# Improved ion-pairing equation of state and its use for various property calculations

Abtin Raeispour Shirazi, Fufang Yang, Tri Dat Ngo, Olivier Bernard,

Jean-Pierre Simonin, Jean-Charles de Hemptinne

► To cite this version:

Abtin Raeispour Shirazi, Fufang Yang, Tri Dat Ngo, Olivier Bernard, Jean-Pierre Simonin, et al. Improved ion-pairing equation of state and its use for various property calculations. Fluid Phase Equilibria, 2025, 595, pp.114396. 10.1016/j.fluid.2025.114396. hal-04989332

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Submitted on 13 Mar 2025

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# Improved ion-pairing equation of state and its use for various property calculations

Abtin Raeispour Shirazi<sup>a,b</sup>, Fufang Yang<sup>c</sup>, Tri Dat Ngo<sup>c</sup>, Olivier Bernard<sup>b</sup>, Jean-Pierre Simonin<sup>b</sup>, Jean-Charles de Hemptinne<sup>a,*<sup>†</sup></sup>

## ARTICLE INFO

**Keywords:**

Ion pairing

ePPC-SAFT

Binding MSA

Mixed-solvent

Bjerrum theory

RSP models

Gibbs energy of transfer

## ABSTRACT

This work utilizes an advanced electrolyte polar perturbed chain SAFT theory (ePPC-SAFT) framework combined with the Binding MSA approach to simulate interactions between ions, solvents, and ion pairs in aqueous and mixed-solvent environments. The model allows a higher flexibility to quantify ion pairing and thus improve the accuracy in predicting key properties, such as mean ionic activity coefficients, solubility, and density. The findings underscore the importance of considering both ion pairing and solvation (here represented through association) in thermodynamic models for both aqueous and mixed solvent systems. The model makes it possible to evaluate the impact of the various molecular interactions for ionic activity coefficients and Gibbs energies of transfer.

## 1. Introduction

Ion pair formation is an important phenomenon in electrolyte solutions, where unlike ions associate to form a stable pair. This phenomenon, is influenced by various factors, including the concentration of ions, the nature of the ions, temperature and the properties of the solvent, particularly its dielectric constant (also called Relative static Permittivity, RSP) [1–6]. Ion pairing can lead to significant changes in the physical properties of the solution. Specifically, the formation of ion pairs reduces the number of free ions in the solution, which in turn affects the solution conductivity [7] and of course the ionic strength [4,5]. Moreover, it can impact the solubility of salts (through the ionic activity coefficients), as well as the solubility of molecular species (salting in and out effects [8]).

In aqueous solutions, the high dielectric constant of the water (approximately 78.5 at room temperature) plays a crucial role in stabilizing ions and preventing ion pair formation. The high dielectric constant reduces the electrostatic forces between oppositely charged ions, allowing them to remain dissociated in solution. However, as the concentration of ions increases, the likelihood of ion pairs forming also increases due to the reduced mean distance between ions, which enhances their electrostatic attraction. According to the results from

previous work [8], it was found that for alkali-chloride salts in aqueous system ion-solvent association strength (solvation) is more important than ion pairing. This was also shown and discussed by the group of Collins [9,10].

In mixed solvent systems, such as those involving water and alcohols [8,11–15], the situation becomes more complex. The presence of an organic solvent, typically with a lower dielectric constant than water, alters the solvation environment and the extent of ion pairing [4,6]. For example, alcohols like methanol and ethanol have dielectric constants of about 32.6 and 24.3, respectively, significantly lower than that of water. When water is mixed with alcohol, the overall dielectric constant of the solvent system decreases. This reduction in the dielectric constant increases the electrostatic attraction between oppositely charged ions, thereby promoting ion pair formation. Indeed, a high dielectric constant indicates that the solvent can effectively shield ions from each other, preventing them from forming pairs. Conversely, a lower dielectric constant means that the solvent is less effective at reducing these forces, leading to a higher propensity for ion pairing.

Historically, ion pairing was overlooked in early electrolyte theories, such as the Debye-Hückel (DH) [16] theory, which assumed that ions in a dilute solution are fully dissociated and interact solely through long-range Coulomb forces. However, the simplicity of this approach was soon challenged by the realization that even in dilute solutions, ions

<footnote><sup>a</sup> IFP Energies Nouvelles, 1 et 4 Avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France</footnote>

<footnote><sup>b</sup> CNRS, Laboratoire PHENIX, Campus P.M. Curie, Sorbonne Universités, F-75005 Paris, France</footnote>

<footnote><sup>c</sup> Laboratoire de Chimie de l'ENS Lyon, CNRS and Université de Lyon, 46 allée d'Italie, 69364 Lyon, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jean-charles.de-hemptinne@ifpen.fr (J.-C. de Hemptinne).</footnote>
Nomenclature

<table><tr><td>A</td><td>Helmholtz energy</td></tr><tr><td>a</td><td>activity</td></tr><tr><td>B</td><td>reducing coefficient in the Bjerrum association model</td></tr><tr><td>c</td><td>molarity</td></tr><tr><td>d<sub>v</sub></td><td>parameter in the Schreckenberg RSP model</td></tr><tr><td>d<sub>T</sub></td><td>parameter in the Schreckenberg RSP model</td></tr><tr><td>e</td><td>electronic charge</td></tr><tr><td>g<sup>hs</sup><sub>ij</sub></td><td>hard sphere i-j radial distribution function</td></tr><tr><td>k<sub>B</sub></td><td>Boltzmann constant</td></tr><tr><td>l<sub>B</sub></td><td>Bjerrum length</td></tr><tr><td>MIAC</td><td>mean ionic activity coefficient</td></tr><tr><td>IIAC</td><td>individual ionic activity coefficient</td></tr><tr><td>N<sub>A</sub></td><td>Avogadro's number</td></tr><tr><td>OF</td><td>objective function</td></tr><tr><td>RSP</td><td>relative static permittivity</td></tr><tr><td>T</td><td>temperature</td></tr><tr><td>x</td><td>mole fraction</td></tr></table>

<table><tr><td>y</td><td>mole fraction in vapor phase</td></tr><tr><td>Z</td><td>valence</td></tr><tr><td>Z'</td><td>effective charge of ions</td></tr><tr><td>X<sup>Ai</sup></td><td>fraction of non-bonded sites of A<sub>i</sub></td></tr><tr><td>G<sub>tr</sub></td><td>Gibbs free energy of transfer</td></tr></table>

Greek letters

<table><tr><td>γ</td><td>activity coefficient</td></tr><tr><td>Δ<sup>AiBj</sup></td><td>association strength</td></tr><tr><td>ε<sup>AB</sup></td><td>association energy parameter</td></tr><tr><td>ε<sub>0</sub></td><td>vacuum permittivity</td></tr><tr><td>ε<sub>r</sub></td><td>relative static permittivity</td></tr><tr><td>κ<sup>AB</sup></td><td>association volume parameter</td></tr><tr><td>ρ</td><td>density</td></tr><tr><td>σ<sup>hs</sup></td><td>hard sphere diameter</td></tr><tr><td>σ<sub>p</sub></td><td>sum of anion and cation diameters</td></tr><tr><td>Δ<sub>ac</sub></td><td>apparent association strength of ions</td></tr><tr><td>Δ<sup>0</sup><sub>ac</sub></td><td>association strength of ions at infinite dilution</td></tr></table>

can form stable pairs. This insight was first formalized by Bjerrum [17], who introduced the concept of ion pairs as entities that form when the Coulomb attraction between ions overcomes the thermal energy that tends to keep them apart. There are also other theories that account for the ion pairing phenomenon such as Binding-MSA (BiMSA) [18–21], and Binding-Debye-Hückel (BiDH) [22] introduced later. These electrolyte models, known as primitive models, were introduced in the MacMillan-Mayer (MM) framework [23] that focuses on describing the interactions between solute particles (more specifically ions) while treating the solvent as a continuous medium of fixed chemical potential which is often considered as equivalent to fixed dielectric constant ($ε_r$=constant) [24]. This approach is distinct from the more conventional Lewis-Randall (LR) framework and considers the solvent explicitly in the mixture. In the MM framework, the independent thermodynamic variables include temperature (T), volume (V), the number of solute particles ($n_{solute}$), and the chemical potential of the solvent ($μ_{solvent}$) which is in contrast to the LR framework, where the number of solvent molecules is explicitly considered, along with temperature, pressure, and the number of all components.

Recently, many researchers [8,12,22,24,25] used the primitive models introduced in the MacMillan-Mayer (MM) framework (T, V, $n_{i}$=ions, $μ_{solvent}$) to the equation of state (EoS) framework (T, V, $n_{i}$=all species) with the aim of incorporating more physical details and enhancing the accuracy of thermodynamic models (in the EoS framework all the species including solvent are considered explicitly in the mixture). This is done in general by adding two different models: one is “Born” for ion-solvent interactions, and one is Debye-Hückel or MSA for ion-ion long range electrostatic interactions. These two interactions (ion-solvent and ion-ion) are discussed and detailed below.

**Ion-solvent:** Most electrolyte models (other than primitive models) consider that ion-solvent interactions have both a non-electrostatic part and an electrostatic part. The physical contribution is generally described using a dispersive-type model [14,26–30]. Some authors [8, 12,13,15,31–36] however dispute this approach, stating that solvation is a chemical phenomenon that cannot be neglected. Therefore, they consider the ion-solvent association for describing this phenomenon. In practice, there is no consensus on which terms should be used for ion-solvent interactions. It is generally recognized that the “Born” contribution is not sufficient (we use quotation marks around the name Born in the same way as Simonin [37,38] in order to point out that Born proposed his famous equation for a lone ion in a solvent, not for an ion in an electrolyte of finite concentration), but it should be complemented by either dispersion or association or both. Yang et al. [34,35] suggested

that in order to reduce the number of adjustable parameters, association is the most adapted contribution. In the IFPEN group [8,12,13,15,31–33, 39], ion-solvent association, Born and MSA terms, are considered for ions, and it is believed that they are necessary terms [9,10]. In other groups [14,25–28,40,41], the ion-solvent association is ignored and only the dispersion term is used to describe the chemical part of the solvation energy. Overall, all these approaches provide acceptable results.

