =

n=+∞

 ∑ exp − ( )

n=−∞



3 2/3 2

R 2n 2   U 2 

B0 =

 3 2 / 3 R 2 (2n + 1) 2   exp ∑  − ( 2 ) 4U 2 n =−∞   n =+∞

C0 =

 3 2 / 3 R 2 (n + 1 / 2) 2   exp ∑  − ( 2 ) 4U 2 n=−∞   n=+∞

(25c)

15

ACCEPTED MANUSCRIPT

1:3 /3:1 Electrolytes: ∂ ln γ ± ∂T

∂ ln γ ± ∂P

m,P

m ,T

Q+Q− 1 ∂d   1 ∂U 1 ∂d  3 2 2  1 1 ∂ε + = − + −   {4(A0 − 1) − 6 A0 B0 + 2 A0 B0 }  ln γ ± 3 U ∂ T 3 d ∂ T    T ε ∂T 3d ∂T  32 π εk BTU

(26a)

 1 ∂U 1 ∂d  1 ∂d  Q+Q−  1 ∂ε + = − −   {4(A03 − 1) − 6 A02 B0 + 2 A0 B02 }  ln γ ± 3 U ∂ P 3 d ∂ P   ε ∂P 3d ∂P  32 π εk BTU 

(26b)

 2 −2 / 3 R 2 n 2  exp ∑  − U 2  n =−∞   n =+∞

B0 =

n = +∞

 2 −2 / 3 R 2 ( 2n − 1) 2   4U 2  

∑ exp −

n = −∞

RESULTS AND DISCUSSION

4.1

Mean activity and osmotic coefficients

M AN U

4.

(26c)

SC

A0 =

RI PT

In Eqs. (26a, 26b) the auxiliary terms are:

Hereinafter and for all the investigated systems, we refer to aqueous solvent; T = 298.15 K and P = 0.1 MPa; values of ε, d0 and their derivatives are given in Refs. [48-50]. Furthermore, in formulae below all quantities refer to c = clim or, equivalently, to m = mlim.

TE D

To use QRL, the key step is the determination of clim, cL,1 and cL,2 (Eq. (8)) for the investigated electrolyte. After that cL will result, established over [0, clim] by Eq. (8), so lnγ± can be calculated over [0, clim] by means of the appropriate form of Eq. (3) for the considered order of electrolyte (Eqs. (21-23)). Regarding

3 ∂ ln γ ± ∂c   8 π εkBTU D ln(k )clim /  Q + Q −  ∂m ∂m  

AC C

cL,1 = 1 −

EP

cL,1, it can be calculated as follows.

A derivation of Eq. (27) is given in the Supporting Information;

(27) ∂ ln γ ± can be obtained by numerical ∂m

differentiation, provided that clim (mlim) and a further value of lnγ± (at a molality m1 sufficiently close to

mlim) are experimentally known. Eq. (27) points out that the number of experimental data required by QRL is very limited at least in comparison with existing models. However, Eq. (27) first requires the knowledge of clim. In this connection, as anticipated in the Theory section, a practical iterative procedure can be implemented - the meta-code is depicted in Figure 2a -. In the first iteration, a concentration c0 is

16

ACCEPTED MANUSCRIPT

set (in practice, c0 corresponding to m0 = 1 mol/kg was found adequate for all the aqueous electrolytes investigated in this article). Then, cL*,0 = cL (c0 , P, T ) is searched for by imposing equality between measured and calculated lnγ± at m=m0, and calculations are done by means of the appropriate form of Eq.(3) (Eqs. (21-23)). The search commences by assigning cL*,0 an arbitrary value, then updating cL*,0 and

RI PT

repeating until convergence (within a suitable tolerance). The final cL*,0 will be used in the next iteration by putting c1= cL*,0 . At the i-th iteration, the concentration ci = ci(mi, P, T) will be chosen either equal or reasonably close to cL*,i-1 , in line with available experimental data. The number I of required iterations

SC

does not usually exceed 5. A practical criterion to stop the procedure is that | lnγ±| must be ≤ 0.15 and |mI-

