# Sulfur in Hydrothermal Fluids

Gleb S. Pokrovski, Kalin Kouzmanov, Stefansson Andri

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Gleb S. Pokrovski, Kalin Kouzmanov, Stefansson Andri. Sulfur in Hydrothermal Fluids. 2025. hal-05091554

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# Chapter 4

## Sulfur in hydrothermal fluids

Gleb S. Pokrovski<fnref n="1" />, Kalin Kouzmanov<fnref n="2" />, and Andri Stefánsson<fnref n="3" />

### Abstract

### 4.1 Introduction

### 4.2 Analytical and spectroscopic methods for sulfur in hydrothermal fluids

4.2.1 Sampling techniques and wet-chemistry analyses of geothermal water

4.2.2 Bulk crush-leach analyses of fluid inclusions for sulfur

4.2.3 LA-ICPMS on individual fluid inclusions in hydrothermal minerals

4.2.4 Raman spectroscopy

4.2.5 X-ray absorption (XAS), emission (XES and PIXE) and fluorescence (XRF) spectroscopy

4.2.6 Other spectroscopic methods for sulfur speciation analysis

### 4.3 Sulfur chemical speciation, kinetics, and thermodynamics in hydrothermal fluids

4.3.1 Sulfur aqueous speciation and redox state at low temperatures

4.3.2 Kinetics of sulfur species transformation reactions in hydrothermal fluids

4.3.3 Overview of thermodynamic models for sulfur of hydrothermal fluids

4.3.4 Thermodynamic properties and distribution of sulfur species in hydrothermal fluids

4.3.5 Sulfidation state concept

### 4.4 Metal speciation and sulfur-metal relationships in hydrothermal fluids

4.4.1 General tendencies of metal-ligand complexation

4.4.2 Base metals - Zn, Pb, and Cu

4.4.3 Silver

4.4.4 Gold

4.4.5 Platinum group elements (PGE)

4.4.6 Iron

4.4.7 Arsenic

4.4.8 Antimony

4.4.9 Bismuth

4.4.10 Molybdenum

4.4.11 Sulfate complexes of metals

4.4.12 Speciation of thioanions in natural hydrothermal waters

### 4.5 Sulfur concentration, speciation and reactions in geological fluids

4.5.1 Modern geothermal on-land and deep-sea sources

4.5.2 Magmatic-hydrothermal fluids

4.5.3 Metamorphic fluids

4.5.4 Sedimentary-basin fluids

### 4.6 Perspectives and challenges

### 4.7 References

<footnote n="1">¹ Experimental Geosciences Team (GeoExp), Géosciences Environnement Toulouse (GET), UMR 5563 of the Centre National de la Recherche Scientifique (CNRS), Université Paul Sabatier Toulouse III, Institut de Recherche pour le Développement (IRD), Centre National d'Etudes Spatiales (CNES), Observatoire Midi-Pyrénées, 14 av. Edouard Belin, 31400 Toulouse, France; e-mails: gleb.pokrovski@get.omp.eu ; gleb.pokrovski@univ-tlse3.fr ; glebounet@gmail.com</footnote>

<footnote n="2">² University of Geneva, Department of Earth sciences, University of Geneva, rue des Maraîchers 13, 1205 Geneva, Switzerland; email: kalin.kouzmanov@unige.ch</footnote>

<footnote n="3">³ Institute of Earth Sciences, University of Iceland, Sturlugata 7, 102 Reykjavík, Iceland; email: as@hi.is</footnote>
## Glossary

### Method acronyms

CL: cathodoluminescence

EPMA: electron probe microanalyzer

EPR: electron paramagnetic resonance

FAA: flame atomic absorption

GS: gas chromatography

GC-MS: gas chromatography-mass spectrometry

IC, IPC: ion chromatography, ion-pair chromatography

ICPAES: inductively coupled plasma atomic emission spectrometry

ICPMS: inductively coupled plasma mass spectrometry

ICP-Q-MS: inductively coupled plasma - quadrupole - mass spectrometry

ICP-SF-MS: inductively coupled plasma - sector field - mass spectrometry

HPLC: high pressure liquid chromatography

(LA-)ICPMS: (laser ablation -) inductively coupled plasma mass spectrometry

(FP)MD: (first principles) molecular dynamics

MS: mass spectrometry

(N)IR: (near) infrared (spectroscopy)

NMR: nuclear magnetic resonance

NRIXS: nonresonant inelastic X-ray scattering

PIXE: proton induced X-ray emission

SEM: scanning electron microscopy

SXRF: synchrotron X-ray fluorescence (spectroscopy)

TOF-MS: time of flight mass spectrometry

TQ-MS: triple quadrupole (or alternatively QQQ or QqQ) mass spectrometer

UV-Vis: ultraviolet-visible (spectroscopy)

XAS: X-ray absorption spectroscopy

XANES and EXAFS: X-ray absorption near edge structure and extended X-ray absorption fine structure

XES: X-ray emission spectroscopy

XPS: X-ray photoelectron spectroscopy

XRF: X-ray fluorescence (spectroscopy)

### Models, databases, software acronyms

AD: Akinfiev Diamond (model)

CHESS: Chemical Equilibrium of Species and Surfaces, https://chess.geosciences.mines-paristech.fr/ (software)

DFT: density functional theory

EQ3/6: https://github.com/llnl/eq3_6 (software)

FDMNES: Finite Difference Method near Edge Structure, https://fdmnes.neel.cnrs.fr/ (software)

GEMS: Gibbs Energy Minimization Software, http://gems.web.psi.ch/ (software)

GW: Geochemists Workbench, https://www.gwb.com/index.php (software)

HKF: Helgeson Kirkham Flowers (model)

HCh: HydroChemistry (software)

LLNL: Lawrence Livermore National Laboratory (database)

MINTEQ: https://vminteq.lwr.kth.se/ (software)

(FP)MD: (first principles) molecular dynamics

PHREEQC: https://www.usgs.gov/software/phreeqc-version-3 (software)

RB: Ryzhenko Bryzgalin (model)

SILLS: Signal Integration for Laboratory Laser Systems (software)

SUPCRT(92): http://geopig3.la.asu.edu:8080/GEOPIG/index.html (database)

SUPCRTBL: https://models.earth.indiana.edu/applications_index.php (database)

THERMODEM: https://thermoddem.brgm.fr/ (database)

### Mineral names and their generic chemical formulas
alunite, KAl₃(SO₄)₂(OH)₆

andalusite, Al₂SiO₅

anhydrite and gypsum, CaSO₄, CaSO₄×2H₂O

apatite, Ca₅(PO₄)₃(F,Cl,OH,S)

argentite, Ag₂S

arsenopyrite, FeAsS

bastnäsite, (La,Ce,Y)(CO₃)F

barite, BaSO₄

biotite, K(Mg,Fe)₃Si₃AlO₁₀(OH,F)₂

bismuthinite, Bi₂S₃

boehmite, AlO(OH)

bornite, Cu₅FeS₄

braggite, (Pt,Pd,Ni)S

calcite, CaCO₃

chalcocite, Cu₂S

chalcopyrite, CuFeS₂

cooperite, PtS

corundum, Al₂O₃

covellite, CuS

digenite, Cu₉S₅

enargite, Cu₃AsS₄

epidote, Ca₂FeAl₂Si₃O₁₂(OH)

fayalite, Fe₂(SiO₄)

fluocerite, (Ce,La)F₃

fluorite, CaF₂

galena, PbS

halite, NaCl

hematite, Fe₂O₃

hornblende, (Ca,Na,K)₂(Mg,Fe,Al)₅[Si₆(Al,Si)₂O₂₂](OH,F)₂

löllengite, FeAs₂

magnetite, Fe₃O₄

microcline, KAlSi₃O₈

molybdenite, MoS₂

monazite, (Ce,La,Nd,Th)PO₄

muscovite, KAl₃Si₃O₁₀(OH)₂

orpiment, As₂S₃

prehnite, Ca₂Al₂Si₃O₁₀(OH)₂

pyrite and arsenian pyrite, FeS₂, Fe(S,As)₂

pyrrhotite, Fe₁₋ₓS

quartz, SiO₂

realgar, AsS

rhenite, ReS₂

sanidine, KAlSi₃O₈

scapolite, (Na,Ca)₄[(Al,Si)₁₂O₂₄](Cl,CO₃,S)

sodalite, ultramarine, (Na,Ca)₈[Al₆Si₆O₂₄](Cl,S,SO₄)₁₋₂

sphalerite, ZnS

sperrylite, PtAs₂

stibnite, Sb₂S₃

sulfosalts, (Ag,Pb,Cu,Fe)ₓ(As,Sb)ᵧSz
**Abstract:**

This chapter overviews sulfur chemical speciation and behavior in different types of hydrothermal fluids across the lithosphere, spanning from active geothermal systems to basinal brines, to metamorphic and magmatic-hydrothermal fluids. The information on sulfur in these fluids stems from a wide range of recent analytical, experimental and computational methods whose advantages and limitations are discussed. A special emphasis is given to in situ approaches that have enabled unprecedented insights into the fascinating sulfur chemistry in hydrothermal fluids. These insights motivated a critical review of the chemical speciation of key base, precious, and critical metals whose behavior is intimately linked to sulfur. The results allow the role played by the different sulfur ligands on hydrothermal transport of metals and ore deposit formation to be evaluated. An outline of major challenges and emerging perspectives for sulfur-related research in geological fluids ends up this chapter.
**4.1 Introduction**

Sulfur, sulfur, sulfur... There is unlikely another volatile element that attracts so much attention and engenders so much debate among geochemists, volcanologists, magmatic petrologists, ore deposit geologists, and planetary scientists. This is because sulfur is a key element in the processes that govern the formation and distribution of ore deposits, magma evolution and degassing, volcanic hazard and isotope fractionation in mineral-melt-fluid systems. For technological applications, sulfur is an important player in chemical catalysis, nanotechnology, and materials science. Critical to all these aspects is understanding the chemical speciation, transport and partitioning of sulfur in aqueous fluids typical of those natural and industrial systems, across a wide range of temperatures (T), pressures (P) and physical-chemical environments. Nowadays, we know fairly well how to analyze total sulfur concentrations, chemical forms and isotopes in materials at ambient conditions and natural fluid, gas, or mineral samples brought to the Earth surface and accessible to collection and observation. All these samples are final products of the fluids (and silicate melts) operating at high temperature and pressure in the Earth's interior or in technological processes requiring elevated temperatures. Therefore, the fundamental question that yet remains unanswered is the following. Do all these samples and data collected therefrom correctly reflect the true sulfur speciation and partitioning in fluid(-melt-mineral) systems at elevated temperatures and pressures? The answer to this key question requires a combination of in situ experimental and analytical approaches coupled with physical-chemical modeling. Over the last 20 years, these approaches have seen significant progress that led to paradigm changes about sulfur chemical speciation in deep geological fluids and its role in metal transport and ore deposit formation. These changes will be particularly emphasized here.

In this chapter, we overview sulfur concentrations, speciation and behavior in different types
of hydrothermal fluids. We obtain this information from a wide range of analytical, experimental
and computational methods developed for sulfur, whose advantages and limitations will be
presented. We further discuss available thermodynamic data and their reliability for predicting
sulfur speciation and solubility of sulfur-bearing minerals across a wide range of T-P parameters
and fluid compositions. We overview the chemical speciation of selected key metals and the role
played by the different sulfur ligands on metal transport and ore mineral precipitation. We discuss
the major controls on sulfur concentration and behavior in different types of hydrothermal fluids
spanning from active geothermal sources to basinal brines, to metamorphic and magmatic-
hydrothermal fluids. We conclude by outlining major near-future challenges and emerging
perspectives for sulfur-related research in the vast domains of hydrothermal fluids both operating in
nature and used in chemistry and materials applications.
**4.2 Analytical and spectroscopic methods for sulfur in hydrothermal fluids**

**4.2.1 Sampling techniques and wet-chemistry analyses of geothermal water**

Sulfur is among the major components in hydrothermal waters and is present both in the liquid phase like hot springs and the gas phase like in fumarole vapor. In natural liquid hydrothermal fluids, sulfate (S<sup>VI</sup>, mainly SO<sub>4</sub><sup>2-</sup>) and sulfide (S<sup>-II</sup>, including H<sub>2</sub>S and HS<sup>-</sup>, and sometimes S<sup>2-</sup>) are commonly referred as the dominant sulfur oxidation states, but other sulfur species with intermediate oxidation states may also be present and in some cases predominate. The sulfur species of intermediate oxidation states observed in geothermal water include sulfite (SO<sub>3</sub><sup>2-</sup>), thiosulfate (S<sub>2</sub>O<sub>3</sub><sup>2-</sup>), polythionates (S<sub>n</sub>O<sub>6</sub><sup>2-</sup>), polysulfides (S<sub>n</sub><sup>2-</sup>), dissolved elemental sulfur (S<sup>0</sup>), and aqueous metal sulfide complexes of Fe, As, Mo, Sb and W (Luther et al. 2001; Keller et al. 2014; Planer-Friedrich et al. 2007, 2020). Detailed studies on sulfur speciation in hydrothermal waters are few and typically only total sulfur or sulfate and sulfide concentrations are analyzed (e.g., Takano 1987; Xu et al. 1998, 2000; Druschel et al. 2003; Nordstrom et al. 2009; Kaasalainen and Stefánsson 2011b).

Samples of liquid geothermal water are commonly collected from hot springs and wells (Fig. 4.1). For dissolved sulfate determination, the water samples are filtered on-site through 0.45 µm (or smaller) pore size without further treatment. Alternatively, dissolved sulfide is precipitated when in solution by addition of divalent metal cation (i.e. Zn<sup>2+</sup> and Cd<sup>2+</sup>) containing solutions to precipitate metal sulfide and prevent interference of dissolved sulfide upon sample storage. In the laboratory, the concentration of dissolved sulfate is analyzed, most commonly as SO<sub>4</sub><sup>2-</sup> applying ion chromatography (IC) but alternatively as total sulfur using inductively coupled plasma atomic emission spectroscopy (ICPAES) (Kaasalainen and Stefánsson 2011a). Alternatively, dissolved sulfate is determined by titration with Ba<sup>2+</sup> to form BaSO<sub>4(s)</sub> precipitate or potentiometric titrations using various types of ion selective electrodes (i.e. Ca or Pb) (APHA 2018a).

Total dissolved sulfide is commonly determined on-site on unfiltered samples by precipitation titration with Hg acetate (Arnórsson et al. 2006; Kaasalainen and Stefánsson 2011a), colorimetrically after formation of methylene blue complex (APHA 2018b) or using direct UV-Vis spectrophotometry (Giggenbach 1974a,b). Other forms of dissolved sulfur species that have been observed in hydrothermal liquid water include for example thiosulfate, polythionates and polysulfides. All these intermediate oxidation states of sulfur are unstable upon sample storage and have, therefore, been analyzed on-site when possible (Fig. 4.1a). Thiosulfate is typically analyzed
using IC equipped with conductivity detector together with dissolved sulfate (Kaasalainen and Stefánsson 2011a) or collected onto ion exchange resin for sample preservation followed by elution in the laboratory and IC analysis (Druschel et al. 2003). Polythionates are also analyzed using IC as well as high pressure liquid chromatography (HPLC) and a spectrophotometric detector (Takano 1987; Kamyshny et al. 2006).

Cyclic voltammetry has also been applied successfully to determine concentrations of various dissolved sulfur species in hydrothermal waters and anoxic waters (e.g., Luther et al. 2001). Voltammetric signals are produced when redox active dissolved or nanoparticulate species interacts with the surface of a working electrode, resulting in electron flow that can be determined as a current that is proportional to concentration (Taillefert and Rozan 2002). A setup of three-electrodes is commonly applied for determination of dissolved sulfur species and consists of a silver/silver chloride reference electrode, a platinum counter electrode, and an Au-amalgam working electrode (Brendel and Luther 1995; Luther et al. 2001). Aqueous and nanoparticulate sulfur species that can be identified by cyclic voltammetry include dissolved sulfide, sulfur, polysulfide, thiosulfate and aqueous iron sulfide (e.g., Luther et al. 2001; Boyd and Druschel 2013).

Sulfur may also be present in hydrothermal vapor, for example in fumarole discharges and two-phase well discharges (liquid and vapor discharges). Hydrogen sulfide gas is the most common gaseous species, but sulfur dioxide ($SO_2$) is sometimes also detected at high temperatures (>350–400°C). Fumarole samples are collected by placing a plastic funnel over or inserting a metal tube into the vapor outlet of a fumarole emission and connecting a gas bottle using a silicone or glass tube (Fig. 4.1b). For two-phase well discharges, the liquid and vapor phase are separated at the wellhead using a Webre separator and vapor samples collected into gas bottles like for fumaroles (Fig. 4.1c). The gas bottle, most commonly of 50–250 mL total volume, is evacuated prior to sampling and contains ~10 mL of ~50% KOH or ~20% NaOH solution per 100 mL of total volume. The non-condensable gases, including hydrogen sulfide gas, are analyzed using Hg-acetate precipitation titration previously described (Arnórsson et al. 2006) or as $SO_4$ using IC after oxidation by hydrogen peroxide. Sometimes, $SO_2$ is also analyzed after precipitation of sulfide using divalent metal solution ($Cd^{2+}$, $Zn^{2+}$) followed by filtration, oxidation and analysis of dissolved sulfate using IC.

