Solubility Product of Vanadinite Pb 5 (VO 4 ) 3 Cl at 25 ◦ C—A Comprehensive Approach to Incongruent Dissolution Modeling

: Although vanadinite (Pb 5 (VO 4 ) 3 Cl) occurs in abundance in various terrestrial geochemical systems of natural and anthropogenic origin and is seriously considered as a potential nuclear waste sequestering agent, its actual application is severely limited by a lack of understanding of its basic thermodynamic parameters. In this regard, the greatest challenge is posed by its incongruent dissolution, which is a pivotal hurdle for effective geochemical modeling. Our paper presents an universal approach for geochemical computing of systems undergoing incongruent dissolution which, along with unique, long-term experiments on vanadinites’ stability, allowed us to determine the mineral solubility constant. The dissolution experiments were carried out at pH = 3.5 for 12 years. Vanadinite has dissolved incongruently, continuously re-precipitating into chervetite (Pb 2 V 2 O 7 ) with the two minerals remaining in mutual equilibrium until termination of the experiments. The empirically derived solubility constant K sp,V,298 = 10 –91.89 ± 0.05 of vanadinite was determined for the ﬁrst time. The proposed modeling method is versatile and can be adopted to other mineral systems undergoing incongruent dissolution.


Introduction
Vanadium is one of the most important metals in modern technology. It is the twentysecond most common element in the Earth's crust and the fifth most common transition metal. Small amounts of vanadium are essential for metabolism of several bacteria, algae, fungi and lichens (e.g., [1][2][3][4][5]). It also partakes in several bodily functions and life processes [6]. Vanadinite Pb 5 (VO 4 ) 3 Cl is a secondary mineral, usually found in the oxidation zones of lead deposits and as a product of corrosion of infrastructure. In nature, it is often associated with high V content in mica and clay sediments subjected to hydrothermal processes, occurring under neutral or slightly alkaline conditions [7][8][9]. However, the mineral was also found in lead pipe water systems and it occurs in many already closed lead mines around the world [10,11]. Overall, there are more than 3000 worldwide locations, where vanadinite is present along with its phosphate and arsenate counterparts: pyromorphite Pb 5 (PO 4 ) 3 Cl and mimetite Pb 5 (AsO 4 ) 3 Cl [12], with which it forms solid solution series [13][14][15]. The minerals belong to the group of lead apatites [16], the composition of which is expressed by the general formula Pb 5 (BO 4 ) 3 X, where BO 4 represents the coordinated tetrahedron oxyanions (e.g., PO 4 3-, AsO 4 3-, VO 4 3-) and X represents the monovalent electronegative anions (e.g., F -, Cl -, Br -, I -, OH -) [17][18][19]. The hexagonal structure of most of the minerals in this group is consistent with the apatite structure

Synthesis of Vanadinite
Synthesis of vanadinite was carried out at 90 • C and pH 3.5 adjusted using 0.1 M HNO 3 (MERCK, supra pure) by dropwise mixing of the aqueous solutions of Pb(NO 3 ) 2 (POCH.SA, pure for analysis), KCl (CHEMPUR, pure for analysis), and NH 4 VO 3 (POCH.SA, pure for analysis), in molar proportions based on the stoichiometry of the vanadinite (Pb:V:Cl = 5:3:1 molar ratio). The temperature and the pH of the synthesis were indicated by preliminary geochemical modeling [64,65] as the conditions favorable for the VO 4 3− presence in the solution, thus favorable for vanadinite precipitation. Suspension of forming precipitate was stirred using a mechanical stirrer for 5 h and then allowed to settle. After 2 weeks, the precipitate was separated from the solutions by decantation, washed thoroughly several times with redistilled water and acetone and dried at 110 • C.