**Ion-ion:** These interactions are not often considered in EoS approach. However, recently, the following authors have introduced the concept explicitly (see Table 1): Bülow et al. [25] for the first time added the Bjerrum model (a primitive model) within a reactive EoS framework (ePC-SAFT model) to predict the salt solubility in non-aqueous solvents with a particular focus on the impact of ion pairing. Their model is a combination of the PC-SAFT EoS, Debye-Hückel and Born terms that include a reactive equilibrium. The Born term was added to account for the energy required to transfer ions from water to a non-aqueous medium, with a composition dependent model for RSP. Additionally, the Bjerrum model was used within the chemical equilibrium constant to address the formation of ion pairs in solvents with low dielectric constants. Later on, Yang et al. [12], added the Bjerrum model through the association term within the ePPC-SAFT framework to investigate the impact of ion pairs (ion-ion association) in aqueous and mixed aqueous-solvent systems. Their model is a combination of PC-SAFT, polar, association, MSA and Born terms. They used the Schreckenberg model to compute the dielectric constant (or relative static permittivity, RSP) in both mixed and aqueous systems. Naseri-Boroujeni et al. [22] introduced a new version of DH model, called Binding-Debye-Hückel, in order to introduce ion-ion association within the DH theory. They then applied their model within the SAFT-VR-Mie framework. Their proposed model was used to calculate the liquid density, mean ionic activity coefficient, and osmotic coefficient for binary NaCl, Na₂SO₄, and MgSO₄ aqueous solutions at 298.15 K. Raeispour-Shirazi et al. [8] applied the Simplified-Binding-MSA and Bjerrum model to ePPC-SAFT framework in order to investigate the impact of ion pairing in aqueous and mixed solvent systems. Müller et al. [42–44] recently introduced the COSMO-RS-ES model to describe ion pairing using the Bjerrum treatment and their model was successfully applied to low permittivity systems.

In all the aforementioned works, the Bjerrum model was added to calculate the strength of the ion-ion association. However, this model is applicable only at infinite dilution and must therefore be corrected for finite salt concentrations. The correction factor, also known as the cavity
<table><caption>**Table 1**
Short summary of ion pairing models introduced in equation of state framework.</caption><thead><tr><th></th><th>Reference</th><th>Electrostatic (long range)</th><th>Born</th><th>Ion-ion association (electrostatic)</th><th>Short-range ion-ion and ion-solvent interactions (non-electrostatic)</th></tr></thead><tbody><tr><td>ePPC-SAFT</td><td>Bülow et al. [25]</td><td>DH</td><td>Yes</td><td>Bjerrum (no cavity function)</td><td>Dispersion for cation-anion and ion-solvent</td></tr><tr><td>ePPC-SAFT</td><td>Yang et al. [12]</td><td>SMSA</td><td>Yes</td><td>Bjerrum (no cavity function)</td><td>Association for ion-solvent(Wertheim)</td></tr><tr><td>SAFT-VR Mie</td><td>Naseri-Boroujeni et al. [22]</td><td>BIDH</td><td>Yes</td><td>Bjerrum+(cavity function from BiDH)</td><td>Dispersion for ion-ion and ion solvent</td></tr><tr><td>ePPC-SAFT</td><td>This work and previous work (Raeispour-Shirazi et al. [31])</td><td>BIMSA</td><td>Yes</td><td>Bjerrum+(cavity function from BIMSA)</td><td>Association for ion-solvent(Wertheim)</td></tr></tbody></table>

function [45–50], is added to the ion-ion association strength in order to consider the non-ideality of the system. In the works of Yang et al. [12] and Bülow et al. [25], this correction factor is neglected, and in the work of Raeispour-Shirazi et al. [8] and Naseri-Boroujeni et al. [22], it was taken from BiMSA or BiDH theories, respectively.

A second difficulty encountered with the Bjerrum model, is that the theory on which it is based contains assumptions that are approximative. In fact, the model that is derived is purely predictive in such a way that the association strength is only a function of the solvent RSP and of the ion-ion distance of closest approach (which can be considered as solvated or not). As a consequence, in our previous work [8], we added an adjustable parameter that made it possible to distinguish the salts that are strongly ion-pairing from the ones that are weakly so. Yet, these parameters were determined for water solvent only. There is no theoretical foundation for stating that in a different solvent, the correction factor should be the same. This is why in the present work the correction factor that used to be a binary interaction parameter (BIP) is now considered to be a ternary interaction parameter (TIP).

In this work, the impact of ion pairs in aqueous and mixed solvent solutions are investigated using the same Simplified-BiMSA-ePPC-SAFT (SBiMSA-ePPC-SAFT) models as that was introduced previously [8]. To this end, the proposed models are augmented using a new adjustable parameter to promote ion pairing in the mixed solvent systems and the obtained results are discussed and compared to previous results [8]. In order to evaluate the predictive capability of the model, new properties such as individual ionic activity coefficient, Gibbs free energy of transfer and association constant of ions from experimental measurements are investigated.

## 2. Framework

In the ePPC-SAFT model [12,13,33–35], the residual Helmholtz energy is expressed as a function of temperature, volume and mole numbers as follows:

$$A^{\text{res}} = A^{\text{hc}} + A^{\text{disp}} + A^{\text{assoc}} + A^{\text{polar}} + A^{\text{SBiMSA}} + A^{\text{Born}} \quad (1)$$

where the superscripts *hc*, *disp*, *assoc* and *polar* denote hard-chain, dispersion, association and multipolar (as presented by Ref [51–53]) effects, respectively, and the Born term describes the electrostatic contribution to solvation [54]. Despite the fact that the Polar term has no major impact on the equation of state behaviour with electrolyte, it was kept in order to keep the predictive capacity of the model, as demonstrated in previous work (see for example [55]) The SBiMSA superscript denotes Simplified-Binding MSA [8]. SBiMSA model is a simplification of the full BiMSA model introduced by Bernard et al. [18, 19,56]. The SBiMSA model consists of two terms, the first one is the $A^{\text{SBiMSA}}$ (presented in Section 2.1) and the second is $A^{\text{MAL}}$ ($A^{\text{ion-ion}}$) which is introduced in the association contribution ($A^{\text{assoc}}$ presented in Section 2.2) of the ePPC-SAFT model. Therefore, the two contributions of $A^{\text{SBiMSA}}$ and $A^{\text{assoc}}$ are described below.

### 2.1. SBiMSA contribution ($A^{\text{SBiMSA}}$)

The BiMSA model is an extended version of the MSA theory that includes the association of ions. In the SBiMSA model [8], the residual Helmholtz energy is expressed as a function of temperature, volume and mole numbers as follows.

$$\frac{A^{\text{SBiMSA}}}{k_B T} = -\frac{e^2}{4\pi\epsilon_0\epsilon_r k_B T} \left\{ \sum_{i=1}^{n_{\text{ion}}} \frac{n_i z_i^2 \Gamma}{1 + \Gamma \sigma_i} \right\} + \frac{\Gamma T^{-3}}{3\pi} \quad (2)$$

where $\Gamma$ is the screening parameter (inverse screening length) that is determined using the following expression [18,19,56] that includes the impact of ion pairing for a single salt:
$$4\Gamma^2 = \frac{e^2}{\varepsilon_0 \varepsilon_r k_B T} \sum_{i=1}^{n_{\text{ion}}} \frac{n_i}{V} \left(\frac{z_i}{1 + \Gamma \sigma_i}\right)^2 + \frac{2e^2}{\varepsilon_0 \varepsilon_r k_B T V} \frac{n_{p_{\text{ac}}}}{\sigma_{p_{\text{ac}}}} \left(\frac{\sigma_a}{1 + \Gamma \sigma_a} + \frac{\sigma_c}{1 + \Gamma \sigma_c}\right) Z_a Z_c^* \quad (3)$$

$$Z_a = \frac{z_a}{1 + \Gamma \sigma_a} \quad \text{and} \quad Z_c = \frac{z_c}{1 + \Gamma \sigma_c} \quad (4)$$

$$\sigma_{p_{\text{ac}}} = \sigma_a + \sigma_c \quad (5)$$

where the subscripts c and a denote cation and anion, respectively. The elementary charge is $e$, $\varepsilon_r$ is the relative permittivity of the medium, $\varepsilon_0$ the vacuum permittivity, $z_i$ is the valence, $\sigma_i$ is the diameter of the ion, $n_i$ is the number of type i ions, $Z_a$ and $Z_c$ are the effective charge of anions and cations respectively (the actual charge is decreased because of the screening due to both the solvent RSP and ion pairing) and $\sigma_{p_{\text{ac}}}$ is the sum of anion and cation diameters. The number of pairs formed between cations and anions is $n_{p_{\text{ac}}}$ and it is obtained from the $A^{\text{MAL}}$ ($A^{\text{ion-ion}}$) contribution described in Section 2.2.1.

## 2.2. Association contribution ($A^{\text{assoc}}$)

Following the Wertheim association theory, sticky spots, called association sites, are located on the associating compounds, allowing them to create chemical bonds. Association bonds are short range and directional. In this work, we consider that they may represent both hydrogen bonding and the strong electrostatic interactions that lead to hydration or ion pairing.

Generally, the association contribution based on the Helmholtz energy using the Wertheim theory [57,58] is described by the following expression:

$$\frac{A^{\text{assoc}}}{k_B T} = \sum_i n_i \sum_{A_i} \left[ \left( \ln X^{A_i} - \frac{X^{A_i}}{2} \right) + \frac{1}{2} M_i \right] \quad (6)$$

$$X^{A_i} = \left[ 1 + N_{Av} \sum_{B_j} (X^{B_j} \Delta^{A_i B_j}) \right]^{-1} \quad (7)$$

where $M_i$ stands for the number of association sites on molecule i, $X^{A_i}$ is the fraction of non-bonded sites of $A_i$ and $\Delta^{A_i B_j}$ is the association strength. In our model the association term is constructed out of the three contributions to consider three types of association. One has.

$$A^{\text{assoc}} = A^{\text{ion-ion}} + A^{\text{ion-solvent}} + A^{\text{solvent-solvent}} \quad (8)$$

where $A^{\text{ion-solvent}}$ and $A^{\text{solvent-solvent}}$ represent the ion-solvent and solvent-solvent contributions that cannot be split in Eq. (8), because the same site on a solvent molecule can associate with either a solvent site or an ion site. The first term, $A^{\text{ion-ion}}$ is the ion-ion association, also called MAL for “Mass Action Law” ($A^{\text{MAL}} = A^{\text{ion-ion}}$). It represents ion pairing. One of the sites on the ions has been labeled such that it can only interact with another labeled ion site, hence their contribution can be separated. The MAL contribution originates from the BIMSA model and is described below.