### 4.2.2 Bulk crush-leach analyses of fluid inclusions for sulfur

Fluid inclusions are tiny volumes of single- or multiphase fluid (usually 100s to 1000s µm³; Bodnar 1983) trapped in both gangue and ore minerals during their growth and posterior
deformation. These microsamples provide the only direct witness of the composition of paleo
hydrothermal fluids that has been recognized since 200 years (e.g., Davy 1822; Sorby 1869) and
more systematically explored since 1950s (e.g., Roedder 1984; Diamond 2003; references therein).
Historically, chemical analysis of the solute content (dissolved cations, anions, and volatiles,
including sulfur) of fluid inclusions, termed crush-leach method, involved bulk extraction of
numerous fluid inclusions from the host crystals, followed by analysis of leachates by various
methods (e.g., conventional solution chemistry, spectrophotometry, ion and gas chromatography,
potentiometry or mass-spectrometry). Both volatile (H₂S and SO₂, gas-bubble hosted) and anionic
(sulfate, liquid/brine-phase hosted) sulfur forms have been analyzed. For example, reduced sulfur,
including H₂S and COS, was analyzed by decrepitating fluid inclusions from vein quartz samples
in an N₂ stream followed by entrapment in NaOH and fluorometric analysis, at ng S levels (Bottrell
and Miller 1989). Potentiometry with Ag/Ag₂S electrode was used to analyze sulfide sulfur from
quartz- and fluorite-hosted fluid inclusions with detection limits of <10⁻⁴ m S (Korsakova et al.
1991). Mass spectrometry (MS) was used to analyze H₂S and SO₂, along with other common
volatiles (CO₂, N₂, H₂, CH₄) after crushing or thermal decrepitation of fluid inclusions (e.g., Norman
and Sawkins 1987). Gas-chromatography mass-spectrometry (GC-MS) analyses of volatile
components of bulk fluid inclusions comprise ~30% of the total analyses published so far according
to the statistical compilation of Naumov et al. (2009). Crush-leach dispositives coupled with ion
chromatography were used for analyses of sulfate ion and other anions (halogens) with detection
limits of less-than-ppm level in brine-type fluid inclusions in quartz, emerald, calcite, and fluorite
from metamorphic and sediment-hosted deposits (e.g., Banks et al. 2000; Giuliani et al. 2003;
Sośnicka et al. 2023). A remarkable suite of sulfur inorganic and organic species (H₂S, COS, CS₂,
organic polysulfanes, and thiols) has been revealed by thermal decrepitation/desorption GC-MS
analyses at 250 °C on fluid inclusions hosted by 3.5-billion-year-old barites from Pilbara Craton,
Australia (Mißbach et al. 2021). Although in general crush-leach techniques for sulfur offer quite
low detection limits for properly selected and processed samples, they are commonly prone to
contamination by impurities and are limited by the availability of sufficient fluid-inclusion mass.
The major general limitation of bulk methods is that such averaged data from potentially different
inclusion generations in one sample provide little information about the time-space-composition
evolution of fluid-mineral systems. Nowadays, these bulk methods stand as rather subordinate for
sulfur (as well as most trace metals) as compared to in situ analyses using LA-ICPMS and Raman
spectroscopy of individual fluid inclusions as will be shown in the following sections.

4.2.3 LA-ICPMS of individual fluid inclusions in hydrothermal minerals
Individual fluid inclusions can be trapped during or after crystal growth of the host mineral and usually classified into primary, secondary and pseudo-secondary inclusions according to their origin (Roedder 1984; Goldstein 2003). Their study by microthermometry techniques can provide direct information about physical parameters and composition of the mineralizing hydrothermal fluids and their evolution in space and time in terms of temperature, pressure, fluid salinity and major volatile content. Goldstein and Reynolds (1994) introduced the use of fluid inclusion assemblages (FIAs), defined as groups of cogenetic fluid inclusions occupying particular petrographic features, such as zones of crystal growth, healed fractures, sector zoning. These features can, in most cases, be unambiguously recognized using classical petrographic or other imaging techniques (e.g., scanning electron microscopy-cathodoluminescence, SEM-CL, or optical CL; Van den Kerkhof and Hein 2001). In fluid inclusion research, the concept of FIAs is crucial to correct inclusion analysis and interpretation of the results. This concept has been largely applied to study ore-forming processes and its caveats have been recognized and thoroughly assessed. For example, Wilkinson (2001) presented a summary of fluid inclusion data from a variety of hydrothermal environments, discussing the uncertainties associated with the determination of timing relationships between fluid inclusion entrapment and ore formation, post-entrapment modifications of inclusions, diffusion of components and isotopic exchange with the host minerals. Most of the fluid inclusion studies relevant to hydrothermal ore deposits are conducted on optically transparent gangue minerals, based on the assumption that they are syngenetic with the intergrown ore minerals. However, there is growing evidence that ore mineralization in hydrothermal systems may form from episodic events of multiple fluid sources, where precipitation of ore and gangue minerals can be decoupled in time (Wilkinson et al. 2009). Near-infrared (NIR) transmitted-light microscopy has been successfully used to study fluid inclusions also in opaque ore minerals, thus providing evidence of distinct fluid inclusion compositions recorded in gangue and ore minerals which display apparent “equilibrium” textural relationships (e.g., Campbell and Robinson-Cook 1987; Campbell and Panter 1990; Kouzmanov et al. 2002, 2010).

An important caveat of microthermometry is about the determination of true entrapment
temperature and pressure. Trapping pressure for fluid inclusions can only be estimated if the exact
temperature of entrapment is known, or if fluid inclusions are trapped during phase separation. Fluid
inclusions that have been entrapped as a single-phase fluid, can provide information, during
microthermometry measurements, on the minimum temperature and pressure of entrapment only
(Roedder 1984). This is because most inclusions have trapped fluids at pressures higher than their
vapor pressures. Thus, to establish the true temperature or pressure of entrapment, in many cases,
an important “pressure correction” is needed. This correction can be determined either using
independent mineral geothermometers or from independent evidence of the depth of the system at the time of trapping. Then the pressure may be determined from P-V-T-X data on the inclusion fluid, which is generally approximated by a water-salt±gas (CO₂, CH₄) system (Roedder and Bodnar 1980). Such a correction may result in differences as high as several hundred degrees between temperature of homogenization and trapping temperature (see section 4.5.3 on metamorphic fluids).

Fluid inclusion microthermometry alone provides only limited information about the composition of studied inclusions and exclusively in terms of major components of the entrapped fluid. In the 1990s, two microanalytical techniques were successfully applied to quantify, for the first time, elemental concentrations in individual fluid inclusions, mainly from magmatic-hydrothermal deposits, laser ablation - inductively coupled plasma mass-spectrometry (LA-ICPMS) and proton-induced X-ray emission spectroscopy (PIXE) (e.g., Ryan et al. 1991; Heinrich et al. 1992; Rankin et al. 1992; Wilkinson et al. 1994; Audôtat et al. 1998; Günther et al. 1998). Despite a destructive character of the analysis compared to PIXE (see section 4.2.5), LA-ICPMS quickly became the most powerful and efficient multi-element technique for the analysis of concentrations of a large range of major to ultra-trace elements, spanning from Li to U, in individual fluid inclusions being single-phase (liquid or gas) or polyphase (liquid+gas±solids) mixtures at the time of measurement. The unique features of the method consist of the simultaneous acquisition of the signal, via optically controlled ablation of the entire fluid inclusion volume, and accurate quantification of the elemental concentrations, using a combination of mostly matrix-independent external calibration procedures along with internal standards usually using the Na concentration determined from thermometric measurements (Pettke et al. 2012).

Early prototype analytical setups used for fluid inclusion analysis as well as their modern analogues have been described in detail by Günther et al. (1997), Günther et al. (1998), Heinrich et al. (2003), Pettke (2008) and Pettke et al. (2012). Both quadrupole and single collector sector field mass spectrometers are commonly used for element concentration analysis, allowing very fast mass jumping between the different isotopes. The high speed is an important advantage for the analysis of relatively short signals arisen from fluid inclusions, thus enabling limits of detection as low as 0.01 ppm for the heavier elements (e.g., Sr, Ba, Rb, Cs, Au, Pb). Recently, promising results using time-of-flight mass spectrometers (TOFMS) for analysis of fluid inclusions were also reported (Harlaux et al. 2015). The laser system used to drill out individual inclusions is an essential component of the setup. The use of a homogenized ArF excimer (193 nm wavelength) laser ablation systems is best suited for the controlled ablation of inclusions, thereby allowing proper ablation of quartz (at energy density of ~25 J·cm⁻²), which is the most common host for fluid inclusions in hydrothermal deposits. The use of Iris diaphragms, allowing either progressive or stepwise opening
of the laser beam diameter (Günther et al. 1998; Guillong and Heinrich 2007), reduces unwanted
common effects during ablation such as cracking of the quartz host near the crater rim that can result
in explosion of the inclusion being analyzed.

As summarized by Pettke et al. (2012), fluid inclusion analysis by LA-ICPMS consists of
the following steps. 1) The fluid inclusion sample (usually small chip of few mm in diameter) is
placed in a small ablation cell. 2) The target fluid inclusion is selected using transmitted-light optical
microscope. 3) The fluid inclusion volume is ablated using an excimer laser system with full optical
control on the position of the pit during ablation. 4) The aerosol is transported out of the ablation
cell using a He carrier gas mixed with Ar gas (and sometimes H₂) and then introduced into the
ICPMS. 5) Aerosol particles are atomized and then ionized and the ions are extracted from the
plasma and filtered based on their mass-to-charge ratio and measured by the mass spectrometer. 6)
Transient signals of several seconds to several 10s seconds duration are produced (Fig. 4.2). 7)
Quantification of the measurements is performed by analyzing external standard reference materials
(SRM) bracketing the fluid inclusion analyses; additionally, the concentration of an internal
standard element has to be independently known (usually Na from microthermometry determination
of fluid salinity prior to LA-ICP-MS analysis). 8) Calculation of absolute element concentrations in
individual fluid inclusions using available specially designed software (e.g., SILLS; Guillong et al.
2008b) or in-house developed spreadsheets and Matlab routines in various laboratories.

Easy access to several laboratories with good in-house knowledge about LA-ICPMS analysis
of fluid inclusions has resulted in a large amount of data on metal concentrations in mineralizing
fluids in various hydrothermal environments to be generated during the last two decades. However,
sulfur concentrations were not routinely measured, mainly due to polyatomic interferences on
isotope ³²S⁺ (95.02% abundance) from ¹⁶O¹⁶O⁺, and on isotope ³⁴S⁺ (4.21% abundance) from
¹⁶O¹⁸O⁺, originating from the ambient air or from the host mineral. Guillong et al. (2008a) reported
the first quantitative analyses of sulfur in individual fluid inclusions using laser ablation inductively
coupled plasma quadrupole mass spectrometry (ICP-Q-MS) and ICP sector field mass spectrometry
(ICP-SF-MS) allowing the sulfur signal to be resolved from those polyatomic interferences.
However, the background level of sulfur remains relatively high (around 10⁵ cps; Fig. 2a,b) in most
LA-ICPMS laboratories, due essentially to contamination from the ambient air, ablation-cell
environment, and previous analyses of different samples. The accuracy on developed in-house
reference materials scapolite-17 and NIST glasses, reported by Guillong et al. (2008a), was within
5% for ICP-SF-MS and 7% for ICP-Q-MS. Precision (~35% RSD, relative standard deviation) and
detection limits reached (~100 ppm) are limited by sometimes enigmatic and poorly reproducible
sources of sulfur contamination.
Figure 4.2b shows typical transient ablation signals from four main types of natural fluid inclusions. In each case, the sulfur signal shows a distinct maximum corresponding to the fluid inclusion opening by the laser ablation system. Due to the intrinsically high background for sulfur, its signal is shorter in time and the signal-to-noise ratio is much lower than the signal of the other major and trace metals and metalloids in the inclusion fluids. Often, the maxima in the sulfur signal correspond to ablation of solids (e.g., sulfides and sulfates; Fig. 4.2a), present as daughter phases inside the inclusion vacuoles. Such solids either formed on cooling due to decrease of sulfur solubility with lowering the temperature or were accidentally trapped during host crystal growth and fluid inclusion formation.

As summarized in Table 4.2, since the work of Guillong et al. (2008a) a number of studies have been published, reporting sulfur concentrations in various fluid inclusion types from different hydrothermal environments. Total sulfur concentrations in natural fluid inclusions are highly variable, ranging from <0.001 wt% up to 12 wt% that may be reached in some hypersaline liquid inclusions from magmatic-hydrothermal systems and basinal brines. The sulfur contents in different types of ore forming fluids will be discussed in section 4.5.

### 4.2.4 Raman spectroscopy

Raman spectroscopy holds the first place in studies of sulfur speciation in fluid inclusions from a large variety of magmatic-hydrothermal-metamorphic settings as well as in model hydrothermal fluids in laboratory experiments. Sulfur compounds have been routinely analyzed using this technique by chemists and materials scientists since 1950s (e.g., see Long 1977; Ferraro et al. 2002 for fundamentals; Singh 2002 for historical background; Lin et al. 2014 for an overview of mineral physics applications). The Raman spectroscopy method as well as its various applications in Geosciences for studying volatile elements in fluids, melts and gas bubbles (H, B, C, O, and S) have been a subject of numerous reviews over the past 20 years (e.g., Burke et al. 2001; Dubessy et al. 2012; Frezzotti et al. 2012; Neuville et al. 2014; Pasteris and Bessac 2020, to name a few); here we focus exclusively on its practical outcomes for sulfur in the fluid phase. Table 4.3 lists the major Raman characteristic wavenumbers, types of vibration and typical cross-sections (relative to sulfate) for inorganic sulfur species that can typically be analyzed in hydrothermal fluids.

#### *Advantages and limitations*

The main advantages of Raman spectroscopy for analyzing sulfur in dissolved forms are the following. First, the characteristic Raman spectral frequencies (i.e., wavenumbers) for the great
majority of sulfur anions, molecules and complexes in the gas, aqueous or solid phase are well known. The three major types of bonds S-S, S-O/C and S-H display distinct wavenumber ranges for their most intense symmetric stretch vibrations, typically 350–600, 700–1400, and 2300–2600 cm⁻¹, respectively, making it possible to identify the different sulfur species in a single spectrum (Fig. 4.3). *Second*, these frequencies evolve little and predictably with temperature and pressure changes making it possible to perform in situ studies of sulfur in the fluid phase at elevated *T-P*. With proper corrections for Raman scattering intensity and background and with the use of standards or relative cross-sections, quantification of sulfur species in the fluid phase becomes possible both in natural fluid inclusions and experimental laboratory solutions (e.g., Pokrovski and Dubessy 2015; Barré et al. 2017; Schmidt and Seward 2017). *Third*, the resonance Raman phenomenon typical for some chromophore S species such as radical sulfur ions (e.g., Clark and Franks 1975; Clark and Cobbold 1978; Pokrovski and Dubessy 2015), which yields selective enhancement of their Raman spectral signals at certain laser wavelengths, allows detection of such species at concentrations 10 to 1000 times lower than for traditional non-resonating sulfur forms (e.g., sulfate or sulfide). *Fourth*, there exist advanced technologies of high *T-P* optical cells enabling studies of sulfur speciation in model hydrothermal fluids in the laboratory, including large-volume autoclaves with optically transparent windows (e.g., Rudolph 1996; Louvel et al. 2015; Dietrich et al. 2018; Testemale et al. 2024), fused silica capillary cells (e.g., Chou et al. 2008; Pokrovski and Dubessy 2015), synthetic fluid inclusions (e.g., Ni and Keppler 2012; Jacquemet et al. 2014), and diamond-anvil cells (e.g., Schmidt et al. 2009; Pokrovski and Dubrovinsky 2011; Schmidt and Seward 2017; Colin et al. 2020). These techniques have enabled over the past 20 years a significant advance in our understanding of sulfur high *T-P* speciation, in particular by constraining the thermodynamic stability of certain aqueous sulfur species that will be discussed further in section 4.3.4. With the development of µm-size laser beams by coupling a microscope to a Raman spectrometer, micro-Raman spectroscopy has become the method of choice since the 1980s for directly probing sulfur species in natural individual gaseous and fluid inclusions trapped in optically transparent minerals under hydrothermal conditions (e.g., see Roedder 1984, 1990; Burke et al. 2001; Frezzotti et al. 2012, for historical background and an overview). Finally, recent progress in fiber optics used to guide laser beam and collect the scattered signal allowed applications of in situ Raman probes to direct quantitative measurement of sulfur species, such as H₂S and HS⁻, in deep-ocean sediment pore waters and sea-floor hydrothermal springs (e.g., Peltzer et al. 2016; Li et al. 2023).

