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Minerals 2022, 12, 190 4 of 12 KCA = [C+](aq)[A−](aq) [CA](aq) KD,N = [N](aq) [N](org) (4) (5) (6) (7) [MAxN2] [C+]x (aq) (aq) Kf = KD,C = [MAxN2](org) [M]x+ [A−]x [N]2 (aq) (aq) (aq) [MAxN2](aq) Based on the previous equations of local thermodynamic equilibrium, a global constant (Kglobal,i) of each component “i” (“i” can be lithium, sodium, or magnesium) summarizes the global extraction process. Kglobal,i = KDx ,CAKCx AK2D,NKf KD,C (8) The algebraic arrangement of Equation (9) results in the expression for Kglobal,i is as follows. [MAxN2] [C+]x (org) (aq) (9) [CA]x [N]2 Kglobal,i = Finally, predicted selectivity is obtained by dividing the global constants of lithium vs [Mx+] the other metal in the following manner: SLi+/Mx+ = Kglobal,Li (10) Kglobal,Mx+ Equation (10) should be compared with experimentally obtained selectivity to validate this model. In order to validate theoretical calculations and compare them with experimental data, experimental data were obtained from previous studies. To compare lithium extraction using ammonium-based IL, data were taken from the work of Zhao et al. [23], in which different cations and anions were used. Finally, to validate the Li+/Mx+ selectivity trends, data were retrieved from Wang et al. using bis[(trifluoromethyl)sulfonyl]imide-based ILs [22]. 2.2. Computation Procedure The molecules of Equations (3)–(7) have to be optimized before being used in COSMO- RS. Molecular optimizations were carried out in Turbomole® (version 7.4., TURBOMOLE GmbH, Karlsruhe, Germany) using a basis set with triple-zeta valence polarization (def- TZVP) [38]. Density functional theory (DFT) with DGA functionality, Becke–Perdew (BP86), is widely used in organometallic complex optimization [39–42] with dispersion correction (D4), as suggested by Grimme et al. [43,44]. After geometry optimization, a vibrational frequencies analysis was performed in Turbomole® for every molecule in order to obtain the chemical potential and the absence of imaginary harmonic frequencies and ensure that all were stable molecules with zero negative frequencies. Subsequently, COSMO-RS calculations were conducted with these molecules using the COSMOThermX® (version C30_1705, COSMOlogic GmbH & Co., Leverkusen, Germany) software. In order to improve COSMO-RS calculations, the conformers of the ILs were consid- ered searching rotatable bonds. To this end, a stochastic search algorithm at the MMX force field level of theory, which is implemented in the PCmodel software [45]. Only molecules with the lowest energy and not exceeding an energy threshold by more than 0.5 kcal/mol were reoptimized in Turbomole®, and then the molecules with the lowest energy (potential chemical in vacuum + solvation free energy) were employed. (aq) (org) (org)PDF Image | Ionic Liquids for the Selective Solvent Extraction of Lithium
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