The standard Gibbs energy proposed by the HMW model (Harvie et al., 1984) and the corresponding solubility constant for the reaction: 2   Mg 2 + + Cl − + 7   H 2 O   ⇌   Mg 2 Cl OH 3 ∙ 4   H 2 O s + 3   H + (2) lg K S , 3 − 1 − 8 ° = − 26.03 is already contained in the THEREDA model and is within the error range, with the overall mean value lg K _{S,3-1-8,25°C} = −26.1 ± 0.2 at 25°C given in Table 2. From this value, ∆_R G _m ° = 149.0 ± 1.1 kJ mol^-1 is calculated. To fix the value within the error range, the invariant point (IP) of Mg(OH)₂(s) and 3-1-8 phase in the Na-Mg-Cl-OH-H₂O system (at NaCl saturation) experimentally determined in Part I was used (none is available from the ternary Mg-Cl-OH-H₂O system). To reproduce this IP at 0.56 ± 0.1 m _MgCl2 by the model, ∆_R G _m °_,25°C narrows to 148.7 ± 0.1 kJ mol^-1, which corresponds to lg Ks = −26.046 ± 0.011 (Figure 3).

No experimental IP of the Mg(OH)₂(s) and 3-1-8 phase is available to refine the ∆_R G _m °_,40°C (148.7 ± 1.2 kJ mol^-1) calculated from the solubility constant at 40 °C (lg K _{S,3-1-8,40°C} = −24.8 ± 0.2). However, according to the solubility data for Mg(OH)₂ and the 3-1-8 phase, the IP is expected to be between 1.6 and 2.0–2.5 m _MgCl2 in the Mg-Cl-OH-H₂O system and between 1.0 and 1.5 m _MgCl2 in the NaCl-saturated solutions (Na-Mg-Cl-OH-H₂O system) at 40°C. To ensure that the model does not slip below 1.0 m _MgCl2 at NaCl saturation and remains in the IP range for the Mg-Cl-OH-H₂O system, ∆_R G _m °_,40°C cannot exceed 147.7 kJ mol^-1, which corresponds to the lower error limit resulting from K _{S,3-1-8,40°C} (Figure 3).

At 60°C, only solubility data in the Mg-Cl-OH-H₂O system are available to fix ∆_R G _m °_,60°C calculated from the mean value of the solubility constant lg K _{S,3-1-8,60°C} = −23.04 ± 0.11 (Table 2) resulting in 147.0 ± 0.7 kJ mol^-1. The IP of Mg(OH)₂(s) and the 3-1-8 phase is expected between 2.6 and 3.0 m _MgCl2, which limits ∆_R G _m °_,60°C to 146.3 ± 0.2 kJ mol^-1.

At 80 °C, no solubility constant is available, so the IP of Mg(OH)₂(s) and the 3-1-8 phase in the Mg-Cl-OH-H₂O system is expected to be between 3.2 and 4.0. In addition, the experimentally determined IP of the 3-1-8 and 2-1-4 phases was found at 5.8 m _MgCl2. This concentration should be reproduced by the model with ±0.1 m _MgCl2 alongside the expected IP range of the Mg(OH)₂(s) and 3-1-8 phase, constraining ∆_R G _m °_,80°C to 144.5 ± 0.2 kJ mol^-1 (re-calculating lg K _{S,3-1-8,80°C} = −21.369 ± 0.011).

Using all the determined values for ∆ _R G _m °_,3-1-8 for the different temperatures (open squares in Figure 3), the THEREDA-conforming temperature function Δ R G m ° = A + B ∙ T + C ∙ T ∙ ln T + D ∙ T 2 + E ∙ T 3 + F T (3) was fitted and extrapolated to 100°C. According to the experimental data at 100°C, the 3-1-8 phase is replaced by the 9-1-4 phase (Pannach et al., 2017). In order to model the metastable range, the ∆ _R G _m ° obtained by extrapolation had to be slightly shifted to ∆_R G _m °_,100°C = 142.8 kJ mol^-1. The finally fitted ∆ _R G _m °_{,3-1-8 (T)} function (Table 3) calculates ∆_R G _m °_,100°C = 142.59 kJ mol^-1 and is shown in Figure 3 together with all single values at the different temperatures.

