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# The role of sulfur in palladium transport and fractionation from platinum by hydrothermal fluids

C. Laskar, E.F. Bazarkina, G.S. Pokrovski

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C. Laskar, E.F. Bazarkina, G.S. Pokrovski. The role of sulfur in palladium transport and fractionation from platinum by hydrothermal fluids. Geochimica et Cosmochimica Acta, 2025, A Tribute to Terry Seward, 406, pp.134-147. 10.1016/j.gca.2024.12.019. hal-05025407

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# The role of sulfur in palladium transport and fractionation from platinum by hydrothermal fluids☆

C. Laskar<sup>a,b,*</sup>, E.F. Bazarkina<sup>c,d</sup>, G.S. Pokrovski<sup>a</sup>

## ARTICLE INFO

Associate editor: Andri Stefansson

### Keywords:

- Platinum group elements

- Hydrothermal fluid

- Solubility

- Palladium sulfide

- Platinum sulfide

- Hydrogen sulfide

- Sulfur

- Palladium hydrosulfide complexes

- Platinum hydrosulfide complexes

- Stability constant

- Thermodynamic properties

- HKF model

- Polysulfide ions

- Trisulfur radical ion

- Pd/Pt ratio

- Fractionation

- Hydrothermal ore deposit

## ABSTRACT

The solubility of palladium sulfide (PdS(s)) has been measured in H₂S/HS⁻ aqueous solutions across wide ranges of sulfur concentrations (0.5–1.2 molal) and pH (5–8) at temperatures from 50 to 300 °C and pressures from 90 to 600 bar, using a hydrothermal flexible-cell reactor allowing controlled fluid injection and sampling. Combined with thermodynamic modeling and analysis of available literature data, our results demonstrate that palladium tetrahydrogensulfide, Pd¹¹(HS)₂⁻, is the dominant complex in sulfide-rich moderate-temperature hydrothermal fluids. The equilibrium constants of PdS(s) dissolution reaction, PdS(s) + 3 H₂S₀(aq) = Pd(HS)₂⁻ + 2H⁺ (K₅₄,Pd), generated in this study can be described by the function log₁₀K₅₄,Pd = (196.4 ± 60.0) – (11384 ± 3567)/T(K) – (71.24 ± 19.52) × log₁₀T(K), valid over the temperature range 25–300 °C and from saturated vapor pressure to 600 bar. Our results, combined with recent analogous PtS(s) solubility data, enabled the derivation of a self-consistent set of thermodynamic properties of Pd(HS)₂⁻ and Pt(HS)₂⁻ in the framework of the HKF model. Solubility predictions of PdS(s) using these properties yield maximum solubility values of 1 ppb Pd, as the tetra-hydrosulfide complex, in H₂S-bearing hydrothermal fluids (>0.01 m S) at moderate temperatures (50–350 °C) and near-neutral pH (6–7), whereas chloride complexes are predominant at acidic pH (<4). The Pd/Pt atomic ratio in typical H₂S-bearing hydrothermal fluids at 300 °C in equilibrium with PdS(s) and PtS(s) varies from > 10⁴ at pH < 2 to 10⁻² at pH > 3, corresponding to the change from chloride- to sulfide-dominated speciation for both metals. However, the absolute solubilities of both chloride and sulfide complexes are too small over the whole pH range to significantly contribute to Pd vs. Pt fractionations observed in hydrothermal environments. Among different factors that may lead to such fractionations, the role of polysulfide sulfur species, including trisulfur radical ions, should be considered in future studies.

## 1. Introduction

Among the platinum group elements (PGE, including Ru, Rh, Pt, Os, Ir, and Pd), palladium together with its congener, platinum, are the most widely used in technological applications owing to their exceptional catalytic properties in a wide range of temperatures (Wang et al., 2015; Zhang et al., 2015; Zientek et al., 2017; Liu et al., 2018; Tan and Chen, 2019; Torres et al., 2020). Economic Pd concentrations come from mafic and ultramafic magmatic settings, such as Bushveld, Norilsk and Sud-bury, which account for > 95 % of the nowadays mined palladium (e.g.,

Mungall and Naldrett, 2008; Barnes and Liu, 2012; Mudd and Jowitt, 2014; Barnes et al., 2017). In such settings, Pd and other PGE are concentrated along with Fe-Ni-Cu sulfide melts and minerals. This accumulation is believed to be controlled by fractional crystallization and segregation of silicate and sulfide melts (e.g., Kerr and Leitch, 2005; Brenan, 2008; Lorand et al., 2008; Palme, 2008). However, the role of aqueous fluids in Pt and Pd mobilization, concentration and fractionation remains a subject of debate (e.g., Kanitpanyacharoen and Boudreau, 2013; Le Vaillant et al., 2016a,b; Boudreau, 2019, and references therein). The present study aims to investigate Pd speciation in sulfide-

<footnote><sup>a</sup> Géosciences Environnement Toulouse (GET), UMR 5563, Centre National de la Recherche Scientifique (CNRS), Observatoire Midi-Pyrénées (OMP), Université Paul Sabatier Toulouse III, Institut de Recherche pour le Développement (IRD), 14, avenue Edouard Belin, 31400 Toulouse, France</footnote>

<footnote><sup>b</sup> Department of Chemistry, KU Leuven, Celestijnenlaan 200F, P.O. Box 2404, 3001 Leuven, Belgium</footnote>

<footnote><sup>c</sup> Institut Néel, UPR 2940 CNRS, Grenoble, France Univ. Grenoble Alpes, 25, avenue des Martyrs, 38042 Grenoble Cedex 9, France</footnote>

<footnote><sup>d</sup> Institute of Resource Ecology, Helmholtz Zentrum Dresden-Rossendorf (HZDR), P.O. Box 510119, 01314 Dresden, Germany</footnote>

<footnote>* This article is part of a special issue entitled: 'A Tribute to Terry Seward' published in Geochimica et Cosmochimica Acta.</footnote>

<footnote>* Corresponding author at: Géosciences Environnement Toulouse (GET), UMR 5563, Centre National de la Recherche Scientifique (CNRS), Observatoire Midi-Pyrénées (OMP), Université Paul Sabatier Toulouse III, Institut de Recherche pour le Développement (IRD), 14, avenue Edouard Belin, 31400 Toulouse, France.
E-mail address: clement.laskar@gmail.com (C. Laskar).</footnote>
bearing hydrothermal fluids and to identify differences with recently investigated corresponding Pt complexes (Laskar et al., 2022b) that might lead to the fractionation of these metals in hydrothermal environments. Indeed, there are numerous instances of transport and accumulation of Pd and Pd in various hydrothermal settings such as porphyry Cu-Au-Mo deposits, black-shale hosted gold deposits, sedimentary brines, and serpentinites, with Pd concentrations up to 1000 times higher than its average crustal abundance of 0.8 ppb (Ballhaus and Stumpfl, 1986; McKibben et al., 1990; Rehkämper et al., 1997; Wood and Cabri, 2002; Hanley, 2005; Luguet et al., 2008; Arai and Miura, 2016; Barnes and Ripley, 2016; Chen et al., 2016; Boudreau, 2019). These observations point to an important, but often overlooked so far, role of moderate-temperature (200–500 °C) hydrothermal fluids in Pd transport (e.g., Wood et al., 1992). Such hydrothermal settings may become potentially exploitable for PGE in the future. Therefore, knowledge of Pd chemical speciation and solubility in the hydrothermal fluid phase is required for quantifying the formation and distribution of Pd resources as well as for optimizing the rapidly growing Pd use in technology.

Chloride and (hydro)sulfide complexes have been considered to be the most important for Pd transport. For saline, acidic sulfide-poor fluids, there is a consensus that Pd-Cl complexes are the dominant Pd species (e.g., Gammons et al., 1992, 1993; Boily and Seward, 2005; Tagirov et al., 2013; Bazarkina et al., 2014; Mei et al., 2015; references therein). However, their exact stoichiometry and stability remain uncertain. For example, palladium tri- and tetrachlorides, PdCl₃ and PdCl₄²⁻, emerge as the major Pd species, according to Bazarkina et al. (2014) who used X-ray absorption spectroscopy to measure both PdS(s) and PdO(s) solubility and local Pd structure, between 200 and 500 °C and below 1 kbar in a wide range of salinity (up to 9 m NaCl, where m is molality, mol/kg H₂O, throughout the article). These data were confirmed by Mei et al. (2015) who used molecular dynamics (MD) simulations complemented by X-ray absorption spectroscopy (XAS) experiments in saline solutions at 300 °C. Earlier experimental studies also privileged PdCl₄²⁻ as the main complex at 300 °C over a wide range of redox conditions, from the sulfide-sulfate (Gammons et al., 1992) to AgCl/Ag-Pdₐlloy (Gammons et al., 1993) equilibrium. In contrast, earlier theoretical (Wood et al., 1992; Sassani and Shock, 1998) and experimental (Boily and Seward, 2005; Tagirov et al., 2013) studies at similar conditions privileged lower ligation number chloride complexes such as PdCl₂₀(aq). The scatter of the formation constant values for Pd chloride complexes among the different studies remains remarkably large, attaining, for example, 7 log units for PdCl₄²⁻ at temperatures of 300–450 °C (see Bazarkina et al., 2014, for detailed comparisons). At higher, closer to magmatic, temperatures, the discrepancies become truly spectacular. For example, at 900 °C and 2 kbar in an acidic fluid of 1 m Cl and oxygen fugacity of the nickel-nickel oxide buffer (NNO), the published HKF model predictions of Pd solubility span over 9 log units, from 1 ppt (Sassani and Shock, 1998) to 1000 ppm (Tagirov et al., 2013), as compared with the experimental value of 0.1 ppm (Sullivan et al., 2022). Despite the absence of quantitative consensus, notwithstanding their exact stoichiometry and stability, chloride complexes are only sufficiently abundant at strongly acidic pH and oxidizing conditions.

Consequently, sulfide complexes have been invoked, but their accurate quantification is equally lacking. Complexes of type Pd(HS)₂⁻(2 ≤ n ≤ 4) have been proposed from PdS(s) solubility experiments between 25 and 300 °C at the saturated vapor pressure of the system (Pₛₐt) in H₂S-bearing fluids by Gammons and Bloom (1993), Pan and Wood (1994), Tagirov and Baranova (2009) and Tagirov et al. (2013). These authors used different techniques such as silica tubes, batch and sampling reactors, and sealed Pyrex glass ampoules. Their interpretations of solubility data suggested Pd(HS)₃⁻ and Pd(HS)₂₀(aq) as the most likely species, with a poorly quantified contribution of Pd(HS)₄²⁻. Additional solubility data were also obtained by Wood et al. (1994) at 25–90 °C, tentatively suggesting Pd(HS)₂₀(aq) though not excluding Pd(HS)₃⁻ and Pd(HS)₄²⁻ as alternative species. In contrast, molecular dynamics (MD) modeling of

Mei et al. (2015) yielded Pd(HS)₄²⁻ as the stable complex in solution at 300 °C and 500 bar. More recently, similar results were obtained for its Pt analog, Pt(HS)₄²⁻, at similar conditions using a combination of solubility, in situ XAS and MD approaches (Laskar et al., 2022b). Thermodynamic estimations based on linear free energy relationships (Hancock et al., 1977) were also attempted for Pd(HS)₂⁻(1 ≤ n ≤ 4) at 25 °C, with tentative extrapolations to 300 °C (Elliott, 1988; Mountain and Wood, 1988; Wood et al., 1989). The available formation constant values for sulfide complexes present impressive discrepancies, spanning over 5 to 12 orders of magnitude for Pd(HS)₂₀, Pd(HS)₃⁻ and Pd(HS)₄²⁻. Such uncertainties might, at least partly, arise from intrinsic limitations of the solubility techniques applied so far to high-temperature sulfur-rich systems.

This study attempts to resolve the existing discrepancies and to provide more accurate experimental constraints on Pd speciation and solubility in sulfide-bearing fluids. We measured PdS(s) solubility in H₂S-NaOH solutions over a wide T-P, pH and S concentration range (50–300 °C, 90–600 bar, pH 5–8, 0.5–1.2 m S). We used a hydrothermal-flexible-cell reactor, allowing controlled sampling and injection of fluid in the presence of volatile compounds such as H₂S, coupled with sensitive analytical techniques. Such a design, which has not been applied before in PdS(s) solubility studies, should yield a more consistent dataset of solubility measurements within the same experiment as a function of P-T and fluid composition, to investigate the identity and stability of key Pd complexes. Our results enable more quantitative constraints on palladium transport, and help to better interpret Pd vs Pt fractionations observed in hydrothermal systems.