### 2.2.1. Association of ion pairs ($A^{\text{MAL}} = A^{\text{ion-ion}}$)

Ion-ion association $A^{\text{ion-ion}}$ which comes from the BIMSA theory is expressed below for a single anion-cation (ac) ion pair:

$$\frac{A^{\text{MAL}}}{k_B T} = \sum_{i=1}^{n_{\text{ion}}} n_i \ln X^{A_i} + n_{p_{\text{ac}}} \quad (9)$$

where $n_{p_{\text{ac}}}$ is the number of pairs obtained from the following expression:

$$n_{p_{\text{ac}}} = \frac{2 \Delta^{\text{ac}} n_a n_c}{V + \Delta^{\text{ac}} n_t + [V^2 + 2 \Delta^{\text{ac}} V n_t + (\Delta^{\text{ac}})^2 (n_c - n_a)^2]^{1/2}} \quad (10)$$

where $\Delta^{\text{ac}}$ and $n_t = n_a + n_c$ are the equilibrium constant of ion pairing and the total number of ions, respectively. It should be mentioned that (Eqs. 9 and 6) are identical. Therefore, Eqs. (6) and (7) (SAFT association term) are used instead of Eqs. (9) and (10).

### 2.2.2. Computation of the association strengths using the SAFT association term

Eqs. (9) and (10) require numerical values for the association strength, $\Delta$. For the association strength between the ion-solvent and solvent-solvent molecules the usual Wertheim association theory is used:

$$\Delta^{A_i B_j} = d_{ij}^3 g_{ij}^{hs} \kappa^{A_i B_j} \left[ \exp \left( \frac{\varepsilon^{A_i B_j}}{k_B T} \right) - 1 \right] \quad (11)$$

where $\varepsilon^{A_i B_j}$ is the energy of association, $\kappa^{A_i B_j}$ denotes the association volume and $g_{ij}^{hs}$ is the hard sphere i-j radial distribution function. It should be noted that the association strength in SAFT-type models is the molar concentration based chemical equilibrium constants. It is therefore expressed in volume unit (m3/mol).

For the calculation of the ion-ion association strength the expression that was introduced in the previous work [8] is used again:

$$\Delta_{ac} = \Delta_{ac}^{0} \beta_{ac} \quad \text{where} \quad \beta_{ac} = \frac{\gamma_{a}^{+} \gamma_{c}^{-}}{\gamma_{ac}} \quad (12)$$

where $\Delta_{ac}^{0}$ is the apparent association strength of ions. $\Delta_{ac}^{0}$ is the association strength of ions at infinite dilution and $\beta_{ac}$ represents the non-ideality of the system: it is the ratio of activity coefficients, and sometimes called cavity function.

The ion-ion association strength at infinite dilution is based on the Bjerrum theory which is an electrostatic theory [17]. In this work, for calculating the Bjerrum association strength, the correlation introduced by Yang et al. [12] is used:

$$\Delta_{ac}^{0} = 4 \pi N_{A} (B_{ac} g_{ac}^{hs})^{3} e^{n_{1}[\ln(B-2)]^{3} + n_{2}[\ln(B-2)]^{2} + n_{3}\ln(B-2) + n_{4}(B-2)^{n_{6} + n_{5}}} \quad (13)$$

where $n_{1-6}$, are universal parameters from ref [12]. B is the reducing coefficient in the Bjerrum model expressed as:

$$B = \frac{2l_B}{\sigma_{ac}^{hs}} \quad (14)$$

$$l_B = \frac{b_{ac} |z_a z_c| e^2}{8 \pi \varepsilon_0 \varepsilon_r k_B T} \quad (15)$$

where $l_B$ is the Bjerrum length and $b_{ac}$ is an adjustable parameter. This parameter is a correction to the Bjerrum association constant, because this theory, being predictive (it only depends on the RSP and distance of closest approach, $\sigma_{ac}^{hs}$), cannot provide accurate results.

As a consequence, we propose this correction parameter to be also solvent-composition dependent as follows:

$$b_{ac} = \frac{\sum_{k}^{\text{solvent}} n_k b_{ac}^{k}}{\sum_{k}^{\text{solvent}} n_k} \quad (16)$$

We thus introduce a ternary parameter $b_{ac}^k$ that is different for every solvent k. Regarding the non-ideality function, $\beta_{ac}$, we use the one that is provided by the BIMSA theory which reads

$$\beta_{ac} = g_{ac}^{hs} g^{el} \quad (17)$$

where $g_{ac}^{hs}$ is the radial pair distribution function at contact, designating the short-range part of the interactions, namely the contributions of hard
<table><caption>**Table 3**</caption><caption>Table 2<br>Summary of model combinations for the mixed solvent systems.</caption><thead><tr><th colspan="11">Contributions:</th></tr><tr><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Hard sphere (hc):</td><td>PC-SAFT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Dispersion (disp):</td><td>PC-SAFT</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Electrostatic solvation (Born):</td><td>Born</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Electrostatic ion-ion interaction (MSA):</td><td>SBiMSA</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Association (Assoc) (ion-solvent and solvent-solvent):</td><td>Wertheim (Eq. 11 for computing Δ<sup>AB</sup>)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mass action law (MAL) (ion-ion association):</td><td>Only Bjerrum (Eq. 13 for computing Δ<sup>ac</sup> = Δ<sup>0</sup><sub>ac</sub>)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RSP:</td><td>Schreckenberg</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Model name:</td><td>BS Previous work [8]</td><td></td><td></td><td></td><td>BS-Tbij (This work)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Ion-specific pseudo-unary parameters of models introduced in Table 2 (BS, BT and BgT models) for chloride salts in aqueous solution [8].

<table><thead><tr><th></th><th>N<sub>site</sub></th><th>σ<sub>Pauling</sub> (Å)<sup>a</sup></th><th>μ<sup>HS</sup></th><th>ε<sup>AB</sup>/k (K)</th><th>μ<sup>MSA</sup></th><th>σ<sub>Born</sub> (Å)<sup>c</sup></th></tr></thead><tbody><tr><td colspan="7">BS model</td></tr><tr><td>Na<sup>+</sup></td><td>7</td><td>1.9</td><td>0.89</td><td>2546</td><td>2.74</td><td>3.23</td></tr><tr><td>K<sup>+</sup></td><td>7</td><td>2.66</td><td>0.89</td><td>1805</td><td>1.80</td><td>3.89</td></tr><tr><td>Rb<sup>+</sup></td><td>7</td><td>2.96</td><td>0.89</td><td>2501</td><td>1.28</td><td>4.16</td></tr><tr><td>Cs<sup>+</sup></td><td>7</td><td>3.38</td><td>0.89</td><td>-191<sup>d</sup></td><td>3.15</td><td>4.48</td></tr><tr><td>Cl<sup>-</sup></td><td>6</td><td>3.62</td><td>0.89</td><td>3237</td><td>1.01</td><td>4.51</td></tr><tr><td colspan="7">BT model</td></tr><tr><td>Na<sup>+</sup></td><td>7</td><td>1.9</td><td>0.94</td><td>1373</td><td>2.86</td><td>3.23</td></tr><tr><td>K<sup>+</sup></td><td>7</td><td>2.66</td><td>0.94</td><td>949</td><td>1.46</td><td>3.89</td></tr><tr><td>Rb<sup>+</sup></td><td>7</td><td>2.96</td><td>0.94</td><td>493</td><td>1.89</td><td>4.16</td></tr><tr><td>Cs<sup>+</sup></td><td>7</td><td>3.38</td><td>0.94</td><td>-155<sup>d</sup></td><td>2.33</td><td>4.48</td></tr><tr><td>Cl<sup>-</sup></td><td>6</td><td>3.62</td><td>0.94</td><td>2766</td><td>1.48</td><td>4.51</td></tr><tr><td colspan="7">BgT model</td></tr><tr><td>Na<sup>+</sup></td><td>7</td><td>1.9</td><td>0.95</td><td>1884</td><td>3.24</td><td>3.23</td></tr><tr><td>K<sup>+</sup></td><td>7</td><td>2.66</td><td>0.95</td><td>948</td><td>1.3</td><td>3.89</td></tr><tr><td>Rb<sup>+</sup></td><td>7</td><td>2.96</td><td>0.95</td><td>1620</td><td>2.15</td><td>4.16</td></tr><tr><td>Cs<sup>+</sup></td><td>7</td><td>3.38</td><td>0.95</td><td>1482</td><td>2.27</td><td>4.48</td></tr><tr><td>Cl<sup>-</sup></td><td>6</td><td>3.62</td><td>0.95</td><td>1928</td><td>1.3</td><td>4.51</td></tr></tbody></table>

a The Pauling diameter values are obtained from Ref [64].

c The Born diameters are calculated using the Gibbs energy of solvation [65] using the procedure introduced in previous work [32]

d The pseudo-pure association energy parameter, ε<sup>AB</sup><sub>Cs<sup>+</sup></sub>, could be negative. However, ε<sup>AB</sup><sub>Cs<sup>+</sup>-w</sub> calculated using Eq. (19) is positive.

<caption>**Table 4**</caption>

Cation-anion association parameters for the models (BS, BT and BgT models) introduced in Table 2 for chloride salts in aqueous solutions.