The Sorel phase 2-1-4 is stable from 60 °C and was found in the Mg-Cl-OH-H₂O system at higher MgCl₂ solution concentrations during the investigations of Pannach et al. (2017). The solubility constant according to the formation reaction in Eq. 4 was determined in Part I of this work at 60 °C (Table 2). 3   Mg 2 + + 2   Cl − + 8   H 2 O   ⇌   Mg 3 Cl 2 OH 4 ∙ 4   H 2 O s + 4   H + (4)

Analogous to the evaluation of the ∆_R G _m ° function of the 3-1-8 phase, further values of ∆_R G _m ° for the 2-1-4 phase were derived from the available solubility data up to 120°C.

The solubility constant lg K _{S,2-1-4,60°C} = −32.95 ± 0.20 (Table 2) corresponds to ∆_R G _m °_,2-1-4,60°C = 210.2 ± 1.3 kJ mol^-1. From the solubility data at 60°C, the IP of the 3-1-8 and 2-1-4 phases is expected to be slightly above 5.4 m _MgCl2 (between 5.4 and 5.7 m _MgCl2). To represent this by the model, the ∆_R G _m °_,2-1-4,60°C needs to be constrained to 211.2 ± 0.1 kJ mol^-1. From the solubility data at 80°C, the experimentally determined IP of the 3-1-8 and 2-1-4 phases in 5.8 m _MgCl2 is available. For a model reproducing the IP with ±0.1 m _MgCl2, ∆_R G _m °_,2-1-4,80°C needs to be 209.5 ± 0.1 kJ mol^-1.

At 100°C, in addition to the 2-1-4 phase, the 9-1-4 phase occurs as a stable Sorel phase in the Mg-Cl-OH-H₂O system (Pannach et al., 2017). The estimation of the crossing point from the trend of the solubility data of the 9-1-4 and 2-1-4 phases at 100°C suggests the IP at 5.4 ± 0.2 m _MgCl2. To represent it by the model, a value of ∆_R G _m °_{,2-1-4,100°C} = 205.6 ± 0.1 kJ mol^-1 needs to be applied.

At 120°C, the IP of the 9-1-4 and 2-1-4 phases was determined at 5.8 m _MgCl2 (Pannach et al., 2017) and adjusts ∆_R G _m °_{,2-1-4,120°C} = 200.86 ± 0.1 kJ mol^-1.

Using all the values determined at different temperatures, a ∆_R G _m °_,2-1-4 temperature function according to Eq. 3 was generated (Figure 4; Table 3).

The 9-1-4 phase (9Mg(OH)₂∙MgCl₂∙4H₂O = 2 Mg₅Cl(OH)₉∙2H₂O) occurs above 80°C. Solubility data for the 9-1-4 phase are available at 100°C and 120°C, including IPs with Mg(OH)₂(s) for both temperatures, as well as with the 2-1-4 phase at 120°C. However, unlike the 3-1-8 or 2-1-4 phases, there are no solubility constants available. For an initial estimation of lg K _S,9-1-4 values, the IPs with Mg(OH)₂(s) were used, as the 9-1-4 phase formation according to reaction (5) is in equilibrium with Mg(OH)₂(s) at the same MgCl₂ concentration (at 100°C: 4.04 m _MgCl2 and 120°C: 4.00 m _MgCl2 (Pannach et al., 2017)). The solubility constants were calculated according to Eq. 6 using the activity coefficients γ _i of Mg²⁺ and Cl⁻ and the activities for water (a _W), and H⁺ (a _H+ = m _H+ · γ _H+ ) calculated by THEREDA for these MgCl₂ concentrations at 100°C and 120°C. This results in lg K _{S,9-1-4, 100°C} = −58.2 and lg K _{S,9-1-4,120°C} = −53.9, which correspond to ∆_R G _m °_,9-1-4,100° = 415.5 kJ/mol and ∆_R G _m °_,9-1-4,120° = 405.9 kJ/mol according to Eq. 1. 5   Mg 2 + + Cl − + 11   H 2 O   ⇌   Mg 5 Cl OH 9 ∙ 2 H 2 O s + 9   H + (5) lg K S , 9 − 1 − 4 ° = lg m H + 9 m M g 2 + 5 ∙ m C l − ∙ γ H + 9 γ M g 2 + 5 ∙ γ C l − ∙ 1 a W 11 . (6)

Both “initial values” were adjusted to the position of the IPs of Mg(OH)₂(s) and the 9-1-4 phase in 4.0 molal MgCl₂ solution at 100°C and 120°C respectively, and to the IP of 9-1-4 and 2-1-4 phase at 120°C in 5.8 m _MgCl2 (Pannach et al., 2017). The final Gibbs energies are ∆_R G _m °_{,9-1-4,100°C} = 415.60 kJ mol^-1 and ∆_R G _m °_{,9-1-4,120°C} = 406.00 kJ mol^-1, respectively, and agree well with the previously calculated values. A linear temperature function was generated for the small temperature range of 100°C–120°C (Table 3).