There are four general limitations of Raman spectroscopy for studies of dissolved species in hydrothermal fluids. *First*, intrinsic weakness of the Raman inelastic scattering signal results in relatively high detection limits for sulfur species both in fluid inclusions and high *T-P* cells, typically
of 100s ppm S (except for the resonant species such as radical ions S₃⁻, if combined with the right choice of the laser wavelength; e.g., Pokrovski and Dubessy 2015), as compared to much more sensitive wet chemistry analyses of geothermal fluids offering ppm-level detection limits (see section 4.2.1). *Second*, quantifying S species concentrations in dense water-rich media requires corrections for background signal, Raman cross-sections, and spectrometer response (Schmidt and Seward 2017 and references therein). This limitation is more severe for natural fluid inclusions for which systematic calibrations procedures are hard to set up. As a result, quantitative estimates of absolute sulfur species concentrations or molar ratios in natural fluid inclusions have been limited mostly to gas-phase dominated and water-poor inclusions (e.g., CO₂-rich; Burke et al. 2001). *Third*, Raman spectroscopy is an excellent tool for in-situ distinguishing protonated *vs* deprotonated sulfur anions (e.g., HSO₄⁻ *vs* SO₄²⁻, H₂S *vs* HS⁻), but it is not for detecting alkali ion pairing that becomes increasingly important in magmatic-hydrothermal fluids (e.g., SO₄²⁻ *vs* NaSO₄⁻, HS⁻ *vs* NaHS). The Raman peak position and signal intensity of sulfur anions are generally weakly affected by complexing with Na⁺, providing detection of the overall sulfur form (e.g., sulfate). *Forth*, the main stretching bands of some sulfur species exhibit an assymetry at the lower-wavenumber peak side that is due to both the development of hot bands with increasing temperature and hydrogen-bond dymanics (proton hopping) in aqueous solution. This is, for the example, the case for the main stretch band of gaseous and aqueous H₂S (~2610 and ~2590 cm⁻¹, respectively; Jiang et al. 2017; Pokrovski and Dubessy 2015; **Table 4.3**), and of the HSO₄⁻ ion (Dawson et al. 1986; Schmidt and Jahn 2024). Such an asymmetry may easily be mistaken for a band of another species (e.g., HS⁻ at 2570 cm⁻¹; Fansang and Zajacz 2025). *Finally*, a specific limitation, in particular for natural fluid inclusions, is laser-induced fluorescence common in the presence of organic impurities (e.g., Burke 2001), which limits data availability from fluid inclusions of sedimentary-basin fluids. This limitation may eventually be avoided by using a different laser wavelength or emerging more advanced Raman acquisition techniques (e.g., coherent anti-Stokes Raman scattering, CARS, e.g., Burruss et al. 2012; Pasteris and Bessac 2020), which have not yet been routinely applied for sulfur. In the following section, we discuss the key data on sulfur speciation obtained by Raman spectroscopy from natural fluid inclusions.

*Sulfur speciation from analysis of natural fluid inclusions*

**Table 4.4** summarizes sulfur species reported so far in natural fluid inclusions using micro-Raman spectroscopy, together with selected references. Hydrogen sulfide (H₂S, and in some cases HS⁻) has been the most commonly detected S species across a large variety of fluid inclusions types (gaseous, aqueous or brine) trapped mostly in quartz and more rarely in topaz, calcite, fluorite,
garnet or pyroxene, from different geological environments spanning from epithermal and sedimentary basin to shallow mantle settings. Cooling down H₂S-rich gas-like fluid inclusions (N₂-CO₂-CH₄) in microthermometry analyses may produce a range of H₂S clathrates and solid-phase polymorphs (α-, β-, γ-H₂S; e.g., Sośnicka and Lüders 2021), which may be relevant to sulfur chemistry on cold extraterrestrial planetary systems such as Uranus and Neptune that are treated in Chapter 20 (Lodders and Figley 2025, This Book). The second, most commonly detected sulfur form is sulfate (SO₄²⁻, HSO₄⁻) in fluid inclusions from sedimentary and metamorphic settings, evaporite formations and volcanic/magmatic rocks. The unique ability of Raman to distinguish between H₂S and HS⁻ and between SO₄²⁻ and HSO₄⁻ that have distinct Raman band wavenumber positions (Table 4.3) makes it possible to estimate the pH value in moderate-temperature fluids (<350°C) by coupling the measured species ratios with the known deprotonation constants in these pairs of species (e.g., Dubessy et al. 1992; Boiron et al. 1999; see also section 4.3.4). In addition to dissolved sulfate and sulfide species, both corresponding types of minerals (e.g., anhydrite, gypsum, barite, pyrite, pyrrhotite, chalcopyrite, and some others, e.g., Frezzotti et al. 2012) have been frequently identified in fluid inclusions from sedimentary to magmatic settings. In particular, such mineral phases can be mapped in inclusions by combining Raman and X-ray fluorescence micro-tomography (e.g., Schialvi et al. 2020). Most often, these minerals are products of fluid cooling and subsequent solid phase precipitation after the inclusion entrapment at high temperatures. In some cases, however, such as in porphyry copper systems, they may also reflect post-entrapment diffusion of copper (Cu⁺) into quartz-hosted H₂S-rich inclusions resulting in formation of chalcopyrite (e.g., Lerchbaumer and Audôtat 2012; Seo and Heinrich 2013), or hydrogen movement in or out the inclusion that modifies the sulfate/sulfide mineral proportions and aqueous species ratios (e.g., Mavrogenes and Bodnar 1994). Even though, for the sake of commodity, most Raman analyses on natural inclusions have been performed at ambient conditions, i.e. on final products of cooled and decompressed fluids, they do attest to the common presence, and in many cases to coexistence, of sulfate and sulfide aqueous forms in geological fluids across the lithosphere.

Intermediate-valence sulfur forms are less commonly detected in fluid inclusions. The main reason is both because most of them are usually less abundant than sulfate and sulfide, in particular at low temperatures, and because their Raman signals overlap with those of the inclusion host (quartz) and other, often more abundant species (e.g., carbonates). For example, the Raman signals of SO₂ and S₈ (Table 4.3) are often obscured by more intense Raman peaks arising from quartz (e.g., Jacquemet et al. 2014). Combined with the (partial) disproportionation of SO₂ in aqueous solution to sulfate and sulfide on cooling (e.g., see section 4.5 and Kouzmanov and Pokrovski 2012), SO₂ has only been detected in rather particular fluid inclusions such as those water-poor and CO₂-rich
inclusions from mantle derived carbonatites (e.g., Frezzotti et al. 2002). The formation of native sulfur in detectable amounts in some inclusions from evaporites and metamorphic rocks (e.g., Giuliani et al. 2003) implies high total S concentrations in the original hydrothermal fluid.

More “exotic” species, such as polysulfanes H₂S*n*, were detected, for the first time, in CH₄/H₂S-rich and S⁰-bearing inclusions in metamorphic quartzite and ruby-bearing marble (Hurai et al. 2017; Huang et al. 2017), but were likely subjected to photolysis upon exposure to the laser beam (Hurai et al. 2017). Carbonyl sulfide, COS, was also detected in the gas phase of inclusions from similar metamorphic settings (Grishina et al. 1992; Giuliani et al. 2003; **Table 4.3**). Finally, aqueous polysulfide ions S*n*²⁻ and polymeric molecular sulfur S*n*⁰ other than S₈, together with the trisulfur radical ion S₃⁺⁻, have recently been detected and semi-quantified in fluid inclusions in fluorite and quartz from the Carnian evaporite formation in French Alps by Barré et al. (2017). These S-rich inclusions (of total concentration up to ~3000 ppm S) contained both SO₄²⁻ and H₂S together with native sulfur, S₈(s,l), at ambient conditions, and displayed the growing formation of the polymeric sulfur species upon heating to 300°C. Their study is the first direct finding of the trisulfur radical ion in natural hydrothermal fluids, in excellent agreement with synthetic fluid inclusion Raman spectroscopy analyses (Jacquemet et al. 2014) and more systematic measurements on analogous model S-bearing solutions in laboratory experiments and derived therefrom thermodynamic properties of S₃⁺⁻ (Pokrovski and Dubessy 2015, see section 4.3.4).

In summary, Raman spectroscopy of natural fluid inclusions reveals that even though sulfate and sulfide are the apparently dominant aqueous sulfur forms, intermediate-valence sulfur species commonly form in various types of geological fluids. Their quantification at elevated *T-P* in natural fluids would require thermodynamic and kinetic constants that are derived from laboratory experiments (both in situ and ex situ) under controlled conditions. Physical-chemical insight into sulfur aqueous speciation from such data will be provided in section 4.3.4.

### 4.2.5 X-ray absorption (XAS), emission (XES and PIXE), and fluorescence (XRF) spectroscopy

These X-ray based spectroscopies for sulfur probe electronic transitions at S K-edge (~2470 eV), i.e. from the core electronic level (1s) and, more rarely, at S L-edge (~163 eV), i.e. from 2p levels, of the sulfur atom to different higher electronic orbital levels (3p, 3s, 4d) defined by S oxidation state and chemical bonding with other atoms. These transitions may be probed by different types of spectroscopy, for determining either the total sulfur concentration by XRF and PIXE, or analyzing S redox and chemical speciation using higher energy resolution offered by XAS and XES.
*X-ray absorption spectroscopy (XAS)*

Synchrotron-based X-ray absorption spectroscopy (XAS) has been used for analyzing the chemical speciation of sulfur and other heavier elements since the mid-1980s (Teo 1986). The XAS spectrum includes the near-edge part, typically around ±10 eV of the absorption edge, called X-ray absorption near-edge structure or XANES, and the extended X-ray absorption fine structure part or EXAFS, extending to 100s eV above the edge. For light elements like sulfur, having low-energy absorption edges, the XANES part is usually used both because it is the most sensitive to S atomic environment and is relatively easy to acquire, whereas the EXAFS part is more commonly used for studying heavier absorbing elements (e.g., Teo 1986; Stöhr 1992). The shape and energy position of XANES features represent a powerful in situ method for studying the chemical and redox state of sulfur in a variety of technological and geological materials. It has been extensively applied for sulfur speciation analysis in minerals (Fleet 2005), silicate glasses (Wilke et al. 2011; Simon and Wilke 2025), organic matter and biological samples (Jalilehvand 2006; Burton et al. 2013; Qureshi et al. 2021), and meteorites (e.g., Bose et al. 2017) to name a few. Excellent reviews discuss fundamentals and practical advantages and limitations of S K-edge and L-edge XANES spectroscopy in mineralogy, environmental and life sciences (Fleet 2005; Jalilehvand 2006). Detailed applications of this type of spectroscopy to S-bearing minerals and silicate glasses are presented in other chapters of this book (Pan and Mi 2025; Simon and Wilke 2025).

Compared to Raman spectroscopy, XANES spectroscopy is a more straightforward approach because i) the energy position (extending for over >10 eV from S<sup>2</sup>− to S<sup>6+</sup>; Table 4.3) and shape of the spectra are both sensitive to sulfur redox state and chemical bonding, and ii) it enables identification and potential quantification of all different sulfur forms owing to the additive nature of the X-ray absorption signal, in contrast to other types of spectroscopy (e.g., EPR, resonance Raman, UV-Vis) that do not have enough redox resolution and/or privilege only certain species (e.g., radical polysulfide ions). Combined with accurate knowledge of the absorption energy position for different S redox states (Table 4.3) and S-bearing reference mineral/organic compounds whose XANES spectra can be accessed from databases (e.g., https://www.esrf.fr/home/UsersAndScience/Experiments/XNP/ID21/php.html), this method offers direct quantitative information on the relative amounts of the different S redox and chemical forms in a sample. The analysis may be complemented by rapidly developing theoretical chemistry modeling tools for simulating XANES spectra (e.g., FDMNES, Joly 2022; FEFF10, Kas et al. 2021), to help relate certain observed spectral features to a particular S bonding molecular environment in organic and mineral samples such as petroleum and ultramarines (e.g., Risberg et al. 2007; Pascal et al. 2015; Rejmak 2018).
The inconvenience of XANES is both weak penetration of the incident X-ray beam into the sample at the low energies of S absorption edges (< few µm at S K-edge, ~2.5 keV) and strong absorbance of both incident and scattered radiation by the sample environment, therefore requiring He atmosphere and/or high-vacuum environment at synchrotron beamlines and weakly absorbing sample holders made of light materials (plastic films, extremely thin-walled capillaries), in particular for studies of liquid samples.

The weak penetration of soft X-rays (at energies <2.5 keV) has been the major limitation in applying this type of spectroscopy to aqueous solutions, especially at elevated temperatures and pressures. We are only aware of studies up to 70 °C. XANES spectra of aqueous solutions of common organic and inorganic sulfur compounds have been used as spectral references for comparison purposes in many studies, but fewer studies were devoted to in-depth characterization of S redox speciation and structure. Those concerned organic aliphatic and aromatic sulfur molecules in biological solutions, petroleum, and aqueous organic matter such as humic substances at ambient conditions (e.g., Qurechi et al. 2021). Inorganic aqueous sulfur species that have been studied by XAS to a significant degree of structural detail include thiosulfate in which sulfur redox state is consistent with -1 and +5 (Vairavamurthy et al. 1993), sulfate and hydrosulfate that have XANES spectra clearly distinguishable from those of their solid-phase counterparts (Pin et al. 2013), as well as sulfite, sulfonate and SO₂. For example, XANES spectra of hydrosulfite aqueous solutions (Risberg et al. 2007) reveal two coordination isomers, HSO₃⁻ and SO₂(OH)⁻ present at 25 °C in a ~1:2 ratio. This ratio slightly increases with temperature, in quantitative agreement with recent Raman measurements (Eldridge et al. 2018) that show clearly distinguishable peaks for the two isomers (Table 4.3). High-temperature low-pressure sulfur vapors, to 300°C at <0.1 bar, were also examined by S K-edge XANES in an exploratory study using a furnace design with Be windows, to show the formation of S₂(g) and possibly other Sₙ species, and their oxidation to SO₂ in the presence of H₂O with increasing temperature (Engel et al. 2014). There are a few examples demonstrating that the S K-edge XANES may, in some cases, be complementary to Raman spectroscopy, revealing structural and speciation features of sulfur in fluids inaccessible by other methods. The major limitation related to the strong attenuation of X-ray signal by the cell material may be overcome in the future by recent developments of perforated diamond anvil cells for high pressure measurements. In such cells, the incoming focused X-ray beam passes through a thin diamond disk (30 µm) affixed to a fully perforated diamond anvil and X-ray fluorescence photons from the sample are collected in back-scattering geometry through the same diamond disk (Wilhelm et al. 2016). Such design holds great promise to allow us one day to study sulfur speciation by in
situ XANES spectroscopy in model hydrothermal fluids as it is almost routinely being done nowadays by Raman spectroscopy (see the section above).

An alternative approach that overcomes the low penetration capacity of soft X-rays is the use of nonresonant inelastic X-ray scattering (NRIXS) or alternatively called Raman X-ray scattering (RXS) spectroscopy. In this method, the sample is excited with a hard X-ray beam (usually ~10 keV) to obtain XANES spectra at sulfur L-edges (at 160–230 eV). The technique is increasingly used at modern synchrotrons to access absorption K-edges of light elements (e.g., O, N, P) and L- and M-edges of 3d transition elements (e.g., Sahle et al. 2015). We are aware of one recent study of concentrated sulfuric acid solutions using this method (Niskanen et al. 2015), but with improvements of detector resolution and sensitivity and synchrotron X-ray beam brightness, the method is promising for studying S aqueous speciation at elevated T-P.

A future approach to access sulfur speciation in natural fluid phases at elevated T-P could be the use of zeolite-family silicate minerals belonging to the sodalite group, (Na,Ca)₈[Al₆Si₆O₂₄](Cl,S,SO₄)₁⁻₂, which are rather common accessory minerals in a wide range of metamorphic and magmatic-hydrothermal settings (e.g., Mi and Pan 2018). These minerals are capable of hosting, in their Al-Si network structural cages, the whole range of sulfur redox forms, from sulfide to sulfate, which may potentially reflect the sulfur redox and speciation in the original fluid. In particular, the blue varieties of sodalite – lazurites or ultramarines – are capable of stabilizing the radical chromophore ions, S₃⁺ and S₂⁻, which confer to these natural minerals, and their synthetic analogs widely used in industry, a specific hue ranging from violet-blue to green-blue (e.g., Reinen and Lindner 1999; Farsang et al. 2023). Red ultramarines whose color is likely due to the red chromophore neutral S₄⁰, rather than radical S₄⁻, also exist (e.g., Rejmak 2020). Sulfur species have been extensively studied in both natural and synthetic ultramarines using different analytical, spectroscopic and quantum-chemistry computational methods such as Raman, UV-visible, EPR, XPS, XAS, DFT, mostly at ambient conditions (e.g., Chivers 1974; Reinen and Lindner 1999; Fleet 2005; Fleet et al. 2005; 2010; Ledé et al. 2007; Tauson et al. 2012; Chivers and Elder 2013; Ganio et al. 2018, to name a few). More detailed account of the speciation and structural features of sulfur in these minerals and the corresponding methods is provided in Chapter 3 (Pan and Mi 2025). The exceptionally rich information on S speciation and redox state offered by S-XANES spectra of lazurites is demonstrated in Figure 4.4. Coupled with appropriate sulfur reference compounds and theoretical calculations, the whole range of S oxidation states may be analyzed, including both radical ions that cannot otherwise be stabilized in quenched aqueous solution. Therefore, lazurites, together with other accessory non-sulfide minerals such as scapolite, Na₄[Al₃S₉O₂₄](Cl,CO₃,S)-Ca₄[Al₆S₆O₂₄](Cl,CO₃,S) (Fleet et al. 2005; Qiu et al. 2021; Hamisi et al.
2023), and apatite, Ca₅(PO₄)₃(F,Cl,OH,S) (Kim et al. 2017; Konecke et al. 2017; Sadove et al. 2019), which also host multiple sulfur forms and oxidation states that may reflect the redox conditions of the formation environment, emerge as promising indicators of redox conditions and S fluid-phase speciation in high T-P settings. However, experimental calibrations of fluid/mineral partition coefficients of each sulfur major redox form and thermodynamic and kinetics constraints on the solubility and stability of these minerals are needed to use them on a quantitative basis.