## 2. Materials and methods

### 2.1. Experimental and analytical setups

Palladium sulfide, PdS(s), was used as the solid phase in our solubility experiment. It was synthesized by reacting sulfur and palladium metal powders (>99.9 % purity) in evacuated sealed silica glass tubes at 600 °C (as per Tagirov and Baranova, 2009). The same solid was used in the solubility experiments of Bazarkina et al. (2014) (Fig. S1). To buffer the redox conditions, the mineral assemblage pyrite-pyrrhotite (PP) was used, inspired by the seminal gold solubility study of Seward (1973). Pure pyrite (FeS₂, Aldrich) and troilite (FeS, Aldrich) were loaded into the reactor, together with H₂S-bearing aqueous solution. Its sulfur concentration was large enough (0.5–1 m, see below) to remain almost constant during equilibration in the pyrrhotite-pyrite system, troilite being rapidly transformed to pyrrhotite ensuring good buffer chemical reactivity. The oxygen fugacity fO₂ is thus controlled by the formal equilibrium reaction between pyrrhotite and pyrite (normalized to Fe = 1):

$$ \mathrm{FeS}_{1+x(s)} + (1-x) \mathrm{H}_{2} \mathrm{S}_{(\mathrm{aq})}^{0} + (1-x)/2 \mathrm{O}_{2(\mathrm{aq})}^{0} = \mathrm{FeS}_{2(s)} + (1-x) \mathrm{H}_{2} \mathrm{O} \quad (1) $$

where x < 0.1. This reaction buffers fO₂ at values of several orders of magnitude lower than those in the H₂S/HS⁻/SO₄²⁻ system, thereby ensuring the predominance of H₂S (and HS⁻) and the stability of PdS(s). At such conditions, there are negligibly low concentrations of polysulfide dianions Sₙ²⁻ (<10⁻⁸ mol/kg H₂O; 2 ≤ n ≤ 8), as well as the S₃⁻ radical ion, which are strong complexing agents for Au and Pt (Pokrovski et al., 2015, 2021, 2022). Scanning electron microscopy coupled with energy-dispersive X-ray spectroscopy (SEM-EDS) analyses of PdS(s) before the experiment showed a Pd:S atomic ratio of 1:1, with no other elements detected (Fig. S1). Platinum sulfide (PtS(s), synthetic, as per Laskar et al., 2022b and analyzed by XRD as PtS-h sample in Laskar et al., 2022a), and molybdenite (MoS₂(s), natural, courtesy of M. Kokh) were also present in our experiment. Their solubilities were, however, too low (<10 ppm of the metal) to significantly modify the composition, pH or redox of the fluid. Platinum solubility data have
been interpreted in Laskar et al. (2022b); those of Mo will be the subject of another paper. Metallic gold as the material of the flexible cell was also present in the experiment. The source of sulfur in the fluid was sodium hydrosulfide, NaHS, whose solutions were freshly prepared by bubbling H₂S(g) through a 1 mol/L NaOH aqueous solution until saturation.

The solubility of PdS(s) was measured in the T-P range of 50–300 °C and 90–600 bar using a flexible-cell hydrothermal reactor (Coretest®) allowing controlled changes in T, P and fluid composition, with variations in pH (5–8) and sulfide concentration (0.5–1.2 m), along with accurate fluid sampling (Tables 1 and S2). The reactor, described in detail in Rosenbauer et al. (1993), is equipped with a 100–200 cm³ flexible inner cell made of gold (99.99 % purity) and a rapid sampling device adapted for S-rich fluids (Kokh et al., 2020). Sampling procedures are detailed in Kokh et al. (2017, 2020) and Laskar (2022). The cell was loaded with the solids (PdS, 0.39 g; PtS, 0.15 g; MoS₂, 1.0 g; FeS, 2.0 g; FeS₂, 1.8 g) and the starting aqueous solution (1.04 m NaHS + 0.12 m H₂S(aq)), under argon atmosphere. The cell was compressed to remove the gas phase; the experiment was conducted in the single-phase liquid domain. The fluid was periodically sampled at each temperature step, from 56 up to 300 °C and down to 50 °C at pressures from 90 to 600 bar depending on the step, using a rigid sampling vial (see Tables 1 and S2). When HS⁻ was the main sulfide species in the fluid (Table 2), the solution was also directly sampled with a gas-tight syringe to directly measure the fluid pH. The fluid composition in the reactor was modified by injections (Fig. 1 and Table S2) of pure water to decrease the sulfur concentration, HCl and NaOH solutions to change the pH, and H₂O₂ solutions to increase the oxygen fugacity.

At each step of the solubility experiment, multiple samples (usually 3 or 4, 1–2 g each) of fluid were extracted from the flexible cell. Each sample was destined for a specific chemical analysis (see Kokh et al., 2017, 2020; Laskar, 2022, for a detailed account of analytical procedures). Briefly, sample 1 removed the residual solution in the sampling tube from a previous injection or sampling to avoid cross-contamination (sample 1 was discarded); sample 2 was titrated for reduced sulfur with an excess of iodine (0.5 m I₂) directly placed in the sampling vial; sample 3 was trapped in ultrapure 30 wt% NH₃ aqueous solution to measure total metal (Pd, Pt, Au, Mo) and sulfur concentrations; sample 4 was trapped in a 1 m aqueous zinc acetate (Zn(CH₃COOH)₂) to precipitate ZnS (titrated afterward with I₂, see below) as well as to check for eventual sulfate (SO₄²⁻) in the supernatant. Vials containing ZnS (sample

4) were acidified with HCl (18 wt%) with an excess of I₂, which was back-titrated by sodium thiosulfate (Na₂S₂O₃, 0.1 mol/L) to determine the total sulfide concentration (H₂S(aq) + HS⁻ + NaHS). Sample 3 was split into two fractions, for metals and total sulfur analyses, respectively. Total sulfur was converted to sulfate by addition of 2 mL 30 wt% H₂O₂ aqueous solution. The fraction for the metal analyses was transferred into a clean Teflon beaker (Savillex®) and processed as follows: i) evaporation of the solution in the vial on a hot plate (70 °C); ii) addition of 5 mL aqua regia (1/3 of 70 wt% HNO₃ plus 2/3 of 37 wt% HCl) in the closed beaker (at 120 °C for 3 h); iii) evaporation of aqua regia on a hot plate (70 °C); iv) dilution of the residue with doubly-distilled acid (0.5 wt% HNO₃ plus 1.5 wt% HCl). The use of aqua regia and an HCl-HNO₃ matrix is indispensable to dissolve the poorly soluble PdS(s) precipitates and to keep Pd in solution for analyses as soluble chloride species (PdCl₂⁻). Concentrations of Pd were measured with inductively coupled plasma – mass spectrometry (ICP-MS). Samples for ICP-MS were further diluted in 2 wt% HNO₃ before the analytical session to reduce the Cl⁻ and SO₄²⁻ ions concentration, which is a source of instability, signal degradation with time, and elevated analytical backgrounds in the solution matrix-sensitive ICP-MS analysis. The maximum total dissolved salt (alkali and heavy metals, chloride and sulfur) load did not exceed 100 ppm in our samples, and the total sulfur content did not exceed 5 ppm. An ICP-MS Element XR spectrometer was used, enabling limits of detection (LoD) as low as 1 ppt Pd and limits of quantification (LoQ) of 7 ppt Pd. Two standard solutions (0.19 ± 0.01 and 18.3 ± 0.1 ppb Pd) were used for calibration and showed excellent signal linearity through zero (R²xy > 0.999). Measurements on the ¹⁰⁰Pd isotope were preferred to those on ¹⁰⁴Pd because of the lower LoQ values for the former (7 vs 20 ppt, respectively). For concentrations above LoQ, similar Pd concentrations (within ± 10 %) were found on both isotopes demonstrating the absence of spectral interferences. ICP-OES was used to analyze total sulfur in a matrix of 0.5 wt% HNO₃ – 1.5 wt% HCl. Chloride (Cl⁻) and sulfate (SO₄²⁻) ion concentrations were determined by high-pressure liquid chromatography (HPLC) in water-diluted samples. Sodium (Na⁺) concentrations were measured using flame atomic absorption spectrometry (FAAS) in a matrix of 2 wt% HNO₃. Sodium and chloride concentrations had very small analytical errors (<±2 %) and remained stable during a given step (within ± 5 %). They, therefore, were used to more accurately determine dilution factors from the injections and the resulting fluid composition at each step.

On the termination of the experiment, the remaining solution was

<table><caption>**Table 1**

Different steps of experiment m35 in a flexible cell reactor at the indicated T-P parameters in chronological order.</caption><thead><tr><th>Run/step</th><th>T, °C</th><th>P, bar</th><th>NaHS, m</th><th>H₂S, m</th><th>HCl, m</th><th>NaOH, m</th><th>H₂O₂, m</th><th>Elapsed time from the start for samplings<sup>a</sup>, days</th><th>Injected aqueous solution and elapsed time from the start for injections, days</th></tr></thead><tbody><tr><td>m35/1</td><td>56</td><td>320 ± 5</td><td>1.04</td><td>0.12</td><td>–</td><td>–</td><td>–</td><td>4</td><td>–</td></tr><tr><td>m35/2</td><td>200</td><td>470 ± 40</td><td>1.04</td><td>0.12</td><td>–</td><td>–</td><td>–</td><td>7, 11</td><td>–</td></tr><tr><td>m35/3</td><td>250</td><td>510 ± 70</td><td>1.04</td><td>0.12</td><td>–</td><td>–</td><td>–</td><td>13, 18, 21, 24</td><td>–</td></tr><tr><td>m35/4</td><td>300</td><td>470 ± 50</td><td>1.04</td><td>0.12</td><td>–</td><td>–</td><td>–</td><td>32, 35, 35</td><td>–</td></tr><tr><td>m35/5</td><td>300</td><td>450 ± 30</td><td>1.01</td><td>0.11</td><td>0.29</td><td>–</td><td>–</td><td>42, 42</td><td>HCl (35)</td></tr><tr><td>m35/6</td><td>300</td><td>490 ± 40</td><td>0.81</td><td>0.10</td><td>0.71</td><td>–</td><td>–</td><td>45, 45, 49, 52</td><td>HCl (42)</td></tr><tr><td>m35/7</td><td>250</td><td>400 ± 100</td><td>0.67</td><td>0.09</td><td>0.62</td><td>–</td><td>–</td><td>56, 60, 64, 67, 69</td><td>H₂O (53)</td></tr><tr><td>m35/8</td><td>200</td><td>300 ± 50</td><td>0.51</td><td>0.07</td><td>0.48</td><td>–</td><td>–</td><td>78, 81</td><td>H₂O (70)</td></tr><tr><td>m35/9</td><td>100</td><td>250 ± 5</td><td>0.51</td><td>0.07</td><td>0.48</td><td>–</td><td>–</td><td>89, 92, 95</td><td>–</td></tr><tr><td>m35/10</td><td>50</td><td>260 ± 10</td><td>0.51</td><td>0.07</td><td>0.48</td><td>–</td><td>–</td><td>106, 109, 113</td><td>–</td></tr><tr><td>m35/11</td><td>50</td><td>440 ± 5</td><td>0.51</td><td>0.07</td><td>0.48</td><td>–</td><td>–</td><td>120, 131</td><td>–</td></tr><tr><td>m35/12<sup>b,c</sup></td><td>50</td><td>90 ± 10</td><td>0.51</td><td>0.07</td><td>0.48</td><td>–</td><td>–</td><td>138, 147</td><td>–</td></tr><tr><td>m35/13</td><td>50</td><td>250 ± 5</td><td>0.44</td><td>0.06</td><td>0.41</td><td>0.04</td><td>–</td><td>162, 168</td><td>NaOH (153)</td></tr><tr><td>m35/14<sup>c</sup></td><td>50</td><td>260 ± 5</td><td>0.15<sup>d</sup></td><td>0.05</td><td>0.35</td><td>0.03</td><td>0.29</td><td>175, 180</td><td>H₂O₂ (169)</td></tr><tr><td>Error</td><td>±1</td><td></td><td>±10 %</td><td>±10 %</td><td>±5%</td><td>±5%</td><td>±5%</td><td></td><td></td></tr></tbody></table>

m denotes the number of moles of each compound (NaHS, H₂S, HCl, NaOH and H₂O₂) per kg of water (mol/kg H₂O); the number in m35/(1 to 14) represents the chronological order of the corresponding step; P is reported as the mean value with 1 standard deviation accounting for pressure variations during a given step; see Table S2 for analytical details.

a Samples used to calculate the reported average mPd, mean value (see Table 2).

b Suspected cell leak at this step.

c Not considered in the thermodynamic analysis.

d Injection of H₂O₂ solution results in the formation of solid sulfur S₈ (0.18 m) in the system.
<caption>**Table 2**</caption>

Palladium concentrations and thermodynamically modeled fluid compositions in the H₂O-NaHS-PdS-PtS-FeS-FeS₂ system at the indicated T-P parameters during the different steps of experiment m35 in a flexible cell reactor.