<table><thead><tr><th></th><th>BS b<sup>w</sup><sub>ac-Cl<sup>-</sup></sub></th><th>BT b<sup>w</sup><sub>ac-Cl<sup>-</sup></sub></th><th>BgT b<sup>w</sup><sub>ac-Cl<sup>-</sup></sub></th></tr></thead><tbody><tr><td>Na<sup>+</sup></td><td>0.73</td><td>0.74</td><td>0.75</td></tr><tr><td>K<sup>+</sup></td><td>0.82</td><td>0.84</td><td>0.86</td></tr><tr><td>Rb<sup>+</sup></td><td>0.86</td><td>0.94</td><td>0.96</td></tr><tr><td>Cs<sup>+</sup></td><td>1.15</td><td>1.12</td><td>0.95</td></tr></tbody></table>

sphere of the ionic species, and it is assumed to be equal to one. Besides, g<sup>el</sup> accounts for electrostatic ion screening. It is expressed as:

$$g^{el} = \exp\left(\frac{2e^2(Z_aZ_c - Z_aZ_c)}{4\pi\epsilon_0\epsilon_r kT\sigma_{Pac}}\right) \qquad (18)$$

This factor (g<sup>el</sup>) considers the effect of long-range electrostatic interactions between pairing ions. The expressions for Z can be found in Eq. (4). Eventually, the final expression for calculating the ion-ion association using the BiMSA model is expressed as:

$$\Delta^{ac} = \Delta^{0}_{ac} g^{el} \qquad (19)$$

## 2.3. Relative static permittivity (dielectric constant)

The relation between the dielectric constant and ion pairing in the equation of state framework was settled in the previous work [8]. It was found that using the Bjerrum or Binding-MSA (BiMSA) models to account for ion pairing, the best RSP functionality is the one that does not decrease with salinity (neither explicitly nor implicitly through its volume dependence). In other words, the best choice for the RSP functionality was found to be temperature and solvent composition. However, for further investigation, we keep the volume and solvent composition functionality as well. This means that, the Schreckenberg model [59] which is temperature, volume and solvent-composition dependent and the Zuo & Fürst [60] and Maryott and Smith [61] models which are only temperature and solvent-composition dependent are used in this work as well.
### 2.3.1. Schreckenberg model

The Schreckenberg model [59] is a volume and solvent-composition dependent model which is expressed by the following equation:

$$ \epsilon_r = 1 + \frac{n_{\text{solv}}}{V} d_v \left( \frac{d_T}{T} - 1 \right) \qquad (20) $$

where $n_{\text{solv}}$ is the number of moles of the solvent in volume V, $d_v$ and $d_T$ are parameters determined using the mixing rules.

$$ d_v = \frac{\sum_{i}^{\text{solvent}} n_i d_{v,i}}{\sum_{i}^{\text{solvent}} n_i} \qquad (21) $$

$$ d_T = \frac{\sum_{i}^{\text{solvent}} n_i d_{T,i}}{\sum_{i}^{\text{solvent}} n_i} \qquad (22) $$

where $d_v$ and $d_T$ values for water, methanol and ethanol are taken from ref [59].

### 2.3.2. Zuo and Fürst and Maryott and Smith models

The Zuo and Fürst [60] model is only temperature-dependent and expressed by the following equation for the pure ethanol and water:

$$ D_s = d_1 + \frac{d_2}{T} + d_3 T + d_4 T^2 + d_5 T^3 \qquad (23) $$

where $d_1-d_5$ are constants are taken from ref [62,63]. For pure methanol, the Maryott and Smith [61] model is used:

$$ \log D_s = 1.514 - 0.00264(T - 298.15) \qquad (24) $$

For the mixed solvents systems, the following mixing rules is used.

$$ \epsilon_r = \frac{\sum_{i}^{\text{solvent}} n_i D_i}{\sum_{i}^{\text{solvent}} n_i} \qquad (25) $$

## 2.4. List of modeling options

As a consequence of the above discussion, several models can be constructed and will be investigated in what follows. They are summarized in Table 2.

Table 2 illustrates that in the ePPC-SAFT framework two different models are used for ion-ion association, both inspired by the Bjerrum theory: (i) the original expression proposed by the BIMSA model (Eq. 19), called g-models, (ii) the Bjerrum model alone $\Delta^{ac} = \Delta_{ac}^0$ (Eq. 13). For the RSP model there will be two different options as well: (i) the Schreckenberg model [59] which is temperature, volume and solvent-composition dependent; (ii) an RSP model that depends only on temperature and solvent composition, like those of Zuo & Fürst [60] or Maryott and Smith [61]. The originality of this work is the use of a ternary interaction parameter (TIP) that is introduced within the Bjerrum length (Eq. 15).

## 2.5. Parameterization method for the mixed solvent systems

The solvent parameters of the ePPC-SAFT model were obtained from

<caption>**Table 5**</caption>

Ion-specific parameterization procedure of models introduced in Table 2 for the mixed solvent chloride solutions.

<table><caption>**Table 6**</caption><thead><tr><th rowspan="2">Models:</th><th colspan="2">Parameters</th><th rowspan="2">Ref</th></tr><tr><th>Binary (BIP)</th><th>Ternary (TIP)</th></tr></thead><tbody><tr><td>BS, BT, BgT</td><td>$W_{Na^+}$-alc, $W_{K^+}$-alc, $W_{Rb^+}$-alc, $W_{Cs^+}$-alc, $W_{Cl^-}$-alc</td><td>—</td><td>Previous work [8]</td></tr><tr><td>BS-Tbij, BT-Tbij, BgT-Tbij</td><td>$W_{Na^+}$-alc, $W_{K^+}$-alc, $W_{Rb^+}$-alc, $W_{Cs^+}$-alc, $W_{Cl^-}$-alc</td><td>$b_{Na^+}$<sup>alc</sup>-cl-, $b_{K^+}$<sup>alc</sup>-cl-, $b_{Rb^+}$<sup>alc</sup>-cl-, $b_{Cs^+}$<sup>alc</sup>-cl-</td><td>This work</td></tr></tbody></table>

Alcohol-ion energy parameters for BS-Tbij, BT-Tbij and BgT-Tbij models introduced in Table 2 for the mixed-solvent electrolyte solutions.

<table><caption>**Table 7**</caption><thead><tr><th rowspan="2">BIP</th><th colspan="3">Methanol</th><th colspan="3">Ethanol</th></tr><tr><th>BS-Tbij</th><th>BT-Tbij</th><th>BgT-Tbij</th><th>BS-Tbij</th><th>BT-Tbij</th><th>BgT-Tbij</th></tr></thead><tbody><tr><td>$W_{Na^+}$-alcohol</td><td>0.18</td><td>0.13</td><td>-0.02</td><td>0.09</td><td>0.18</td><td>0.20</td></tr><tr><td>$W_{K^+}$-alcohol</td><td>0.52</td><td>0.66</td><td>0.64</td><td>0.09</td><td>0.53</td><td>0.68</td></tr><tr><td>$W_{Cl^-}$-alcohol</td><td>-0.09</td><td>0.07</td><td>0.08</td><td>0.81</td><td>0.40</td><td>0.21</td></tr><tr><td>$W_{Rb^+}$-alcohol</td><td>0.11</td><td>-0.08</td><td>0.07</td><td>—</td><td>—</td><td>—</td></tr><tr><td>$W_{Cs^+}$-alcohol</td><td>—</td><td>—</td><td>—</td><td>-0.08</td><td>-0.06</td><td>0.05</td></tr></tbody></table>

Cation-anion association parameters for the BS-Tbij, BT-Tbij and BgT-Tbij models introduced in Table 2 for the mixed-solvent electrolyte solutions.

<table><thead><tr><th rowspan="2">TIP</th><th colspan="3">Methanol</th><th colspan="3">Ethanol</th></tr><tr><th>BS-Tbij</th><th>BT-Tbij</th><th>BgT-Tbij</th><th>BS-Tbij</th><th>BT-Tbij</th><th>BgT-Tbij</th></tr><tr><th></th><th>$b_{cat-Cl-}^{alc}$</th><th>$b_{cat-Cl-}^{alc}$</th><th>$b_{cat-cC-}^{alc}$</th><th>$b_{cat-Cl-}^{alc}$</th><th>$b_{cat-Cl-}^{alc}$</th><th>$b_{cat-Cl-}^{alc}$</th></tr></thead><tbody><tr><td>Na<sup>+</sup></td><td>0.57</td><td>0.80</td><td>1.02</td><td>0.98</td><td>1.25</td><td>1.64</td></tr><tr><td>K<sup>+</sup></td><td>0.71</td><td>1.01</td><td>1.46</td><td>1.13</td><td>1.53</td><td>2.34</td></tr><tr><td>Rb<sup>+</sup></td><td>0.64</td><td>1.06</td><td>1.12</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Cs<sup>+</sup></td><td>—</td><td>—</td><td>—</td><td>0.88</td><td>1.18</td><td>2.24</td></tr></tbody></table>

previous works by Yang et al. and Ahmed et al. [13,31]. The aqueous unary and binary parameters are kept from previous publication [8] and they are presented in Tables 3 and 4.

In the previous work [8], the association energy parameter related to water solvation ($\epsilon_i^{AB}$ in Eq. 19) was considered as an adjustable parameter. The CR1 combining rule [66] was used, such that the parameter to adjust is the “pseudo-unary” ion association energy, $\epsilon_i^{AB}$.

$$ \epsilon_{w-i}^{AB} = \frac{\epsilon_{w}^{AB} + \epsilon_{i}^{AB}}{2} \qquad (26) $$

where the subscript w denotes water, the subscript i denotes ion. On the other hand, the ion-pairing association strength at infinite dilution ($\Delta_{ac}^0$) was computed from Eq. (13), and the binary parameter $b_{ac}^w$ (presented in Table 4) was used to adjust the association strength.

The objective of this work was to further improve the existing models with a minimum number of adjustable parameters. As was already pointed out in previous work [8], the ion pairing has a very significant impact on the MIAC in mixed solvents electrolyte solutions. This is why, for every salt, the adjustable parameter that regulates this phenomenon was modified so as to become a ternary parameter ($b_{ac}^k$), to be used in Eq. 16. Its value for water solvent ($b_{ac}^{water}$) is kept unchanged (it is taken from previous work [8], see Table 4), which means that only its value with alcohol ($b_{ac}^{alcohol}$) needs to be regressed. In addition, the binary alcohol-ion solvation energy (that is used in Eq. 27) is also regressed which makes two new parameters for every ion. Recall that the ion-alcohol solvation energy is obtained from the unary ion and alcohol energy parameters using a binary ion-alcohol parameter $w_{i-alc}$:

$$ \epsilon_{i-alc}^{AB} = \frac{\epsilon_{w}^{AB} + \epsilon_{alc}^{AB}}{2} (1 - w_{i-alc}) \qquad (27) $$

The results using these model combinations (BS-Tbij, BT-Tbij and BgT-Tbij) are compared with those obtained from previous work [8] (BS, BT, BgT) for the mixed solvent systems only (chloride salts in water-ethanol and chloride salts in water-methanol). The number and types of adjustable parameters in this and previous work are presented in Table 5.