M AN U

mI-1|/mI must be≤10%, at least at the I-th or (I-1)-th iteration. Since mI and mI-1 are sufficiently close to mlim, the criterion also allows ignoring cL,2 in Eq. (8) such that the following system can be solved yielding both clim and cL,1.

c*,L I−1 ≈ clim + cL,1 [cI−1 − clim ] c*,L I ≈ clim + cL,1 [cI − clim ]

(28)

TE D

In summary, clim and cL,1 can be obtained from the experimental clim plus Eq.(27) or, alternatively, from Eq. (28). Application to various systems has shown that the two methods are in excellent agreement with

EP

each other, although Eq. (28) is preferred since there is no longer a need for first-order derivatives required by Eq. (27). Concerning cL,2 (Eq. (8)), it might be determined from the knowledge of clim,

AC C

∂ 2 ln γ ± ∂ ln γ ± and , however, an alternative way to obtain cL,2 without using derivatives is based on the ∂m 2 ∂m experimental knowledge of the osmotic coefficient at m = mlim (this method has been used for all the cases investigated in this article). The Gibbs-Duhem Equation reads as follows.

Φ(mlim, P, T)=1-

mlim

∫ ln γ

±

dm /mlim

(29)

0

After determining clim and cL,1, we search for cL,2 by imposing equality between measured and calculated osmotic coefficient at m=mlim, through Eq. (29). This search commences by assigning cL,2 an arbitrary

17

ACCEPTED MANUSCRIPT

value, then updating cL,2 - the meta-code is depicted in Figure 2b - and repeating until convergence

EP

TE D

M AN U

SC

RI PT

(within a suitable tolerance).

AC C

Figure 2. Iterative procedure for QRL parameterization. Mean activity and osmotic coefficients. (a) Meta-code representing the procedure to determine clim and cL,1 (Eq. (8)) by means of Eq. (28). (b) Meta-code representing the procedure to determine cL,2 (Eq. (8)) by means of Eq. (29).

A software version (MATLAB) of the iterative procedure (Figure 2) is given in the Supporting Information. Figures 3 and 4 show results of application of the procedure to La(NO3)3. In this case, cL,1 was calculated by means of Eq. (27) and clim = 4.689 M was taken from experimental literature [42, 5153] (mlim= 7.98 mol/kg, supersaturated solution; Φ(mlim)=1.944). Numerical differentiation, performed at

m=mlim, yielded

∂c ≈ 0.32 kg/dm3 (densities in the super-saturation range were extrapolated from ∂m

18

ACCEPTED MANUSCRIPT

∂ ln γ ± available data [42]) and ≈ 0.26 kg/mol. Eq. (27) yielded cL,1 = -0.06, the iterative method depicted ∂m

M AN U

SC

RI PT

in Figure 2b yielded cL,2= -0.275 M-1.

AC C

EP

TE D

Figure 3. Mean activity coefficient of aqueous La(NO3)3. Red symbols, experimental values (25 °C) [42, 51-52]. Black line, QRL model, predicted values (parameters of Eq. (8) are given in the main text).

Figure 4. Osmotic coefficient of aqueous La(NO3)3. Red symbols, experimental values (25 °C) [42,51-52]. Black line, QRL model, predicted values (parameters of Eq. (8) are given in the main text). Green line, 3-parameter Hückel Equation [26]; blue line, 3-parameter Pitzer Equation [29].

Figure 4 shows a comparison with Hückel Equation [26] and Pitzer Theory [29]. With both theories regression techniques are applied to three-parameter equations, for concentrations up to 3 mol/kg (Hückel Equation) and 1.55 mol/kg (Pitzer Theory). The figure shows that experimental data are accurately reproduced based upon the concentration ranges above, however, values of parameters are not reliable if

19

ACCEPTED MANUSCRIPT

used to predict data at concentrations outside the optimized ranges. Figures 5-9 show calculated mean activity and osmotic coefficients in comparison with measured data for NaCl, CaCl2 and LaCl3. For these salts, the iterative procedure consisted of determining clim and cL,1 by means of the method depicted in

M AN U

SC

RI PT

Figure 2a, and cL,2 by means of the method depicted in Figure 2b.