<table><thead><tr><th>Run/step</th><th>T, °C<sup>a</sup></th><th>P, bar<sup>a</sup></th><th>log<sub>10</sub>(a(H<sub>2</sub>S))<sup>b</sup></th><th>pH<sup>b</sup></th><th>log<sub>10</sub>f<sub>O<sub>2</sub></sub><sup>b</sup></th><th>log<sub>10</sub>(mPd<sub>mean</sub>)</th><th>log<sub>10</sub>K<sub>s4,Pd</sub></th></tr></thead><tbody><tr><td>m35/1</td><td>50</td><td>300</td><td>-1.32</td><td>7.8</td><td>-78</td><td>-5.6</td><td>-18.0</td></tr><tr><td>m35/10</td><td>50</td><td>300</td><td>-0.33</td><td>5.3</td><td>-80</td><td>-7.7</td><td>-17.2</td></tr><tr><td>m35/11</td><td>50</td><td>300</td><td>-1.15</td><td>5.3</td><td>-80</td><td>-7.3</td><td>-18.0</td></tr><tr><td>m35/12</td><td>50</td><td>300</td><td>-0.33</td><td>5.4</td><td>-80</td><td>-6.5</td><td>-<sup>c</sup></td></tr><tr><td>m35/13</td><td>50</td><td>300</td><td>-0.45</td><td>5.8</td><td>-80</td><td>-6.7</td><td>-17.6</td></tr><tr><td>m35/9</td><td>100</td><td>300</td><td>-0.33</td><td>5.0</td><td>-67</td><td>-7.2</td><td>-16.9</td></tr><tr><td>m35/2</td><td>200</td><td>500</td><td>-1.03</td><td>7.1</td><td>-49</td><td>-6.3</td><td>-18.5</td></tr><tr><td>m35/8</td><td>200</td><td>500</td><td>-0.33</td><td>5.0</td><td>-50</td><td>-7.3</td><td>-17.2</td></tr><tr><td>m35/3</td><td>250</td><td>500</td><td>-0.98</td><td>7.3</td><td>-42</td><td>-6.4</td><td>-19.3</td></tr><tr><td>m35/7</td><td>250</td><td>500</td><td>-0.21</td><td>5.3</td><td>-43</td><td>-6.7</td><td>-18.0</td></tr><tr><td>m35/4</td><td>300</td><td>500</td><td>-0.95</td><td>7.5</td><td>-36</td><td>-6.3</td><td>-19.9</td></tr><tr><td>m35/5</td><td>300</td><td>500</td><td>-0.44</td><td>6.8</td><td>-37</td><td>-6.5</td><td>-20.3</td></tr><tr><td>m35/6</td><td>300</td><td>500</td><td>-0.13</td><td>5.7</td><td>-38</td><td>-7.0</td><td>-19.3</td></tr><tr><td>Error</td><td></td><td></td><td>±0.04</td><td>±0.1</td><td>±1</td><td>±0.1</td><td>±0.5</td></tr></tbody></table>

<sup>a</sup> Temperature and pressure values used in the thermodynamic analysis.

<sup>b</sup> pH, f<sub>O<sub>2</sub></sub> and H₂S(aq) activities are calculated from the composition of the experiment at different steps using the HCh software.

<sup>c</sup> Not calculated because not considered in thermodynamic analysis.

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Concentrations of dissolved Pd and Pt in the fluid of experiment m35 (H₂O-NaHS-HCl-NaOH) as a function of time. The solid phases at equilibrium with the fluid are PdS(s) and PtS(s). The injections (inj HCl, H₂O, NaOH, H₂O₂) are indicated by red vertical dashed lines. The temperature changes are indicated by grey vertical dashed lines. Experimental Pd concentrations below the limit of detection (LoD) are plotted as their LoD value (empty down-head arrows) and experimental Pd concentrations below the limit of quantification (LoQ) are plotted as their LoQ value (filled down-head arrows). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

filtered and the recovered solid was washed 3 times with deionized water, dried at 50 °C, and analyzed by powder X-ray diffraction (XRD) and SEM, demonstrating well-defined peaks for molybdenite, pyrite, sulfur and gold (Fig. S2). Only very small peaks of PdS(s) and PtS(s) were observed because of their low initial amount compared to the other solids (<5 wt%). The peaks of PdS(s) overlapped with those of pyrrhotite, which are also very small because the increase in oxygen fugacity during the final step (m35/14, Fig. 1) caused pyrrhotite to convert to pyrite. Despite these difficulties of the PdS(s) and PtS(s) composition analysis after our solubility experiment, the following independent arguments point in favor of their stability and stoichiometry. First, the excellent agreement with the solubility data for PtS(s) measured independently using a XAS hydrothermal cell in which only PtS(s) was present as the solid phase (Laskar et al., 2022b). Second, the very low solubility of both solids in our experiment, along with the intrinsically slow diffusion kinetics in the solid phase at such moderate temperatures (<300 °C), prohibits any formation of significant (Pt,Pd)S solid solutions within the

experimental duration. Third, it has been shown that even at much higher temperatures (1000 °C), Pd and Pt behave incompatibly during the formation of monosulfide solid solution from sulfide liquid, showing a weak affinity to Fe sulfide solids (Li et al., 1996). Finally, the presence of up to 10,000 ppm of one element in the sulfide solid of another would have a negligible effect on the solid-phase activity (e.g., from a(PdS) = 1 in the pure solid to a(PdS) ≈0.99 in a PdS solid containing 1 wt% of Pt). Such a small change in activity has a negligible effect on the PdS(s) solubility.

## 2.2. Thermodynamic modeling of solubility data

Equilibrium thermodynamics calculations were used to quantify the fluid chemical speciation and the solubility-controlling parameters (pH, f<sub>O<sub>2</sub></sub>, the activity of sulfur ligands). The sources of thermodynamic data used in these calculations are reported in Table S1. Data for major aqueous species were taken from the updated version of the Supcrt92
database (Johnson et al., 1992; Oelkers et al., 2009), based on the revised and extended Helgeson-Kirkham-Flowers model (HKF; Tanger and Helgeson, 1988) for the common fluid species such as Na⁺, Cl⁻, OH⁻, H⁺, NaClO₃⁻, NaOH₃⁻ (Sverjensky, 2019), and HClO₃⁻ (Tagirov et al., 1997). The properties of aqueous volatile non-electrolyte species (H₂Sₐ₃⁻, SO₃²⁻ (aq), H₂O₃⁻ (aq), and O₃²⁻ (aq)) were adopted according to the equation of state of Akinfiev and Diamond (2003), which performs better than HKF (Supcrt92) for low to moderate pressures (<1 kbar). Recent HKF sets of parameters were taken for polysulfide dianions (S₂⁻, 2 ≤ n ≤ 8; Barré et al., 2017) and S₃²⁻ radical ion (Pokrovski and Dubessy, 2015). The NaHS₀₃⁻ ion pair, which is significant at pH > 7, was also included assuming that its dissociation constant is equal to that of NaClO₃⁻ at all T-P. The PdS(s) phase shows large disparities in its thermodynamic data, as overviewed by Bazarkina et al. (2014). For example, Tagirov et al. (2013) reported Gibbs energy of formation from elements (ΔfO₂₉₸K. 1bar) values (–82 kJ/mol) of 10–15 kJ/mol lower compared to the other studies (–67 kJ/mol, Sassani and Shock, 1998; –72 kJ/mol, Karzhavin, 2007; –68 kJ/mol, Bazarkina et al., 2014). This difference translates to a PdS(s) solubility difference of more than 1 log unit. In light of these discrepancies, we believe that Bazarkina et al.'s (2014) thermodynamic data set for PdS(s) is the most coherent one because it is based on in situ solubility measurements of both PdO and PdS solids in the same saline fluids, in which PdCl₂⁻ was the dominant species at 350 and 450 °C as directly constrained by X-ray absorption spectroscopy. The same PdS solid was used in our study adding additional consistency to the solubility data. The HCh software (Shvarov, 2008) was used for equilibrium thermodynamics calculations. The standard state of solid phases and H₂O is unit activity at all temperatures and pressures; the standard state of aqueous species is 1 m aqueous solution whose behavior is ideal; activity coefficients of charged species are calculated using the extended Debye-Hückel equation (Shvarov, 2008), those of neutral species are assumed to be 1. The module OptimA connected to HCh (Shvarov, 2015) was used to identify the possible dominant Pd-HS species by optimizing the given species Gibbs energy by regression of the experimental datapoints of Pd equilibrium concentrations obtained at a given T-P condition and taking into account the datapoint relative weights (equal to 1/σ, where σ is the standard deviation in log units). The overall error on the resulting reaction constants of PdS(s) solubility as Pd-HS complexes is within 0.5–1.5 log units. This error stems both from chemical analyses of intrinsically low Pd concentrations, the experimental datapoint scatter, and the uncertainties in the thermodynamic properties (see also Pokrovski et al., 2015, 2021, for a more detailed assessment).

## 2.3. Thermodynamic analysis of literature data

The interpretation of PdS(s) solubility data in terms of Pd-HS complexes as a function of pH, redox and sulfide ligand activity is strongly dependent on the exact values of those variables. These values stem from various thermodynamic data sources that differed among the previous studies. To provide more thermodynamically consistent comparisons between the stability constants of Pd-HS complexes derived in this study and in previous work, the same thermodynamic framework must be used. Using our set of thermodynamic data sources (Table S1), we reanalyzed the large set of PdS(s) solubility data acquired in the seminal work of Gammons and Bloom (1993) with the help of robust thermodynamic analysis tools (HCh and OptimA; Shvarov, 2015), which were not available at the time of their original work. These authors measured PdS(s) solubility, in silica tubes quenched after the experiment, between 200 and 300 °C at Psat in NaHS-NaOH-HCl aqueous solutions with sulfide concentrations of 0.4–1.0 m in the pH range 6.4–8.4. Their conditions were thus very similar to those used in our study. The set of solubility datapoints from Gammons and Bloom (1993) at each temperature (16, 13 and 14 datapoints at 200, 250, and 300 °C, respectively), has been processed, using the OptimA software, in terms of Pd-

HS species stoichiometry found in our experiments (in particular Pd (HS)₂⁻, see section 3), to derive species stability constants. In our calculations, we also took into account the amount of H₂S in the vapor phase occupying 20 to 30 % of the total experimental silica tube volumes used by these authors in their experiments (as per our discussion with C. Gammons).

No attempt was made to reprocess, in a similar way, the data of Pan and Wood (1994), Wood et al. (1994), Tagirov and Baranova (2009), and Tagirov et al. (2013) because of their different conditions (and resulting speciation). The studies of Wood et al. (1994), Tagirov and Baranova (2009) and Tagirov et al. (2013) concerned much more dilute H₂S solutions (<0.1 m) in which low ligation number complexes (Pd(HS)₂₀₃⁻ (aq) and Pd(HS)₃) might also contribute. Pan and Wood (1994) employed sulfatesulfide solutions in which Pd complexes with the trisulfur radical ion or polysulfide dianions might make a significant contribution. To provide closer comparisons with our speciation scheme, we manually recalculated their reaction constant values using the reported Pd concentrations, H₂Sₐ₃⁻ activities and pH (from Table 4 in Pan and Wood, 1994), because their original K values likely contained misprints. Among the theoretical sources of data, those from Mei et al. (2015) were also compared with our experimental results. The past theoretical estimations of Wood et al. (1989) were disregarded, because they differ significantly from the experiments of Wood's own group that followed (Pan and Wood, 1994; Wood et al., 1994). The values from the previous studies are summarized in Table S3 and compared in section 4.

## 3. Results

### 3.1. Results from solubility experiments in flexible-cell reactor

At each corresponding step of the solubility run, reduced sulfur concentrations analyzed with iodine and zinc acetate were identical within uncertainties (Table S2), demonstrating that sulfur forms other than sulfide were negligible in solution. The total aqueous sulfur concentration (Stot) was also similar to that of sulfide within analytical errors (±10 %). Both pH and fO₂ equilibrated within < 2 days as was monitored by the Au concentration changes sensitive to those parameters (see Laskar et al., 2022b for more details). The gold concentrations rapidly attained a steady state at all experimental conditions, in good agreement with Au(HS)₂ predicted concentrations in equilibrium with metallic gold (Pokrovski et al., 2014).

The PdS(s) solubility results are reported in Tables 1, 2 and S2 and Fig. 1. Measured Pd concentrations range between 0.05 and 2.7 (×10⁻⁶) m (Fig. 1). They show systematic variations with changes in temperature (50–300 °C), pH (5–8), and sulfide concentration (0.5–1.2 m). Within a given step, a steady state was attained after a few days following the temperature change or the modification of the fluid composition via injection, suggesting that fluid equilibrium is likely to be reached with PdS(s). In our thermodynamic modeling, the Pd concentrations below the LoD were assumed equal to this limit, and the concentrations that fell between the LoD and LoQ were taken equal to the LoQ value (Fig. 1 and Table S2). Our thermodynamic analysis (see below) thus defines the upper limit of Pd solubility. The pressure effect on PdS(s) solubility was undetectable within the analytical errors, which is consistent with the very low compressibility of aqueous solutions at moderate temperatures (<250 °C). Pressure values are taken equal to 500 bar in our thermodynamic analysis at 200–300 °C and 300 bar at 50–100 °C (Table 2), for consistency with monitored pressure values (Tables 1, S2). The concentrations of Pt, detailed in Laskar et al. (2022b), were about 1 log unit higher than those of Pd (Fig. 1). In step m35/14 at 50 °C and 260 bar, hydrogen peroxide was injected to examine the effect of redox on PdS(s) solubility. Thermodynamic calculations predicted native sulfur precipitation, in agreement with sulfur loss found by fluid analyses, and an fO₂ value increase by 15 log units (log₁₀fO₂ = −65). Palladium concentrations in the fluid increased by 1 to 2 orders of magnitude compared to the more reduced previous steps (log₁₀(mPdmean) = −5.9). This increase
would suggest the formation of additional complexes, most likely with polysulfide ions produced via H₂S(aq) oxidation, which may be important ligands for Pd, as they are for Pt (e.g., Pokrovski et al., 2021). This solubility step was ignored in the derivation of stability constants for hydrosulfide complexes (described below).