*X-ray Emission Spectroscopy (XES)*

Both S Kα and Kβ XES, originating from 2p-1s and 3p-1s transitions, respectively, are valuable tools, complementary to XAS, to extract information about the oxidation state and bonding structure of sulfur in various sulfur-containing compounds, including silicate glasses and minerals (e.g., Alonso-Mori et al. 2009, 2010; Wilke et al. 2011; Hettmann et al. 2012; Qureshi et al. 2021). Measurements of emitted S Kα X-rays wavelengths are performed during exposure of the solid sample to a high-energy electron beam, typically using electron probe microanalysers (EPMA). These wavelengths exhibit a systematic shift toward higher energies by about 2 eV from sulfide to sulfate. These shifts, calibrated using sulfate and sulfide standards, have been routinely used in EPMA to determine the apparent S oxidation state in glasses and minerals (e.g., see Wilke et al. 2011 for detailed examples). Aqueous and non-aqueous solutions and some organic sulfur compounds were also measured using an X-ray excitation beam instead of an electron beam to determine reduced/oxidized sulfur ratios (e.g., Holden et al. 2018), but the intrinsically low energy resolution due to relatively deep atomic electron transitions (2p), does not provide a sufficient degree of sulfur speciation details compared to XAS.

In contrast, high-resolution laboratory or synchrotron-based Kβ XES, probing shallower electronic levels (3p) much more sensitive to S chemical bonding, shows considerably greater chemical sensitivity, with sulfide-to-sulfate energy shifts extending over >10 eV. This large range is sometimes accompanied by spectral fine structure allowing distinction between sulfur individual chemical species, in addition to S apparent redox states (Deluigi and Riveros 2006; Kavčič et al. 2007; Alfonso-Mori et al. 2010; Qureshi et al. 2021). In general, however, S Kβ XES shows lesser sensitivity than S K-edge XANES, with rare exceptions such as, for example, organosulfur molecules with different types of benzoic rings (e.g., Qureshi et al 2021). Even though, in principle, XES may be made with both laboratory sources and synchrotron facilities, it requires specialized instrumentation, while XANES may be collected using fairly standard equipment available on many beamlines. At present, these limitations make the XES method generally less convenient and less
sensitive than XANES for determining S speciation, but its potential contribution to S analyses in
model hydrothermal fluids remains to be seen when high T-P optical cells, transparent enough at S
K-edge X-ray energies, have been developed.

Synchrotron X-ray Fluorescence (SXRF) and Proton Induced X-ray emission (PIXE) spectroscopies

Fundamentally, these two types of spectroscopy probe X-rays emitted during electronic transitions to K- and L-edges of moderately heavy elements (>Na) from excited states created by either X-ray (SXRF) or proton (PIXE) beams of high energy generated by synchrotron or nuclear sources, respectively. Coupled with adequate standards and X-ray absorption matrix coefficient corrections, these methods enable quantifying bulk element concentrations without providing chemical speciation information. While both methods are quite sensitive for heavy elements having high energies of their absorption edges (e.g., Vanko et al. 2001; Ménez et al. 2002; Ryan et al. 2005; references therein), sulfur has rarely been probed by this type of spectroscopy in fluid inclusions due to both its relatively low concentrations and low K-edge energy engendering strong absorption of X-rays by the host mineral. For example, the X-ray penetration length at the S K-edge (i.e., 1/e of the transmission signal ~ 37% of total X-ray flux at the sulfur Kα₁ fluorescence line of 2.3 keV) of quartz is less than 10 µm (Vanko et al. 1993), making fluorescence detection challenging. We are only aware of a couple of reports of S qualitative detection by PIXE in magmatic brine and vapor inclusions from Mole Granite (Heinrich et al. 1992) and in metamorphic tungsten district of the French Massif Central (Gama 2000). Sulfur is very rarely mentioned in SXRF studies of host minerals and their fluid inclusions in which it is commonly below detection limit (e.g., Tang et al. 2005). With the development of the LA-ICPMS method for fluid inclusions offering greater versatility, penetration and analytical sensitivity at lower cost (see section 4.2.3), both SXRF and PIXE stand as less attractive for sulfur concentration analyses in fluid inclusions and other geological samples.

**4.2.6 Other spectroscopic methods for sulfur speciation analysis**

Infrared spectroscopy (IR)

Infrared spectroscopy is complementary to Raman due to symmetry selection rules for
molecules that make some chemical-bond vibrational bands active in IR, some other in Raman and
some in both types of excitation. IR spectroscopy is used for remote sensing of SO2 and H2S in
volcanic gases (e.g., Clarisse et al. 2011), as well as in material sciences for investigation, in
combination with Raman, of structural and electronic properties of sulfur species mostly in water-
free solvents and mineral matrices (e.g., Sakashita et al. 2000; Song et al. 2005; Gobeltz et al. 2011).
Compared to Raman spectra, IR absorption bands of the same species are generally much weaker
in intensity and larger (e.g., typically with FWMH of the same band of ~100s cm⁻¹ for IR vs ~10
cm⁻¹ for Raman), rendering spectral resolution weaker. In addition, they are more affected by the
spectral background and, in particular, by the presence of water that has a very strong IR absorption
signal. This fundamental limitation explains the much lower amount of IR vs Raman studies of
sulfur in aqueous solution. The strong water absorption in the IR range also poses limitations for the
spectroscopic cell dimensions, with optical paths of 10s µm lengths. An advantage of IR
spectroscopy (as well as UV-Vis, see below) compared to Raman is that the species concentration
is linearly proportional to its absorption signal according to the Beer-Lambert law, making IR
spectroscopy more quantitative than Raman in some simple aqueous systems. IR spectroscopy on
aqueous solutions has been used mostly in the sulfite-bisulfite system for determining both
structures and deprotonation and dimerization equilibrium constants of S<sup>IV</sup> species to 50 °C (e.g.,
Herlinger and Long 1969, Zhang and Ewing 2002; Ermatchkov et al. 2005; Risberg et al. 2007).

*UV-Visible (UV-Vis) spectroscopy*

UV-Vis spectroscopy is complementary to IR, but instead of chemical bond vibrations it
probes outer-shell electronic transitions of molecules, ions and complexes as well as their
interactions with the solvent. This property makes UV-Vis spectroscopy sensitive to a whole
hydrated ion or complex rather than to a specific chemical bond - like Raman or IR, thereby making
interpretation of UV-Vis spectra in complex systems more challenging. Like IR, UV-Vis
spectroscopy obeys the Beer-Lambert law of absorption. This fundamental property makes it a
quantitative method provided the identity and cross-section of the electronic transitions are known
or estimated. This type of spectroscopy generally provides a more sensitive detection of aqueous
species, because water is almost transparent in the UV-Vis region. This property allows quantitative
application of UV-Vis spectroscopy at millimol levels of sulfur species concentrations, in contrast
to Raman and IR that require far more concentrated solutions. However, the UV-Vis absorption
bands are much larger than those of IR and Raman spectroscopy, and more strongly temperature-
dependent. For example, the HS⁻ ion in aqueous solution between 50 and 350 °C has an absorption
band spreading over the range of 220–300 nm (Suleimenov and Seward 1997), corresponding to a
FWMH value of the UV-Vis band of >6000 cm⁻¹, vs only 20–30 cm⁻¹ for the HS⁻ Raman band
(~2570 cm⁻¹, Table 4.3) in the same temperature range (Pokrovski and Dubessy 2015). Similarly,
the absorption band of the blue S₃⁻ ion in water or other solvents spreads over the 500–700 nm
region (e.g., Steudel and Chivers 2019), which is >3000 cm⁻¹ in FWHM, whereas its S-S-S Raman stretch band (~530 cm⁻¹) has typical FWHM values of 10–50 cm⁻¹ (depending on the spectrometer resolution, temperature and laser wavelength; Pokrovski and Dubessy 2015; Colin et al. 2020).

The major application of UV-Vis for sulfur analysis in geosciences is ultraviolet sensing of volcanic emissions of SO₂ and, less frequently, H₂S that both absorb below 300 nm (Oppenheimer 2010; references therein). In aqueous and non-aqueous solutions, UV-Vis spectroscopy has been used in chemistry since a longtime, in particular for colored sulfur species such as polysulfide dianions and radical sulfur ions that absorb in the visible and near-UV range (e.g., Baer and Carmack 1949; Giggenbach 1972; Chivers and Elders 2013). An interesting application of UV-Vis spectroscopic properties of sulfur is found in chemiluminescence detectors for gas and liquid chromatographic separation and analysis of a number of sulfur species in chemistry (e.g., Steudel et al. 1987; Baudouin et al. 2022). Sulfur chemiluminescence is a two-stage detection method in which the sample undergoes combustion to form sulfur monoxide (SO), which is then carried to a reaction chamber where it reacts with ozone (O₃) to generate sulfur dioxide (SO₂) and photons produced in the reaction.

The high transparency of water in the UV-Vis region allowed building, since the 1970s, of high T-P optical cells for studying sulfur species equilibria in model hydrothermal fluids. Those studies enabled measurements of the H₂S dissociation constant to 350 °C (e.g., Tsonopoulos et al. 1976; Ellis and Giggenbach 1971; Suleimenov and Seward 1997), polysulfide ions equilibria to 240 °C (Giggenbach 1974a,b), and sulfite equilibria to 50 °C (e.g., Beyad et al. 2014; references therein). Apart from sulfur species, UV-Vis spectroscopy allowed a plethora of studies of the identity and stability of chloride and hydroxide complexes for many metals under hydrothermal conditions (e.g., As, Mo, Co, Ni, Fe, Cu, Pb). Some of those remarkable works will be reviewed in section 4.4. Continued progress in optical high T-P cell design and software for spectra processing makes UV-Vis spectroscopy a promising tool for experimental studies of hydrothermal fluid speciation, particularly when combined with complementary spectroscopies such as Raman (for sulfur species) and XAS (for metals).

## Nuclear magnetic resonance (NMR)

Among the four stable sulfur isotopes (32, 33, 34, and 36), only ³³S, which has a nuclear spin of 3/2, is spectroscopically active for this method. However, its nuclear spin properties coupled with the low natural abundance make this method far from ideal (Gerothanassis and Kridvin 2024; Wilke et al. 2011 for detailed discussion), even though the range of chemical shifts for the different sulfur oxidation states is very large, up to 1000 ppm. For reduced low-symmetry sulfur species the method
generally provides weak and broad features, whereas for sulfates and other oxidized and more
symmetrical S species the signals are somewhat better resolved (Jalilehvand 2006; references
therein). Even though ³³S NMR has been applied to mineral compounds, biological samples and
glasses, its ability to provide structural information is yet largely inferior to those of Raman or XAS.
A comprehensive account of the method advantages and limitations and applications to silicate
glasses and other geological materials with up-to-date literature can be found in Wilke et al. (2011)
and Simon and Wilke (2025). We are not aware of studies of S-bearing aqueous solutions by this
method. However, with the development of high-magnetic field spectrometers (>20 T) that allow
increasing spectral quality both in signal-to-noise ratio and energy resolution, this method could, in
the future, be used as a complementary to Raman and XAS for studying sulfur speciation in
hydrothermal fluids.

X-ray photoelectron spectroscopy (XPS)

X-ray photoelectron spectroscopy (XPS) applied to sulfur provides information about 2p
electronic levels of the element, and as such it is complementary to XANES and XES. Typically,
Al Kα excitation (1487 eV) is used as the laboratory X-ray source or a monochromatic X-ray beam
at < 2 keV at synchrotrons, and 2p binding energies in the range from ~160 to ~175 eV are measured
for S redox states spanning from sulfide to sulfate. Because of the low X-ray energies, XPS is
notoriously a surface probe and requires high-vacuum conditions (e.g., Hüfner 1995; Papp and
Steinrück 2013). As a result, while XPS has been widely used in geosciences for studies of sulfur
on mineral surfaces, in particular including sulfides and sulfates in weathering processes and ore
processing (e.g., Chandra and Gerson 2010; Mikhlin 2020), it cannot directly be applied to aqueous
solutions. As in the case of XAS, sulfur species trapped in the structural cages of lazurite minerals
have been extensively studied in both natural and synthetic samples using XPS (e.g., Fleet et al.
2005; Tauson et al. 2012; Cato et al. 2018) mostly as a complement to XAS and Raman, to better
constrain the relative distribution of sulfur oxidation states. Such minerals may potentially serve as
proxies for fluid sulfur speciation in the future. While XPS is a relatively easily accessible laboratory
technique, its strong sensitivity to the quality and state of the sample surface and corresponding
sample handling requirements, combined with lower than XAS or Raman spectral resolution details
as to the particular sulfur chemical species, make XPS alone potentially less attractive than XAS or
Raman for studies of sulfur speciation; it should be used in combination with those techniques.

Electron paramagnetic resonance (EPR)
Electron Paramagnetic Resonance (EPR) or Electron Spin Resonance (ESR) spectroscopy, which probes only species with an unpaired electron, has a limited use in geosciences compared to other types of spectroscopy such as Raman, XAS or XPS. Its applications are mostly for paramagnetic transition metals in minerals and glasses (e.g., Pan and Nilges 2014) and in the petroleum domain to study organic radicals including those of sulfur (e.g., Bakr et al. 1991). In chemistry, by contrast, EPR has widely been used since the 1960s for studying organic sulfur radicals in different organic and metalorganic synthesis reactions (e.g., Sullivan 1968; Hadley and Gordy 1975). Among inorganic sulfur species, only polysulfide radical ions yield an electron resonance spectrum, which was particularly useful for measuring absolute S₃⁻ concentrations and dynamic behavior in ultramarine pigments (e.g., see Chivers and Elders 2013; Steudel and Chivers 2019 for overview; and Gobeltz et al. 1998; Arieli et al. 2004; Goslar et al. 2009 for specific case studies). In addition, S₂⁻, which is poorly resolved by traditional EPR, was identified in green ultramarine pigments using a specific acquisition mode (field sweep echo detected; Raulin et al. 2011). EPR spectra of polysulfides in non-aqueous solvents and lithium-sulfur batteries have been used since longtime to detect and quantify S₃⁻ (Chivers 1974). An EPR signal of the third polysulfide radical, S₄⁻, in low concentrations was also reported in Li₂Sₕ solutions and ultramarine red pigments, but spectral assignment remains elusive (Steudel and Chivers 2019). We are not aware of studies of sulfur aqueous solutions, but like for XAS, EPR holds promise for its future use for direct in situ qualification of S₃⁻ in hydrothermal fluids, at least from laboratory experiments.
## 4.3 Sulfur chemical speciation, kinetics and thermodynamics in hydrothermal fluids

### 4.3.1 Sulfur aqueous speciation and redox state at low temperatures

Of all elements of the Periodic Table, sulfur likely exhibits the greatest multitude of formal oxidation states ranging from -2 to +6 that co-exist under terrestrial conditions. This remarkable redox versatility results in a plethora of corresponding species that may form at near-ambient conditions in aqueous solution, non-aqueous solvents, glasses, and solid phases known in chemistry since a long time (e.g., Cotton et al. 1999; Steudel 2003, 2020). The most important groups of these sulfur-bearing species are shown in Figure 4.5. Interestingly, the great majority of them are either molecules (i.e. of zero electrical charge) or anions (i.e. of negative charge). This property directly reflects the non-metallic chemistry of sulfur in general, even though (poly)cationic forms such as S<sub>8</sub><sup>2+</sup>, S<sub>4</sub><sup>2+</sup>, H<sub>3</sub>SO<sub>4</sub><sup>+</sup> are known to form in specific chemical environments such as oleum (i.e. concentrated H<sub>2</sub>SO<sub>4</sub> solutions; Cotton et al. 1999). Another fundamental property of sulfur is a strong tendency to polymerization via formation of [-S-S-] chemical bonds. This tendency is expressed in a variety of polysulfides and zero-valent sulfur molecules, as well as in oxygen bearing species, such as thiosulfate, disulfite and polythionates in which two S atoms of different oxidation state are bound one to the other. An amazing example is thiosulfate, (H)S<sup>-</sup>S<sup>2-</sup>O<sub>3</sub>(H), in which reduced sulfur (called sulfane sulfur) and oxidized sulfur (called sulfonate sulfur) atoms coexist (e.g., Vairavamurthy et al. 1993).