AC C

EP

TE D

Figure 5. Mean activity and osmotic coefficients of aqueous NaCl. Red symbols, experimental values (25 °C) [42,53]. Black lines, QRL model, predicted values (parameters of Eq. (8) are shown in Table III, last column). Blue lines, QRL model in the original formulation [6,7].

Figure 6. Mean activity coefficient of aqueous CaCl2. Red symbols, experimental values (25 °C) [42, 54]. Black line, QRL model, predicted values (parameters of Eq. (8) shown in Table III, last column). Blue symbols, SiS Theory [20] (values are extrapolated from plotted data [20] at rounded abscissas). To compare results, the Ionic Strength I (molal scale) [4] is used.

20

SC

RI PT

ACCEPTED MANUSCRIPT

EP

TE D

M AN U

Figure 7. Osmotic coefficient of aqueous CaCl2. Symbols, experimental values at (25 °C) [42, 54]. Black line, QRL model, predicted values (parameters of Eq. (8) shown in Table III, last column). Blue line, 3-parameter Hückel Equation [26].

AC C

Figure 8. Mean activity coefficient of aqueous LaCl3. Red symbols, experimental values (25 °C) [42, 55]. Black line, QRL model, predicted values (parameters of Eq. (8) are shown in Table III, last column). Blue symbols, SiS Theory [20] (values are extrapolated from plotted data [20] at rounded abscissas). To compare results, the Ionic Strength I (molal scale) [4] is used.

21

SC

RI PT

ACCEPTED MANUSCRIPT

M AN U

Figure 9. Osmotic coefficient of aqueous LaCl3. Red symbols, experimental values (25 °C) [42, 55]. Black line, QRL model, predicted values (parameters of Eq. (8) are shown in Table III, last column). Green line, 3-parameter Hückel Equation [26]; blue line, 3-parameter Pitzer Equation [29].

Figures 6 and 8 show a comparison with SiS Theory [20], whilst Figures 7 and 9 compare with Pitzer and Hückel equations [26, 29]. In terms of model reliability and accuracy of results, the SiS Theory seems fitting as most models developed for dilute solutions, e.g., DHX [9, 56]. Concerning Pitzer and Hückel

TE D

equations, both used with three adjustable parameters, osmotic coefficients of CaCl2 and LaCl3 (Figures 7 and 9) are predicted nearly as accurately as QRL, though in these cases, optimized ranges are wider than

AC C

CaCl2 and LaCl3.

EP

[0, clim]. Table 3 summarizes main iteration steps and results of the QRL procedure applied to NaCl,

22

ACCEPTED MANUSCRIPT Table 3. Experimental data used for QRL parameterization

m0=1 mol/kg (c0 = 0.979 M)

measured property ln γ ± =-0.418

m1=6.148 mol/kg (c1=5.419 M) ln γ = 0.007 ± satd. solution m2=5.5 mol/kg (c2= 4.913 M) ln γ =-0.072 ±

cL*,3 = 5.631 M (4.75%dev.)

mlim =6.094 mol/kg

Φ =1.276

cL = clim

5.8 mol/kg

A2 =-1973.3 J/mol At m=mlim: A2 =-1989.0 J/mol A2 = -1994 J/mol,

6.148 mol/kg

A2 =-1997.2 J/mol

5.54 mol/kg

d=1.181 gr/cm3

5.83 mol/kg

d=1.189 gr/cm3

6.02 mol/kg

d=1.194 gr/cm3 0 2

V =

m0=1 mol/kg (c0=0.974 M)

16.62 cm3/mol ln γ± = -0.689

m1=2.5 mol/kg (c1=2.334 M)

ln γ± = 0.067

m2=2.25 mol/kg (c2=2.116 M)

ln γ± = -0.084

m3=2 mol/kg (c3=1.895 M)

ln γ± =-0.227

mlim =2.395 mol/kg

Φ = 1.528

2 mol/kg

A2 = 5.29 kJ/mol

2.25 mol/kg

A2 = 5.68 kJ/mol A2 =6.10 kJ/mol

1.67 mol/kg

d=1. 133 gr/cm3

2 mol/kg

d=1.157 gr/cm3

2.5 mol/kg

d=1.192 gr/cm3

Infinite dilution

V20 =17.6 cm3/mol ln γ± = -1.022

EP

m0=1 mol/kg (c0= 0.970 M)

A2 =-2317 J/mol,

∂A2 = 635.2 Jmol-2kg. ∂m

At m=mlim: V2=21.37 cm3/mol,

∂c L =-0.01 M/MPa ∂P

P = 9.9x10-5 1/ MPa cL,1

P =10-4 M-1/MPa cL,2

cL*,0 = 2.305 M (2.76 % dev.)