## 3.2. Choice of the main complex stoichiometry

The solubility results of this study and that of Gammons and Bloom (1993) were analyzed in terms of Pd(HS)₂⁻(aq), Pd(HS)₃⁻ or Pd(HS)₄²⁻, to identify the major complexes in the fluid at m(sulfide) = 0.4–1.0 m, pH = 5–8 and T = 50–300 °C (Figs. S3 and S4). The standard deviation (σ, i.e. a fit-quality measure), between experimental and calculated Pd solubility (log₁₀mPdᵢexp-log₁₀mPdᵢcalc) at the different temperatures, both for our experiment and those of Gammons and Bloom (1993), is reported in Table S4. The Pd(HS)₂⁻(aq) complex does not match the combined datasets. Discriminating between Pd(HS)₃⁻ and Pd(HS)₄²⁻ is more difficult, given the σ values obtained. Both Pd(HS)₂⁻ and Pd(HS)₃⁻ show comparable σ values without clear systematics. Including both complexes in modeling simultaneously results, however, in a statistical overfit and, more importantly, yields inconsistent dependences on pH and H₂S concentration. Therefore, a single dominant complex would be more likely. These results can also be viewed using the slope analysis method (Fig. 2), which provides a rough graphical representation of the stoichiometry of the major complex(es) at a given temperature (provided that experimental points at different pH and/or H₂S concentration values are available). The slope is expected to be indicative of the main complex present in the fluid; with ideal values of 1 for Pd(HS)₂⁻(aq), 2 for Pd(HS)₃⁻, or 3 for Pd(HS)₄²⁻ (see Gammons and Bloom, 1994, for more details). In these graphs, both a(H₂S(aq)) and pH were calculated using thermodynamic data (see Table 2). However, it can be seen in Fig. 2 that the uncertainties associated with the slope values that amount to almost 50% of the slope value itself prevent any quantitative assessment. Thus, the slope analysis, within the available data point statistics, appears insufficient (if not misleading) to discriminate between Pd(HS)₃⁻ and Pd(HS)₄²⁻.

Given the impossibility of the species discrimination between Pd(HS)₃⁻ and Pd(HS)₄²⁻ on rigorous statistical criteria, stability constants for each complex tentatively assumed to be the dominant one are reported in Table S3. However, we believe that several independent arguments strongly point in favor of Pd(HS)₄²⁻. First, the MD theoretical work of Mei et al. (2015) independently supports the conclusion that Pd(HS)₄²⁻ is the dominant species. Their values of constants at 300 °C and 500 bar, estimated using the free energy integration method, imply that tetrahydrosulfide dominates over trihydrosulfide at HS⁻ concentrations above 10⁻⁵ m in a wide pH range, which covers most natural and experimental fluid compositions examined here and in previous work. The dihydrosulfide complex appears to be at least 6 orders of magnitude less abundant than trihydrosulfide at such conditions. Second, both PdCl₄²⁻ and Pt(HS)₄²⁻ complexes are the major species in the temperature range of 25–350 °C according to in situ XAS experiments (Bazarkina et al., 2014; Laskar et al., 2022b). The same tetrahydrosulfide species is expected to be the most plausible one for Pd because of the same Pt¹¹ and Pd¹¹ outer shell electronic configuration (Cotton et al., 1988). Third, the chemical literature survey demonstrates that both Pt and Pd stable sulfide complexes in inorganic and metal–organic compounds are square planar with HS (or S) ligands or mixed HS with other soft ligands such as of amine and phosphine-types, rather than of O-type (Pluth and Tonzetzich, 2020). The lack of stable HS-Pd-O(OH, H₂O) configurations in organometallic chemistry of Pd suggests that complexes such as Pt(HS)₂(H₂O)₂ and Pt(HS)₃(H₂O) hypothesized in hydrothermal fluids should not be stable. Fourth, the analogous chalcophile metals, such as Mo (MoS₄²⁻; Liu et al., 2020) or Zn (Zn(HS)₄²⁻; Tagirov and Seward, 2010) show the most stable sulfide complexes when the metal is fully coordinated by sulfide ligands, thus lending additional independent support for our tetrasulfide stoichiometry for both Pd et Pt. Finally, the high stability of complexes that are symmetrical and fully coordinated with the same HS⁻ ligand, at least at temperatures below 350 °C, is in line with i) the strong chemical affinity of the metal to reduced sulfur ligands, and ii) the relatively high dielectric constant of aqueous solution at such temperatures (ε > 10) stabilizing negatively charged species

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Slope analysis of the PdS(s) solubility data. The H₂S(aq) activity and pH values are calculated for each experimental datapoint (see Table 2). The average slopes are expected to illustrate the stoichiometry of the dominant hydrosulfide Pd complex: ideally 1 for Pd(HS)₂⁻(aq), 2 for Pd(HS)₃⁻ or 3 for Pd(HS)₄²⁻. However, the intrinsic errors in slope determination do not allow reliable discrimination between Pd(HS)₃⁻ and Pd(HS)₄²⁻. Note that such level of slope uncertainties is equally typical of previous solubility studies. Therefore, we have preferred an alternative, a more statistically sound method, using the sum of deviations of each experimental datapoint value with its analytical uncertainty from the calculated solubility value using a given Pd complex (see Table S4 and Figs. 3 and S4).</figcaption></figure>
of high symmetry (e.g., Brimhall and Crerar, 1987). These arguments collectively point out that Pd(HS)₂⁻, rather than Pd(HS)₃⁻, is the most plausible major complex in the H₂S/HS⁻ fluids investigated in this and previous studies, over a wide T-P range, at least to 350 °C, and is used for further calculations.

### 3.3. Derivation of the stability constant

Based on the arguments above, we derived the stability constant of Pd(HS)₂⁻ from PdS(s) solubility data in the temperature range 50–300 °C (Table 3):

$$ \text{PdS}_{\text{(s)}} + 3 \text{H}_2\text{S}_{\text{(aq)}}^0 = \text{Pd(HS)}_4^{2-} + 2 \text{H}^+, \text{K}_{\text{s4,Pd}} \quad (2) $$

This stability constant of Pd(HS)₂⁻ allows the generation of a self-consistent set of thermodynamic properties in the framework of the revised HKF equation of state. The derivation procedure is resumed in 4 steps.

First, for each temperature, the optimized value of the apparent molal Gibbs energy (G⁰ₜₚ) of the hydrosulfide complex is determined from experimental solubility measurements by fitting the solubility data at each experimental temperature using the OptimA software (Fig. 3). Then, the stability constant of reactions K₅₄,Pd (Eqs. (2)) is calculated from this Gibbs energy and those of the other reaction constituents from thermodynamic data sources reported in Table S1. The differences

<caption>**Table 3**</caption>

Values of the equilibrium constants of PdS(s) dissolution reaction to form the Pd (HS)₂⁻ complex at the indicated T-P conditions derived in this study.

<table><thead><tr><th rowspan="2">T, °C</th><th rowspan="2">P, bar</th><th>log<sub>10</sub>K<sub>s4,Pd</sub> (T,P)</th></tr><tr><th>PdS<sub>(s)</sub> + 3 H<sub>2</sub>S<sub>(aq)</sub><sup>0</sup> = Pd(HS)<sub>4</sub><sup>2-</sup> + 2H<sup>+</sup></th></tr></thead><tbody><tr><td>50</td><td>300</td><td>-17.7 ± 0.6</td></tr><tr><td>100</td><td>300</td><td>-16.9 ± 0.8</td></tr><tr><td>200</td><td>500</td><td>-18.4 ± 0.6</td></tr><tr><td>250</td><td>500</td><td>-19.0 ± 0.6</td></tr><tr><td>300</td><td>500</td><td>-19.9 ± 0.6</td></tr></tbody></table>

Uncertainties correspond to the standard deviation determined from the Gibbs energy value variation in the fit performed with the OptimA software, taking into account the experimental uncertainties (±1σ).

between experimental and regressed solubility datapoints at 50, 100, 200, 250 and 300 °C do not exceed 1 log unit for most points. Such a deviation may be considered a decent agreement in light of the measured ppb-level absolute Pd concentrations.

Second, a logK-T(K) regression, using a weighted least-squares method that takes into account each experimental datapoint uncertainty, is applied to the log₁₀K₅₄,Pd values of reaction (2). A 3-parameter function was found to yield the most stable fit of the derived log₁₀K₅₄,Pd values as a function of the absolute temperature between 50 and 300 °C:

$$ \log_{10} K_{s4,Pd} = A + B/T + C \times \log_{10} T \quad (3) $$

where T is the temperature in Kelvin, and A, B and C are T-P independent regression coefficients. Only 3 parameters were considered because of the limited temperature range and uncertainties in log₁₀K₅₄,Pd-values. The effect of pressure was ignored in this analysis because no detectable changes in Pd solubility were observed at a given T, but at different P, from 100 to 500 bars (Table S2). Note that the 3-parameter equation adopted in this study also provides a good match of the K₅₄,Pd values from Pan and Wood's (1994) and Gammons and Bloom's (1993) work, with a maximum between 50 and 200 °C (Fig. 4). Note also that equation (3) provides a statistically better fit of the data within the range and uncertainties of the experimental dataset compared to a simple linear fit (log₁₀K₅₄,Pt = A + B/T), which was previously used by Laskar et al. (2022b) for the analogous reaction of formation of the Pt(HS)₂⁻ complex by PtS(s) dissolution:

$$ \text{PtS}_{\text{(s)}} + 3 \text{H}_2\text{S}_{\text{(aq)}}^0 = \text{Pt(HS)}_4^{2-} + 2 \text{H}^+, \text{K}_{\text{s4,Pt}} \quad (4) $$

For consistency, a similar 3-parameter fit was also used in this study to refit the data of Laskar et al. (2022b) (Fig. S5):

$$ \log_{10} K_{s4,Pd} = (196.4 \pm 60.0) - (11384 \pm 3567)/T(K) - (71.24 \pm 19.52) \times \log_{10} T(K) \quad (5) $$

$$ \log_{10} K_{s4,Pt} = (119.0 \pm 47.0) - (7259 \pm 2789)/T(K) - (45.27 \pm 15.28) \times \log_{10} T(K) \quad (6) $$

The regression equations (5) and (6) correspond to the curves shown in Figs. 4 and S5, respectively.

Third, these equations are used to extrapolate logK₅₄,Pd and logK₅₄,Pt

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

Standard molal thermodynamic properties at 25 °C and 1 bar and parameters of the revised HKF equation of state for Pd(HS)₂⁻ and Pt(HS)₂⁻ complexes retrieved in this study.

<table><thead><tr><th>Thermodynamic property</th><th>Pd(HS)<sub>4</sub><sup>2-</sup></th><th>Pt(HS)<sub>4</sub><sup>2-</sup></th></tr></thead><tbody><tr><td>Gibbs energy, Δf<sub>G</sub><sup>0</sup>, cal/mol <sup>a</sup></td><td>-11724</td><td>-14980</td></tr><tr><td>Entropy, S<sup>0</sup>, cal/(mol K) <sup>a</sup></td><td>52.919</td><td>45.303</td></tr><tr><td>Heat capacity, C<sub>p</sub><sup>0</sup>, cal/(mol K) <sup>a</sup></td><td>-3.050</td><td>48.502</td></tr><tr><td>Volume, V<sup>0</sup>, cm<sup>3</sup>/mol <sup>a</sup></td><td>125.4</td><td>126.9</td></tr><tr><td>a<sub>1</sub> × 10<sup>-2</sup>, cal/(mol bar) <sup>b</sup></td><td>19.729</td><td>19.9782</td></tr><tr><td>a<sub>2</sub> × 10<sup>-2</sup>, cal/mol <sup>b</sup></td><td>40.3900</td><td>40.9985</td></tr><tr><td>a<sub>3</sub>, cal K/(mol bar) <sup>b</sup></td><td>-10.1248</td><td>-10.364</td></tr><tr><td>a<sub>4</sub> × 10<sup>-4</sup>, cal K/mol <sup>b</sup></td><td>-4.4487</td><td>-4.4739</td></tr><tr><td>c<sub>1</sub>, cal/(mol K) <sup>b</sup></td><td>-6.4810</td><td>50.4082</td></tr><tr><td>c<sub>2</sub> × 10<sup>-4</sup>, cal K/mol <sup>b</sup></td><td>12.5693</td><td>10.5321</td></tr><tr><td>ω × 10<sup>-5</sup>, cal/mol <sup>c</sup></td><td>2.40</td><td>2.53</td></tr></tbody></table>

<sup>a</sup> Calculated from reaction (2) and an equivalent reaction for Pt, using the Δf<sub>G</sub><sup>0</sup>, Δf<sub>H</sub><sup>0</sup>, S<sup>0</sup> and C<sub>p</sub><sup>0</sup> values at 298.15 K and 1 bar of the reaction constituents from the sources cited in Table S1. Typical uncertainties are ± 1000 cal/mol, ±30 cal/mol K, and ±40 cal/mol K for Δf<sub>G</sub><sup>0</sup> (and Δf<sub>H</sub><sup>0</sup>), S<sup>0</sup> and C<sub>p</sub><sup>0</sup> values, respectively. The reaction (2) volume change was assumed to be 0 in the absence of detectable pressure dependence of the solubility.