Although sulfate (+6) and sulfide (-2) are by far the major forms in most natural environments, intermediate-valence sulfur species commonly occur in aqueous solution in a variety of marine and continental sedimentary environments where they contribute to the (bio)geochemical cycle of sulfur as addressed in detail in other chapters (Turchyn et al. 2025; Bradley et al. 2025). For example, colloidal polymeric zero-valent sulfur is the third most abundant S form after sulfate and sulfide in marine sediment pore waters (Jorgensen 2021; references therein) and is quite common, together with polysulfides, in geothermal springs and fumaroles (e.g., Migdisov and Bychkov 1998; Kamyshny et al. 2008, 2014; references therein). Sulfur oxyanions, and in particular thiosulfate and polythionates, have been commonly reported both in sedimentary-basin and low-temperature (<100°C) geothermal waters since the 1970s, with concentrations sometimes reaching 20–30% of total dissolved sulfur (e.g., Boulëgue 1978a; Takano 1987; Xu et al. 1998, 2000; Kaasalainen and Stéfansson 2011; references therein). More detailed account of the formation and
transformation pathways of such S species in hydrothermal fluids will be given in Section 4.5. These
and other intermediate-valence S species also play an important role as reaction intermediates in the
biological sulfate-sulfide redox cycle (e.g., Milucka et al. 2012; Bradley et al. 2025). Polysulfide
radical ions (S₂⁻, S₃⁻ and, possibly, S₄⁻) are important constituents of chemical engineering
products (e.g., lithium-sulfur batteries, color pigments, glasses; Chivers 1974; Chivers and Elder
2013; references therein), natural zeolite minerals (e.g., Pan and Mi 2025) and S-rich magmatic-
hydrothermal fluids (see below and Pokrovski and Dubessy 2015; Colin et al. 2020). To the
multitude of inorganic sulfur forms shown in Figure 4.5 is added a near infinite number of organic
thiol compounds (i.e., having S–C–C– bonds, hereafter ‘S–Corg’) both molecular and anionic, along
with C–S-type radical ions, all of which are generated by microbial activity in (sub)surface aqueous
environments (e.g., Amend and Shock 2001; Cody 2004) and, at higher temperatures, in
thermochemical sulfate reduction processes by organic matter in sedimentary rocks (Schulte and
Rogers 2004; Oduro et al. 2011). Among well-known inorganic C-bearing sulfur species,
thiocyanate (SCN⁻, e.g., Patai 1977) and carbonyl sulfide (COS, e.g., Svoronos and Bruno 2002)
are worth mentioning. The former is a minor product of bacterial metabolism in aquatic
environments (e.g., Jiang et al. 2017), an important analytical reagent for transition metals, and a
chemical reagent used in hydrometallurgy for metal recovery from ore and technological waste (e.g.,
Au, REE; Jha et al. 2016; Ding et al. 2019; Azizitorghabeh et al. 2021). Thiocyanate has been
detected in geothermal and volcanic springs in sub-µM concentrations (e.g., Mukhin 1974;
Kamyshny et al. 2014) and at 10-µM levels in Red Sea brines (Dowler and Ingmanson 1979). COS
is a minor, but ubiquitous, sulfur species occurring at nM-levels in both fresh and sea water (e.g.,
Kamyshny et al. 2003a); furthermore, it is an important S compound in the modern terrestrial
atmosphere (Brimblecombe and Norman 2025), having been probably more abundant in the
Archean atmosphere (e.g., Ueno et al. 2009) where it might have mediated prebiotic synthesis of
aminoacids (e.g., Leman et al. 2004). The abundance and geochemical significance of COS in
hydrothermal fluids is much less known, but it has been reported a couple of times by Raman
spectroscopy in natural fluid inclusions from metamorphic settings (see section 4.2.4).

According to their thermodynamic properties, the great majority of intermediate-valence S
species should transform almost completely to sulfide and sulfate at equilibrium under the
conditions of most terrestrial aquatic environments including both surficial and crustal fluids, at
least at temperatures below 150 °C. Their equilibrium concentrations should therefore be much
lower than those analyzed in these natural settings. In fact, their higher than thermodynamically
allowed concentrations in such environments are due to intrinsically slow rates of sulfur redox
transformations at low-to-moderate temperatures. With increasing temperature, in general these
rates largely increase, yet they remain extremely variable amongst different S species types, being
strongly dependent upon fluid composition and pH and controlled by different reaction
intermediates. The following section discusses the fundamentals of sulfur species transformation
kinetics in aqueous solution with an emphasis on hydrothermal conditions.

4.3.2 Kinetics of sulfur species transformation reactions in hydrothermal fluids

The application of thermodynamic data to the interpretation of mineral assemblages in
hydrothermal sulfide ore deposits and fluid-rock interaction phenomena, as well as redox conditions
($f_{O_2}$, $f_{H_2}$), acidity (pH), chemical speciation in the hydrothermal fluid phase, and the related metal
transport, depends critically on the assumption that chemical equilibrium is attained. However, rates
of reactions amongst sulfur species in geological fluids vary over an extremely large range, which
is primarily defined by the extent of electron transfer between the reacting species.

Very fast rates are typical for reactions between sulfur species of the same or a close
oxidation state, such as those involving proton transfer or ion pairing between cations and anions in
aqueous solution, e.g., $H_2S = HS^- + H^+$, $HSO_4^- = SO_4^{2-} + H^+$, or $Na^+ + SO_4^{2-} = NaSO_4^-$. Such
reactions typically attain equilibrium within the millisecond range even at ambient temperatures.
Their reaction kinetics is controlled by the transport (i.e. diffusion) of reacting species in aqueous
solution, which is very fast (Chen and Irish 1971; Weston and Schwartz 1972; Stumm and Morgan
1996; Lasaga 1998).

The rates are slower for the reactions between sulfur species of different redox states that
imply an electron transfer. In general, the more electrons transferred between the reactants and
products, the slower the overall reaction rate. Thus, disproportionation reactions involving $H_2S/HS^-$
and polysulfide dianions and/or radical sulfur ions both in aqueous solution and organic solvents,
such as $S_n^{2-} + HS^- + OH^- = S_x^{2-} + S_y^{2-} + H_2O$ ($n + 1 = x + y$) or $S_6^{2-} = 2 S_3^{·-}$, are pretty fast, being
at second-to-minute scales for 90% of reaction equilibrium attainment ($t_{90}$) at near-neutral pH
conditions and millimol-levels of S species concentration (e.g., Hartler et al. 1967; Chen and Morris
1972; Kamyshny et al. 2003b; Chivers and Elder 2013). Reactions involving molecular sulfur $S^0$
and polysulfides and sulfides, such as $(1-n) S^0 + HS^- = S_n^{2-} + H^+$ or $2 S_3^{·-} + 2H^+ = 5 S^0 + H_2S$,
which require transfer of a small number of electrons ($\le 2$ e per reaction) without complete breaking
of S–S chemical bonds in $S^0$ and polysulfides, have rates ranging from minutes to months (in terms
of $t_{90}$) for geologically relevant conditions, such as marine sediments or geothermal sulfur-bearing
fluids. The exact rate would depend on the state of $S^0$, e.g., dissolved, colloidal or particulate,
crystalline or amorphous (Hartler et al. 1967; Boulëgue and Michard 1977; Avetisyan et al. 2019;
references therein). Formation of S⁰ in colloidal or particulate (i.e. solid) state is systematically observed both in natural geothermal settings and laboratory experiments during cooling of H₂S-SO₄-bearing fluids. This phenomenon directly reflects the decreasing S₈⁰(s,l) equilibrium (thermodynamic) solubility with decreasing temperature (see below) along with fast kinetics of H₂S-Sn²⁻-S⁰ exchange reactions (Kaasalainen and Stéfansson 2011; Pokrovski and Dubessy, 2015; Kokh et al. 2020). Similarly, sulfite oxidation to sulfate, SO₃²⁻ + ½ O₂ = SO₄²⁻, with transfer of 2 electrons, is relatively fast, typically in the minute-to-hour range in seawater, the rate being also dependent on the presence of transition metals that may act as catalysts (e.g., Zhang and Millero 1991).

Reactions that require more electron transfer between the reacting S species, such as between sulfate and sulfide, with 8 electrons to exchange and to completely reconstruct S chemical bonds and coordination from reactants to products, are fundamentally much slower and involve different intermediate steps and precursors depending on fluid temperature, acidity, salinity, and sulfur speciation. The detailed discussion of their mechanisms is beyond the scope of this chapter; here we provide a brief overview of the major kinetic trends between sulfide and sulfate relevant to hydrothermal fluids. Indeed, at least at moderate temperatures of ≤350 °C, sulfate and sulfide are the major redox forms of sulfur (Fig. 4.5) and, therefore, determine fluid redox potential, the abundance of other more subordinate S redox forms as well as S stable isotope fractionations (e.g., Ohmoto and Lasaga 1982; Eldridge 2025). Experimental studies of sulfide-sulfate reaction kinetics under hydrothermal conditions were carried out since the 1930s by exploring the radioactive <sup>35</sup>S isotope exchange in sulfide-thiosulfate-sulfate aqueous mixtures and sulfate reduction by organic matter (e.g., Voge and Libby 1937; Voge 1939; Pryor 1960; Toland 1960; Ciuffarin and Pryor 1964). These studies recognized the complexity of the reaction mechanism, involving different intermediate-valence sulfur species such as thiosulfates and polysulfides as transition complexes, and a strong effect of temperature and pH on the reaction rates. Subsequent works explored different starting sulfur compounds, such as thiosulfate, native sulfur or mixtures of sulfide and sulfate, by analyzing the evolution versus time of the ratio of most abundant sulfur stable isotopes <sup>34</sup>S/<sup>32</sup>S in the produced sulfate and sulfide (e.g., Grinenko et al. 1969; Robinson 1973; Igumnov 1976; Sakai and Dickson 1978). These studies have confirmed the key dependence of the reaction rates on temperature and pH inferred in earlier work. An important recognition of those studies was also that the rate of attainment of chemical equilibrium between aqueous sulfide and sulfate can be assessed from the information on the rate of sulfur isotopic exchanges between them. This is because both types of reactions, chemical and isotopic, imply breaking of sulfur bonds and exchanges of sulfur atoms in molecular structures of sulfide and sulfate.
This concept was explored by Ohmoto and Lasaga (1982) to build the first comprehensive model of sulfate-sulfide reaction rates, based on those experimental studies coupled with general knowledge of sulfate, sulfide and thiosulfate speciation as a function of pH that varies from dominant protonated to deprotonated species with increasing pH. According to those authors, the overall rate of sulfate-sulfide reaction (R) may adequately be described as

$$R = k_r \times [\Sigma S^{6+}] \times [\Sigma S^{2-}] \qquad (4.1)$$

where ΣS6+ and ΣS2- are total concentration of sulfate-type (e.g., SO42-, HSO4-, NaSO4-) and sulfide-type (e.g., H2S, HS-, NaHS0) species, respectively, and k_r is the rate constant, which is a function of the solution temperature, pH, and composition. Ohmoto and Lasaga's (1982) model, in accordance with most available experimental data, indicates that, at a given temperature, the rate constant is strongly dependent on pH, as exemplified schematically in Fig. 4.6 for typical hydrothermal conditions of 300 °C. These rates are fastest in strongly acidic solutions (pH <2), decrease with increasing pH to ~4, stay almost constant between pH ~4 and ~7, and then further decrease at more basic pH, becoming extremely slow at pH >9. Such a pattern was quantitatively interpreted by Ohmoto and Lasaga (1982) by suggesting intermediate thiosulfate species depending on pH, with H2S2O30 + HS2O3-, S2O32-, and NaS2O3- acting as the rate-controlling transition complexes at acidic, neutral, and basic pH, respectively. In these thiosulfate species, intra-molecular exchange of sulfur atoms is therefore the rate determining step. Using the thiosulfate model fitted to available experimental data at different temperatures and pH values, the time necessary to attain sulfate sulfide equilibrium may be estimated at different temperatures and fluid pH as outlined in Fig. 4.7. It can be seen that in the typical pH range of hydrothermal fluids (4–7), both chemical and isotopic equilibrium between sulfate and sulfide species may be attained in most cases above 200 °C within geologically reasonable time scales of <100 years. Such time scales are generally shorter than the duration of most events relevant to the circulation of geological fluids, ore-deposit formation and the evolution of geothermal systems. However, at lower temperatures and/or more alkaline pH, significant disequilibrium in the sulfate-sulfide chemical and isotopic system may persist in the absence of bacteria-mediated processes. More discussion about the impact of sulfur redox kinetics on ore deposit formation and sulfur isotope fractionation may be found in Ohmoto and Lasaga (1982), Ohmoto and Goldhauer (1998) and Eldridge (2025).

Ohmoto and Lasaga's (1982) kinetic model was later reconsidered by Chu et al. (2004) who based their analysis on the extensive experimental data of S isotope exchange in thiosulfate-sulfide-sulfate solutions obtained by Uyama et al. (1985) coupled with earlier identification of polysulfide ions (S_nS2-) in such solutions using UV-Vis spectroscopy by Giggenbach (1974a,b). Chu et al. (2004) suggested that the formation and disproportionation of S_nS2- control the exchange rates
between the S<sup>VI</sup> and S<sup>II</sup> atoms in thiosulfate through intermolecular exchange with polysulfides, rather than intramolecular exchanges within the thiosulfate molecule itself as originally proposed by Ohmoto and Lasaga (1982). However, both models were based on limited thermodynamic data available for those intermediate species and lacked unambiguous in situ evidence.

The in situ Raman spectroscopy identification of the radical S<sub>3</sub><sup>-</sup> ion in aqueous solution above 150-200 °C at the sulfide-sulfate transition (Pokrovski and Dubrovinsky 2011; Jacquemet et al. 2014; Truche et al. 2014; Pokrovski and Dubessy 2015; Barré et al. 2017) provided a fresh update to the long history of the sulfate-sulfide reaction mechanisms. This very reactive species whose maximum abundance lies within the slightly acidic to neutral pH range (pH 4-6, see below) may therefore act as the rate controlling intermediate. In contrast, no thiosulfate species or polysulfide dianions were identified by Raman spectroscopy in Pokrovski and Dubessy's (2015) high-temperature experiments using thiosulfate or mixtures of sulfate and sulfide. The S<sub>3</sub><sup>-</sup> ion was also systematically detected in the thermochemical sulfate reduction (TSR) experiments involving various organic compounds at temperatures 100-300 °C, and was hypothesized to be a key intermediate in TSR processes (Truche et al. 2014). At more acidic pH (<4) where the rates are fastest, molecular sulfur polymeric species (S<sub>n</sub><sup>0</sup>; Tables 3 and 4) may also play a role in the sulfate-sulfide exchange reaction at moderate temperatures (<300 °C). Such species were identified by Raman spectroscopy in experimental sulfate-sulfide solutions (Pokrovski and Dubessy, 2015; Wan et al. 2024) and confirmed in natural fluid inclusions (Barré et al. 2017). In alkaline solutions, polysulfide dianions are more abundant and would likely be the rate-controlling species, in agreement with Chu et al.'s (2004) model.

Notwithstanding the exact mechanism and chemical intermediates in the sulfide-sulfate
redox reactions, the key fundamental point seems to be the formation of -S-S- chemical bonds to
allow electron transfer between sulfur atoms. More accurate knowledge of the abundances of such
intermediate-valence S-S-type species in hydrothermal fluids would be necessary to build a
comprehensive kinetic model of sulfur redox reactions that would be consistent with the whole
sulfur speciation picture. The sulfur species abundances may be assessed in two ways. The most
direct one is to use in situ spectroscopic methods, but they remain rare and are currently restricted
to a limited T-P-compositional range. Alternatively, equilibrium thermodynamics approaches
become more universal, in particular at elevated temperatures at which the kinetic limitations
discussed above are minor due to the overall increase of reaction rates both within the fluid phase
and between fluid and minerals. The following section overviews the major thermodynamic models
and discusses thermodynamic data of the key sulfur aqueous species in hydrothermal fluids.
### 4.3.3 Overview of thermodynamic models for sulfur in hydrothermal fluids

Thermodynamic data for aqueous sulfur species at elevated temperatures and pressures are necessary for predicting the species equilibrium concentrations and solubility of S-bearing minerals in hydrothermal fluids. These data, together with those for other major and trace elements, can nowadays be assessed from thermodynamic databases (e.g., SUPCRT, SUPCRTBL, MINTEQ, EQ36, LLNL). Such databases are integrated in computer codes allowing thermodynamic equilibrium calculations in multicomponent fluid-mineral-gas systems at elevated temperatures and pressures (e.g., EQ3/6, CHESS, HCh, PHREEQC, GEMS, GWB to name a few). A detailed account of the existing databases, associated software, and advantages and limitations of the thermodynamic models for aqueous species and minerals in general may be found in Oelkers et al. (2009) and Pokrovski (2025a). Below we provide basic thermodynamic relationships and briefly overview major thermodynamic models used nowadays for S aqueous species and their complexes with metals. Then we discuss in more detail the origin of thermodynamic data for different classes of sulfur species and reliability of their use at the conditions of crustal fluids. Finally, using these data we quantify sulfur solubility and speciation in major types of geological fluids across a wide T-P range.