-

cL*,2 = 2.264 M (0.93 % dev.) cL*,3 = 2.301 M (2.58 % dev.)

clim= 2.243 M cL,1= -0.166 cL,2= -0.13 M-1

cL = clim

At m=mlim: A2= 5922 J/mol,

A2 = 9989 J/mol,

∂A2 = 4734 Jmol-2kg. ∂m

At m=mlim: V2 = 27.03 cm3/mol, V2 =31.99 cm3/mol, ∂V2 = 2.28 cm3kg/mol2. ∂m

∂c L =0.007 M/K ∂T T = 0.0033 K-1 cL,1 T = 6.05 x10-4 M-1K-1 cL,2

∂c L =-0.0025 M/MPa ∂P P = cL,1

-1.9x10-4 1/MPa

P = 4.6x10-4 M-1/MPa c L,2

cL*,0 = 1.868 M (-6.5 % dev.)

m1=2 mol/kg ( c1=1.871 M)

ln γ± = -0.141

cL*,1 = 1.994 M (-0.2 % dev.)

m2=1.8 mol/kg (c2=1.697 M)

ln γ± = -0.335

cL*,2 = 1.988 M (-0.5 % dev.)

mlim=2.148 mol/kg

Φ = 1.836

cL = clim

AC C

T = 0.0092 K-1 cL,1

T =0.0012 M-1K-1 cL,2

V2 = 23.54 cm3/mol,

∂V2 =0.527 cm3kg/mol2. ∂m

∂ c L = -0.015 M/K ∂T

SC

6 mol/kg

clim=5.378 M cL,1=-0.292 cL,2=-0.05 M-1

cL*,2 = 5.514 M (2.5% dev. )

ln γ ± =-0.130

2.5 mol/kg

LaCl3 c

-

m3=5 mol/kg (c3 = 4.513 M)

Infinite dilution

CaCl2 b

cL*,0 = 5.805 M (7.9% dev. )

TE D

NaCl a

Calculated parameter f

calculated parameter d,e

RI PT

Concentration Used

M AN U

aqueous electrolyte

1.69 mol/kg

A2 =14.77 kJ/mol

1.97 mol/kg

A2 =16.55 kJ/mol

At m=mlim: A2 = 17.75 kJ/mol,

2.25 mol/kg

A2 =18.47 kJ/mol

∂A2 =17.32 kJmol-2kg. ∂m

1.85 mol/kg

d= 1.368 gr/cm3

2.08 mol/kg

d= 1.408 gr/cm3

2.28 mol/kg

d= 1.443 gr/cm3

Infinite dilution

V20 =14.3 cm3/mol

At m=mlim: V2= 33.08 cm3/mol,

A2 = 32.57 kJ/mol,

V2 = 41.75 cm3/mol,

∂V2 = 5.4 cm3kg/mol2. ∂m

clim= 1.998 M cL,1= 0.034 cL,2=-0.205 M-1 ∂c L = 0.009 M/K ∂T T = 0.0049 1/K cL,1 T =8.5 x10-4 M-1K-1 cL,2

∂cL =-1.7x10-3 M/MPa ∂P P =-5.09x10-4 1/MPa cL,1 P =3x10-4 M-1/MPa cL,2

a

Refs. [42, 53, 57]. b Refs.[42, 54, 58]. c Refs. [42, 55]. d Eq. (8). Percentage deviations indicated in parentheses: (estimated value-clim)/clim (%). e Eqs. (32, 33). Measured data refer to 25°C and 0.1 MPa. Apparent volumes are calculated from density data. Apparent properties at m=mlim are obtained by numerical interpolation of measured values, partial properties are then obtained by numerical differentiation. f Eqs. (8, 19, 20, 31-34).