The 5-1-8 phase forms from OH⁻ supersaturated MgCl₂ or NaCl-MgCl₂ solutions before the stable 3-1-8 phase crystallizes (Pannach et al., 2017; 2023). Within the time window of metastable existence, the lg K _S,5-1-8 according to Eq. 7 could be determined at 25°C from pH_m measurements in the Na-Mg-Cl-OH-H₂O system by Xiong et al. (2010) and in the Mg-Cl-OH-H₂O system in Part I of this work (Pannach et al., 2023). The mean values of both, which agree within the error range (see Table 2), were implemented in the THEREDA model as ∆_R G _m °_,5-1-8,25°C calculated according to Eq. 1; (Table 3). 3   Mg 2 + + Cl − + 9   H 2 O   ⇌   Mg 3 Cl OH 5 ∙ 4   H 2 O s + 5   H + (7)

To improve the model, attempts were made to adjust Pitzer coefficients that affect the OH⁻ solution concentrations in MgCl₂ and NaCl-MgCl₂ solutions. According to the HMW model, THEREDA already contains the binary coefficients β⁽⁰⁾ and β⁽¹⁾ between MgOH⁺ and Cl⁻, and the mixing coefficient ψ, between MgOH⁺, Mg²⁺, and Cl⁻ at 25°C (Table 1). Only these act in the two systems Mg-Cl-OH-H₂O and Na-Mg-Cl-OH-H₂O and are not relevant for the well-described binary Mg(OH)₂-H₂O system; also there are no other complex Mg, Cl, and OH-containing systems for which experimental data are available, so adjustments were made. Based on the solubility data in the Mg-Cl-OH-H₂O system at 25°C (Pannach et al., 2017), systematic fit tests were performed. Both β⁽⁰⁾ alone and with β⁽¹⁾, as well as with and without ψ_{MgOH⁺-Mg²⁺-Cl⁻} (and also ψ alone), were fitted. Furthermore, the parameters C^ϕ _{MgOH⁺-Cl⁻} and ψ_{MgOH⁺-OH⁻-Cl⁻} not included in the dataset were also tested. However, no results could be obtained that sufficiently described the analyzed total OH⁻ concentrations. When fitting the data at higher temperatures (available up to 120°C for the Mg-Cl-OH-H₂O system), the deviation was even greater than at 25°C.

Since the modeling of OH⁻ concentrations is not improved by Pitzer coefficients, it is assumed that, besides MgOH⁺, further Mg-OH species that influence OH⁻ concentration must exist in the solution.

As known from other aqueous metal oxide or hydroxide salt systems, such as Be, Cu, Ni, and Zn (Feitknecht and Schindler, 1963; Mesmer and Baes, 1990; Plyasunova et al., 1997; 1998; Zhang and Muhammed, 2001), the existence of complex Mg-OH species containing more than one cation Mg_x(OH)_y ^(2x-y)+ has been already discussed (D’Ans and Katz, 1941; Ved et al., 1976; Mesmer and Baes, 1990). Stability constants (Lewis, 1963; Einaga, 1977) alongside MgOH⁺ (McGee and Hostetler, 1977; Palmer and Wesolowski, 1997) have also been proposed. Einaga (1977) derives the species Mg₂(OH)₂ ²⁺ and Mg₃(OH)₄ ²⁺ from emf measurements in nitrate solutions at 25°C with the stability constants log β = −22.0 and log β = −39.0, respectively. However, the determination of complex Mg-OH solution species lacks sufficient resolution to precisely determine the formulas, especially when more than one species coexists over a range of solution compositions. As Mg²⁺ is exclusively octahedrally coordinated by OH⁻ (partly together with OH₂) in these solids (Mg(OH)₂, Sorel phases 3-1-8, 5-1-8, and 9-1-4 (De Wolf and Walter-Lévy, 1953; Sugimoto et al., 2007; Dinnebier et al., 2010)), multiple OH⁻ coordinated Mg ions, in addition to MgOH⁺ = [Mg(H₂O)₅OH]⁺, could be expected to exist in appropriate equilibrium solutions.