<sup>b</sup> HKF parameters were generated from G<sub>T,P</sub><sup>0</sup> values calculated from experimental reaction constants in Table 3 for Pd(HS)₂⁻ (this study) and Pt(HS)₂⁻ (Laskar et al., 2022b) and using HKF model correlations for ionic complexes (Sverjensky et al., 1997) as implemented in the OptimB program (Shvarov, 2015). Parameters c₁ and c₂ were optimized while keeping the theoretical relation C<sub>p</sub><sup>0</sup> = f(c₁, c₂).

<sup>c</sup> Calculated based on the S<sup>0</sup>–ω correlation equation (40) in Sverjensky et al. (1997). Note that the use of the default equation (39) implemented in OptimB, yields negative, i.e. unphysical, ω values that are in contradiction with the fundamental Born equation for ion solvation energy in the HKF model formalism.
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Difference between $PdS_{(s)}$ solubility measured in our experiments (exp.) and $PdS_{(s)}$ solubility modeled with $Pd(HS)_{2}^{2-}$ as the main complex in solution (calc.) as a function of pH at the indicated temperatures and pressures. Statistical analysis of these differences is given in Table S4.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Values of $log_{10}K_{s4,Pd}$ as a function of $1000/T$ for reaction (2). Data of this study were derived from results of solubility experiment m35 at 300 °C, 500 bar; 250 °C, 500 bar; 200 °C, 500 bar; 100 °C, 300 bar and 50 °C, 300 bar; then fitted using the equation $log_{10}K_{s4,Pd} = (196.4 + 60.0) - (11384 \pm 3567)/T(K) - (71.24 \pm 19.52) \times log_{10}T(K)$ (dashed line). Datapoints from other studies were not considered in this regression to maintain coherence, because of their much larger associated uncertainties. The X-axis positions of the literature datapoints are slightly shifted horizontally from their nominal temperature values to avoid error bar overlap.</figcaption></figure>

values to 25 °C and calculate the $\Delta_{f}G^{0}_{T,P}$, as well as the $\Delta_{f}S$ and $\Delta_{f}C_{p}$ value as derivatives, for reactions (5) and (6) at the reference conditions of 25 °C and 1 bar (see details in Pokrovski et al., 1995; Pokrovski and

Dubessy, 2015). The reaction volume change was assumed to be zero given the absence of detectable solubility dependence on pressure. Combining these values with the known corresponding thermodynamic functions (entropy $S^{0}$, heat capacity $C_{p}^{0}$, volume $V^{0}$) of the other reaction constituents, well-constrained thermodynamic functions of each hydrosulfide complex at the reference T-P of 25 °C and 1 bar were produced.

Finally, these values were used to derive the HKF parameters for each complex using the OptimB software that regresses the apparent molal Gibbs energy ($G^{0}_{T,P}$) values of the complex (obtained in step 1) and integrates the HKF parameter correlations depending on the type of species (Shvarov, 2015). The $a_{1-4}$ and $\omega$ HKF parameters were determined by using the established correlations for negatively charged metal complexes (Sverjensky et al., 1997), whereas the $c_{1}$ and $c_{2}$ parameters were optimized while keeping the theoretical $C_{p}(c_{1},c_{2})$ relationship by regressing $G^{0}_{T,P}$ values of the complex (see Table 4 for details).

The procedure of deriving and constraining the species standard thermodynamic functions at 25 °C, 1 bar described above allows eliminating the inevitably large errors on thermodynamic parameters that would have occurred if the HKF equation were directly applied to regress the $G^{0}_{T,P}$ values of the complex, due to our relatively low number of datapoints. This is because the HKF parameters ($a_{1-4}$, $c_{1-2}$, and $\omega$) are bound by correlation equations to the $S^{0}$, $C_{p}^{0}$ and $V^{0}$ values at reference conditions of 25 °C and 1 bar. Constraints on those values are necessary to produce both consistency and stability in the HKF parameters values, by avoiding the generation of unrealistic thermodynamic functions at 25 °C, 1 bar and resulting unphysical HKF parameters for aqueous complexes (see Pokrovski et al., 2024 for numerical examples). Given the uncertainties associated both with the experimental $K_{s4}$ values and
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Values of $log_{10}K_{8,4}$ as a function of 1000/T for reactions (2) and (4). Symbols show the data derived from this study for $PdS_{(s)}$ solubility (red diamonds) and from Laskar et al. (2022b; La22) for $PtS_{(s)}$ solubility (blue squares). Curves are calculated at 500 and 1000 bar using the HKF equation of state parameters from Table 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

the HKF correlations, the generated parameters allow prediction of the $G_{T,P}^0$ values of $Pd(HS)_4^-$ and $Pt(HS)_4^-$ within $\pm$ 10 kJ/mol over the T-P range covered by the experimental data (25–300 °C, $\le$600 bar), and within $\pm$ 30 kJ/mol to 500 °C and 2 kbar (Fig. 5).

# 4. Discussion

## 4.1. Stability constants of Pd hydrosulfide complexes

Overall, we would like to emphasize that the extraction of species stoichiometries from solubility data in such complex systems with multiple variables is an unmanageable task without a priori constraints from independent approaches such as in situ spectroscopy, fundamental coordination chemistry or molecular modeling. Note that Pd-HS complexes concentrations being low, prohibit an easy isolation of such species or crystallization of their corresponding solids – the approach commonly adopted in coordination chemistry (e.g., Pluth and Tonzetich, 2020). We emphasize that it is difficult to discriminate between $Pd(HS)_3^-$ and $Pd(HS)_4^-$ based on our bulk solubility data alone, but according to the independent arguments above (see section 3.2), the tetrahydrosulfide complex, $Pd(HS)_4^{2-}$, is very likely to be favored. The equilibrium constants of this and other Pd-HS species reported in the literature are summarized in Table S3 and compared in terms of $PdS_{(s)}$ solubility in Fig. 6.

The Pd speciation in terms of $Pd(HS)_4^{2-}$ adequately reconciles the Gammons and Bloom (1993) and Pan and Wood (1994) experimental results, with a difference between $log_{10}K_{8,4,Pd}$ values less than 1 log unit at 250 and 300 °C, and 2 log units at 200 °C. In particular, our results are also nearly identical (within < 0.5 log unit) to those of Gammons and Bloom (1993) at 200 and 250 °C, even though the experimental approach is different in the two studies: we used a single experiment for all the T-P conditions and fluid compositions, in addition, our systems also contained other metals (Pt, Mo, Au), allowing consistent comparisons and reducing systematic errors. Gammons and Bloom (1993) used an individual silica tube for each T-P-composition datapoint and each chemical element (Pt or Pt), which might be prone to more dispersion that was partly compensated by a larger number of datapoints.

In contrast, our stability constants at $T \le 100$ °C are up to 10 log units lower than those of Wood et al. (1994), assuming $Pd(HS)_4^{2-}$ to be the major Pd-HS complex. This discrepancy is harder to reconcile. Among the possible factors, there are analytical issues, in the first place, when measuring Pd concentrations in previous low-temperature experiments. In addition, lower $H_2S$ concentrations ($<0.1$ m) and more alkaline pH

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Solubility of $PdS_{(s)}$ in aqueous solution of 0.7–1.0 m total sulfide ($H_2S + HS^-$) as a function of pH at 300 °C and 500 bar shown by the experimental points of this study (symbols). These data are compared with solubility predictions of this study according to reaction (2) (solid red curve) and those of Gammons and Bloom (1993) (GB93), Tagirov et al. (2013) (Ta13) and Mei et al. (2015) (Me15), all calculated for an average sulfide concentration of 0.90 m (dotted or dashed curves) with the solubility constants for $PdS_{(s)}$, $PdS_{(s)} + (n-1) H_2S_{(aq)}^0 = Pd(HS)_n^-$, $n + (n-2) H^+; 2 \le n \le 4$. Redox is buffered by the pyrite–pyrrhotite–$H_2S$ equilibrium ($log_{10}f_{O_2}(bar) \approx -38$ at 300 °C). Solubility predictions as per Pan and Wood's (1994) originally reported constants were omitted, as judged to be in error. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

$(>10)$ in Wood et al.'s study might favor other sulfide complexes such as $Pd(SH)_2^-$ as suggested in other work (e.g., Tagirov and Baranova, 2009). For example, $Pd(HS)_3^-$ was tentatively chosen as the dominant species by Gammons and Bloom (1993), but the large scatter intrinsic to solubility-pH and solubility-$a(HS^-)$ dependences in their study render such a stoichiometry uncertain. As shown in our study, their data would equally be consistent with $Pd(HS)_4^{2-}$. Similarly, Pan and Wood (1994) and Wood et al. (1994) failed to derive a well-constrained Pd complex stoichiometry from their solubility measurements; rather they reported a large range of possible alternative complexes, spanning from $Pd(HS)^+$ to $Pd(HS)_4^{2-}$.

Our directly measured $PdS_{(s)}$ solubility at 300 °C and 500 bar is 2–3 log units lower than that calculated in Mei et al.'s (2015) MD-predicted value of the reaction constant (Fig. 6). Note that MD-derived reaction constants usually exhibit significant uncertainties, typically of > 1 log unit per stepwise complex formation reaction (Mei et al., 2018; Guan et al., 2020; Mei, 2020; Laskar et al., 2022b), resulting in an overall uncertainty of at least 4 log units for the $K_{8,4,Pd}$ value (reaction (2)). Given this uncertainty, the formation constant of $Pd(HS)_4^{2-}$ from Mei et al. (2015) does overlap with our experimental value.

## 4.2. Palladium hydrosulfide complexes in sulfur-poor acidic conditions

In sulfur-poor acidic solutions and with increasing temperature above 300 °C, $Pd(HS)_4^{2-}$ is expected to become less abundant and may be overcompeted by lower ligation number complexes and/or associated with alkali metals by forming stable ion pairs, thus making it difficult to distinguish the different Pd-HS complexes uniquely from solubility experiments versus pH. A tendency of lowering ligation number was reported for Pd chloride complexes with increasing temperature toward magmatic conditions (e.g., Bazarkina et al., 2014; Sullivan et al., 2022). Indeed, the log $K_{8,4,Pd}$ value decreases with increasing $T$ above 200 °C (Fig. 5). This pattern follows the decrease of both fluid density and dielectric constant with increasing temperature favoring both ion association and the stability of uncharged and weakly-charged species of lower entropy (Seward, 1984; Ruaya and Seward, 1986; Walther and
Schott, 1988; Seward and Barnes, 1997; Da Silva et al., 2009; Hou et al., 2020). Alternatively, ion pairs, such as Na₂Pd(HS)₄²⁻ which have the same zero electrical charge as Pt(HS)₂⁰⁻, for example, could be stable in the fluid. Indeed, such ion pairs were observed for other chalcophile metals, as NaₙCuCl₂ⁿ⁻¹ and NaₙAuCl₂ⁿ⁻¹ at 1000 °C and 1500 bar, both in the experiment (Zajacz et al., 2010, 2011) and MD simulation (Mei et al., 2014). Combining in situ experiments and MD modeling may help better distinguish between different types of possible complexes.

# 5. Role of sulfide and chloride in PGE transport and fractionation in hydrothermal fluids

## 5.1. Palladium transport

The stability constants of Pd(HS)₄²⁻ generated in this study allow improved predictions of Pd transport in moderate-temperature (<400 °C) S-bearing hydrothermal fluids. Fig. 7a and 7b show PdS(s) solubility in a typical saline fluid (10 wt% NaCl) under redox conditions corresponding to those of the conventional oxygen mineral buffer Ni-NiO (NNO), as a function of pH and H₂S(aq) content from 0.01 to 1 m, at 300 °C (500 bar) and 50 °C (300 bar), respectively. It can be seen that the solubility is largely dominated by Pd(HS)₄²⁻ from slightly acidic to alkaline pH, whereas the contribution of PdCl₄²⁻ is limited to strongly acidic (pH < 3)

and sulfur-poor (<0.01 m S) fluids. The Pd concentrations are about two orders of magnitude higher at 50 °C than at 300 °C, demonstrating the higher solubility of PdS(s) at lower temperatures. A similar trend was observed by Gammons and Bloom (1993) with a decrease in PtS(s) and PdS(s) solubility above 200 °C and by Tagirov et al. (2013) in less concentrated S-bearing solutions (0.05 m Stot, pH = 7, Psat) with PdS(s) solubility decreasing by 3 log units between 25 and 300 °C. Note that the contribution of hydroxide and sulfate complexes at most hydrothermal conditions is vanishingly small, less than 10⁻²⁴ mol of Pd per kg of fluid (i. e. < 1 Pd atom), according to available thermodynamic data (Sassani and Shock, 1998; Tagirov et al., 2013). Despite the formation of chloride complexes, the absolute PdS(s) solubility remains very low (<1 ppt Pd) at acidic conditions at the pyrite-pyrhotite coexistence. The maximum solubility is attained at near-neutral pH (6–7) as Pd(HS)₄²⁻ (Fig. 7a). However, except for very high sulfur concentrations (e.g., 10 ppb Pd at 1 m S; Fig. 7a,b), the absolute dissolved Pd concentrations are yet too small (<0.1 ppb Pd at 0.1 m S and pH 5 as buffered with alkali aluminosilicate mineral assemblages at such T-P; Kouzmanov and Pokrovski, 2012) to account for significant Pd transport in epithermal environments. Therefore, it is unlikely that Pd-HS types of complexes may significantly contribute to the known cases of the metal re-mobilization as observed in many hydrothermal environments (Boudreau, 2019; Economou-Eliopoulos, 2010; Hanley, 2005; Wilkinson, 2013; Arai and Miura,

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Solubility of PdS(s) as Pd(HS)₄²⁻ and PdCl₄²⁻ in an aqueous fluid of 10 wt% NaCl at the oxygen fugacity of the nickel-nickel oxide buffer (NNO, log₁₀fO₂₃₀₀°c (bar) = -34, log₁₀fO₂₅₀°c = -67) as a function of pH at the indicated total sulfide concentrations at (a) 300 °C, 500 bar and (b) 50 °C, 300 bar. Comparison of the solubility of PdS(s) and PtS(s) in a hydrothermal fluid of total sulfide concentration of 0.1 m, dominated by the indicated Pd and Pt species at (c) 300 °C, 500 bar and (d) 50 °C, 300 bar. Note that Pd-Cl and Pt-Cl complexes concentrations are < -19 log units in (b) and (d). Thermodynamic data sources are listed in Table S1.</figcaption></figure>
2016; Mungall and Naldrett, 2008; Holwell et al., 2017).