23

ACCEPTED MANUSCRIPT

In cases where the mean activity coefficient holds <1 over the whole available concentration range (including possible super-saturation), the iterative procedure is still valid provided that Eq. (8) is used with cL,2 = 0, since Eq. (29) is no longer applicable. For example, Figure 10 shows mean activity and osmotic coefficients for KCl and CsCl. For these cases, setting m0 = 1 mol/kg yielded c1 = 9.50 M (KCl)

RI PT

and c1= 27.36 M (CsCl), thereby suggesting exploring the highest concentrations available [42]. The iterative method (Figure 2a) yielded clim = 19.22 M (cL,1 = 0.025) for CsCl, and clim = 11.57 M (cL,1 =0.011) for KCl. Figure 10 also shows results concerning NaClO4 (clim=9.292 M; cL,1= -0.02; cL,2 =-0.025

SC

M-1). In this case, available experimental literature extends up to 17.2 mol/kg [42], however, data series appear somewhat inconsistent with each other for concentrations > 6 mol/kg. Consequently, experimental

M AN U

values for NaClO4 were preliminarily averaged based on two different literature sources [59-60] for

AC C

EP

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molalities > 6 mol/kg.

Figure 10. Mean activity and osmotic coefficients of aqueous 1:1 electrolytes. Experimental values (25 °C) [42,59-60]. Red symbols: NaClO4, experimental clim=9.292 M (mlim= 17.02 mol/kg); blue symbols: CsCl, estimated clim =19.22 M; green symbols: KCl, estimated clim= 11.57 M. Continuous lines, QRL model (this article). For all cases, values of parameters of Eq. (8) are given in the main text.

Although satisfactory representations of cL can usually be obtained by means of Eq. (8), improvement of the cL representation can be important in cases where relative deviations of cL with respect to clim are

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much larger than normally observed, occurring in particular with respect to ZnBr2 and ZnCl2 (Figure 11). These Zinc halides are known to show significant “swings” in their osmotic coefficients [42] and

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represent a serious modeling challenge for many theories [29].

Figure 11. cL values for Zinc halides. Aqueous solutions (25 °C) [42]. Red symbols, ZnBr2, clim= 4.755 M (mlim=6.28 mol/kg); blue symbols, ZnCl2, clim= 7.532 M (mlim= 10.7 mol/kg). Dashed lines, Eq. (8); continuous lines, Eq. (30). Large symbols refer to cL values used to estimate clim according to the iterative procedure that correlates parameters of Eqs. (8, 30) (given in the main text) to experimental data.

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In addition to clim, cL,1 and cL,2 as obtained by the iterative procedure described before, a few further cL values, indicated with c~L , are to be determined via the same procedure considering concentrations that

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lie within the range 0.1 - 0.5 mol/kg. In such a range, and for all cases so far investigated, the cL curve shows a clear minimum (Figures 1 and 11) that can be detected numerically on the basis of the c~L values.

cL≈

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The cL curve will then result well described by Eqs. (30a-30b) below. 2 clim − cL,1clim / 2 + cL,2clim / 8 − cL (c*)

(2σ 1 / clim )

2a

− (σ 1 / c *) − (2 / clim ) + (1 / c*) 2a

a

cL ≈ clim + cL,1 (c − clim ) + cL,2 (c − clim ) / 2 2

a

((σ

1

)

/ c) 2 a − (σ 1 / c*)2 a − c − a + (1 / c*)a + cL (c*) c ≤ clim / 2 (30a)

clim / 2 ≤ c ≤ clim

(30b)

In Eqs. (30a, 30b), σ 1 = 2 −1 / a c * , where c* indicates the concentration that corresponds to the minimum of cL. The exponent a is found by fitting Eqs. (30a, 30b) over the { c~L } set of values. With ZnBr2, clim =4.755 M (mlim=6.28 mol/kg), cL,1 = 0.15, cL,2 = -0.275 1/M, c*= 0.23 M, cL(c*)=1.51 M, and a =-0.275. The { c~L } values were determined at 0.1 mol/kg, 0.2 mol/kg, 0.3 mol/kg and 0.4 mol/kg using measured