Solid Characterization
The X-ray powder diffraction (XRD) pattern of the solids was obtained using a Philips PW 3020 X'Pert-APD Diffractometer system (with a Cu anode and a graphite monochromator) using a step scan mode at a step size of 0.02 • 2Θ and a rate of 1 s per step. The scanning electron microscopy with elemental microanalysis (SEM/EDS) was performed using a variable pressure field emission FEI QUANTA 200 FEG microscope on goldcoated and uncoated samples at 20 kV. For the wet chemical analysis, -small portions of synthetic vanadinite (~50 mg) were digested in 0.02 M EDTA (2,2 ,2 ,2 -(ethane−1,2diyldinitrilo)tetraacetic acid) (POCH.SA, pure for analysis) and analyzed for Pb, V and Cl, as described in the next section. Compared to other reagents e.g., HNO 3 , the EDTA is an effective solvent for mineralization of Pb-apatites yielding relatively quick and complete dissolution of the solid [14,36,37,39,55].

Solution Analysis
A Total Reflection X-ray Fluorescence was used to determine the Pb and V content in the solutions resulting from dissolution experiments. The samples were inoculated with a nickel stock solution (1000 mg dm −3 , MERCK, supra pure) as an internal standard, and analyzed at 50 kV and 12 mA with a Nanohunter II spectrometer equipped with a Mo lamp (t = 3000 s). Atomic Absorption Spectroscopy AAS (SavantAA GBC Scientific Equipment) was used to determine the concentration of Pb and V in the solutions resulting from digestion of solids in EDTA for wet chemical analysis of vanadinite. The turbidimetric method, with silver nitrate at 380 nm, was used to determine the concentration of Cl in all solutions with a HITACHI U-1800 UV-Vis Spectrophotometer. The universal meter CX-505 with a glass pH electrode ERH-13-6 was used to measure the pH. All analyses were performed in triplicate.

Dissolution Experiments
Dissolution experiments were performed in triplicate in 250 mL polycarbonate bottles at 25 • C ± 1 • C. Equal portions (150 mg) of synthetic vanadinite were washed thoroughly with redistilled water and introduced as a suspension to the bottles containing 250 mL of 0.05 M KNO 3 solution. The KNO 3 (POCH.SA, pure for analysis) was used as the background electrolyte because both, K + and NO 3 − ions are not involved in the reactions in the studied experimental system. At this concentration of the background electrolyte, the increase in ion concentration caused by the dissolution of the mineral during the experiment does not cause significant changes in the total ionic strength of the solution, and thus does not affect the experimental conditions. At the same time, the ionic strength is low enough that the Debye-Hückel equation can be used to calculate ion activity. The experiments were performed in triplicate at pH = 3.5, adjusted using 0.1 M HNO 3 (MERCK, supra pure). This pH value was indicated in our previous studies as adequate for the potential thermodynamic modeling of vanadinite stability [55]. The dissolution was carried out for 12 years. For the first 8 months, the bottles were immersed in a closed water bath with thermostatic control and manually stirred at least two times a week. After this time, the bottles were transferred to a room temperature of 22 • C ± 2 • C and kept for 9 years in darkness; the reactors were occasionally mixed and checked for leaks. For the last 3 years of the experiments, the temperature was raised to 25 • C ± 1 • C and the reactors were manually stirred at least once a month. During the experiments, 4 mL aliquots of the solutions were collected periodically, filtered using a 0.2 µm polycarbonate filter and analyzed in terms of pH and the total concentration of Pb, V and Cl. The evolution pattern of [Pb], [V] and [Cl] was observed to determine the equilibrium in the suspensions; the system was considered in equilibrium when the results of three consecutive solution analyses were equal at 95% confidence level. The residual solids were syringe-sampled, washed with acetone, air-dried, and characterized with SEM/EDS and XRD.

Statistics
The statistical software Statistica 13 (StatSoft Polska) was used to process the solution composition data. The normal distribution was verified with the use of the Kolmogorov-Smirnov (K-S) test with Lilliefors correction (K-S-L) and the Shapiro-Wolf test (S-W). The Wald-Wolfowitz, the U Mann-Whitney and the Kołmogorow-Smirnow tests and the Student's t test were applied accordingly to verify whether the elemental contents of the solutions change significantly over time at 95% confidence level.