#### Basic thermodynamic relationships and models for aqueous solution

Thermodynamic equilibrium calculations use either the standard molal Gibbs energies for aqueous species, minerals or gases or, alternatively, reaction equilibrium constants. The Gibbs energies of individual species are defined as:

$$G_{i,T,P} = \Delta G_{i,T,P}^{0} + RT \ln a_{i} \quad (\text{for solid and aqueous species}) \quad (4.2)$$

$$G_{i,T,P} = \Delta G_{i,T,P}^{0} + RT \ln f_{i} \quad (\text{for gases}) \quad (4.3)$$

where $\Delta G_{i,T,P}^{0}$ = standard molal Gibbs energy of i-th species, $a_{i}$ = activity (aqueous or solid species) and $f_{i}$ = fugacity (gaseous species). The standard states commonly adopted in geochemistry of fluid-rock-gas interactions (e.g., Oelkers et al. 2009) are the pure substance ($a_{i}$ = 1) for solid phases, pure gas substance at 1 bar pressure having ideal-gas behavior ($f_{i}$ ≈ $P_{i}$) for gases, and 1 molal (1 mole per kg of H$_{2}$O) solution whose behavior is ideal ($a_{i}$ ≈ $m_{i}$) for aqueous species. The standard Gibbs energy is related to the equilibrium constant of a reaction as

$$\Delta_{r} G_{T,P}^{0} = \sum_{i} n_{i} \times G_{i,T,P}^{0} = -2.3026RT \times \log_{10} K_{T,P} \quad (4.4)$$
where R is the ideal gas constant = 8.314 J/(mol K), T = temperature in Kelvin = T °C + 273.15, nᵢ is the stoichiometric coefficient of each reaction constituent, and Kᵢₜₚ is the equilibrium constant, at given T and P, of the reaction

$$wW + xX = yY + zZ; \text{ with } K = \frac{Y^y \times Z^z}{W^w \times X^x} \qquad (4.5)$$

where W, X and Y, Z are activities of the products and reactants, respectively, and w, x and y, z are their stoichiometric coefficients in the reaction. Note that both G° and K depend on T and P, but do not depend on the system composition at given T and P. At equilibrium, the total Gibbs energy of the system is minimal: ΣGᵢₜₚ → min. Depending of the computer program, thermodynamic equilibrium calculation algorithms either search to minimize the overall Gibbs energy of the system (reactions 4.2 and 4.3) or to solve the system of reaction equations (equation 4.5), in both cases by finding a corresponding set of the species activities. Both algorithms apply mass and electrical charge balance constraints.

The major caveat of such calculations, often disregarded by many users of geochemical computer codes, is the reliability of estimations of K and G° numerical values at elevated T-P. If these values at near ambient conditions are known for major sulfur species with sufficient precision, this precision worsens dramatically with increasing T and P. This situation contrasts with that for mineral phases for which variations in K and/or G° values with T and P are much smaller and far more regular than for aqueous species for which interactions with the solvent (hydration) represent the major contribution to the species energy. The variation of apparent standard Gibbs energy with T and P of any substance or species may be obtained by integrating the heat capacity and volume functions over the T-P interval:

$$\Delta G_{i,T,P}^{0} = \Delta G_{i,T_{r},P_{r}}^{0} - (T - T_{r}) \times S_{i,T_{r},P_{r}}^{0} + \int_{T_{r}}^{T} C p_{i}^{0} dT - T \int_{T_{r}}^{T} (C p_{i}^{0} / T) dT + \int_{P_{r}}^{P} V_{i,T}^{0} dP \quad (4.6)$$

where ΔGᵢₜₚ = molal Gibbs free energy of formation from the elements at the reference temperature and pressure Tᵣ and Pᵣ, which are 25°C (298.15 K) and 1 bar, respectively; Sᵢₜₚ = molal entropy of the i-th species at Tᵣ and Pᵣ, and Cpᵢ and Vᵢ = molal isobaric heat capacity and volume of the i-th species. Note that this way of expressing the Gibbs energy of a species/phase at T and/or P different from the reference values above (Tᵣ, Pᵣ), called an apparent Gibbs energy, is different from the Gibbs energy of formation from elements sensu stricto. In the latter case, equation (4.6), instead of the Sᵢ⁰, Cpᵢ⁰ and Vᵢ⁰ terms of the species itself, would contain the terms ΔrS⁰, ΔrCp⁰ and ΔrV⁰ of the reaction of formation of the species from elements in their stable states (whose ΔG⁰ₜₚ is zero by definition). Such values of energies were tabulated in some thermodynamic
databases (e.g., Robie and Hemingway 1995) and are numerically different, at *T* and *P* other than the reference values, from the apparent Gibbs energies from equation (4.6), which is currently adopted in most aqueous species model formulations and databases (see below). The omission of the thermodynamic parameters of the elements in equation (4.6) greatly simplifies the Δ*G*⁰ expression and eliminates potential ambiguities related to the choice of element stable states at elevated *T* and *P* (e.g., solid vs gas). Naturally, these different numerical expressions of *G*⁰ values have no effect on calculations of reaction constants (e.g., equation 4.5) or thermodynamic equilibria in multicomponent systems, provided that the same formulation is applied to all species and phases and their standard state definition is coherent (equations 4.2 and 4.3). Knowledge of *Cp*⁰ and *V*⁰ as a function of *T* and *P* is thus required for accurate calculation of the Δ*G*⁰ values (or equilibrium constants). The thermodynamic models for aqueous species mostly differ in the way they treat these functions. The most common models for solutes in aqueous fluids are briefly overviewed as follows.

## *Van't Hoff equation and its extensions*

Direct information on *Cp* and *V* values at elevated *T*-*P* is often absent for many species including sulfur-bearing ones. Therefore, thermodynamic equations used for describing such aqueous species assumed that the standard molal heat capacity of a reaction is zero, which allows a simple linear extrapolation of log₁₀*K* (hereafter log*K*, for simplicity) as a function of the reciprocal of absolute temperature:

$$ \log K_{T,P} = \frac{-\Delta_r H_{298}^0}{2.3026RT} + \frac{\Delta_r S_{298}^0}{2.3026R} \quad (4.7) $$

where *T* is in Kelvin and Δ*H*⁰ and Δ*S*⁰ are assumed constant, which translates to a linear equation

$$ \log K = a/T + b \quad (4.8) $$

Such a simple relationship, the so called van't Hoff approximation (e.g., Anderson 2005; Oelkers et al. 2009), is generally valid over a relatively small *T*-range (typically from ambient to ~100 °C) but may be extended to ~200–250 °C for certain isocoulombic reactions, i.e. those having similar ionic charges at both sides of the reaction. For some of such reactions, the equations above may further be simplified assuming that the entropy change is also zero:

$$ \Delta_r G^0_T = \Delta_r G^0_{298K} = -2.3026RT \times \log K_T = -2.3026R \times 298.15 \times \log K_{298} \quad (4.9), $$

which translates to

$$ \log K_T = \log K_{298} \times 298.15/T(K) \quad (4.10) $$

This simplification allows straightforward extrapolations above 25 °C (Gu et al. 1994). Equations (4.7–4.10) have been used, for example, for describing equilibria among polysulfide ions from 60 to 260 °C by Giggenbach (1971b) and, more recently, the reaction constants between polysulfide
ions and HS⁻ measured from 20 to 80°C by Kamyshny et al. (2007), and solubility of S₈(s) in water from 4 to 80 °C by Kamyshny (2009). When the ΔₜCₚ of the reaction is different from zero, but may be approximated by a constant value over the fitted T range, a third term is added to equation (4.8) such as:

$$ \log K = a + b/T + c \times \log T \quad (4.11) $$

When ΔₜCₚ of the reaction changes with temperature, additional terms like d×T, e/T² or f×T² may be added for data regression.

$$ \log K = a + b/T + c \times \log T + d \times T + e/T^2 \quad (4.12) $$

The latter equation or similar is used, for example, to express reaction constants in the PhreeqC or LLNL databases. In the case of sulfur species, such equations have been used for describing the formation constant of the S₃⁻ from sulfate and sulfide from 0 to 500 °C (see below and Pokrovski and Dubessy 2015), HSO₄⁻ and H₂S dissociation constants from 25 to 350 °C (Marshall and Jones 1966; Rudolph 1996; Suleimenov and Seward 1997), as well as a number of metal-hydrosulfide complex formation reactions (see section 4.5). Obviously, such polynomial equations should be used mostly for interpolation purposes because they lack predictive capabilities beyond the regressed T-P range.

### Density model

For more extended T ranges, the deviation of logK from the linear equation (4.8) with 1/T at high temperatures is observed for many reactions including aqueous species (e.g., Oelkers et al. 2009), which is mostly because of the strong and non-linear variations with temperature of the key solvent properties, such as density (ρH₂O) and dielectric constant (εH₂O). The combined effect of T and water density on the solubility of oxide and silicate minerals in steam (e.g., Morey and Hesselgesser 1951) and on the dissociation constants of water, acids and salts in aqueous solution (e.g., Frank 1956, 1961) was observed more than 70 years ago, but it is only since the 1980s that it has been applied in a more systematic way to describe the dissociation constants of water, acids, and bases in aqueous solution and mineral solubility over a large T-range (e.g., Marshall and Franck 1981; Mesmer et al. 1988; Anderson et al. 1991; references therein), using equations involving T and ρH₂O:

$$ \log K = A + \frac{B}{T} + \frac{C}{T^2} + \frac{D}{T^3} + (E + \frac{F}{T} + \frac{G}{T^2}) \times \log \rho_{H_2O} \quad (4.13) $$

where A to G are T-P-independent constants, T is temperature in Kelvin, and ρH₂O is the water density. Depending of the available T-P data range, simplified equations are also used
$$ \log K = A + \frac{B}{T} + E \log \rho_{H_2O} \qquad (4.14) $$

$$ \log K = A + \frac{B}{T} + \frac{F}{T} \log \rho_{H_2O} \qquad (4.15) $$

Such density model equations have been very efficient for describing the solubility of quartz (e.g., Manning 1994; references therein) and a few silicate minerals (Dolejs and Manning 2010) in supercritical fluid over a wide hydrothermal-magmatic T-P range, from hydrothermal steams to subduction-zone fluids, as well some metalloid and metal oxides (e.g., GeO₂, Pokrovski et al. 2005a; CuO, Palmer et al. 2004) over more limited T-P windows. For S-bearing species, the density model has been used to describe data on the first dissociation constant of H₂SO₄ in the supercritical T-P range (400–800°C and 500–5000 bar; Quist et al. 1965, see below). More recently, the model has been applied to the solubility reactions of Au and Pt metals and platinum sulfide, respectively in the form of AuCl₂⁻, PtCl₄²⁻ and Pt(HS)₂⁰ or Pt(HS)₃⁻ aqueous species, in hydrothermal fluids (Zotov et al. 2018; Tagirov et al. 2019a,b; Filimonova et al. 2021). The density model offers a straightforward description of simple solubility reactions for cases where aqueous speciation of the element is dominated by a single species, potentially in a wide T-P range using the best-known parameter of the fluid – water density. However, the density-model equations are rarely directly integrated in common thermodynamic databases that use either more elaborate models for G⁰ values of aqueous species (such as HKF, see below) or polynomial least-square fits as a function of T for reaction constants (e.g., EQ3/6). In addition, in case of aqueous speciation strongly changing with T-P and/or involving multiple major species (e.g., sulfide and sulfate, multiple metal complexes), each individual reaction requires its own density model equation. Finally, the model is prone to large uncertainties when extrapolating beyond the fitted experimental data of a limited density range, for instance in liquid-like hydrothermal fluids at moderate T-P, where the water density changes are relatively small (e.g., in the range 25–300 °C and pressures from Psat to a few kbar, the change in ρH₂O is only from 1.0 to 0.7 g/cm³).

## Ryzhenko-Bryzgalin model

For the liquid-like type of fluids, another relatively simple equation, the so called Ryzhenko-Bryzgalin (RB) model (Ryzhenko 1981; Bryzgalin and Rafalsky 1982; Borisov and Shvarov 1992) has shown its efficiency:

$$ pK_{T,P} = pK_{298K,1bar} \times 298.15 / T(K) + f(T,P) \times (zz / a)_{\text{eff}} \qquad (4.16) $$

where pK = -logK, f(T,P) is a species-independent function as computed from the dissociation constant of water as function of T and P, and (zz/a)eff is a property of the complex, which is described
by a linear equation as a function of the reciprocal of absolute temperature, $(zz/a)_{\text{eff}} = A + B/T(K)$,
where *A* and *B* are *T*-independent constants intrinsic to each complexation reaction. The function
$f(T,P)$ is computed from the dissociation constant of water according to Marshall and Franck (1981)
using equation (4.12) with $(zz/a)_{\text{eff}} = 1.0107$.

The RB model requires only three adjustable parameters, pK<sub>298K,1bar</sub>, *A*, and *B* – the latter being optional and may be omitted when the data to be described are in a limited *T*-range. The RB equation is integrated in the HCh computer code (Shvarov 2008). The associated user-friendly OptimB module is offered to help the user to find the values of these adjustable parameters by regressing experimental pK<sub>T,P</sub> data points. More details about the model may be found in Shvarov (2015). The RB model over performs the density model particularly for liquid-like fluids where the water density variations are small (see above). The performance of the RB model with 3 adjustable parameters in some cases may even be comparable with that of the HKF model with 7 adjustable parameters (see below). However, the RB model, having been published in the Russian literature, has unfortunately received a rather limited recognition in the international geochemical community, despite its practical simplicity coupled with predictive capability. It was employed to accurately describe the stability constants of hydrosulfide complexes of Zn (Tagirov and Seward 2010) and Au (Shvarov 2015), sulfate complexes of uranyl (Kalintsev et al. 2019) or chloride complexes of Ni (Liu et al. 2012) and Y (Guan et al. 2020), derived from either experimental or molecular dynamics simulation data at ≤300 °C. Its major practical limitation is a lack of integration into widely-used thermodynamic software and databases, mostly because of the necessity for coding the *f*(T,P) function of water. In addition, uncertainties when extrapolating the modeled stability constants beyond the *T*-P range within which the regression coefficients have been obtained are hard to assess for different types of aqueous species that exhibit *T*-trends and amplitudes of dissociation different from that of water. In contrast to the density model, the RB model has not been validated on or directly applied to reactions involving solid phases and thus currently remains applicable only to dissociation of acids, ion pairs or complexes of metal cations in aqueous solution.

HKF model

The greatest advance in our understanding of the solubility and speciation of metals and volatiles, including sulfur-bearing species, in the hydrothermal fluid phase was with the development of the Helgeson-Kirkham-Flowers (HKF) model in early 1980s (Helgeson et al. 1981). This model describes the Gibbs energy of an aqueous species up to 1000 °C and 5 kbar by functions of the dielectric constant of water and considering two distinct contributions, the solvation (s) and non-solvation (n) terms, to the values of the partial molal volume $V^0$ and isobaric heat capacity $Cp^0$
of an aqueous species: $V^0 = V^0_s + V^0_n$ and $Cp^0 = Cp^0_s + Cp^0_n$. The model was subsequently revised by Tanger and Helgeson (1988) and Shock et al. (1992). A detailed development of the approach may be found in those original papers and an overview summary in Oelkers et al. (2009). The overall equation for the Gibbs energy is fundamentally similar to the general equation (4.6) and describes the variation of the $Cp^0$ and $V^0$ properties of an aqueous species as a function of T and P using 7 HKF adjustable parameters and 2 solvent (i.e. water) constants:

$$ 
\begin{aligned}
G_{P,T}^o - G_{P_r,T_r}^o = & -S_{P_r,T_r}^o (T-T_r) - c_1 [T \ln(\frac{T}{T_r}) - T + T_r] + a_1 (P-P_r) + a_2 \ln(\frac{\Psi+P}{\Psi+P_r}) - \\
& c_2 \{[(\frac{1}{T-\Theta}) - (\frac{1}{T_r-\Theta})](\frac{\Theta-T}{\Theta}) - (\frac{T}{\Theta^2}) \ln(\frac{T_r(T-\Theta)}{T(T_r-\Theta)})\} + (\frac{1}{T-\Theta}) [a_3(P-P_r) + a_4 \ln(\frac{\Psi+P}{\Psi+P_r})] + \\
& \omega(\frac{1}{\epsilon}-1) - \omega_{P_r,T_r}(\frac{1}{\epsilon_{P_r,T_r}}-1) + \omega_{P_r,T_r} Y_{P_r,T_r}(T-T_r)
\end{aligned}
\quad (4.17) $$

where $a_1, a_2, a_3, a_4$, and $c_1, c_2$, are the HKF coefficients for $i^{th}$ species, $\omega$ and $\omega_{P_r,T_r}$ are, respectively, the effective Born coefficients of $i^{th}$ species at the given temperature and pressure and at the reference temperature and pressure ($T_r = 298.15$ K, and $P_r = 1$ bar), $\Psi$ and $\Theta$ are the solvent constants, 2600 bar and 228 K, respectively, $\epsilon$ is the water dielectric constant, and $Y$ is the derivative of $\epsilon$ over temperature: $Y_{P_r,T_r} = \frac{1}{\epsilon_{P_r,T_r}}[(\frac{\partial \ln \epsilon}{\partial T})_{P_r,T_r}]$. The $a_{1-4}$ and $c_{1-2}$ coefficients describe the non-solvation $V^0_n$ and $Cp^0_n$ contributions, respectively, whereas the Born coefficient ($\omega$) accounts for the solvation contributions $V^0_s$ and $Cp^0_s$. The solvation contribution to the Gibbs energy of a species is described by the Born equation (Born 1920)

$$ G_s^0 = \omega(\frac{1}{\epsilon}-1) \quad (4.18) $$

whose differentiation, respectively over P and T, results in the expressions for $V^0_s$ and $Cp^0_s$ in equation (4.17). The Born parameter for neutral species is assumed to be constant at all T-P, whereas that for charged species is a function of the ion radius and the so called g-function, which, in turn, is a T-P dependent solvent function retrieved from experimental data on the NaCl dissociation constant (Tanger and Helgeson 1998; Shock et al. 1992).