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mean activity coefficients at such molalities [42]. With ZnCl2, clim = 7.532 M (mlim= 10.7 mol/kg), cL,1 = -0.23, cL,2 = -0.05 1/M, c*=0.36 M, cL(c*)= 3.1 M, and a = 0.2. The { c~L } values were determined at 0.2 mol/kg, 0.3 mol/kg, 0.4 mol/kg and 0.5 mol/kg, using mean measured activity coefficients at such molalities [42]. Figures 12 and 13 show results for ZnBr2 and ZnCl2, and compare with three-parameter

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Pitzer Equation [29]. Despite the accuracy of results over the optimized ranges (from 0 to 2 mol/kg), yet again, best-fit values of parameters from Pitzer Equation do not give satisfactory results if used outside such ranges. Figure 12 also shows mean activity and osmotic coefficients of ZnBr2 calculated with

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cL=clim, according to previous QRL parameterization [6-10], so as to appreciate current advances of the

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QRL model.

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Figure 12. Mean activity and osmotic coefficients of aqueous ZnBr2. Red symbols, experimental values (25°C) [42]. Black lines, QRL model, predicted values (parameters of Eq. (30) are given in the main text). Green lines, QRL model in the original formulation [6,7]. Blue lines, 3-parameter Pitzer Equation [29].

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Figure 13. Mean activity and osmotic coefficients of aqueous ZnCl2. Red symbols, experimental values (25°C) [42]. Black lines, QRL model, predicted values (parameters of Eq. (30) are given in the main text). Blue lines, 3-parameter Pitzer Equation [29].

4.2

Volumetric and thermal properties

From the evaluation of Eqs. (17, 18) in the limit c → clim, the following equations are derived (a

3

∂cL 8 π εkBTU D ln(k ) A2 clim ∂d = clim + ∂T Q + Q − νRGT 2 d ∂T 3

(

c=clim

)

∂cL 8 π εkBTU D ln(k ) V20 − V2 clim ∂d = clim + ∂P Q + Q− νRGT d ∂P

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c=clim

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derivation of Eqs (31-32) is given in the Supporting Information).

(31)

(32)

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As already said, the approximating of d derivatives (with respect to T or P) with those of solvent density affects results very little. In order to calculate cLT ,1 and cLP,1 (Eqs. (19, 20)), second-order partial derivatives at m=mlim

∂ 2 ln γ ± ∂ 2 ln γ ± − 1 ∂A2 1 ∂V2 are also needed, that is, = and = . For the sake of 2 ∂m∂T νRGT ∂m ∂m∂P νRGT ∂m

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compactness, in Eq. (33) below the symbol X stands for either T or P, and all quantities and derivatives refer to c=clim (m=mlim). Calculations yielding Eq. (33) are reported in the Supporting Information.

X cL,1 =−

∂c Q+ Q_ ∂m

∂ 2 ln γ ± ∂m∂X − ∂ ln(εTUD ) (1 − c ) − c ∂ ln(U D ) ∂s + 2 / k ∂s (1 − c ) + c ∂  1 ∂d  + cL,1 ∂cL   L,1 lim L,1 lim ∂X ∂c ∂X ∂X ∂c  d ∂X  clim ∂X

(33)

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clim 8 π εk BTUD ln(k )

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T P Finally, the remaining unknowns cL,2 and cL,2 (Eqs. (19, 20)) are determined by an analogous procedure

to that concerning cL,2 (Figure 2b). Volumetric and thermal apparent properties at m=mlim are now used [43]. − νRG T 2 mlim

mlim

∫ 0

∂ ln γ ± dm = A2 ( mlim , P , T ) ∂T

ν RG T mlim

mlim

∫ 0

∂ ln γ ± dm = V2 ( mlim , P, T ) − V20 ∂P

(34)

T P The procedure - the meta-code is depicted in Figure 14 - commences by assigning cL,2 ( cL,2 ) an arbitrary

value, imposing that A2 (V2) calculated with Eq. (34) be equal to the measured property at m=mlim, then

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T P updating cL,2 ( cL,2 ) and repeating until convergence (within a suitable tolerance). A software version

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(MATLAB) of the iterative procedure (Figure 14) is given in the Supporting Information.