Synthesized Solid
The synthesis yielded very fine, crystalline vanadinite with crystals smaller than 2 µm ( Figure 1). Lead, vanadium, chlorine, and oxygen were the only elements detected by the SEM/EDS elemental microanalysis (data not shown). No other crystalline or amorphous components were observed. Comparison of XRD patterns with data reported by Dai and Hughes [66] and Trotter and Barnes [67] yielded good agreement, indicating vanadinite Pb 5 (VO 4 ) 3 Cl as the sole reaction product (Table A1). No other phases were found within the detection limit of the method (equal to ca. 0.1 wt%). The wet chemical analysis of synthesized solid confirmed the purity of the solid (mass fraction purity = 0.995).
Minerals 2021, 11, x FOR PEER REVIEW 4 of 14 last 3 years of the experiments, the temperature was raised to 25 °C ± 1 °C and the reactors were manually stirred at least once a month. During the experiments, 4 ml aliquots of the solutions were collected periodically, filtered using a 0.2 μm polycarbonate filter and analyzed in terms of pH and the total concentration of Pb, V and Cl. The evolution pattern of [Pb], [V] and [Cl] was observed to determine the equilibrium in the suspensions; the system was considered in equilibrium when the results of three consecutive solution analyses were equal at 95% confidence level. The residual solids were syringe-sampled, washed with acetone, air-dried, and characterized with SEM/EDS and XRD.

Statistics
The statistical software Statistica 13 (StatSoft Polska) was used to process the solution composition data. The normal distribution was verified with the use of the Kolmogorov-Smirnov (K-S) test with Lilliefors correction (K-S-L) and the Shapiro-Wolf test (S-W). The Wald-Wolfowitz, the U Mann-Whitney and the Kołmogorow-Smirnow tests and the Student's t test were applied accordingly to verify whether the elemental contents of the solutions change significantly over time at 95% confidence level.

Synthesized Solid
The synthesis yielded very fine, crystalline vanadinite with crystals smaller than 2 μm ( Figure 1). Lead, vanadium, chlorine, and oxygen were the only elements detected by the SEM/EDS elemental microanalysis (data not shown). No other crystalline or amorphous components were observed. Comparison of XRD patterns with data reported by Dai and Hughes [66] and Trotter and Barnes [67] yielded good agreement, indicating vanadinite Pb5(VO4)3Cl as the sole reaction product (Table A1). No other phases were found within the detection limit of the method (equal to ca. 0.1 wt%). The wet chemical analysis of synthesized solid confirmed the purity of the solid (mass fraction purity = 0.995).

Dissolution Experiments
The dissolution experiments were carried out over an exceptionally long period-12 years. Standard empirical thermodynamic studies of apatites last several weeks, as this has been widely accepted to be sufficiently long for the solids to equilibrate with the solutions [36,37,39,68]. However, vanadates tend to polymerize, inducing re-precipitation of vanadinite Pb5(VO4)3Cl into chervetite Pb2V2O7; the system is highly pH and temperature dependent and the kinetics of these reactions are poorly understood [55]. This caused difficulties in the interpretation of dissolution data and in unambiguous determination of the equilibrium of the system with the use of the dissolution-precipitation method (e.g., [37]). Thus, we decided to run the experiments over an exceptionally long period, recog-