The same data sourced from previous work [8] on mixed-solvent systems [13,15,34–36,39] are utilized in here as well. The objective function for the mixed-solvent system is:
<figure><img src="image_3.png" /><figcaption>Fig. 1. $b_{ac}^k$ values for the BS-Tbij, BT-Tbij and BgT-Tbij models introduced in Table 2 for the mixed-solvent electrolyte solutions.</figcaption></figure>

<caption>**Table 8**</caption>

Average absolute percentage deviations (AADs*) and maximum absolute percentage deviations (MADs*) for MIAC, VLE, and density of the models introduced in Table 2 for chloride salts in water-ethanol systems. The MIAC, VLE *p*, and density deviations are in percentage. The VLE *y* absolute deviations are absolute deviations multiplied by 100.

<table><thead><tr><th rowspan="3">Model name<br>Water+Ethanol +</th><th colspan="4">MIAC</th><th colspan="4">VLE p</th><th colspan="4">VLE y</th><th colspan="4">Density</th></tr><tr><th colspan="2">BS-Tbij</th><th colspan="2">BS</th><th colspan="2">BS-Tbij</th><th colspan="2">BS</th><th colspan="2">BS-Tbij</th><th colspan="2">BS</th><th colspan="2">BgT-Tbij</th><th colspan="2">BS</th></tr><tr><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th></tr></thead><tbody><tr><td>NaCl</td><td>4.2</td><td>17.0</td><td>6.2</td><td>21.3</td><td>2.6</td><td>11.4</td><td>2.6</td><td>14.4</td><td>2.5</td><td>6.2</td><td>2.6</td><td>6.4</td><td>0.7</td><td>2.5</td><td>0.7</td><td>2.5</td></tr><tr><td>KCl</td><td>2.1</td><td>3.5</td><td>4.2</td><td>6.5</td><td>1.7</td><td>4.5</td><td>1.5</td><td>4.3</td><td>3.6</td><td>6.0</td><td>3.6</td><td>6.3</td><td>0.9</td><td>8.6</td><td>0.9</td><td>8.6</td></tr><tr><td>CsCl</td><td>4.9</td><td>35.1</td><td>6.9</td><td>28.4</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.2</td><td>2.7</td><td>0.4</td><td>3.1</td></tr><tr><td></td><td>BT-Tbij</td><td></td><td>BT</td><td></td><td>BT-Tbij</td><td></td><td>BT</td><td></td><td>BT-Tbij</td><td></td><td>BT</td><td></td><td>BT-Tbij</td><td></td><td>BT</td><td></td></tr><tr><td></td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td></tr><tr><td>NaCl</td><td>5.5</td><td>23.6</td><td>12.1</td><td>33.0</td><td>2.4</td><td>12.2</td><td>2.5</td><td>15.0</td><td>2.4</td><td>6.6</td><td>2.6</td><td>8.5</td><td>0.7</td><td>2.2</td><td>0.7</td><td>2.2</td></tr><tr><td>KCl</td><td>2.8</td><td>5.0</td><td>8.8</td><td>20.1</td><td>1.3</td><td>5.0</td><td>1.2</td><td>4.8</td><td>2.8</td><td>5.1</td><td>3.3</td><td>5.5</td><td>0.5</td><td>6.5</td><td>0.5</td><td>6.5</td></tr><tr><td>CsCl</td><td>4.5</td><td>22.1</td><td>4.5</td><td>15.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.6</td><td>3.2</td><td>0.5</td><td>3.1</td></tr><tr><td></td><td>BgT-Tbij</td><td></td><td>BgT</td><td></td><td>BgT-Tbij</td><td></td><td>BgT</td><td></td><td>BgT-Tbij</td><td></td><td>BgT</td><td></td><td>BgT-Tbij</td><td></td><td>BgT</td><td></td></tr><tr><td></td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td><td>AAD (%)</td><td>MAD (%)</td></tr><tr><td>NaCl</td><td>8.1</td><td>34.4</td><td>17.4</td><td>56.4</td><td>2.4</td><td>14.5</td><td>3.0</td><td>18.4</td><td>2.6</td><td>8.3</td><td>5.0</td><td>20.7</td><td>0.8</td><td>2.2</td><td>0.8</td><td>2.2</td></tr><tr><td>KCl</td><td>2.8</td><td>4.0</td><td>11.4</td><td>28.7</td><td>1.8</td><td>4.6</td><td>2.7</td><td>5.7</td><td>2.2</td><td>6.1</td><td>4.1</td><td>10.1</td><td>0.6</td><td>5.8</td><td>0.6</td><td>5.8</td></tr><tr><td>CsCl</td><td>6.1</td><td>28.7</td><td>26.3</td><td>92.4</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.3</td><td>2.6</td><td>0.2</td><td>2.6</td></tr></tbody></table>

*For the vapor composition (VLE y): $AD\% = \left| \frac{\sum_{i=1}^{n_{dp}^x} (\mathbf{y}_i \text{ cal} - \mathbf{y}_i \text{ exp})}{n_{dp}^y} \right| \cdot 100$, $MAD\% = \max(|\mathbf{y}_i \text{ cal} - \mathbf{y}_i \text{ exp}|) \cdot 100$ and for MIAC, density or vapor pressure (VLE p):

*$AD\% = \frac{\left| \frac{\sum_{i=1}^{n_{dp}^y} \frac{\Psi_{i,cal} - \Psi_{i,exp}}{\Psi_{i,exp}}}{n_{dp}^y} \right|}{n_{dp}^y} \cdot 100$, $MAD\% = \max\left| \frac{\Psi_{i,cal} - \Psi_{i,exp}}{\Psi_{i,exp}} \right| \cdot 100$, $\Psi$ is MIAC, density or vapor pressure (VLE p).

$$ \text{OF} = \frac{\sum_{i=1}^{n_{dp}^{\gamma+}} \left( \frac{\gamma_i^{\gamma+} - \gamma_i^{\gamma-}}{\gamma_i^{\gamma-}} \right)^2}{n_{dp}^{\gamma+}} + \frac{\sum_{i=1}^{n_{dp}^{\gamma-}} \left( \frac{p_{i,cal} - p_{i,exp}}{p_{i,exp}} \right)^2}{n_{dp}^{\gamma-}} + \frac{\sum_{i=1}^{n_{dp}^{\gamma+}} (y_i^{\gamma+} - y_i^{\gamma-})^2}{n_{dp}^{\gamma+}} + \frac{\sum_{i=1}^{n_{dp}^{\gamma-}} (p_{i,cal} - p_{i,exp})^2}{n_{dp}^{\gamma-}} \quad (28) $$

where $\gamma$ is the mole fraction of the alcohol in the vapor phase, and it is the absolute deviation. The $\gamma_{\pm}$ points here to the mean ionic activity coefficients (MIAC), that are molality based and use a mixed solvent reference state.

The resulting parameters from the BS-Tbij, BT-Tbij and BgT-Tbij models (using the ternary $b_{ac}$) are presented in Tables 6 and 7.

Fig. 1 compares graphically the values for $b_{ac}^{\text{alcohol}}$ (using the BS-Tbij, BT-Tbij and BgT-Tbij models regressed with mixed solvent data) and

$b_{ac}^{\text{water}}$ (obtained from regressing the aqueous system data in the previous work [8], where $b_{ac}^{\text{alcohol}} = b_{ac}^{\text{water}}$). As can be seen for the three models for ethanol, the values for $b_{ac}^{\text{ethanol}}$ are systematically larger than those for water ($b_{ac}^{\text{water}}$) and methanol ($b_{ac}^{\text{methanol}}$). Because ethanol has the lower dielectric constant in comparison to water and methanol, we have already observed in previous work [8] that the ion pairing association constant is much larger than in the other solvents. Yet, the size of the correction factor points to the fact that the Bjerrum model is not sufficient and that in practice the ion pairing constant must be even larger. For $b_{ac}^{\text{methanol}}$ the plot shows no such regularity.

### 3. Results and discussion

In this section, the three model combinations (BS-Tbij, BT-Tbij and BgT-Tbij models) using a ternary $b_{ac}$ (TIP) are discussed and compared with the three models from previous work [8] (BS, BT and BgT) when $b_{ac}$ is a BIP. The aim of this comparison is to check the impact of this TIP on
<caption>**Table 9**</caption>

Average absolute percentage deviations (AADs*) and maximum absolute percentage deviations (MADs*) for MIAC, VLE, and density of all models introduced in Table 2 for chloride salts in water-methanol systems. The MIAC, VLE *p*, and density deviations are in percentage. The VLE *y* absolute deviations are absolute deviations multiplied by 100.