To reduce the number of adjustable parameters, different sets of empirical correlations among the HKF parameters and $S^0$, $Cp^0$ and $V^0$ values at 298 K and 1 bar for different types of species, such as cations and anions of different charges (Shock and Helgeson 1988; Shock et al. 1997), metal inorganic complexes (Hass et al. 1995; Shock et al. 1997; Sverjensky et al. 1997; Sassani and Shock 1998), metal organic complexes (Shock and Koretsky 1993, 1995), inorganic neutral species and volatiles (Shock et al. 1989; Plyasunov and Shock 2001; Schulte et al. 2001),
aqueous organic molecules (Shock and Helgeson 1990), and thiols (Schulte and Rogers 2004), have
been established and, in some cases, further revised and extended when more high T-P data became
available. These correlations are largely based on common electrolytes and few aqueous species of
different chemical types for which the range of experimental G°, Cp° and V° values were available
in a sufficiently wide T-P range to allow the full set of their HKF values to be derived directly via
regressions (e.g., Tanger and Helgeson 1988). It is these correlations that provide a predictive basis
for the HKF equations when little or no experimental data are available, which is the case for the
majority of aqueous species.

The major advantage of the model compared to the models discussed above is its great
versatility, being able to reproduce the experimental data to 1000 °C and 5 kbar and water densities
above ~0.4 g/cm³. Since the late 1980s, HKF parameters of a large number of aqueous solutes,
including sulfur species and their complexes with metals, have been generated and incorporated in
free-access thermodynamic databases such as SUPCRT92 (Johnson et al. 1992) and its updated
versions including SUPCRTBL (Zimmer et al. 2016). Some equilibrium thermodynamics computer
codes and associated databases for calculating mineral-fluid-gas equilibria at elevated T-P include
the HKF model (e.g., HCh, Shvarov 2008; GEMS, Kulik et al. 2013; modified EQ3 with DEW
model, Huang and Sverjensky 2019) In addition, user-friendly codes are available to allow
generation and evaluation of the HKF parameters from experimental data (e.g., OptimA, Shvarov
2015; GEMFITS, Miron et al. 2015). The theoretical applicability of HKF has recently been
extended to higher pressure and temperature conditions, up to 60 kbar and 1200 °C, corresponding
to water densities of 1.3 g/cm³, using revised and updated models for extrapolating the water
dielectric constant. This HKF model extension allows equilibrium calculations of mineral solubility
and element aqueous speciation under conditions of deep subduction zones and mantle
metasomatism (Sverjensky et al. 2014; Huang and Sverjensky 2019; Pokrovski et al. 2022a).
Furthermore, the HKF model has been successfully used for solubility and speciation calculations
in supercritical mixed H₂O-CO₂ fluids ubiquitous in metamorphic settings, by using the dielectric
constant of H₂O-CO₂ mixtures instead of that of pure water (e.g., Akinfiev and Zotov 1999; Kokh
et al. 2017).

The model performances are very good at elevated pressures and fluid densities, but its
practical application for charged species is limited to water densities in between 0.35 and 1.3 g/cm³
and “exclude” the T-P region around the water critical point (350<T<400 °C at P<500 bar; Johnson
et al. 1992; Sverjensky et al. 2014). The low-density limit reflects the corresponding limit of
experimentally-based supercritical dissociation constants of NaCl from which Shock et al. (1992)
retrieved the values of the g-function necessary for calculating the Born coefficients for ions and
charged complexes. In contrast, for neutral species whose Born coefficient is T-P independent, the model may in principle be applied to a larger range of densities, down to 0.05 g/cm³. This practical validity range refers to that of the equations used to calculate the dielectric constant and other Born functions of H₂O (Johnson et al. 1992; references therein). However, given the “inaccurate” treatment of the neutral species solvation energy using the Born equations, which are strictly applicable a priori only to electrically charged species, the performance of the HKF model for neutral solutes in the near- and supercritical fluids has been criticized. Improvements in deriving the HKF parameters have been proposed based on more recent experimental data on vapor-liquid partitioning and direct Cp and V measurements at elevated temperatures (e.g., Plyasunov and Shock, 2001; Perfetti et al. 2008; Oelkers et al. 2009). However, for most metal complexes, including those with sulfur and chloride ligands, direct Cp and V data are not available; consequently, the accuracy of HKF predictions for the great majority of metal complexes is difficult to evaluate towards magmatic or deep metamorphic conditions and therefore requires validation by direct experimental data at elevated temperatures (>300 °C).

Nevertheless, the model performance for moderate-to-high density fluids, flexibility of parameterization and large potential for re-tuning the parameters when new experimental data become available, largely overcome these limitations. Therefore, we recommend the HKF model for most ionic S-bearing species and their complexes with metals for liquid-like geological fluids.

## Akinfiev-Diamond model

The intrinsic limitations of the HKF model for describing nominally uncharged species using the Born electrostatic equation and its inability to properly account for the near-critical region of water make this model perform poorly for aqueous nonelectrolyte species at supercritical and near-critical conditions. A potentially powerful alternative equation of state for volatile species has been proposed by Akinfiev and Diamond (2003), hereafter the AD model. In this model, the standard Gibbs energy of an aqueous species is described using the Gibbs energy value of the equivalent ideal-gas species and three adjustable parameters accounting for the interactions of the gaseous species with water, based fundamentally on a classical virial equation of state describing the solubility of a gas in water (e.g., Prausnitz et al. 1986) and the thermodynamic properties of pure water from Hill (1990):

$$G_{aq,P,T}^{0} = G_{g,T}^{0} - RT \ln N_{w} + (1-\xi)RT \ln f_{l}^{0} + RT\xi \ln\left(\frac{\tilde{R}T}{M_{w}}\rho_{l}^{0}\right) + RT\rho_{l}^{0}\left[a+b\left(\frac{10^{3}}{T}\right)^{0.5}\right] \quad (4.19)$$
where $G_{g,T}^0$ is the standard Gibbs free energy of the species in ideal gas state, $M_w$ is the molecular mass of water (18.0152 g/mol), $N_w$ is the number of moles of water in 1 kg of water (=1000/$M_w$), $f_1^0$ and $\rho_1^0$ are the fugacity (in bar) and the density (in g/cm$^3$) of pure water at given T-P, R and $\tilde{R}$ are the gas constants expressed in energy (1.9872 cal/(mol·K)) and volume (83.1441 cm$^3$·bar·mol/K) units, respectively, and a, b et $\xi$ are empirical T-P independent coefficients for each species. The model is fully compatible with the HKF model for aqueous electrolytes, in terms of standard and reference thermodynamic states, and therefore may be used, along with the HKF model, for describing reactions involving gases, aqueous species and minerals. The AD model performs better than HKF for describing vapor-liquid partition coefficients of volatile species and their thermodynamic properties in the near-critical region where $C_p$ and V values of solutes exhibit drastic changes (e.g., Akinfiev and Diamond 2003; Perfetti et al. 2008; references therein). With only 3 adjustable parameters it shows an equally good predictive power as HKF that uses 7 parameters (e.g., Fig. 4.10). However, the AD model requires knowledge of the thermodynamic properties of the gaseous counterparts, which are usually not available for many poorly volatile aqueous metal complexes. The HKF model is generally expected to perform better than the AD model at densities well above that of the water critical point ($\rho_{H_2O} > 0.33$ g/cm$^3$), and is therefore more suitable for deep metamorphic fluids. For the four key volatile species, $H_2S$, $SO_2$, $O_2$ and $H_2$ in hydrothermal fluids, the use of the AD model parameters may be recommended at pressures below 1–2 kbar relevant to shallow magmatic-hydrothermal systems, while the HKF model should be privileged for the same species at higher pressures of metamorphic and subduction-zone contexts.

### 4.3.4 Thermodynamic properties and distribution of sulfur species in hydrothermal fluids

Any thermodynamic equation of state describing the stability constants and other thermodynamic properties requires validation and/or parameterization using experimental data. Acquisition of the thermodynamic properties of sulfur species in aqueous solution from experimental measurements has a more than 100-year history. Lewis and Randall's series of studies in the early 1900s reported determination of the reaction constants and Gibbs free energies in aqueous vapor and liquid phases at elevated temperatures of some key sulfur forms such as native sulfur, $SO_2$, $H_2S$ and sulfate (e.g., Lewis et al. 1918; Lewis and Randall 1918). Since then, numerous studies have tackled the most common sulfur aqueous forms such as sulfate and sulfide in a wide range of temperatures, using a plethora of methods such as solubility, calorimetry, colorimetry,
potentiometry, electrochemistry, UV-Vis and Raman spectroscopy. Intermediate valence sulfur forms have been covered to a much more limited degree. Nowadays, the extent of experimental datasets in the T-P space of hydrothermal fluids is highly variable depending on the given sulfur aqueous species. The amount of our knowledge of the stability of different classes of sulfur species in hydrothermal solutions and the degree of confidence we can grant to these data is evaluated.

## Sulfate species

Sulfate-group species include sulfuric acid, H₂SO₄, and the anions of sulfate, SO₄²⁻, and hydrosulfate (or bisulfate), HSO₄⁻, together with their alkali and alkali-earth metal ion pairs. The sulfate ion is no doubt the most studied sulfur species in aqueous solution given its stability under terrestrial atmosphere conditions and the ease of handling and preservation of its salts. The thermodynamic properties of SO₄²⁻ at near-ambient conditions stem from extensive calorimetric studies of soluble sulfate salts of Na, K, NH₄, and Mg and solubility measurements of some less soluble sulfate solids of Ca, Ag and Ba (see Criss and Cobble, 1964a,b; Helgeson et al. 1981; references therein). These data have been critically revised by the Committee on Data for Science and Technology (CODATA, 1978; Cox et al. 1989) to recommend the standard molal enthalpy of formation (ΔfH⁰) and entropy (S⁰) at the reference conditions of 298.15 K and 1 bar. The recommended values from these two reference compilations differ by less than 0.4 kJ/mol and 0.4 J/mol K for ΔfH⁰ and S⁰. The resulting Gibbs energy (ΔfG⁰) of the sulfate ion is constrained to date to better than ±0.5 kJ/mol, making SO₄²⁻ the best studied sulfur-bearing aqueous species, at least at near-ambient conditions. The V⁰ and Cp⁰ values of SO₄²⁻ stem from heat capacity and density measurements of sulfate electrolytes up to 200 °C. Helgeson et al. (1981) analyzed these data and extracted HKF model coefficients for SO₄²⁻, which were later revised by Shock and Helgeson (1988) and included in the SUPCRT92 (Johnson et al. 1992) and SUPCRTBL (Zimmer et al. 2016) databases, allowing predictions of the Gibbs energy values of SO₄²⁻ at elevated T-P.

In saline fluids, the sulfate ion is accompanied by its ion pairs with alkali and alkali-earth cations that become increasingly predominant over free SO₄²⁻, with increasing temperature and salinity and decreasing pressure according to the general reaction:

$$ \mathrm{Cation}^{n+} + \mathrm{SO}_{4}^{2-} = \mathrm{Cation}(\mathrm{SO}_{4})^{n-} \qquad (4.20) $$

The formation constants of such ion pairs have been assessed using a variety of methods, including potentiometry, spectroscopy, conductivity, ion-exchange techniques, and densitometry, calorimetry, and solubility measurements. The major caveat is that most of these techniques do not "see" the ion pair directly. Its abundance and thermodynamic properties are derived from the difference of those of the bulk electrolyte system and those of the free cations and anions. This
fundamental limitation engenders significant discrepancies in the reported stability constants
depending of the analytical or experimental method used. Moreover, the definition of an ion pair
itself depends on the activity coefficients model used. For example, the use of the classical Debye-
Hückel model (e.g., Helgeson and Kirkham 1974) for describing anhydrite (CaSO₄) solubility in
H₂O-NaCl solutions requires, in addition to the Ca²⁺ and SO₄²⁻ ions, consideration of all potential
ion pairs and complexes with their respective stability constants (e.g., CaSO₄⁰, CaCl⁺, CaCl₂⁰,
CaOH⁺, etc). In contrast, the Pitzer model (Pitzer 1991), for example, utilizes a set of binary and
ternary interaction parameters between simple ions (so called virial coefficients), and only the
strongest ion-ion binary interactions are accounted by ion pairs of explicit entity such as CaSO₄⁰.
Overall, the association constant values for some alkali and alkali-earth sulfate ion pairs (e.g.,
CaSO₄⁰, NaSO₄⁻ or KSO₄⁻) typically vary over at least 0.5 log unit at ambient conditions and up to
1-2 log units at moderate temperatures (~300 °C), depending on the literature source (e.g., see
Pokrovski et al. 1995; Monnin 1999; Dai et al. 2014; Kokh et al. 2020 and references therein for
selected numerical examples). It is impossible to provide here a more comprehensive account of the
thermodynamic data of sulfate ion pairing, given the ambiguity of an ion pair as probed by the
different experimental methods or regarded by the different models. We adopt here the explicit
speciation model for aqueous solution as the most versatile and flexible one for multicomponent
natural fluids (Helgeson 1969 and following studies of his group). Each ion pair is regarded as an
individual metal-ligand complex with its own formation constant and other thermodynamic
parameters. The HKF-model parameters of sulfate ion pairs have been compiled in the SUPCRT92
and SUPCRTBL databases for NaSO₄⁻, KSO₄⁻, MgSO₄⁰, CaSO₄⁰, and KHSO₄⁰ (Sverjensky et al.
1997). Many other aqueous species databases also use *K* values generated from those parameters.
Among them, NaSO₄⁻ is by far the best constrained. Its original HKF predictions are well confirmed
by more recent experimental studies (e.g., Pokrovski et al. 1995; Hnedkovsky et al. 2005;
Scheuermann et al. 2019), with *K*₂₀ values being within <0.2 log unit at 350 °C and *P*sat. Other,
potentially important, ion pairs contributing to S-bearing minerals solubility at elevated *T*-*P* (e.g.,
NaHSO₄⁰ and also NaHS⁰, see below) are absent in most commonly used databases. For such pairs,
analogies with the well-known NaCl⁰ ion pair may be used. This is because, in a first approximation,
the Cl⁻ ion may be regarded, in terms of electrostatic properties, to be similar to HSO₄⁻ and HS⁻.
Therefore, one may assume the constant of the isocoulombic exchange reaction such as NaCl⁰ +
HSO₄⁻ = NaHSO₄⁰ + Cl⁻ to be equal to one at any *T* and *P* (e.g., Kokh et al. 2020). For example,
such analogies yielded *K* values of NaHSO₄⁰ very similar to those from very recently reported HKF
coefficients for this ion pair (Baudry and Sverjensky 2024). For others (e.g., SrSO₄⁰, BaSO₄⁰,
FeSO₄⁰), simple analogies are harder to find, and thus new experimental and thermodynamic data
are required to be able to explicitly use these species in modeling of fluid-rock interactions at elevated T-P. When approaching the salt saturation and liquid-liquid or liquid-vapor immiscibility in the alkali sulfate - water systems, more complex ion 'triples', such as, Na₂SO₄⁰ and K₂SO₄⁰ likely form, as indicated by recent IR and Raman spectroscopic measurements and molecular dynamics simulations (e.g., Reimer et al. 2015). The formation constants of Na₂SO₄⁰ were first derived by Hnedkovsky et al. (2005) from electrical conductivity measurements around the temperatures of the critical point of water (350–390 °C) and fitted using the density model. These data were later refitted by Scheuermann et al. (2019) using a slightly different equation. In both cases, the limited T-range of the original datapoints makes extrapolations rather uncertain beyond the near-critical T-range.

In parallel with ion pairing, the sulfate ion undergoes increasing protonation both with decreasing pH and increasing T, by forming the hydrosulfate ion, HSO₄⁻ (=HO)SO₃⁻ The equilibrium constant of the reaction

$$HSO_4^- = SO_4^{2-} + H^+ \qquad (4.21)$$

has been measured in numerous studies since the 1930s, using a variety of methods ranging from conductivity (Hamer 1934; Davis et al. 1952; Ryzhenko et al. 1964; Hnedkovsky et al. 2005; Conrad et al. 2023) and potentiometry (Matsushima and Okuwaki 1988; Dickson et al. 1990) to calorimetry (Readnour and Cobble 1969; Oscarson et al. 1988), solubility (Lietzke et al. 1961; Marshall and Jones 1966) and to Raman spectroscopy (Rudolph 1996), covering a T range of 0–400 °C from P₃sat to ~300 bar. Despite the multitude of methods and studies, the values of K₂₁ stay in good agreement (Fig. 4.8). Even more importantly, the HKF-model predictions by Shock and Helgeson (1988) based on some of these 1950–1960s data below 250 °C are surprisingly well confirmed by more recent studies at higher temperatures (Fig. 4.8), lending credence to the HKF parameters for both sulfate and hydrosulfate ions reported in SUPCRT92. Therefore, we recommend these thermodynamic properties for both ions in high T-P speciation calculations. The distinguishing feature of reaction (4.21) is shifting equilibrium to the right with increasing temperature. As a result, the HSO₄⁻ to SO₄²⁻ transition is displaced from pH~2 at ambient conditions to pH~7 at 300 °C. The pressure exerts an opposit trend, favoring dissociation (Fig. 4.8). Thus SO₄²⁻ and its ion pairs with alkali and alkali earth cations predominate in very high-pressure (10s kbars) subduction-zone fluids.

