(574a) A Unified Model for the Thermodynamic, Transport, and Physical Properties of Associative Electrolyte Solutions

Electrolyte solutions are ubiquitous in chemical engineering applications, ranging from energy storage devices to separation processes. Modeling the thermodynamic, electrical, and dielectric properties of electrolyte solutions is challenging due to the complex interactions between charged and neutral species in the solution. In this study, we present a comprehensive modeling approach using the revised version of the electrolyte cubic plus association (eCPA-Revised) equation of state for thermodynamic properties, Naseri Boroujeni et al.¹ model based on the Debye-Hückel-Onsager (DHO) theory for the electrical conductivity, and the modified Maribo-Mogensen et al.² model based on the Kirkwood theory for the static permittivity. The modeling approach presented in this study has the ability to predict the properties of electrolyte solutions across a wide range of ion-ion association strengths, from very weak to very strong.

The eCPA-Revised equation of state (eq. ( 1 )) is a perturbation theory-based approach that considers both charged and neutral species in the solution. The perturbation terms account for the long-range electrostatic interactions between charged species and the short-range interactions between all species in the solution. This model has been extended to account for the association between ions in the solution using the Binding Debye-Hückel theory (BiDH)³. BiDH is a self-consistent mean-field theory developed based on the Wertheim two-density formulation, reference cavity approximation, and the Debye-Hückel equation that accounts for ion-ion interactions and association in the electrolyte solutions. The mean ionic activity coefficient⁴, osmotic coefficient, and volumetric properties are among the thermodynamic properties of electrolyte solutions that can be predicted by this equation of state across a broad range of temperature, pressure, and composition of the system. In addition, the fraction of unbounded ions can be predicted by this equation of state which is useful for the prediction of electrical conductivity and dielectric constant of associative electrolyte solutions.

A^r/Nk_BT=(A^SRK+A^Assoc+A^BiDH+A^Born)/Nk_BT (1)

The decrement of the relative static permittivity of the electrolyte solutions with increasing ionic strength caused from dielectric saturation has been modeled by Maribo-Mogensen et al.². They considered the cancellation of oppositely directed dipolar molecules around the ions because of electro-restriction (ion-solvent association) and extended their previously developed model for the static permittivity of dipolar molecules to mixtures containing salts and dipolar molecules (eq. ( 2 )). This equation has been used in the long-range part of the eCPA-Revised equation. And, for the Kirkwood g-factor, the association part of the eCPA-Revised has been employed. Thus, an iterative approach should be considered.

(2ε_r+ε_∞)(ε_r-ε_∞)/ε_r=(ε_∞+2)²(N_A/9ε₀k_BT)∑_i(ρ_ig_iΘ_iμ_i,0) (2)

The relative static permittivity of electrolyte solutions is not always decreasing. The presence of ion pairs with a high dipole moment results in reduction of the decrement of electrolyte solutions. In highly associative electrolyte solutions, such as MgSO₄-water, the static permittivity increases up to a maximum and then decreases. For these solutions, the contribution to the static permittivity of the solution has been divided into contribution from dipolar molecules and contribution from dipolar ion pairs. Although eq. ( 2 ) is perfectly capable of predicting the static permittivity from dipolar molecules, it cannot capture the contribution from the dipolar ion pairs.

As a result, we have extended this model for associative electrolyte solutions using the fraction of ion pairs from eCPA-Revised and an average dipole moment for contact, solvent-shared, and solvent-separated ion pairs.

For the electrical conductivity of electrolyte solutions, we have developed a model¹ based on the solution of the Fuoss-Onsager continuity equation and Debye-Hückel theory assuming full dissociation of salts in the solution. In this model, the ionic conductivity can be predicted as a reduction of ionic conductivity at infinite dilution (λ_i⁰) (no ionic screening) due to the relaxation (δk_i/k_i) and electrophoretic (δυ_i/Ï…_i) effects (eq. ( 3 )).

λ_i = λ_i⁰ (1 - δk_i/k_i) (1 - δυ_i/Ï…_i) (3)

In this model, the effect of ion-solvent interactions has been considered by employing a concentration-dependent dielectric constant predicted from eq. ( 2 ) as suggested by Naseri Boroujeni et al.^5,6. Furthermore, ion-ion association significantly affects the electrical conductivity of electrolyte solutions since ion-pairs does not carry a charge (in case of symmetrical electrolytes) or carry a reduced charge (in case of asymmetrical electrolytes). As a result, the effect of ion-ion association should be considered in the prediction of the electrical conductivity. In this study, we have used the fraction of unbounded ions to calculate the number density of free ions in the solution. Then, the ionic conductivity of free ions (λ'_i) has been calculated using eq. ( 3 ). Finally, the molar conductivity of the electrolyte solution has been calculated using eq. ( 4 ).

Λ = ∑_i (ν_i α_i |Z_i| λ'_i) (4)

We have used this unified framework to investigate the thermodynamic (mean ionic activity coefficient, density, and osmotic coefficient), transport (electrical conductivity), and physical (relative static permittivity) properties of electrolyte solutions with varying degrees of ion-ion association strengths. Our results indicate that this modeling framework is highly adept at capturing the underlying physics of electrolyte solutions, thereby enabling the prediction rather than the mere correlation of their properties.

In conclusion, our modeling approach provides a comprehensive understanding of the thermodynamic, electrical, and dielectric properties of electrolyte solutions, taking into account the extent of ion-ion association. The combination of the electrolyte CPA equation of state^7–10, Debye-Hückel-Onsager theory¹, and Kirkwood theory¹¹ enables us to predict the properties of electrolyte solutions in a wide range of industrial and environmental applications. The results demonstrate the importance of considering the association between ions in the solution when modeling the properties of electrolyte solutions. This approach has significant implications for designing and optimizing chemical processes involving electrolyte solutions.

Acknowledgment

We thank the ERC Advanced Grant Project No. 832460.

References

1 S. Naseri Boroujeni, B. Maribo-Mogensen, X. Liang and G. M. Kontogeorgis, Physical Chemistry Chemical Physics (submitted).

2 B. Maribo-Mogensen, G. M. Kontogeorgis and K. Thomsen, Journal of Physical Chemistry B, 2013, 117, 10523–10533.

3 S. Naseri Boroujeni, B. Maribo-Mogensen, X. Liang and G. M. Kontogeorgis, unpublished work.

4 S. Naseri Boroujeni, B. Maribo-Mogensen, X. Liang and G. M. Kontogeorgis, Journal of Molecular Liquids (submitted).

5 S. Naseri Boroujeni, B. Maribo-Mogensen, X. Liang and G. M. Kontogeorgis, Fluid Phase Equilib, 2023, 567, 113698.

6 S. Naseri Boroujeni, X. Liang, B. Maribo-Mogensen and G. M. Kontogeorgis, Ind Eng Chem Res, 2022, 61, 3168–3185.

7 G. M. Kontogeorgis and G. K. Folas, Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories, 2009.

8 A. Schlaikjer, K. Thomsen and G. M. Kontogeorgis, Fluid Phase Equilib, 2018, 470, 176–187.

9 B. Maribo-Mogensen, K. Thomsen and G. M. Kontogeorgis, AIChE Journal, 2015, 61, 2933–2950.

10 G. M. Kontogeorgis, A. Schlaikjer, M. D. Olsen, B. Maribo-Mogensen, K. Thomsen, N. von Solms and X. Liang, Int J Thermophys, 2022, 43.

11 B. Maribo-Mogensen, Development of an Electrolyte CPA Equation of state for Applications in the Petroleum and Chemical Industries, 2014.