Dissolution Experiments
The dissolution experiments were carried out over an exceptionally long period-12 years. Standard empirical thermodynamic studies of apatites last several weeks, as this has been widely accepted to be sufficiently long for the solids to equilibrate with the solutions [36,37,39,68]. However, vanadates tend to polymerize, inducing re-precipitation of vanadinite Pb 5 (VO 4 ) 3 Cl into chervetite Pb 2 V 2 O 7 ; the system is highly pH and temperature dependent and the kinetics of these reactions are poorly understood [55]. This caused difficulties in the interpretation of dissolution data and in unambiguous determination of the equilibrium of the system with the use of the dissolution-precipitation method (e.g., [37]). Thus, we decided to run the experiments over an exceptionally long period, recognizing the long-term stability of the solutions' composition as the sufficient base to estimate the missing thermodynamic parameters for vanadinite. The analytical methods (TXRF), requiring small amounts of aliquots, as well as the proper sealing of the reactors assured the quality of the long-term experiments, and the effect of potential evaporation was negligible. The SEM image of the dissolution residue after 12 years of aging in the suspension is presented in Figure 2. During the experiments, apart from the very fine, hexagonal particles of primary vanadinite, (the experimental material subject to dissolution; please compare with Figure 1), notably larger (>10 µm) prismatic crystals with the monoclinic symmetry appeared in the solid fraction. The EDS (data not shown) and the XRD analysis indicated that the prismatic crystals of the secondary mineral phase are chervetite Pb 2 V 2 O 7. [69,70]. All X-ray diffraction peaks were assigned to either vanadinite or chervetite, confirming beyond doubt the incongruent nature of the dissolution process in which vanadinite re-precipitates partially into chervetite (Table A1). The possible mechanism behind the appearance of chervetite in the solid residue was that dissolving vanadinite released Pb and V ions in concentrations exceeding the solubility product of chervetite inducing its precipitation as the secondary mineral. However, the recrystallization of vanadinite into chervetite due to adsorption of polymerized vanadates or their direct formation on the vanadinite surface cannot be excluded.
nizing the long-term stability of the solutions' composition as the sufficient base to estimate the missing thermodynamic parameters for vanadinite. The analytical methods (TXRF), requiring small amounts of aliquots, as well as the proper sealing of the reactors assured the quality of the long-term experiments, and the effect of potential evaporation was negligible.
The SEM image of the dissolution residue after 12 years of aging in the suspension is presented in Figure 2. During the experiments, apart from the very fine, hexagonal particles of primary vanadinite, (the experimental material subject to dissolution; please compare with Figure 1), notably larger (> 10 μm) prismatic crystals with the monoclinic symmetry appeared in the solid fraction. The EDS (data not shown) and the XRD analysis indicated that the prismatic crystals of the secondary mineral phase are chervetite Pb2V2O7. [69,70]. All X-ray diffraction peaks were assigned to either vanadinite or chervetite, confirming beyond doubt the incongruent nature of the dissolution process in which vanadinite re-precipitates partially into chervetite (Table A1). The possible mechanism behind the appearance of chervetite in the solid residue was that dissolving vanadinite released Pb and V ions in concentrations exceeding the solubility product of chervetite inducing its precipitation as the secondary mineral. However, the recrystallization of vanadinite into chervetite due to adsorption of polymerized vanadates or their direct formation on the vanadinite surface cannot be excluded.  Table 1 and additionally visualized in Figure A1 (Appendix B) The pH was systematically measured; however, its value remained stable for the whole 12 years, oscillating at 3.5 ± 0.05, hence the data was not included in the table. The applied statistical analysis yielded that at the 95% confidence level the solution samples collected within the 1st, 2nd, 5th, 8th and 19th week as well as the 116th, 132th and 147th month of the experiment were identical in terms of Cl concentration. This indicated that Cl had reached its close-to-equilibrium level in the solutions relatively quickly, i.e., within the first week of the experiment. In contrast, the concentrations of Pbtot and Vtot systematically increased in the solutions and only the last three samples collected during the final 3 years of the experiment were statistically identical at 95% confidence level (Table 1). It is worth mentioning that vanadium concentration in most of the samples collected at the beginning of the experiment was below or at its detection limit (0.5 μmol dm −3 ). The results suggested that the system had reached significant, long-term steady state conditions: three years of statistically stable solution parameters. Hence, we concluded that the final levels (in μmol dm −3 ) of Cl = 24.4 ± 3.2, Pb = 45.9 ± 7.00 and V = 2.4 ± 1.6, expressed as the mean value of  Table 1 and additionally visualized in Figure A1 (Appendix B) The pH was systematically measured; however, its value remained stable for the whole 12 years, oscillating at 3.5 ± 0.05, hence the data was not included in the table. The applied statistical analysis yielded that at the 95% confidence level the solution samples collected within the 1st, 2nd, 5th, 8th and 19th week as well as the 116th, 132th and 147th month of the experiment were identical in terms of Cl concentration. This indicated that Cl had reached its close-to-equilibrium level in the solutions relatively quickly, i.e., within the first week of the experiment. In contrast, the concentrations of Pb tot and V tot systematically increased in the solutions and only the last three samples collected during the final 3 years of the experiment were statistically identical at 95% confidence level (Table 1). It is worth mentioning that vanadium concentration in most of the samples collected at the beginning of the experiment was below or at its detection limit (0.5 µmol dm −3 ). The results suggested that the system had reached significant, long-term steady state conditions: three years of statistically stable solution parameters. Hence, we concluded that the final levels (in µmol dm −3 ) of Cl = 24.4 ± 3.2, Pb = 45.9 ± 7.00 and V = 2.4 ± 1.6, expressed as the mean value of concentrations determined in the solutions during the last three years of the experiment, adequately represented the equilibrium in the solutions. The final molar ratios of the Pb tot to Cl tot in the solutions varied between 1.8 and 1.9, while the ratios of Pb tot to V tot fell in the range between 17 and 20. The proportions were neither typical for vanadinite (Pb:V;Cl = 5:3:1), nor for chervetite (Pb:V = 1:2), indicating dissolution in which the properties of both involved phases simultaneously affect the composition of the solution.