<table><thead><tr><th rowspan="2">Model name</th><th colspan="4">MIAC</th><th colspan="4">VLE p</th><th colspan="4">VLE y</th><th colspan="4">Density</th></tr><tr><th colspan="2">BS-Tbij</th><th colspan="2">BS</th><th colspan="2">BS-Tbij</th><th colspan="2">BS</th><th colspan="2">BS-Tbij</th><th colspan="2">BS</th><th colspan="2">BS-Tbij</th><th colspan="2">BS</th></tr><tr><th>Water+Methanol+</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th><th>AAD (%)</th><th>MAD (%)</th></tr></thead><tbody><tr><td>NaCl</td><td>2.8</td><td>17.8</td><td>3.9</td><td>18.5</td><td>2.2</td><td>9.5</td><td>2.3</td><td>10.4</td><td>1.1</td><td>4.8</td><td>1.2</td><td>4.7</td><td>1.9</td><td>8.5</td><td>1.9</td><td>8.8</td></tr><tr><td>KCl</td><td>2.0</td><td>12.6</td><td>2.0</td><td>13.6</td><td>2.6</td><td>6.0</td><td>2.6</td><td>6.8</td><td>0.7</td><td>3.5</td><td>0.8</td><td>4.2</td><td>6.0</td><td>13.7</td><td>6.0</td><td>13.7</td></tr><tr><td>RbCl</td><td>1.8</td><td>7.9</td><td>3.6</td><td>21.7</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td></td><th colspan="2">BT-Tbij</th><th colspan="2">BT</th><th colspan="2">BT-Tbij</th><th colspan="2">BT</th><th colspan="2">BT-Tbij</th><th colspan="2">BT</th><th colspan="2">BT-Tbij</th><th colspan="2">BT</th></tr><tr><td>NaCl</td><td>3.3</td><td>9.9</td><td>3.5</td><td>11.3</td><td>2.3</td><td>11.1</td><td>2.3</td><td>12.2</td><td>1.1</td><td>4.3</td><td>1.1</td><td>4.6</td><td>1.2</td><td>4.7</td><td>1.3</td><td>4.7</td></tr><tr><td>KCl</td><td>2.4</td><td>11.0</td><td>3.2</td><td>9.5</td><td>3.1</td><td>9.2</td><td>3.1</td><td>8.7</td><td>1.1</td><td>5.5</td><td>1.1</td><td>5.1</td><td>3.8</td><td>9.0</td><td>3.8</td><td>9.0</td></tr><tr><td>RbCl</td><td>2.4</td><td>8.3</td><td>2.4</td><td>8.4</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td></td><th colspan="2">BgT-Tbij</th><th colspan="2">BgT</th><th colspan="2">BgT-Tbij</th><th colspan="2">BgT</th><th colspan="2">BgT-Tbij</th><th colspan="2">BgT</th><th colspan="2">BgT-Tbij</th><th colspan="2">BgT</th></tr><tr><td>NaCl</td><td>3.0</td><td>9.4</td><td>5.4</td><td>19.3</td><td>2.1</td><td>10.5</td><td>2.1</td><td>7.9</td><td>1.0</td><td>3.8</td><td>0.9</td><td>4.0</td><td>1.2</td><td>4.0</td><td>1.2</td><td>4</td></tr><tr><td>KCl</td><td>2.1</td><td>11.1</td><td>5.0</td><td>19.7</td><td>2.8</td><td>6.7</td><td>2.5</td><td>8.4</td><td>1.2</td><td>5.8</td><td>2.1</td><td>9.0</td><td>2.9</td><td>6.5</td><td>2.9</td><td>6.5</td></tr><tr><td>RbCl</td><td>2.1</td><td>6.0</td><td>3.5</td><td>18.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

*For the vapor composition (VLE y): $AD\% = \left| \frac{\sum_{i=1}^{n_{dp}^y} (\mathbf{y}_{i,cal} - \mathbf{y}_{i,exp})}{n_{dp}^y} \right| \ast 100$, MAD\% = $\max(|\mathbf{y}_{i,cal} - \mathbf{y}_{i,exp}|) \ast 100$ and for is MIAC, density or vapor pressure (VLE p):

$$ *AAD\% = \frac{\sum_{i=1}^{n_{dp}^y} \left| \frac{\Psi_{i,cal} - \Psi_{i,exp}}{\Psi_{i,exp}} \right|}{n_{dp}^y} \ast 100, MAD\% = \max \left| \frac{\Psi_{i,cal} - \Psi_{i,exp}}{\Psi_{i,exp}} \right| \ast 100, \Psi \text{ is MIAC, density or vapor pressure (VLE p).} $$

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Comparisons of the MIAC of (water + ethanol + NaCl) calculated using BS-Tbij, BT-Tbij and BgT-bij models at 298 K (straight lines are models result. Circles in black, gray, gold and green are experimental data related to pure water, 20 %, 40 %, and 80 % of ethanol, respectively).</figcaption></figure>

the accuracy of the model and ion-ion association strength in the mixed solvent systems. In order to check the prediction capability of the new models, new properties such as individual ionic activity coefficient (IIAC) and Gibbs energy of transfer ($\Delta G_{tr}$) are investigated.

### 3.1. Comparison of models in the mixed solvent systems

In this section the results from BS, BT and BgT models are compared to those of BS-Tbij, BT-Tbij and BgT-Tbij models for the modeling of chloride salts in mixed solvents systems (water-ethanol and water-methanol) to investigate the impact of the new TIP parameter on the accuracy of the models. The results are presented in Tables 8 and 9.

Tables 8 and 9 show the average absolute percentage deviations

(AADs) and maximum absolute percentage deviations (MADs) for MIAC, bubble pressure (*p*), vapor composition (*y*), and density of models introduced in Table 2, for chloride salts in water-ethanol and water-methanol systems, respectively. As can be seen in Table 8, introducing $b_{ac}$ as a ternary interaction parameter (TIP), systematically improves the accuracy of all three models. This improvement is most noticeable for the MIAC, and it is much more visible for BT-Tbij and BgT-Tbij models. For the other properties (bubble pressure (*p*), vapor composition (*y*), and density) the accuracy of the new models is almost the same as previously (when $b_{ac}$ is a binary parameter). Overall, for the water-ethanol system, the highest accuracy belongs to the BS-Tbij model where the Schreckenberg model is used for the RSP (lowest value of the MAD). Table 9 illustrates the results of the models introduced in Table 2 for the chloride
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Comparisons of the MIAC of (water + methanol + NaCl) calculated using BS-Tbij, BT-Tbij and BgT-bij models at 298 K (straight lines are models result. Circles in black, blue, gray, gold and green are experimental data related to pure water, 40 %, 60 %, 80 % and 100 % methanol, respectively). Data for 100 % methanol were taken from ref. [67] and, they were not included in regression (extrapolation results). The obtained AAD% for 100 % of Methanol using the BS-Tbij, BT-Tbij and BgT-Tbij models is 9.0, 3.3 and 7.2 respectively.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** 4a and 4d show the ratio of the ion-ion association constant ($\Delta^{0}_{AB}$) of BS, BT and BgT models to their corresponding Tbij models in pure ethanol and methanol respectively. 4b and 4e show the ratio of ion-water association constant to that of ion-ethanol and ion-methanol respectively (including both anion and cation). 4c and 4f show the ratio of anion-water association constant to that of cation-water and the ratio of anion-alcohol association constant to that of cation-alcohol.</figcaption></figure>

salts in water-methanol system. Again, a systematic improvement is observed for the three models, which is most noticeable for the MIAC. However, here, the improvement in terms of accuracy is less than for the water-ethanol system. This may be related to the fact that the larger modification in $b_{ac}$ value observed for ethanol than for methanol (Fig 1). Overall, for the water-methanol system, the BgT-Tbij model using a temperature dependent RSP model (Maryott and Smith [61]) presents the best accuracy.

Regarding the MIAC, the results for NaCl in ethanol-water and methanol-water mixtures using the new models, BS-Tbij, BT-Tbij and BgT-Tbij, are shown in Figs. 2 and 3. All new models (BS-Tbij, BT-Tbij and BgT-Tbij) using the new parameter (TIP) are accurate for both pure water and mixed solvent systems. Experimental data for pure methanol were taken from ref. [67] and, they were used to check the extrapolation capability of the new models (i.e. they are not included in the parameter regression). The AAD% for pure methanol using the
<figure><img src="image_7.png" /><figcaption>Fig. 5. 5a compares the obtained dielectric constant using the BS, BT and BgT models with the experimental data [68–70] for pure water from 293 to 573 K. 5b compares the ion-ion association constant obtained from the BS, BT and BgT models with the experimental data from literature [3,71] for sodium chloride in water.</figcaption></figure>

BS-Tbij, BT-Tbij and BgT-Tbij models are 9.0, 3.3 and 7.2 respectively. This contrasts with, using the BS, BT and BgT model (when $b_{ac}$ was a binary parameter) where the AAD% values were 16.43, 3.24, and 3.04 respectively. It shows that the introduction of the ternary parameter also improves the extrapolation capability of the BS-Tbij and BT-Tbij models.

## 3.2. Comparison and impact of the association strengths: solvation and ion pairing

The model provides some tools to evaluate the relative strengths of the various interactions that occur in an electrolyte solution. The analysis below has no intention to prove any physical reality, but rather it illustrates what type of information can be gathered from the model and its specific mode of regression.

Fig. 4a and 4d illustrate the ratios of the infinite dilution ion-ion association constant ($\Delta_{AB}^0$) of Tbij models to their corresponding non-Tbij models in pure ethanol and methanol respectively. It can be seen that for ethanol systems, adding this TIP significantly increases the ion-ion association strength, and more specifically for the BgT-Tbij model (this ratio is more than one in all three Tbij models). In contrast, for methanol systems, this increase is less than that of ethanol system. This was also observed in Fig. 1: the correction has a much smaller impact on methanol than on ethanol solvent (methanol was in fact rather well represented with the binary $b_{ac}$). In fact, in the methanol system this ratio for the BS-Tbij model is less than one which means that adding this Tbij parameter reduced ion-ion association.

Fig. 4b and 4e illustrate the relative solvation strength of ions in water and in alcohol. It is expected to be larger than one. This is indeed

<figure><img src="image_8.png" /><figcaption>Fig. 6. The comparison of the $\Delta G_{ir}^{\text{ion}}$ using the Tbij models introduced in Table 2 with the experimental data (recommended values by Marcus [73,74] are used as experimental data) for $\text{Na}^+$, $\text{K}^+$, and $\text{Cl}^-$ ions in methanol-water and ethanol-water mixtures.</figcaption></figure>
the case for ethanol cosolvent (except for BgT), but none of the models provide the expected trend with methanol as cosolvent.

Fig. 4c and 4f illustrate the relative solvation strengths of anions (Cl⁻) vs cations. Clearly, the anion solvation is always stronger than that of cations, both in water and in alcohol. Only the BS-Tbij model with ethanol is an exception. Note that there are no data in the regression database that make it possible to distinguish anions from cations. The only information that may distinguish them comes from the fact that several cations are investigated at once.