Therefore, additional species were postulated based on the following considerations. In contrast to the solid structures with up to six OH⁻ ions in the octahedral coordination sphere of Mg²⁺, the OH⁻ concentration in the equilibrium solution is relatively very low compared to the Mg²⁺ (and Cl⁻) concentration. Thus, besides MgOH⁺, additional solution-Mg²⁺ should be only slightly more highly coordinated with OH⁻. Hence, species with a systematically increasing ratio of OH⁻ to Mg²⁺ up to a maximum of two were taken into account—Mg₃(OH)₄ ²⁺ (ratio = 1.33), Mg₂(OH)₃ ⁺ (ratio = 1.5), and Mg(OH)₂ ^±0 (ratio = 2)—and were tested for their effect. Estimated ∆_R G _m ° values were used, in orientation to the value of MgOH⁺ and the stability constant for Mg₃(OH)₄ ²⁺ given by Einaga (1977), to first see the general effects. Mg₂(OH)₂ ²⁺ was not taken into account because the Mg:OH ratio is the same as in MgOH⁺ (tests have shown that the different charge does not have an effect). Calculations with the current THEREDA model show that MgOH⁺ determines the total OH⁻ concentration, so the fraction of free OH⁻ is negligible. Fitting studies with the additional three postulated species were performed both singularly and in combinations.

Finally, the fitting tests to the 25°C isotherm in the Mg-Cl-OH-H₂O system (Pannach et al., 2017) showed that the total OH⁻ concentrations can be reproduced when including additional Mg-OH species. It was found that the species Mg₃(OH)₄ ²⁺ alone was sufficient. The other species, alone or in combination, worsened the agreement with experimental results. Therefore, the final fits to all isotherms in the system Mg-Cl-OH-H₂O were performed with Mg₃(OH)₄ ²⁺ additionally in the THEREDA dataset, turning off all ion interactions for MgOH⁺ (see Table 1). ∆_R G _m °_,Mg3(OH)4 ²⁺ (according to the formation reaction: 3Mg²⁺ + 4H₂O ↔ Mg₃(OH)₄ ²⁺ + 4H⁺) was adjusted so that the isothermal branch of Mg(OH)₂(s) up to the invariant point with the 3-1-8 phase will be reproduced in agreement with the experimental data. With increasing MgCl₂ concentration, the total OH⁻ concentration in the presence of the 3-1-8 phase is calculated to be increasingly too high until MgCl₂ saturation is reached. This overestimation was subsequently adjusted with the final Pitzer coefficients (Section 4.2.3). The resulting values of ∆_R G _m °, _{Mg3(OH)4 ²⁺} for each temperature are shown in Figure 5, together with the fitted temperature function (Table 3).

To remove the over-calculated total OH⁻ concentrations following the IP of Mg(OH)₂(s) and the 3-1-8 phase caused by the implementation of Mg₃(OH)₄ ²⁺, ion interactions for that species were introduced. Fitting tests were carried out with binary as well as mixing Pitzer coefficients. Finally, the mixing parameter ψ _{Mg-Mg3(OH)4-Cl} was adjusted together with ψ _Mg-MgOH-Cl. The binary coefficients β⁽⁰⁾ and β⁽¹⁾ between MgOH⁺ and Cl⁻ previously included beside the mixing ψ _Mg-MgOH-Cl proved unnecessary; they were removed from the dataset. The parameterization was again performed separately on each isotherm. The obtained values for ψ _{Mg-Mg3(OH)4-Cl} and ψ _Mg-MgOH-Cl are shown in Figures 6A, B. The THEREDA conformal temperature function (Eq. 8) was fitted to the systematic variation of the parameters with T obtained from the isothermal fits (Figure 6). P = a T + b + c ∙ ln T + d ∙ T + e ∙ T 2 + f T 2 . (8)

Due to the polynomial type of the function (Eq. 8), the course for ψ _{Mg-Mg3(OH)4-Cl} with the strong decrease from the 25°C value to the only slightly decreasing values up to 120°C can only be adjusted with turning points (Figure 6B). However, this does not show any noticeable effects on the model results within the experimental data trend and reproduction (Section 4.2).

For the modeling of the two solubility isotherms in the system Na-Mg-Cl-OH-H₂O at 25°C and 40°C, the implementation and adjustment of the mixing parameters ψ _Na-MgOH-Cl and ψ _{Na-Mg3(OH)4-Cl} were still necessary. The fitted values with linear progression over the two temperatures are shown in Figure 7.

All Pitzer coefficients with their temperature functions are listed in Table 4.