## 5.2. Palladium-platinum fractionation

Multiple factors may contribute to Pd/Pt fractionations that will be

<figure><img src="image_11.png" /></figure>

**Fig. 8.** Atomic Pd/Pt ratio in aqueous fluid of 0.1 m total sulfide concentration and 10 wt% NaCl at the oxygen fugacity of the nickel-nickel oxide buffer (NNO, $log_{10}f_{O_{2,300}°C}$ = -34) as a function of pH at 300 °C, 500 bar. At pH > 4, where sulfide complexes are dominant, the Pd/Pt ratio stays constant and does not depend on the sulfide concentration. Data for $Pd(HS)_{2-}$ and $Pt(HS)_{2-}$ – this study; $PdCl_{4}^{2-}$ – Sassani and Shock, 1998 (SS98), Tagirov et al., 2013 (Ta13) and Bazarkina et al., 2014 (Ba14); $PtCl_{4}^{2-}$ – Sassani and Shock, 1998 (for SS98 curve) and Tagirov et al., 2019 (for Ta13 and Ba14 curves). The SS98 and Ta13 curves equally flatten (no shown) at Pd/Pt ratios and pH outside the present graph axis margins.

considered here. At pH > 4, where hydrosulfide complexes are dominant for both metals, our speciation model predicts a constant atomic Pd/Pt ratio of 0.02 in the fluid at 300 °C, independently of pH and sulfide concentration (Fig. 8). Such a ratio value is up to 5 orders of magnitude lower than those encountered in sulfide minerals that are primary hosts of PGE in bulk rock in natural settings showing evidence of a remobilization of Pd and Pt by hydrothermal fluids (Fig. 9). Such settings include, for example, volcanogenic massive sulfide (VMS) deposits (e.g., Pan and Xie, 2001; Pašava et al., 2007; Le Vaillant et al., 2016b; Li et al., 2023), hydrothermally altered magmatic sulfide ores (e.g., Holwell et al., 2017), porphyry copper deposits (e.g., Eliopoulos et al., 2014), as well as komatiite rock-hosted ores (e.g., Barnes and Liu, 2012). Other settings, involving hydrothermal fluids such as serpentinites (Keays et al., 2021) or black shales (Xu et al., 2013), show scattered but generally lower Pd/Pt ratios in sulfide minerals (Fig. 9). Thus, most hydrothermal environments cited above show a large range of Pd/Pt values in hydrothermally formed or altered sulfide-bearing rocks. In contrast, Pd/Pt ratios in the nominally magmatic sulfide ores are less variable (Pd/Pt (at.) = 5.7 ± 3.8; Barnes and Ripley, 2016) and close to the average Pd/Pt ratio in the upper continental crust (Pd/Pt(at.) = 1.8; Chen et al., 2016).

In H₂S-poor solutions, palladium fractionation relative to other PGE might partially be controlled by chloride complexing in acidic solutions. Despite discrepancies between available studies on Cl complexes (Sassani and Shock, 1998; Tagirov et al., 2013; Bazarkina et al., 2014; Tagirov et al., 2019), the Pd/Pt ratio is predicted to strongly increase from 0.01 to > $10^4$ at pH below 4 to 2 (depending of the data source chosen; Fig. 8). The switch of speciation from chloride-dominated to sulfide-dominated with increasing pH was found for example to contribute to Cd/Zn fractionations in hydrothermal fluids (Bazarkina et al., 2023). However, this effect is expected to be less important for Pd/Pt given the low solubility of both metals in strongly acidic S-bearing solutions at typical NaCl contents of ≤ 10 wt% (Fig. 8).

Multiple factors at different scales may contribute to these fractionations, depending on the particular environment, assuming that Pd

<figure><img src="image_12.png" /></figure>

**Fig. 9.** Pd/Pt ratios analyzed in sulfide minerals from various natural settings: volcanogenic massive sulfide (VMS) deposits (Pan and Xie, 2001 – PX01; Vikentyev et al., 2004 – Vi04; Pašava et al., 2007 – Pa07; Le Vaillant et al., 2016b – LV16; Li et al., 2023 – Li23), hydrothermally altered magmatic sulfide ores (Holwell et al., 2017), porphyry deposits (Eliopoulos et al., 2014 – El14), black shales (Xu et al., 2013 – Xu13), serpentinites (Keays et al., 2021 – Ke21), komatiites (Barnes and Liu, 2012 – Ba12). The Pd/Pt ratio in the upper continental crust (Chen et al., 2016 – Ch16) is also reported for comparison. Elevated Pd/Pt ratios could be caused by the existence of polysulfide and radical sulfur ion complexes that require further investigation (red arrows). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
and Pd are from the same source, so as not particularly depleted or enriched in one of the two metals (e.g., Zientek, 2012). Those factors include, to name a few, different sources and geochemical pathways for the two metals, their differential intake and release by the major sulfide minerals, contrasting vapor-liquid partitioning, different oxidation states of Pt (+II and + IV) compared to Pd (+II), formation of strongly acidic fluids where Pd chloride complexes might be favored (e.g., caused by SO₂ hydrolysis in porphyry-epithermal systems, or oxidation of sulfides by meteoric waters), interactions with organic matter (in sedimentary basins), or the formation of PGE minerals other than sulfides (e.g., arsenides or native metals at reducing conditions typical of serpentinization). Among them, the differences in aqueous speciation of Pd and Pt revealed in this study may yield a measurable contribution. For example, assuming that the Pd/Pt ratio in the fluid reflects the ratio in the sulfide minerals that precipitate therefrom, our Pd and Pt speciation scheme at pH > 3–4 (Fig. 8) might roughly account for Pd/Pt atomic ratios of < 1 found in serpentinites and black shales. Indeed, these environments are commonly characterized by near-neutral to alkaline, reducing, H₂S-dominated fluids similar to those considered in Fig. 8. In contrast, porphyry and volcanogenic massive sulfide (VMS) deposits, as well as some hydrothermally altered magmatic sulfide ore bodies, generally show higher Pd/Pt ratios (up to 10⁴). In this regard, complexes with polysulfides (H₂–nSₙ⁻, at T < 250 °C; e.g., Barré et al., 2017) and radical di- and trisulfur ions (S₂²⁻, S₃²⁻, at higher temperatures; Pokrovski et al., 2021) might also play a role in the Pd/Pt fractionations shown in Fig. 9. Indeed, these ligands are particularly abundant under redox conditions of the sulfate-sulfide coexistence (Pokrovski and Dubessy, 2015), which are common for fluids operating in VMS and porphyry deposits and during hydrothermal alteration of magmatic bodies in general. Thus, accurate knowledge of aqueous speciation is required to evaluate the different factors leading to PGE fractionations. Work is currently in progress to assess the role of Pd and Pt complexes with the polysulfide and sulfur radical ions over a larger T-P-compositional range of hydrothermal-magmatic fluids.

## 6. Concluding remarks

We have investigated palladium speciation in sulfide-bearing hydrothermal fluids. Our data allow the generation of a consistent set of stability constants for Pd(HS)₂⁻ and its analog, Pt(HS)₄²⁻, within the framework of the HKF equation of state. More consistency compared to previous studies was achieved. The derived thermodynamic dataset for Pd(HS)₂⁻ and Pt(HS)₄²⁻ allows more accurate predictions of the solubility of Pd-bearing phases. Under typical hydrothermal conditions, the solubility of PdS(s) as chloride or hydrosulfide complexes does not exceed 1 ppb, which is too low to explain the natural cases of hydrothermal re-mobilization of Pd. The change of speciation from HS-dominated to Cl-dominated complexes at pH < 4 might partially account for Pd/Pt fractionation in S-poor, acidic and oxidizing fluids at closer-to-magmatic temperatures, pending more accurate data on Pd-Cl complexes. However, for the majority of sulfur-bearing near neutral-pH fluids, the contribution of other ligands such as polysulfides and sulfur radical ions could be more important.

### CRediT authorship contribution statement

**C. Laskar:** Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Conceptualization. **E.F. Bazarkina:** Writing – review & editing, Visualization, Resources. **G.S. Pokrovski:** Writing – review & editing, Visualization, Validation, Supervision, Resources, Project administration, Funding acquisition.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence

the work reported in this paper.

### Acknowledgments

This work was funded by the French National Research Agency (grant RadicalS – ANR-16-CE31-0017), the Institut Carnot ISIFO R (grants OrPet and AsCOCrit), and the CNRS through the MITI interdiscip- ciplinary programs MetalloMix 2021 (grant PtS3), and Conditions Ex- tremes 2024 (grant ExtremeS). C.L acknowledges support from the University of Toulouse (PhD fellowship of the Ministère de l’Enseigne- ment Supérieur, de la Recherche et de l’Innovation, MESRI). We thank K. Jaouen, P. Gisquet, A. Castillo, P. Bénézeth, C. Causserand, P. Besson, K. Alloncle, L. Menjot, M. Henry, A. Marquet, J. Chmeleff, H. Müller, L. Bourcet, and G. Berger for their advice and help with experiments and analyses. This article is a tribute to Terry Seward whose remarkable pioneering experimental studies on metals in hydrothermal fluids have always been a great source of inspiration for the authors. We are grateful to C. Gammons and one anonymous referee for comments and Jean-François Boily for editorial handling.

## Appendix A. Supplementary material

Supplementary material includes additional tables (Table S1–S4) and figures (Fig. S1–S5) referenced in the main text. Table S1 gives sources of thermodynamic species used in our modeling, Table S2 gives analytical details of experimental samples, Table S3 gives a summary of dissolution reactions constants for PdS(s) from our study and literature, Table S4 gives statistical fitting data. Fig. S1 gives SEM-EDS spectra of the initial PdS(s), Fig. S2 gives the diffractogram of the final powder after experiment, Fig. S3 and S4 give fitting comparison between calculated and experimental data for the different hydrosulfide species considered and Fig. S5 gives stability constant fitting for Pt hydrosulfide species. Supplementary material to this article can be found online at [http://doi.org/10.1016/j.gca.2024.12.019](http://doi.org/10.1016/j.gca.2024.12.019).

### Data availability

Data are available through Mendeley Data at [https://doi.org/10.17632/v52mgp7mmm.1](https://doi.org/10.17632/v52mgp7mmm.1).