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Figure 14. Iterative procedure for QRL parameterization. Apparent and partial molal volumes and enthalpies. (a) T Meta-code representing the procedure to determine cL,2 , Eq. (19), by means of Eq. (34). (b) Meta-code representing the P procedure to determine cL,2 , Eq. (20), by means of Eq. (34).

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Figures 15-17 show QRL results concerning apparent and partial molal properties of NaCl, CaCl2 and

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LaCl3 by means of Eqs. (24-26). Related parameterization is summarized in Table 3.

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Figure 15. Apparent and partial molal properties of aqueous NaCl. (a) Apparent and partial molal volumes. (b) Apparent and partial molal enthalpies. Red symbols and lines, experimental values (25 °C), mlim=6.094 mol/kg [42, 53, 57]. Black lines, QRL model, predicted values (parameters of Eqs. (8, 19, 20) are shown in Table III, last column). Green lines, 2-parameter Hückel Equation [25]; blue lines, 3-parameter Hückel Equation [26]. Continuous lines, apparent properties; dashed lines, partial properties.

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Figure 16. Apparent molal properties of aqueous CaCl2. (a) Apparent molal volumes. (b) Apparent molal enthalpies. Symbols, experimental values (25 °C), mlim=2.395 mol/kg [42, 54, 58]. Black lines, QRL model, predicted values (parameters of Eqs. (8, 19, 20) are shown in Table III, last column). Green lines, 3-parameter Hückel Equation [26]; blue lines, Pitzer Theory [28].

29

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Figure 17. Apparent and partial molal properties of aqueous LaCl3. (a) Apparent and partial molal enthalpies. (b) Apparent and partial molal volumes. Red symbols and lines, experimental values (25 °C), mlim=2.148 mol/kg [42]. Black lines, QRL model, predicted values (parameters of Eqs. (8, 19, 20) are shown in Table III, last column). Continuous lines, apparent properties; dashed lines, partial properties.

In Figure 15, apparent and partial properties of NaCl calculated via Hückel equations are also shown. In going from two-parameter [25] to three-parameter [26] Hückel equations, common parameters are subject to relatively small changes, they do however significantly affect width of optimized ranges. Further comparison with Hückel and Pitzer equations is given in Figure 16 (CaCl2). Further application of the QRL model to volumetric and thermal properties of electrolytes is described by plotting for KOH,

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Ca(NO3)2 and Lu(NO3)3 in Figures 18 to 19. Experimental information used [42, 57, 61-63] and

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parameterization for these salts are summarized in the Supporting Information.

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Figure 18. Apparent and partial molal volumes of aqueous electrolytes. Experimental values at 25 °C [42, 57, 61-63]. Red symbols and dashed lines: Lu(NO3)3, mlim= 3.29 mol/kg. Brown symbols and short-dashed lines: Ca(NO3)2, mlim= 9.62 mol/kg. Green symbols and dashed lines: KOH, mlim= 2.74 mol/kg. Black, blue, and brown lines: QRL model, respectively for Lu(NO3)3, Ca(NO3)2 and KOH. Symbols and continuous lines: apparent volumes; dashed lines: partial volumes. For all cases, parameters of Eqs. (8, 19, 20) are given in the Supporting Information.

Figure 19. Apparent and partial molal enthalpies of aqueous electrolytes. Experimental values at 25 °C [42, 57, 61-63]. Red symbols and dashed lines: Lu(NO3)3, mlim= 3.29 mol/kg. Brown symbols and light-brown dashed lines: Ca(NO3)2, mlim= 9.62 mol/kg. Green symbols and dashed lines: KOH, mlim= 2.74 mol/kg. Black, blue, and brown lines: QRL model, respectively for Lu(NO3)3, Ca(NO3)2 and KOH. Symbols and continuous lines: apparent enthalpies; dashed lines: partial enthalpies. For all cases, parameters of Eqs. (8, 19, 20) are given in the Supporting Information.

5.