### 3.3. Comparison of the ion-ion association constant with experimental data

Experimental data on ion pairing for the strong salts are not very abundant in literature. In addition, they are only available in aqueous system, as for example in ref [3], where they are obtained from spectroscopic measurements, and in [63] where they are derived from conductivity data. Fig. 5a compares the obtained dielectric constant using the BS, BT and BgT models with the experimental data [68–70] for pure water from 293 to 573 K. In Fig. 5b, the infinite dilution association strengths (Δ₀<sub>AB</sub>) are compared with the experimental data taken for ref [3,71] for NaCl in water. For the BS model the Schreckenberg [72], and for the BT and BgT models the Zuo & Fürst [60] RSP models are used. According to the Fig. 5b, the Zuo & Fürst RSP model extrapolates badly to temperatures beyond 600K in comparison to the Schreckenberg model. The high temperature behaviour is not shown because it extrapolates very badly, and because the focus of this work is on moderate temperatures. As a result, the corresponding infinite dilution association constant of ions (Δ₀<sub>AB</sub>) using the BT and BgT models increases too fast with temperature, in comparison to that using the BS model. In contrast, the Schreckenberg RSP follows the experimental trend much better which can also be observed for the association strength. Note that the association constant at infinite dilution for the BT and BgT models are almost the same: the RSP models are identical. The slight difference lies in the parameterization of the hard sphere diameters that enter into Eq. (14). This comparison shows that the Bjerrum provides a good relationship between the RSP and the ion-pairing constant.

### 3.4. Gibbs energy of transfer (ΔG<sub>tr</sub>)

The Gibbs energy of transfer (ΔG<sub>tr</sub>) refers to the change in Gibbs free energy when a solute, typically an ion, is transferred from one solvent (usually water) to another solvent or solvent mixture. It is defined as the difference in chemical potential of the ion between the mixed solvent reference state and the water reference state:

$$ \Delta G_{tr} = \mu_{ion}^{ref.s} - \mu_{ion}^{ref.w} \qquad (29) $$

This energy change reflects the difference in solvation of the ion between the two solvents and is important in understanding solute behavior, which has an impact on solubility, reaction kinetics, and equilibrium in various solvents. It explains how stable the ion is in the target solvent compared to the reference solvent (usually water). The larger the ΔG<sub>tr</sub> is, the less stable the ions are. The ΔG<sub>tr</sub> is rarely investigated when developing thermodynamic models. Notice that we did not use these data for regressing our model parameters. Yet, we believe that this property deserves more attention, because data exist, and it shows how the ions tend to distribute themselves, in particular when liquid-liquid phase splits exist. It also provides a complementary picture of the ion activity compared to MIAC. In this section, the property is investigated first for the ions and then for the salts.

#### 3.4.1. Comparison of the model results with the data of ΔG<sup>ion</sup><sub>tr</sub>

We have attempted to reproduce the data (reported and interpreted by Marcus [73,74]) using our models. The Gibbs energy of transfer on molarity basis (reported by Marcus [73,74]) is calculated from the equation of state using the following expression.

$$ \Delta G_{tr}^{i} = \mu_{i}^{s} - \mu_{i}^{w} = RT \ln\left(\frac{\varphi_{i}^{s}}{\varphi_{i}^{w}}\right) + RT \ln\left(\frac{v^{s}}{v^{w}}\right) \qquad (30) $$

where the superscripts s and w denoted the solvent (including the alcohol) and pure water respectively. Here, φᵢ is the fugacity coefficient of ion i and v is the molar volume. The calculation is performed at zero mole fraction of ions (in the reference state). The volume comes into the expression because of the molarity-based reference state.

Fig. 6 shows the comparison of the ΔG<sup>ion</sup><sub>tr</sub> using three Tbij models with the data (recommended values by Marcus [73,74] are used as experimental data) for Na⁺, K⁺, and Cl⁻ ions in methanol-water and

<figure><img src="image_9.png" /><figcaption>Fig. 7. Contributions to ln(γ<sub>tr</sub>) of the terms for Na⁺ and Cl⁻ in water-ethanol mixture using the BT-Tbij model at 0.1 MPa and 298 K (Hard chain (HC) in blue, Dispersion (Disp) in orange, Association (Assoc) in gray, Polar in gold, Born in green, MSA in yellow and MAL in dark blue).</figcaption></figure>
ethanol-water mixtures. As can be seen in this figure, for all ions in methanol-water and ethanol-water systems, the $\Delta G_{tr}^{ion}$ (experimental data are dots) increases with the mole fraction of alcohol, indicating that the transfer of ions from water to mixed solvents is less favorable as the alcohol concentration increases (ions are more stable in water than in alcohol). The increase is stronger in ethanol, indicating weaker solvation in ethanol in comparison to methanol. Clearly, none of the models are in full agreement with the data. This was expected because the models were not trained on such data. Yet, we observe that the $\Delta G_{tr}^{ion}$ in alcohol is always positive. The behavior for all models using methanol solvent is qualitatively acceptable (except for the curvature). For ethanol, one can notice that while the cation $\Delta G_{tr}$ shows a minimum, that of the anion (Cl⁻) overshoots the experimental trend. For further investigations the contribution of the terms to the property will be plotted and discussed in the next section.

### 3.4.2. Model Contributions of to the Gibbs energy of transfer

In order to analyze the results of the model, we evaluate here what are the contributions that are the most important for the property of interest. As shown in Eq. (30), the Gibbs energy of transfer is related to the ratio of fugacities (or fugacity coefficients) in the reference state of mixed solvent (s) and pure water (w). Considering that the fugacity coefficients are calculated as:

$$RT\ln(\phi_t) = \sum \left( \frac{\partial A^X}{\partial n_{ion}} \right) - RT\ln Z \quad (31)$$

where the sum (X) points to the different terms of Eq. (1), and $Z = Pv/RT$. The ion's Gibbs energy of transfer is then:

$$\Delta G_{tr}^{ion} = \sum \left( \frac{\partial A^{s \times}}{\partial n_{ion}} - \frac{\partial A^{w \times}}{\partial n_{ion}} \right) - RT\ln \frac{Z^s}{Z^w} + RT\ln \left( \frac{v^s}{v^w} \right) \quad (32)$$

where X refers to the different contributions from the EoS, as introduced in Eq. (1). Pressure and temperature are identical in both solvent and water reference states, which means that the volume terms then cancel out and the activity coefficient of transfer is obtained as follows:

$$\ln(\gamma_{tr}) = \frac{\Delta G_{tr}^{ion}}{RT} = \frac{1}{RT} \sum \left( \frac{\partial A^{s \times}}{\partial n_{ion}} - \frac{\partial A^{w \times}}{\partial n_{ion}} \right) \quad (33)$$

As mentioned earlier, the behavior of all models for methanol solvent is qualitatively acceptable. However, for ethanol, a minimum was

observed that deviates significantly from the data. Therefore, the contributions to the Gibbs energy of transfer for Na⁺ and Cl⁻ in water-ethanol mixtures were investigated to identify the cause of this large deviation. It should be noted that the BT-Tbij, BgT-Tbij models show similar trends and accuracy levels. Therefore, only the BT-Tbij model is used to plot the contributions of terms for Na⁺ and Cl⁻ in the water-ethanol mixture.

Fig. 7 shows the contributions to ln($\gamma_{tr}$) of the EoS terms for Na⁺ and Cl⁻ in ethanol solvent, for the model BT-Tbij. One may notice at first that several contributions are exactly zero (MAL) or very small. This is because the calculation is performed at zero salt content. In contrast, the most important contributions are Born, association and HC. The “Born” term is expected to help describing the Gibbs energy of transfer, because it is specifically designed for this purpose. Yet, the results show that it is not sufficient, which is in accordance with the observations of Simonin [37,38]. The next most important contribution is association, which describes solvation forces. The hard chain (that represents entropic effects) results here in a negative contribution.

For Na⁺, the association term shows a minimum, which points to a higher solvation for the mixed solvent rather than either of the pure solvents (a phenomenon that was observed by Marcus for some ions). It is strikingly different from the behavior of Cl⁻. The reason may possibly be found in the very strong difference in solvation strength between the two ions, as was shown in Fig. 6c: The association strength of Cl⁻ with either of the two solvents is close to 100 times larger than that of Na⁺ which is related to ion-solvent and ion-cosolvent associations. The position of the endpoints of the association contribution is related to the ratio of association strength in water vs in alcohol. This ratio is made visible according to Fig. 6b: Cl⁻-water association strength is much stronger than that of Cl⁻-ethanol. Therefore, Cl⁻ requires a lot of energy to be transferred into ethanol resulting in a large value for its contribution to the Gibbs energy of transfer. On the other hand, Na⁺-water association strength is much weaker than that of Cl⁻-water. Therefore, Na⁺ requires less energy to be transferred into ethanol and it is more solvated than it should.

To address this issue (minimum of the association term), a larger association contribution (in value) is needed, which requires a stronger cation-water or weaker anion-ethanol association strength. This is because as the anion-ethanol association strength decreases (or cation-water association strength increases), more energy is required to transfer ions into ethanol (increasing the association contribution in

<figure><img src="image_10.png" /><figcaption>Fig. 8. The comparison of the $\Delta G_{tr}$ using the models introduced in Table 2 and BT-Tbij (including the $\Delta G_{tr}$ data for regression) with data (Average values of experimental data plotted in Fig. 9) for Na⁺ and Cl⁻ in ethanol-water.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 9. The comparison of the $\Delta G_{tr}^{salt}$ using the three Tbij models introduced with the experimental data (Sum of the average values of data in Fig. 6) for Na⁺, K⁺, and Cl⁻ ions in methanol-water and ethanol-water mixtures.</figcaption></figure>

value) which will eliminate the minimum in the association term.

This points to a wrong balance of anion vs cation solvation behavior in the model. This balance was made visible in Fig. 6, and it was mentioned that no specific information regarding this balance was introduced in the parameterization.

### 3.4.3. Model parameterization using the Gibbs energy of transfer experimental data

As was concluded in the previous section, in order to show the correct trend for association contribution and increase the model accuracy, an improved balance between the anion and cation solvation energies should be established.

A new parameterization test was performed on the BT-Tbij model with the water-ethanol mixed solvent system. This time the $\Delta G_{tr}^{ion}$ data are included in the parameterization procedure (recommended values of experimental data by Marcus [73,74]). The results are presented in Fig. 8 for the case studies of Na⁺ and Cl⁻ in ethanol-water system. As can be seen, including the $\Delta G_{tr}^{ion}$ data in the parametrization approach significantly improves the model accuracy for both Na⁺ and Cl⁻ in water-ethanol system (however, the model accuracy for the MIAC and VLE reduces (results are presented in supplementary document). For Na⁺ the minimum is less visible. We can also notice that the obtained values for the $w_{Na-ethanol}$ (see Eq. 27) is close to 1 which means the Na⁺-ethanol association strength is zero. This leads to a significant increase in the $\Delta G_{tr}^{ion}$ of Na⁺ and improves the model accuracy. However, even including the $\Delta G_{tr}$ data was not efficient enough, and the model still shows a large deviation with reference data. On the other hand, for the Cl⁻ the $w_{Cl-ethanol}$ was found to be negative ($w_{Cl-ethanol} = -0.1$). This means Cl⁻-ethanol association strength increases which leads to a

decrease of the Gibbs energy of transfer for this anion.