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Supplementary material for

# The role of sulfur in palladium transport and fractionation from platinum by hydrothermal fluids

C. Laskar, E.F. Bazarkina, and G.S. Pokrovski

This file contains:

- Supplementary Tables S1 to S4

- Supplementary Figures S1 to S5

- Supplementary references
<table><caption>**Table S1.** Sources of thermodynamic data for aqueous species and minerals used in this study.</caption><thead><tr><th>Chemical species and phases</th><th>Data source</th></tr></thead><tbody><tr><td><strong>Aqueous species</strong></td><td></td></tr><tr><td>H<sub>2</sub>O, H<sup>+</sup>, OH<sup>-</sup>, Cl<sup>-</sup>, Na<sup>+</sup>, NaCl<sup>0</sup>, NaOH<sup>0</sup>, NaHSO<sub>4</sub><sup>0</sup>, HS<sup>-</sup>, H<sub>2</sub>S<sub>2</sub>O<sub>3</sub><sup>0</sup>, HS<sub>2</sub>O<sub>3</sub><sup>-</sup>, S<sub>2</sub>O<sub>3</sub><sup>2-</sup>, HSO<sub>3</sub><sup>-</sup>, SO<sub>3</sub><sup>2-</sup>, HSO<sub>4</sub><sup>-</sup>, SO<sub>4</sub><sup>2-</sup></td><td>Jo92</td></tr><tr><td>NaSO<sub>4</sub><sup>-</sup></td><td>Po95</td></tr><tr><td>S<sub>(2 to 8)</sub><sup>2-</sup></td><td>Ba17</td></tr><tr><td>HCl<sup>0</sup></td><td>Ta97</td></tr><tr><td>H<sub>2</sub>S<sup>0</sup>, SO<sub>2</sub><sup>0</sup>, H<sub>2</sub><sup>0</sup>, O<sub>2</sub><sup>0</sup></td><td>AD03</td></tr><tr><td>NaHS<sup>0</sup>, NaHSO<sub>4</sub><sup>0</sup></td><td>Jo92<sup>a</sup></td></tr><tr><td>S<sub>8</sub><sup>0</sup>, S<sub>3</sub><sup>2-</sup></td><td>PD15</td></tr><tr><td>PtCl<sub>4</sub><sup>2-</sup></td><td>Ta19</td></tr><tr><td>Pt(HS)<sub>4</sub><sup>2-</sup></td><td>La22</td></tr><tr><td>Pd<sup>2+</sup></td><td>SS98<sup>b</sup></td></tr><tr><td>PdCl<sub>3</sub><sup>-</sup>, PdCl<sub>4</sub><sup>2-</sup></td><td>Ba14</td></tr><tr><td>Pd(HS)<sub>4</sub><sup>2-</sup></td><td>This work</td></tr><tr><td><strong>Solids and liquids</strong></td><td></td></tr><tr><td>S<sub>(s)</sub>, S<sub>(l)</sub>, Na<sub>2</sub>SO<sub>4</sub><sub>(s)</sub></td><td>Ch98</td></tr><tr><td>Pd, FeS<sub>2</sub>, Fe<sub>0.875</sub>S, Fe<sub>2</sub>O<sub>3</sub>, Fe<sub>3</sub>O<sub>4</sub></td><td>Ro95</td></tr><tr><td>PdS</td><td>Ba14<sup>c</sup></td></tr><tr><td>PtS</td><td>SS98</td></tr><tr><td>FeS</td><td>Jo92</td></tr></tbody></table>

Jo92 – Johnson et al., 1992; PW94 – Pan and Wood, 1994; Ro95 – Robie et al., 1995; Po95 – Pokrovski et al., 1995; Ta97 – Tagirov et al., 1997; Ch98 – Chase Jr, 1998; SS98 – Sassani and Shock, 1998; AD03 – Akinfiev and Diamond, 2003; Ba14 – Bazarkina et al., 2014; PD15 – Pokrovski and Dubessy, 2015; Ba17 – Barré et al., 2017; Ta19 – Tagirov et al., 2019; La22 – Laskar et al., 2022.

<sup>a</sup> Formation constant [cation + ligand = ion pair] is chosen equal to NaCl<sup>0</sup>, and KHSO<sub>4</sub><sup>0</sup>, respectively, from Johnson et al. (1992).

<sup>b</sup> Used for K<sub>s4</sub>Pd calculations from constants reported by Mei et al. (2015) for reaction: Pd<sup>2+</sup> + 4 HS<sup>-</sup> = Pd(HS)<sub>4</sub><sup>2-</sup>.

<sup>c</sup> See Bazarkina et al. (2014) for a detailed comparison of PdS<sub>(s)</sub> thermodynamic properties from the literature.
<table><caption>**Table S2.** Fluid composition measured in the H₂O-NaHS-PdS-PtS-FeS-FeS₂ system at the indicated T-P conditions using a gold flexible cell reactor (experiment m35; 171 g of initial aqueous solution of 0.96 mol/kg soln NaHS + 0.1 mol/kg soln H₂S(aq)).</caption><thead><tr><th>Elapsed time from the start, days</th><th>T °C</th><th>P bar</th><th>Pd ppm</th><th>Pt ppm</th><th>S<sub>tot</sub> mol/kg soln</th><th>Na mol/kg soln</th><th>Cl mol/kg soln</th><th>S<sub>red</sub> (with I<sub>2</sub>) mol/kg soln</th><th>S<sub>red</sub> (with ZnAc) mol/kg soln</th><th>S<sub>red</sub> (mean) mol/kg soln</th><th>SO<sub>4</sub><sup>2-</sup> mol/kg soln</th></tr></thead><tbody><tr><td>#m35/1</td><td>4</td><td>56</td><td>320</td><td>0.288</td><td>2.5</td><td>-<sup>a</sup></td><td>1.12</td><td>0.00</td><td>1.03</td><td>0.94</td><td>0.99</td><td>&lt;0.01</td></tr><tr><td>#m35/2</td><td>7<sup>e</sup></td><td>200</td><td>498</td><td>0.068</td><td>0.61</td><td>0.94</td><td>0.91</td><td>0.00</td><td>0.97</td><td>0.99</td><td>0.98</td><td>&lt;0.01</td></tr><tr><td></td><td>11<sup>e</sup></td><td>200</td><td>432</td><td>0.051</td><td>0.64</td><td>0.99</td><td>0.93</td><td>0.00</td><td>0.95</td><td>1.03</td><td>0.99</td><td>&lt;0.01</td></tr><tr><td>#m35/3</td><td>13<sup>e</sup></td><td>250</td><td>503</td><td>0.028</td><td>0.45</td><td>1.00</td><td>0.93</td><td>0.00</td><td>0.96</td><td>0.93</td><td>0.95</td><td>&lt;0.01</td></tr><tr><td></td><td>18<sup>e</sup></td><td>250</td><td>442</td><td>0.023</td><td>0.46</td><td>1.01</td><td>0.93</td><td>0.00</td><td>0.89</td><td>-</td><td>0.89</td><td>&lt;0.01</td></tr><tr><td></td><td>21</td><td>250</td><td>421</td><td>0.096</td><td>0.91</td><td>1.14</td><td>1.12</td><td>0.00</td><td>0.89</td><td>-</td><td>0.89</td><td>&lt;0.01</td></tr><tr><td></td><td>24</td><td>250</td><td>483</td><td>0.036</td><td>0.48</td><td>1.13</td><td>1.11</td><td>0.00</td><td>0.93</td><td>0.94</td><td>0.94</td><td>&lt;0.01</td></tr><tr><td>#m35/4</td><td>28</td><td>300</td><td>523</td><td>0.080</td><td>0.95</td><td>1.11</td><td>1.06</td><td>0.00</td><td>0.92</td><td>0.95</td><td>0.93</td><td>&lt;0.01</td></tr><tr><td></td><td>32</td><td>300</td><td>464</td><td>0.143</td><td>0.94</td><td>1.10</td><td>1.01</td><td>0.00</td><td>0.90</td><td>0.95</td><td>0.93</td><td>&lt;0.01</td></tr><tr><td></td><td>35</td><td>300</td><td>425</td><td>0.047</td><td>0.65</td><td>1.10</td><td>0.87</td><td>0.00</td><td>0.98</td><td>0.94</td><td>0.96</td><td>&lt;0.01</td></tr><tr><td></td><td>35<sup>e</sup></td><td>300</td><td>414</td><td>0.021</td><td>0.23</td><td>0.96</td><td>0.90</td><td>0.01</td><td>0.98</td><td>0.94</td><td>0.96</td><td>&lt;0.01</td></tr><tr><td>#m35/5, Injection HCl (10.1 g; 3.2 mol/kg soln), t=35 days</td><td>40</td><td>300</td><td>478</td><td>0.862</td><td>5.4</td><td>1.18</td><td>1.03</td><td>0.27</td><td>0.84</td><td>0.87</td><td>0.85</td><td>&lt;0.01</td></tr><tr><td></td><td>42</td><td>300</td><td>416</td><td>0.041</td><td>0.92</td><td>1.21</td><td>1.11</td><td>0.28</td><td>0.83</td><td>0.84</td><td>0.84</td><td>&lt;0.01</td></tr><tr><td></td><td>42</td><td>300</td><td>462</td><td>&lt;0.023<sup>b</sup></td><td>0.51</td><td>1.09</td><td>1.00</td><td>0.25</td><td>0.80</td><td>0.87</td><td>0.83</td><td>&lt;0.01</td></tr><tr><td>#m35/6, Injection HCl (21.5 g; 2.9 mol/kg soln), t=42 days</td><td>45</td><td>300</td><td>524</td><td>0.171</td><td>0.62</td><td>-<sup>a</sup></td><td>0.84</td><td>-<sup>a</sup></td><td>0.71</td><td>0.68</td><td>0.69</td><td>&lt;0.01</td></tr><tr><td></td><td>45</td><td>300</td><td>499</td><td>0.039</td><td>0.20</td><td>-<sup>a</sup></td><td>0.85</td><td>-<sup>a</sup></td><td>0.71</td><td>0.68</td><td>0.69</td><td>&lt;0.01</td></tr><tr><td></td><td>49</td><td>300</td><td>477</td><td>&lt;0.012<sup>b</sup></td><td>0.12</td><td>-<sup>a</sup></td><td>0.84</td><td>-<sup>a</sup></td><td>0.74</td><td>0.72</td><td>0.73</td><td>&lt;0.01</td></tr><tr><td></td><td>52</td><td>300</td><td>434</td><td>&lt;0.003</td><td>0.05</td><td>-<sup>a</sup></td><td>-<sup>a</sup></td><td>-<sup>a</sup></td><td>0.74</td><td>0.69</td><td>0.72</td><td>&lt;0.01</td></tr><tr><td>#m35/7, Injection H<sub>2</sub>O (14.5 g), t=53 days</td><td>56</td><td>250</td><td>502</td><td>&lt;0.026<sup>b</sup></td><td>0.09</td><td>-<sup>a</sup></td><td>-<sup>a</sup></td><td>-<sup>a</sup></td><td>0.64</td><td>0.60</td><td>0.62</td><td>&lt;0.01</td></tr><tr><td></td><td>60</td><td>250</td><td>443</td><td>0.437</td><td>0.24</td><td>0.68</td><td>0.66</td><td>0.65</td><td>0.62</td><td>0.53</td><td>0.58</td><td>&lt;0.01</td></tr><tr><td></td><td>64</td><td>250</td><td>383</td><td>&lt;0.015<sup>b</sup></td><td>0.18</td><td>0.67</td><td>0.68</td><td>0.64</td><td>0.67</td><td>0.58</td><td>0.62</td><td>&lt;0.01</td></tr><tr><td></td><td>67</td><td>250</td><td>303</td><td>&lt;0.002</td><td>&lt;0.02<sup>a</sup></td><td>0.71</td><td>0.71</td><td>0.69</td><td>0.61</td><td>0.67</td><td>0.64</td><td>&lt;0.01</td></tr><tr><td></td><td>69</td><td>250</td><td>467</td><td>&lt;0.014<sup>b</sup></td><td>0.028</td><td>0.67</td><td>0.75</td><td>0.69</td><td>0.59</td><td>0.60</td><td>0.60</td><td>&lt;0.01</td></tr></tbody></table>
Table S2 continued

#m35/8, Injection H₂O (24.1 g), t=70 days

<table><tbody><tr><td></td><td>78</td><td>200</td><td>345</td><td>&lt;0.012<sup>b</sup></td><td>0.07</td><td>0.56</td><td>0.55</td><td>0.52</td><td>0.50</td><td>0.48</td><td>0.49</td><td>&lt;0.01</td></tr><tr><td></td><td>81</td><td>200</td><td>269</td><td>&lt;0.002</td><td>&lt;0.02<sup>a</sup></td><td>0.56</td><td>0.59</td><td>0.55</td><td>0.51</td><td>0.49</td><td>0.50</td><td>&lt;0.01</td></tr><tr><td>#m35/9</td><td>89</td><td>100</td><td>249</td><td>&lt;0.012<sup>b</sup></td><td>&lt;0.02<sup>a</sup></td><td>0.55</td><td>0.58</td><td>0.54</td><td>0.50</td><td>0.48</td><td>0.49</td><td>&lt;0.01</td></tr><tr><td></td><td>92</td><td>100</td><td>245</td><td>&lt;0.012<sup>b</sup></td><td>0.038</td><td>0.53</td><td>0.53</td><td>0.50</td><td>—<sup>a</sup></td><td>0.47</td><td>0.47</td><td>&lt;0.01</td></tr><tr><td></td><td>95</td><td>100</td><td>251</td><td>&lt;0.002</td><td>&lt;0.02<sup>a</sup></td><td>0.54</td><td>0.57</td><td>0.51</td><td>—<sup>a</sup></td><td>0.49</td><td>0.49</td><td>&lt;0.01</td></tr><tr><td>#m35/10</td><td>106</td><td>50</td><td>255</td><td>&lt;0.002</td><td>0.025</td><td>0.54</td><td>0.56</td><td>0.51</td><td>—<sup>a</sup></td><td>—<sup>a</sup></td><td>0.47<sup>c</sup></td><td>&lt;0.01</td></tr><tr><td></td><td>109</td><td>50</td><td>269</td><td>&lt;0.002</td><td>0.011</td><td>0.52</td><td>0.54</td><td>0.49</td><td>—<sup>a</sup></td><td>—<sup>a</sup></td><td>0.47<sup>c</sup></td><td>&lt;0.01</td></tr><tr><td></td><td>113</td><td>50</td><td>267</td><td>&lt;0.002</td><td>0.023</td><td>0.54</td><td>0.56</td><td>0.52</td><td>—<sup>a</sup></td><td>—<sup>a</sup></td><td>0.47<sup>c</sup></td><td>&lt;0.01</td></tr><tr><td>#m35/11</td><td>120</td><td>50</td><td>441</td><td>&lt;0.002</td><td>0.063</td><td>0.54</td><td>0.56</td><td>0.52</td><td>—<sup>a</sup></td><td>—<sup>a</sup></td><td>0.47<sup>c</sup></td><td>&lt;0.01</td></tr><tr><td></td><td>131</td><td>50</td><td>444</td><td>0.012</td><td>0.042</td><td>0.53</td><td>0.56</td><td>0.52</td><td>—<sup>a</sup></td><td>—<sup>a</sup></td><td>0.47<sup>c</sup></td><td>&lt;0.01</td></tr><tr><td>#m35/12</td><td>138<sup>d</sup></td><td>50</td><td>92</td><td>&lt;0.010<sup>b</sup></td><td>0.28</td><td>0.47</td><td>0.49</td><td>0.44</td><td>—<sup>a</sup></td><td>—<sup>a</sup></td><td>0.47<sup>c</sup></td><td>&lt;0.01</td></tr><tr><td></td><td>147<sup>d</sup></td><td>50</td><td>83</td><td>0.065</td><td>1.1</td><td>0.50</td><td>0.55</td><td>0.49</td><td>—<sup>a</sup></td><td>0.47</td><td>0.47</td><td>&lt;0.01</td></tr><tr><td>#m35/13, Injection NaOH (15.0 g; 0.5 mol/kg soln), t=153 days</td><td>162</td><td>50</td><td>262</td><td>0.047</td><td>0.76</td><td>0.42</td><td>0.55</td><td>0.40</td><td>—<sup>a</sup></td><td>0.37</td><td>0.37</td><td>&lt;0.01</td></tr><tr><td></td><td>168</td><td>50</td><td>248</td><td>&lt;0.009<sup>b</sup></td><td>0.094</td><td>0.39</td><td>0.49</td><td>0.37</td><td>0.42</td><td>0.41</td><td>0.41</td><td>&lt;0.01</td></tr><tr><td>#m35/14, Injection H₂O₂ (13.5 g; 1.7 mol/kg soln), t=169 days</td><td>175</td><td>50</td><td>255</td><td>0.173</td><td>0.58</td><td>0.15</td><td>0.45</td><td>0.33</td><td></td><td>0.13</td><td>0.13</td><td>&lt;0.01</td></tr><tr><td></td><td>180</td><td>50</td><td>266</td><td>0.115</td><td>0.43</td><td>0.16</td><td>0.47</td><td>0.35</td><td>0.15</td><td>0.14</td><td>0.14</td><td>&lt;0.01</td></tr><tr><td>Error</td><td></td><td>±2</td><td>±10%</td><td>±10%</td><td>±10%</td><td>±10%</td><td>±5%</td><td>±5%</td><td>±10%</td><td>±10%</td><td>±10%</td><td></td></tr></tbody></table>