CONCLUDING REMARKS

In this article, the generalized QRL model has been presented as applied to a variety of electrolytes, for calculating their mean activity and osmotic coefficients, and their apparent and partial molal enthalpies

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and volumes. Further properties, such as apparent molal heat capacities, and further electrolytes will be considered in future research. The generalization of QRL has yielded a very competitive representation of electrolyte data over well defined concentration ranges. The experimental information needed by QRL is generally very modest compared to the number of data points used in usual regression techniques for

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best-fit purposes. The thermodynamic consistency of QRL has been illustrated by the level of agreement with measured properties in the mPT frame. The semi-predictive character of QRL leads to the exploration of model potentialities basically connected to the knowledge of the microscopic nature of clim.

mesoscopic / macroscopic properties is demanded,

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Besides agreement with experiments, though encouraging, correlation between microscopic and and a capillary investigation of model

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parameterization is needed to substantiate the proposed approach. Indeed, it may happen that a model (or a parameter) reasonably able to describe some “target” properties can reveal unfounded hypotheses or it can fail to describe other properties. Interpretation of clim and cL availing the language of Mean-Field theories [2, 4], and comparison with Molecular Dynamics simulations will be the subject of future research. It is outlined here that clim corresponds to a mean inter-ionic distance R belonging to the range

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4Å - 8Å for most aqueous salts. This is a significant range from the point of view of Radial Distribution Function (RDF) Theory [2]. In addition, cL looks like a “dual concentration” showing a curious trend

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(Figures 1 and 11) that becomes insensitive to clim when c approaches the strong-dilution range.

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SUPPORTING INFORMATION

Additional material available: derivation of Eqs. (27,31-33); experimental information and parameterization for aqueous solutions of KOH, Ca(NO3)2 and Lu(NO3)3; software implementation (MATLAB) of the iterative procedures depicted in Figures 2,14.

ACKNOWLEDGMENTS

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The author wishes to thank Dr. Wilma Kirsten (London, UK) for her proofreading support, and Emeritus Professor Bruno Bianco (DITEN Department, University of Genoa, Genoa, Italy) for his scientific comments.

V2 , V 2 V

0 2

T T c L,1 , c L,2 P P c L,1 , c L,2

Quasi-Random Lattice Model mean inter-ionic distance [m] Molar, molal concentration [mol/dm3, mol/kg] Mean molal activity coefficient Stoichiometric numbers Molecular weight of solvent, molecular weight of solute Solution volume Number of solute molecules in a volume V N/V Absolute density of solution Ion (±) charge [Coulomb] Dielectric permittivity of solvent Boltzmann Constant Absolute temperature [K] Pressure Distance between lattice points A and B Madelung Constant of a ionic crystal QRL constant of a reference ionic-lattice frame Space coordinates of lattice point A Lin. stand. dev. of a Gaussian density of charge (effective carrier) Debye-Hückel Screening Length Error Function Molar (molal) concentration corresponding to γ± = 1 Generalized concentration parameter Coefficients of cL Osmotic coefficient Gibbs Energy of a real (G) and of an ideal (Gid) solution Apparent, partial molal enthalpy of solute

Eq. (8) Eq. (9) Eq. (10) Eq. (11)

Apparent, partial molal volume of solute

Eq. (11)

Partial molal volume of solute at infinite dilution ∂c Coefficients of L ∂T ∂cL Coefficients of ∂P

Eq. (11)

Eq. (1) Eq. (2) Eq. (3)

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QRL R c, m γ± ν±,ν M1 , M2 V N ρ d Q± ε kB T P RAB α k RA U UD erf clim (mlim) cL cL,1, cL,2 Φ G; Gid A2, A2

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NOMENCLATURE

Eq. (3)

Eq. (3) Eq. (4a) Eq. (4a) Eq. (3) Eq. (4a) Eq. (3)

Eq. (19) Eq. (20)

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1812–1832.

39

ACCEPTED MANUSCRIPT The article evaluates the most relevant thermodynamic properties of ionic solutions

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The article is built on a very competitive model for the mean activity coefficient

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Regression techniques are not required by the model

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The model parameterization requires a small number of experimental data

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The article advances toward a truly predictive model of the electrolytic behavior

AC C

EP

TE D

M AN U

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RI PT

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