### 3.4.4. Salt Gibbs energy of transfer ($\Delta G_{tr}^{salt}$)

So far, the Gibbs energy of transfer of ions were investigated. Yet, experimentally, we know that only salt properties are obtained. For example, if the solubility product, $K_{SP}$ is compared in pure water and in the mixed solvent, one finds:

$$ \Delta G_{tr}^{salt} = \Delta G_{tr}^{anion} + \Delta G_{tr}^{cation} = RT\ln\left(\frac{K_{SP}^{S}}{K_{SP}^{w}}\right) \quad (34) $$

Salt Gibbs energy of transfer is equal to the sum of the Gibbs free energy of transfer of the respective ions. If the Gibbs free energy of transfer of one ion is known, then with the respective experimental $K_{SP}$ values for the salt, the value for the other ion might be calculated.

In Fig. 9, the three Tbij models result were compared with the data for the $\Delta G_{tr}^{salt}$ for NaCl and KCl salts in water-ethanol and water-methanol systems. It should be noted that the average values of experimental data plotted in Fig. 6 for anion and cation were summed (see Eq. 34) and therefore the $\Delta G_{tr}^{salt}$ is obtained and compared with the model results. As can be seen in the figure, the sum of ions Gibbs energy of transfer obtained from the models are in a good agreement we the data without any regression. All the models agree qualitatively with the data.

### 3.5. Analysis of ionic activity coefficients

What is generally called the ionic activity coefficient is related to the difference between two ion chemical potentials, but in contrast to Gibbs
<figure><img src="image_12.png" /><figcaption>Fig. 10. Contributions to lnγ± of the terms for sodium chloride in pure ethanol using the models introduced in Table 2 at 0.1 MPa and 298 K (Hard chain (HC) in blue, Dispersion (Disp) in orange, Association (Assoc) in gray, Polar in gold, Born in green, MSA in yellow and MAL in dark blue).</figcaption></figure>

energies of transfer, one refers to the true chemical potential, while the other refers to that of the ion in its solvent reference state:

$$ \ln(\gamma_{ion}) = \frac{\mu_{ion} - \mu_{ion}^{ref.s}}{RT} \qquad (35) $$

This is the individual ion activity coefficient (IIAC). In the same way as for the Gibbs energies of transfer, the individual property cannot be measured without extra-thermodynamic assumptions. In practice, it is its mean value that is measured experimentally (MIAC), which is defined as (using the molality reference state for a symmetric salt):

$$ \mathrm{MIAC} = \sqrt{\gamma_{cation}\gamma_{anion}} \qquad (36) $$

In this section, both MIAC and IIAC are discussed. Regarding the first one, the objective is to compare with the former work [8]; the second is a new analysis that brings additional understanding on the model behavior.

### 3.5.1. Contributions to MIAC

Fig. 10 shows the contributions to lnγ± (mole fraction based), as calculated in the same way as proposed in [8] for sodium chloride in pure ethanol, calculated using the 6 models introduced in Table 2 at 0.1
<figure><img src="image_13.png" /><figcaption>Fig. 11. Comparison of the IIAC of the aqueous NaCl and KCl solutions calculated using BS, BT and BgT ods with the experimental data from ref [80,81].</figcaption></figure>

MPa and 298 K. The $x_{\text{ion}}$ range exceeds the solubility of NaCl in pure ethanol and is presented here for illustration of the contribution trends. We focus on the analysis of the association and MAL contributions, which are the contributions that have most impact and have changed the most by including the solvent-specific $b_{C-A}$. In the three Tbij models, the MAL contributions are larger in magnitude compared to the corresponding BS, BT, and BgT models, demonstrating that including the solvent-specific $b_{C-A}$ increases the ion-pairing association at high alcohol content (as demonstrated here in pure alcohol). However, in the BS-Tbij model, the association and MAL contributions are larger in magnitudes compared to the BS, BT, and BgT models. The increased magnitudes of the positive association contribution and the negative MAL contribution cancel each other out. Consequently, the total $\ln\gamma_{\pm}$ is very close for the BS-Tbij and BS models. On the other hand, in the BT-Tbij and BgT-Tbij models, the positive association contribution remains almost unchanged compared to the BT and BgT models, while the increased magnitude of the negative MAL contribution decreases the total $\ln\gamma_{\pm}$. Thus, these models agree better with the experimental MIAC data at high ethanol composition. Furthermore, the curvature of the MAL contribution in the Tbij models is stronger compared to the BS, BT, and BgT models, and resembles the curvature of the MSA contribution. The alcohol composition dependence of the MAL contribution is much stronger compared to that of the MSA contribution. This behavior of the MAL contribution is exactly the reason why the new approach (Tbij) improves the representation of MIAC without using an effective RSP as in the work by Yang et al. [12,13]. The MAL contribution in the BgT model is larger in magnitude compared to the BS and BT models. This strong dependence of the MAL contribution on alcohol composition by introducing $g^{el}$ significantly improves the MIAC result at high alcohol composition.

### 3.5.2. Individual ion activity coefficient (IIAC)

In addition to the mean properties, individual ionic activity coefficient (IIAC) is an important property that can be used for further testing the model. IIAC is important, because it appears in the Nernst equation for the electrode potential in a half-cell [75]. Although there is some debate about the validity of the experimental measurement of the activity coefficient of individual ions [76–78], recent Monte-Carlo simulations [79] qualitatively agree with the experimental data. Therefore, IIAC could be used as a qualitative test. Experimental [80,81] and simulation investigations show that the relative magnitudes of the cation and anion IIAC for the aqueous NaCl and KCl solutions are opposite, as shown on Fig. 11. Considering that ion pairing affects both ions equally, it may be asserted that the larger value of the cation

activity coefficient compared with the anion in NaCl results from a strong solvation (ion-water interaction strength) of Na+, while in KCl, the solvation of the cation $K^+$ is smaller than that of the anion Cl. Fig. 11, compares the IIAC of the aqueous NaCl and KCl solutions calculated using BS, BT and BgT models with the experimental data from Refs. [80,81]. This property is not included in the regression. Thus, these results are pure predictions. Only the BgT model correctly predicts the opposite relative magnitudes of the cation and anion IIAC of the aqueous NaCl and KCl solutions, while the BT and BS fail.

## 4. Conclusion

This work has focused on improving the SBiMSA model to provide a more accurate description of ion-ion and ion-solvent associations in mixed solvent electrolyte systems. It is then used to analyze the impact of solvation and ion-pairing on new properties that are Gibbs energies of transfer and individual ion activity coefficients. The key points to remember are:

* Ion-ion association (formation of ion pairs) is crucial in systems where the dielectric constant of the medium is altered by the presence of co-solvents such as alcohols. This type of association affects the thermodynamic properties of the solution such as MIAC. The correction that was added to the model (introducing a solvent specific $b_{ac}$) in this work provides a more accurate calculation of ion-ion association strength that leads to a significant improvement on the accuracy of the model for calculation of MIAC in ethanol.

* Ion-solvent association (solvation effects) are equally critical as they determine how ions interact with the solvent molecules, influencing the overall stability and properties of the electrolyte solution: it has been shown that, when investigating the Gibbs energies of transfer, the correct description of these interactions is essential.

* The Born term is not sufficient to describe ion-solvent interactions, which is the reason why it is generally used. We have shown that its impact depends on the property investigated, but in all cases it needs to be complemented by a physical (non-electrostatic) contribution, which, in our case, is the ion-solvent association. Using our approach, the Born term has a minor impact on the MIAC, but its role on the Gibbs energy of transfer is key.

Overall, many models were compared and discussed in this work and the previous one [8]. To designate the best approach for the properties and the systems that were investigated, we split the models results for aqueous and mixed solvent systems as follows:
a. Aqueous systems: The three models (BS, BT and BgT) have the same level of accuracy (BS model is slightly worst) for the MIAC, VLE and density. For the estimation of association strength, it was found that BS model is in a better agreement with experimental data. Because the Schreckenberg RSP model is in fact more accurate for the estimation of water dielectric constant. For the IIAC, among the BS, BT and BgT models only the BgT model has the correct trend with the experimental data.

b. For the mixed-solvent system: All Tbij models have the same level of accuracy for the water–methanol systems (BgT-Tbij model is slightly better). For the water-ethanol system, BgT model with or without the TIP is the least accurate for MIAC, VLE and density. For the ions Gibbs energy of transfer (ΔG ion TB ), BS (with and without the TIP) model is least accurate. However, for the salt Gibbs energy of transfer (ΔG salt TB ), all the models are in a good agreement with the experimental data.

No final unambiguous selection of the best model is possible at this stage: they have to be evaluated on different systems and properties, which will be the topic for further work.

CRediT authorship contribution statement

**Abtin Raeispour Shirazi:** Writing – review & editing, Writing – original draft, Software, Methodology, Investigation. **Fufang Yang:** Methodology, Investigation. **Tri Dat Ngo:** Visualization, Supervision, Software. **Olivier Bernard:** Visualization, Validation, Supervision, Formal analysis, Conceptualization. **Jean-Pierre Simonin:** Visualization, Validation, Supervision, Formal analysis, Conceptualization. **Jean-Charles de Hemptinne:** Validation, Supervision, Software, Methodology, Investigation, Funding acquisition, Conceptualization.

Declaration of competing interest

The authors declare the following financial interests/personal re-
lationships which may be considered as potential competing interests:

Jean-Charles de Hemptinne reports financial support was provided by IFP Energies nouvelles. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Financial support from electrolyte thermodynamic (Elether) chair is
gratefully acknowledged.

Supplementary materials

Supplementary material associated with this article can be found, in
the online version, at doi:10.1016/j.fluid.2025.114396.

Data availability

The authors do not have permission to share data.

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