Discussion
Thermodynamic modelling of incongruent dissolutions poses significant challenges. In this regard, few approaches for the apatite group have been proposed [71][72][73][74]. In this study, the geochemical speciation model PHREEQC (Phreeqc Interactive 3.5.0-14000) [64] was used to support the calculations of thermodynamic parameters. The MINTEQA2 [70] was chosen as the PHREEQC database containing, in our opinion, the most comprehensive dataset related to vanadium compounds. The database did not include referenced thermodynamic data for vanadinite, since, to date, no reliable report has been proposed in these terms; however, the equilibrium constant of chervetite dissolution reaction (1) was indicated there as 10 −0.95 . PbVO 3.5 + 3H + = Pb 2+ + VO 2+ + 1.
The compositions of the experimental solutions were proceeded with the use of PHREEQC [64] to verify their possible saturation with chervetite. The results presented as the logarithm of chervetite saturation index (SI) are shown in Table 1. The log SI values indicated supersaturation with chervetite for the last three years of the vanadinite dissolution. Even though the log SI values only slightly exceeded 0, the experimental conditions were favorable for vanadates polymerization as indicated by the PHREEQC modeling [64] and the precipitation of chervetite occurred as confirmed by the SEM/EDS and XRD analysis. Hence, there were two possible approaches to further the thermodynamic modeling of vanadinite's solubility product, K sp : (i) acknowledging the equilibrium with chervetite or (ii) neglecting the presence of a secondary phase as having little effect on the equilibrium [71][72][73][74]. Both approaches were exercised using PHREEQC [64].
If the dissolution reaction of vanadinite is written as in the Equation (2), then the logarithm of the ion activity product (IAP) for the reaction (2) is given by: where {} denotes activities of the ions. At equilibrium, the Ion Activity Product (IAP) calculated for the dissolution reaction of a mineral is equal to its solubility product, K sp . The activities of aqueous Pb 2+ , VO 4 3− , and Cl − were calculated considering the two different approaches: (i) and (ii) mentioned above with the use of the geochemical speciation model PHREEQC [64]. All reactions were modeled as open to the air with P CO2 = 10 −3.5 atm, at pH = 3.5 and temperature = 25 • C. The experimental concentrations of Pb tot , V tot and Cl tot were used as the input data and the calculated log IAPs are listed in Table 1. For all equilibrated solutions, the obtained values were of similar magnitude; however, the IAP values that neglected the presence of chervetite were slightly higher. At equilibrium, the average log IAP for vanadinite calculated based on the approach (i) was: −91.89 ± 0.05, whereas the average log IAP value obtained using the alternative approach (ii) was: −91.43 ± 0.08 (uncertainties represented by the double standard deviation from three values calculated at equilibrium).
To validate the calculations made upon the "inverse modeling" based on the three-step empirical approach with the use of the geochemical speciation model PHREEQC [64] was applied. In the first step of the inverse approach, the MINTEQA2 [65] thermodynamic database available in the PHREEQC [64] package was modified by the amendment of the dissolution reaction for vanadinite (4): where "log_k" expresses solubility product of vanadinite (log K sp,V,298 ). In the second step, the value of the vanadinite solubility product (log K sp,V,298 ) expressed by X in the Equation (4) was modified in the database with the use of experimentally determined