Samples were diluted by 700 to 3800 times depending on their S concentration before being analyzed by ICP-MS to enable total sulfur concentration of <5 ppm in the analyzed samples to avoid sulfur interferences in the analysis. Pt concentrations are from Laskar et al. (2022).

a ‘—’ means not measured.

b Values below the limit of quantification (LoQ as reported) but above the limit of detection (LoD = 0.002 ppm).

c Assumed to be equal to 0.47 mol/kg soln for sampling points at 92 and 147 days, as there was no injection meanwhile.

d Samples are probably not representative because of a potential leak at this stage.

e Samples for metal analysis were extracted from the reactor with a gas-tight syringe.*
<table><caption>**Table S3.** Summary of the dissolution reaction constants of PdS<sub>(s)</sub> to form the different Pd-HS complexes at the indicated T-P conditions from this and existing studies.</caption><thead><tr><th rowspan="3">Reaction</th><th rowspan="3">T, °C</th><th rowspan="3">P, bar</th><th>log<sub>10</sub>K<sub>sn,Pd</sub> (T,P)</th><th colspan="2">log<sub>10</sub>K<sub>sn,Pd</sub> (T,P)</th><th colspan="4">log<sub>10</sub>K<sub>sn,Pd</sub> (T,P)</th></tr><tr><th rowspan="2">Measured</th><th colspan="2">Recalculated data</th><th colspan="4">Originally reported data</th></tr><tr><th>This study</th><th>GB93</th><th>PW94</th><th>GB93</th><th>Wo94</th><th>PW94<sup>a</sup></th><th>Ta13</th><th>Me15<sup>b</sup></th></tr></thead><tbody><tr><td rowspan="7">PdS<sub>(s)</sub> + 3 H<sub>2</sub>S<sup>0</sup><sub>(aq)</sub> = Pd(HS)<sub>4</sub><sup>2-</sup> + 2 H<sup>+</sup></td><td>25</td><td>1</td><td>—</td><td>—</td><td>—</td><td>—</td><td>-14.3</td><td>—</td><td>—</td><td>—</td></tr><tr><td>50</td><td>300</td><td>-17.7±0.6</td><td>—</td><td>—</td><td>—</td><td>-12.9</td><td>—</td><td>—</td><td>—</td></tr><tr><td>90</td><td>1</td><td>—</td><td>—</td><td>—</td><td>—</td><td>-8.3</td><td>—</td><td>—</td><td>—</td></tr><tr><td>100</td><td>300</td><td>-16.9±0.8</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>200</td><td>500</td><td>-18.4±0.6</td><td>-16.3±0.5</td><td>-18.1±1.6</td><td>-18.2±0.6</td><td>—</td><td>-26.0±1.3<sup>c</sup></td><td>—</td><td>—</td></tr><tr><td>250</td><td>500</td><td>-19.0±0.6</td><td>-19.0±0.5</td><td>-19.4±1.6</td><td>-20.8±0.5</td><td>—</td><td>-28.1±1.6<sup>c</sup></td><td>—</td><td>—</td></tr><tr><td>300</td><td>500</td><td>-19.9±0.6</td><td>-20.8±0.5</td><td>-21.2±1.6</td><td>-22.7±0.5</td><td>—</td><td>-29.9±1.1<sup>c</sup></td><td>—</td><td>-17.2</td></tr><tr><td rowspan="7">PdS<sub>(s)</sub> + 2 H<sub>2</sub>S<sub>(aq)</sub> = Pd(HS)<sub>3</sub><sup>-</sup> + H<sup>+</sup></td><td>25</td><td>1</td><td>—</td><td>—</td><td>—</td><td>—</td><td>-10.9</td><td>—</td><td>—</td><td>—</td></tr><tr><td>50</td><td>300</td><td>-11.8±0.6</td><td>—</td><td>—</td><td>—</td><td>-10.1</td><td>—</td><td>—</td><td>—</td></tr><tr><td>90</td><td>1</td><td>—</td><td>—</td><td>—</td><td>—</td><td>-6.8</td><td>—</td><td>—</td><td>—</td></tr><tr><td>100</td><td>300</td><td>-11.7±0.8</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>200</td><td>500</td><td>-11.6±0.6</td><td>-9.5±0.5</td><td>-11.2±0.6</td><td>-11.2±0.5</td><td>—</td><td>-17.0±0.5<sup>c</sup></td><td>—</td><td>—</td></tr><tr><td>250</td><td>500</td><td>-12.1±0.6</td><td>-11.9±0.5</td><td>-12.1±0.6</td><td>-12.5±0.5</td><td>—</td><td>-17.8±0.5<sup>c</sup></td><td>—</td><td>—</td></tr><tr><td>300</td><td>500</td><td>-12.7±0.6</td><td>-13.2±0.5</td><td>-13.1±0.6</td><td>-14.2±0.5</td><td>—</td><td>-19.4±0.5<sup>c</sup></td><td>-11.6</td><td>-15.2</td></tr><tr><td>PdS<sub>(s)</sub> + H<sub>2</sub>S<sub>(aq)</sub> = Pd(HS)<sub>2</sub><sup>0</sup><sub>(aq)</sub></td><td>300</td><td>500</td><td>-6.1±0.6</td><td>-6.4±0.5</td><td>-5.2±0.5</td><td>-7.2±0.5</td><td>—</td><td>-4.9±0.4</td><td>-8.4</td><td>-14.2</td></tr></tbody></table>

Uncertainties correspond to the standard deviation determined from the Gibbs energy value variation in the fit performed with the OptimA software, taking into account the experimental uncertainties (±1σ). Wood et al. (1994) - Wo94 (solubility at 1 bar); Gammons and Bloom (1993) - GB93 (solubility at P<sub>sat</sub>); Pan and Wood (1994) - PW94 (solubility at P<sub>sat</sub>); Tagirov et al. (2013) - Ta13 (HKF model predictions); Mei et al. (2015) - Me15 (molecular dynamics simulations); ‘-’ means not available.

<footnote><sup>a</sup> Calculated using thermodynamic data from Table S1 from originally reported constants of the reaction: PdS<sub>(s)</sub> + (n-1) HS<sup>-</sup> + H<sup>+</sup> = Pd(HS)<sub>n</sub><sup>2-n</sup> (2≤n≤4).</footnote>

<footnote><sup>b</sup> Calculated using thermodynamic data from Table S1 from originally reported constants of the reaction: Pd<sup>2+</sup> + n HS<sup>-</sup> = Pd(HS)<sub>n</sub><sup>2-n</sup> (2≤n≤4).</footnote>

<footnote><sup>c</sup> Reaction constant values originally reported in Pan and Wood (1994) appear to be in error, given the large differences (1-9 log units) between these values and those recalculated in this study.</footnote>
<table><caption>**Table S4.** Standard deviation (σ) between PdS(s) solubility from experiment m35 of this study and those of Gammons and Bloom (1993), GB93, and PdS(s) solubility calculated with Pd(HS)₄²⁻, Pd(HS)₃⁻, or Pd(HS)₂₀ as the main complex in solution.</caption><thead><tr><th>T, °C</th><th>P, bar</th><th>σ(Pd(HS)<sub>4</sub><sup>2-</sup>)</th><th>σ(Pd(HS)<sub>3</sub><sup>-</sup>)</th><th>σ(Pd(HS)<sub>2</sub><sup>0</sup>)</th><th>Number of datapoints</th><th>Reference</th></tr></thead><tbody><tr><td>50</td><td>300</td><td>0.34</td><td>0.62</td><td>1.09</td><td>8</td><td>This study</td></tr><tr><td>100</td><td>300</td><td>0.37</td><td>0.37</td><td>0.37</td><td>3</td><td>This study</td></tr><tr><td>200</td><td>500</td><td>0.90</td><td>0.34</td><td>1.20</td><td>4</td><td>This study</td></tr><tr><td>200</td><td>P<sub>sat</sub></td><td>0.34</td><td>0.26</td><td>0.39</td><td>16</td><td>GB93</td></tr><tr><td>250</td><td>500</td><td>1.18</td><td>0.62</td><td>0.83</td><td>9</td><td>This study</td></tr><tr><td>250</td><td>P<sub>sat</sub></td><td>0.25</td><td>0.16</td><td>0.28</td><td>13</td><td>GB93</td></tr><tr><td>300</td><td>500</td><td>0.54</td><td>0.44</td><td>0.74</td><td>8</td><td>This study</td></tr><tr><td>300</td><td>P<sub>sat</sub></td><td>0.11</td><td>0.22</td><td>0.43</td><td>14</td><td>GB93</td></tr></tbody></table>

$\sigma = \frac{1}{\sqrt{N}} \times \sum_{i}^{N} \sqrt{(\log_{10}P_{\text{exp}} - \log_{10}P_{\text{calc}})^2}$, where N is the number of experimental points; standard deviation is calculated from data reported in Figures S1 and S2; the σ value at 100 °C is the same because of a single datapoint.
<figure><img src="image_13.png" /><figcaption>**Fig. S1.** SEM photograph of PdS(s) initial powder and SEM-EDS analysis (targeted on the red cross), with the K-edge (for sulfur) and L-edge (for palladium) fluorescence peaks. The elemental ratio obtained is Pd/S=1.0.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. S2.** Diffraction pattern (in black color) of the solid mixture after the termination of experiment m35. The major peaks of PdS(s) are marked with a star. They overlap with the peaks of pyrrhotite, which shows only</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. S3.** The difference between $PdS_{(s)}$ solubility measured in experiments performed by Gammons and Bloom (1993) (GB93) and $PdS_{(s)}$ solubility calculated with $Pd(HS)_{4}^{2-}$, $Pd(HS)_{3}^{-}$, or $Pd(HS)_{2}^{0}$ as the main complex in solution vs pH at (a) 300 °C, (b) 250 °C, (c) 200 °C, all at $P_{sat}$.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Fig. S4.** The difference between $PdS_{(s)}$ solubility measured in our experiments and $PdS_{(s)}$ solubility calculated with $Pd(HS)_{4}^{2-}$, $Pd(HS)_{3}^{-}$, or $Pd(HS)_{2}^{0}$ as the main complex in solution as a function of pH at (a) 300 °C, 500 bar, (b) 250 °C, 500 bar, (c) 200 °C, 500 bar, (d) 100 °C, 300 bar, and (e) 50°C, 300 bar.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. S5.** Values of log₁₀K₅₄Pt as a function of 1/T. Data are from the study of Laskar et al. (2022b) at 300 °C, 500 bar; 250 °C, 500 bar; 200 °C, 500 bar; 100 °C, 300 bar; 50 °C, 300 bar. The fit was performed in this study for the datapoints by using a 3-parameter regression to log₁₀K₅₄Pt = (119±47)– (7259±2789)/T(K)–(45.27±15.28)×log₁₀T(K).</figcaption></figure>
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