IAP values calculated at equilibrium (Table 1) In the third step, the output data of each modeling were compared with the results of the dissolution experiment, in terms of the solution parameters such as: pH, total Pb, Cl and V concentrations and saturation index of other potentially precipitated phases. The obtained set of selected data is presented in Table 2. Table 2. Comparison of the selected solution parameters determined in dissolution experiments of vanadinite and via computing with geochemical model PHREEQC [64]. The PHREEQC [64] calculations with the assumed vanadinite solubility products log K sp,V,298 equal −90.87 and 90.91 yielded solution parameters that were statistically identical (at 95% confidence level) with the empirical results provided by the dissolution experiment. This suggested that the calculations acknowledging the presence of chervetite were more adequate in this case. Thus, we recommend the value (K sp,V,298 = 10 −91.89 ± 0.05 ) as the first empirically determined solubility product of vanadinite which can be used for further thermodynamic calculations, modeling and experimental approximations. The uncertainty (±0.05) corresponds to a double standard deviation of the IAP values calculated for equilibrium in this study. The presented approach involving "inverse geochemical modeling" is a universal method to validate the assumptions made during thermodynamic modeling. The method is simple and in case of a complex incongruent dissolution systems it may refine uncertainties or even act as the straightforward calculations without intricate raw data processing. The obtained value (K sp,V,298 = 10 −91.89 ± 0.05 ) is notably lower than the value (K sp,V,298 = 10 −86 ) proposed by Gerke et al. [10], however, their data was just an approximation based on presumptions of water parameters in lead-containing pipes and did not consider incongruent nature of vanadinite dissolution, precipitation of chervetite or equilibrium evidence in the solution. Nevertheless, regarding the general tendency of Pb-apatites for extremely low solubility products and the relative value of vanadinite parameter when compared to other Cl-Pb-apatites e.g., mimetite K sp,V,298 = 10 −79.11 ± 0.33 [36] and pyromorphite K sp,V,298 = 10 −79.60 ± 0.33 [39], our results were self-consistent.

Conclusions
For the first time, the experimentally determined solubility product of vanadinite equal to K sp,V 10 −91.89 ± 0.05 is presented. Vanadinite is an exceptional mineral among Pb-apatite series; it dissolves incongruently, inducing precipitation of chervetite as the sole secondary phase. The incongruent mechanism of vanadinite's dissolution poses a challenge to the direct experimental determination of the solubility constant; however, a new calculation approach, along with the unique, long term experiments proposed here, made it possible to overcome this difficulty. The approach to modeling with a use of the presented inverse method is novel, universal and can be applied to other systems.  Table A1. Powder X-ray diffraction peaks from synthetic Pb 5 (VO 4 ) 3 Cl used in this study compared with data reported in the literature: Miller indices h k l, distance d between adjacent (h k l) planes and the peak intensity ratio I/Io. Appendix B Figure A1. The evolution of Pbtot, Cltot and Vtot concentrations in the experimental solutions w time.

Appendix C
Exemplary input data for PHREEQC simulation used for "Inverse modeling".

Appendix C
Exemplary input data for PHREEQC simulation used for "Inverse modeling". SOLUTION

Appendix C
Exemplary input data for PHREEQC simulation used for "Inverse modeling". SOLUTION