The HSO₄⁻ ion undergoes further protonation by forming uncharged H₂SO₄⁰ (structurally equivalent to (HO)₂SO₂⁰), with further increase in temperature towards water critical point and above and at relatively low pressures corresponding to vapor-like fluids, whereas at ambient conditions sulfuric acid is a very strong electrolyte, being virtually fully dissociated. As a result, published experimental measurements of its dissociation constant

$$H_2SO_4^0 = HSO_4^- + H^+ \qquad (4.22)$$
are available only at temperatures >150°C (Oscarson et al., 1988), with most studies being conducted between 350 and 400 °C (Xiang et al. 1996; Sue et al. 2004; Hnedkovsky et al. 2005). These studies used calorimetry, pH-metry or conductance methods. An exceptional, and the oldest known to us, work has been that of Quist et al. (1965) who performed conductance measurements of H₂SO₄ aqueous solutions over an extraordinary large (for an experimental study) range of temperature, from 400 to 800 °C, and pressure, from 1 to 4 kbar. The available K₂₂ values and their extrapolations to lower temperatures made in some studies are summarized in Fig. 4.9a versus reciprocal temperature. For the aqueous liquid phase and supercritical fluid phases under pressures well above the water critical point (>400 bar), the dissociation of H₂SO₄ generally remains quite strong, so that at the pH of the water neutrality point (shown in Fig. 4.9a as a dashed blue curve at 1 kbar) H₂SO₄⁰ species would be virtually “fully” dissociated to HSO₄⁻. However, the dissociation is strongly pressure-dependent, becoming weaker with decreasing pressure. Therefore, H₂SO₄⁰ is expected to become the increasingly dominant sulfate species in the magmatic vapor-like fluids (e.g., Ni and Keppler 2012). Being a neutral species and thereby much more volatile than its dissociated anionic counterparts (Pokrovski et al. 2013), H₂SO₄⁰ may significantly contribute to sulfate degassing during magma-fluid separation at shallow depths and subsequent fluid boiling. An HKF-model parameter set of SO₃⁰(aq) (HKF model convention for anhydrous H₂SO₄) have recently been published by Beaudry and Sverjensky (2014) who regressed the Quist et al. (1965) experimental datapoints (Fig. 4.9a). Since the pioneering work of Quist et al. (1965), it has been recognized that the K₂₂ value trends can be significantly linearized as a function of water density as shown in Fig. 4.9b. Nevertheless, values for similar T-P-ρH₂O from different studies yet remain discrepant of up to 2 log units at elevated T (>350 °C), which yields to large errors, over 4 log units, when extrapolating to the reference conditions of 25 °C and 1 bar (Fig. 4.9a,b). As a result, consistent thermodynamic parameters of H₂SO₄⁰, and even empirical equations as function of T and ρH₂O that would cover the whole range of measured data, are currently lacking, leaving room for future thermodynamic studies.

## Sulfide species

Sulfide-type species comprise H₂S, HS⁻, S²⁻ and, to a much lesser extent, their ion pairs with alkali metals.

H₂S(aq). The thermodynamic properties of H₂S(aq) under hydrothermal conditions in liquid-like fluids stem mostly from solubility measurements of gaseous H₂S in water to 350 °C at Psat (Suleimenov and Krupp 1994 being the most recent; see Fig. 4.10 for complete reference list).
Another type of measurement deals with the solubility of aqueous H₂S in fluids in equilibrium with common mineral buffers such as pyrite-pyrrhotite-magnetite and pyrite-magnetite-hematite performed to 500 °C and 1 kbar (Kishima 1989). All these measurements allowed derivation of the Henry constant as compiled in Fig. 4.10

$$H_2S_{(aq)} = H_2S_{(g)} \qquad (4.23)$$

Coupled with more recent direct calorimetric and densimetric measurements of Cp and V of H₂S (aq) (Hnedkovsky et al. 1996; Hnedkovsky and Wood 1997), these data enabled generation of robust thermodynamic parameters within the framework of the HKF (Schulte et al. 2001) and AD (Akinfiev and Diamond 2003) models. It can be seen in Fig. 4.10 that both models are in decent agreement, with the AD model performing slightly better at least for the so far experimentally covered pressure range of <1 kbar (see section 4.3.3 for general comparisons between these models). Using these models, H₂S(aq) concentrations in geological fluids in equilibrium with common sulfide minerals may be predicted within a plausible uncertainty of <0.5 log units to ~600 °C and ~5 kbar, thereby covering a major part of magmatic-hydrothermal and metamorphic fluids in the crust. For the vapor-like fluids with densities below 0.2 g/cm³, generated by boiling processes in hydrothermal systems, H₂O-H₂S mixing rules have also been established through PVTX measurements up to 400 °C (Zezin et al. 2011 and references therein). However, at higher pressures of deep subduction zone fluids, H₂S, similarly to H₂O, is expected to become increasingly dissociated thereby adding more uncertainty to the current thermodynamic predictions of sulfide-type species concentrations.

**HS⁻ ion.** The thermodynamic properties of the hydrosulfide ion, HS⁻, which is the H₂S deprotonated counterpart, stem from measurements or estimations of the dissociation constant whose available studies are compiled in Fig. 4.11:

$$H_2S_{(aq)} = HS^- + H^+ \qquad (4.24)$$

These data have been acquired using potentiometry, calorimetry or UV-Vis spectroscopy. Most of them were limited to temperatures of 150–200 °C, with the exception of the Ellis and Giggenbach (1971) and Suleimenov and Seward (1997) works. These experimentalists managed to acquire direct UV-Vis data on H₂S-NaOH aqueous solutions to, respectively, 276 and 350 °C at Pₐ₉₅, thereby enabling much better constraints on H₂S dissociation. Some of those data have been regressed by the HKF model in Helgeson and Shock (1988) that remains the unique set of HKF parameters for the HS⁻ ion to date. We are not aware of more recent than 1997 experimental studies of H₂S dissociation constants under hydrothermal conditions. This lack is due to notorious technical and analytical difficulties of working with toxic and aggressive sulfide solutions at elevated temperatures. The development of molecular dynamics simulation methods in the recent years made possible the first theoretical estimations, such as by Sulpizi and Sprik (2008) who reported a logK₂₄
value of -7±1, in perfect agreement with the numerous experimental data. However, the intrinsic uncertainties of MD methods in aqueous media are yet much greater than those of experimental measurements. The HKF and AD model calculations shown along with the whole set of experimental data in Fig. 4.11, are expected to predict the H₂S dissociation constant to 400–500 °C and 1–2 kbar with a decent accuracy (i.e. within 0.5 log units). We are not aware of experimental data at pressures above Pₐ₉₅ in the hydrothermal domain (<500-600 °C) except a single conductivity study of Sretenskaya (1977) to 90 °C and to 3 kbar. The HKF model that did not consider her work in its parameterization yet realistically predicts an increasing dissociation of H₂S at elevated pressures (Fig. 4.11). The values of -log K₂₄ are comparable with those of the pH of the neutrality point of pure water (i.e. [H⁺] = [OH⁻] = ½ log Kₐ₉₅(H₂O)) only below 100°C where dissolved hydrogen sulfide would significantly be ionized as HS⁻ at near-neutral pH. In contrast, at higher temperatures, the H₂S ionization becomes much less significant at near-neutral pH imposed by the increasing dissociation of water. For example, only a few percent of H₂S will be ionized at 500 °C and 5 kbar in an aqueous fluid of neutral pH (because pK₂₄ is ~2 log units lower than the pH of water neutrality point, Fig. 4.11). In high-temperature magmatic vapors (>700–800 °C, <2 kbar) H₂S is predicted by all thermodynamic models to be negligibly dissociated. However, recent Raman analyses of S-bearing synthetic fluid inclusions at such conditions were interpreted by the dominant presence of HS⁻ (Farsang and Zajacz 2025), in sharp contrast to the predictions above as well as known dissociation trends of other weak acids. This discrepancy is likely due to the growing lower-frequency asymmetry of the H₂S Raman stretching peak with temperature, which was mistaken with the characteristic Raman peak of HS⁻ situated just 20 cm⁻¹ to the lower wavenumbers (Table 4.3). Further experimental validation of T-P-dependent predictions would require more high-pressure and high-temperature spectroscopic and molecular modeling data. This gap is especially important to be filled for the interpretation of metal complexation because HS⁻ is the key ligand for chalcophile metals (see section 4.5).

S²⁻ ion. The sulfide ion, S²⁻, was commonly considered in many general chemistry treatises and aquatic geochemistry textbooks (see an excellent compilation of May et al. 2018 for references). Its abundance and stability in aqueous solution is controlled by the dissociation reaction

$$HS^{-} = S^{2-} + H^{+} \qquad (4.25)$$

However, this reaction constant, both at low and high temperatures, remains a matter of debate since more than 120 years. The major data sources are compiled in Fig. 4.12 (additional references may be found in May et al. 2018). Pioneering works of Küster and Heberlein (1905) and Knox (1906) who reported logK₂₅ values of -13.2 and -14.9 at 25°C from, respectively, potentiometric pH and HgS solubility/Hg electrode measurements in NaHS/Na₂S aqueous solutions. Other scarce studies
of first half of the 20th century also reported values within this range (as summarized by Maronny
1959). The following experimental and theoretical studies after the 1950s can be divided into two
distinct groups as to the K25 value measured at near-ambient conditions.

The first, relatively earlier, group of studies that used pH-metry (e.g., Maronny 1959; Kryukov et al. 1974 and some previous studies), HgS solubility (Dickson 1966), UV spectroscopy (Ellis and Milestone 1967) and calorimetry (Stephens and Cobble 1971), report values between –15 and –13. These data were adopted in many old textbooks and used both in earlier (Helgeson 1969) and more recent (Phillips and Phillips, 2000) theoretical T-dependence extrapolations as well as in common databases such as LLNL that ignored subsequent data. It can be seen in Fig. 4.12 that these estimations (Helgeson 1969 and LLNL that likely use similar data) as well as some earlier experimental data (e.g., Kryukov et al. 1974) show a very strong increase of logK₂₅ values with temperature rise, with values reaching –7 to –8 at 300 °C, which are above those of the H₂S first dissociation constant (reaction 4.24, Fig. 4.11), making HS⁻ subordinate and S²⁻ largely predominant sulfide species in near-neutral pH moderate-temperature hydrothermal fluids (e.g., Helgeson 1969). However, such predicted values at moderate temperatures, which are still being used nowadays in some thermodynamic modeling studies of S-bearing systems, are in strong contradiction with i) reaction (4.24) studies that are far better constrained and ii) the large set of gold solubility and complexation data fully consistent with the formation of Au(HS)₂⁻ in circa-neutral (pH 6–8) hydrothermal fluids over a wide T-P range (see Pokrovski et al. 2014 and section 4.5 for details).

The second group of datasets comprises the seminal work of Ellis and Giggenbach (1971) and Giggenbach (1971a) using improved UV-Vis spectrometry methods and of Meyer et al. (1983) by Raman spectroscopy, and later on, Licht and Manassen (1987) by potentiometry. These works reported consistently much lower logK<sub>25</sub> values at ambient conditions, around –17. Such values mean that the HS<sup>−</sup> ion still remains dominant even in 10 molal NaOH solutions at ambient conditions. A similar range of values has been reported by less direct methods using analogues with polysulfide ions and elemental S<sup>0</sup> surface protonation data (Schoonen and Barnes 1988; Migdisov et al. 2002). More recently, May et al. (2018) re-examined HS<sup>−</sup> in concentrated CsOH and NaOH solutions by Raman spectroscopy and found no significant changes in HS<sup>−</sup> peak intensity (note that Raman probes S-H bond vibrations and cannot directly detect the S<sup>2−</sup> ion). They concluded that any value of logK<sub>25</sub> in the range of –17 to –19 or lower would match the Raman spectra, and that there is no evidence of S<sup>2−</sup> formation in solutions as concentrated as 20 molal NaOH/CsOH. These authors thus proposed to exclude the S<sup>2−</sup> ion from any consideration in aqueous solution. Such spectacular discrepancies among the different methods and studies are due to intrinsic experimental/analytical
difficulties of handling highly reactive and prone to oxidation sulfide solutions, so that general preference should be given to more direct in situ spectroscopic data. Finally, a fresh simulation study using molecular dynamics (Queizán et al. 2022) qualitatively confirmed that S²⁻ rapidly protonates in aqueous solution in the presence of H⁺ by forming HS⁻, even though the simulation conditions were rather far from realistic aqueous solution concentrations.

If this second group of studies looks more robust at ambient conditions, the question still remains open as to the S²⁻ stability at elevated temperatures and pressures. Indeed, most available studies, both experimental and theoretical, whatever the initial ambient-temperature K₂₅ value, indicate an increase of the reaction (4.25) constant with temperature, even though highly variable in amplitude (Fig. 4.12). For example, Migdisov et al. (2002) reported an increase by ~3 orders of magnitude of HS⁻ dissociation degree within a very narrow T-range (25–70°C, Fig. 4.12). Furthermore, by analogy with the H₂O dissociation, greatly increasing with pressure, it cannot be completely excluded that the S²⁻ ion might be a non-negligible sulfur species in fluid of subduction zone conditions, and thereby dramatically affect the aqueous speciation of soft metals for which it has a very strong affinity. It is thus premature to say a definite goodbye to the S²⁻ ion in aqueous solution, and more direct in situ spectroscopic data are urgently needed for high T-P geological fluids. In the meantime, considering the large discrepancy in the K₂₅ value, it is much safer to write solubility products of sulfide minerals in terms of HS⁻ or H₂S, rather than S²⁻ as still commonly used in chemistry (e.g., CuS(s) + H⁺ = Cu²⁺ + HS⁻, or CuS(s) + 2H⁺ = Cu²⁺ + H₂S, rather than CuS(s) = Cu²⁺ + S²⁻).

## Sulfite species

Sulfite-type aqueous species are those with sulfur in a formal oxidation state of IV, and include sulfur dioxide (SO₂), sulfite and bisulfite (SO₃²⁻ and HSO₃⁻), and their dimer (S₂O₅²⁻) with its protonated counterpart (HS₂O₅⁻). Their stabilities are constrained to date, at least at near-ambient conditions. The interest in quantifying aqueous speciation of sulfite was primarily motivated by atmospheric sciences because SO₂ and their products are important constituents of atmospheric aerosols (e.g., Brandt and Eldik 1995; Brimblecombe and Norman 2025). Sulfite species are important products and intermediates in microbial sulfate reduction and associated S isotope fractionation (e.g., Eldridge et al. 2016; Eldridge 2025). Moreover, SO₂, along with H₂S, has long been considered to be important constituents of magmatic fluids produced by magma degassing in the crust (e.g., Burnham and Ohmoto 1980; Wallace 2005; Wallace and Edmonds 2011). The chemical literature about sulfite species kinetics and equilibria in aqueous solution measured at laboratory conditions is too large to be overviewed in detail here, we focus here on key existing
compilations that are currently used for sulfite species calculations at elevated temperatures along with some more recent data relevant to geological fluids.

**Sulfur dioxide.** The solubility and hydration structures of sulfur dioxide in aqueous solution, SO₂(aq), have been studied in chemistry since the early 1900s. In contrast to sulfuric acid, the hypothetical sulfurous acid (H₂SO₃) that might serve as an intermediate in the dissolution of SO₂ in water, similar to carbonic acid (H₂CO₃), has not spectroscopically been detected in aqueous solution (e.g., Connick et al. 1982; Zhang and Ewing 2002; Wang et al. 2022). Even though some earlier theoretical DFT calculations indicated that the formation of hydrated H₂SO₃ might be thermodynamically favorable over hydrated SO₂ (Steudel and Steudel 2009a), it was not confirmed so far by more recent studies using more advanced computational tools (e.g., Wang et al. 2022). In the absence of direct data, H₂SO₃ will be disregarded here in favor of dissolved SO₂(aq). The thermodynamic properties of SO₂(aq) stem from measurements of its solubility in water described by its Henry's constant

$$SO_{2(aq)} = SO_{2(g)} \qquad (4.26)$$

Figure 4.13 compiles the key available high-temperature data and major compilations that also include numerous near-ambient temperature data from older chemical literature that can be found in Goldberg and Parker (1985) and Pereda et al. (2000). In their critical compilation, Goldberg and Parker (1985) have revised the K₂₆ values and related thermodynamic properties of reaction (4.21) to at least 100 °C. These values are in good agreement with more recent data (e.g., Rumpf and Maurer 1992) obtained to 120 °C, which is the highest temperature at which K₂₆ has ever been measured, to the best of our knowledge. Similarly, these data are in good agreement with revised HKF model coefficients (Schulte et al. 2001) that include experimental data on the heat capacity of dissolved SO₂ to 350 °C (Sharygin et al. 1997). Another independent set of HKF coefficients (Plyasunov and Shock 2001) yields rather similar predictions to at least 350 °C (not shown). Finally, the AD model (Akinfiev and Diamond 2003) provides a good match of the available data, but deviates from the HKF model above 200 °C. These deviations are, however, rather minor (within less than 0.3 log units, Fig. 4.13) and both models may equally be used to at least 500 °C and 2 kbar for modeling shallow-crust fluids. Greater differences among these three predictions are observed at higher T-P relevant to deep subduction zones, e.g. up to 2 log units at 700 °C and 30 kbar. These differences translate to a minimal error of 2 log units for a redox or protonation reaction involving SO₂(aq). To this minimal error, will be added uncertainties related to the other S-bearing reaction components, thereby resulting in overall uncertainties of at least 2–3 orders of magnitude for SO₂ predicted concentrations in the fluid at such conditions.
**Sulfite and bisulfite ions.** Sulfur dioxide in water undergoes dissociation depending on pH according to the formal reactions:

$$SO_{2(aq)} + H_2O = HSO_3^- + H^+ \qquad (4.27)$$

$$HSO_3^- = SO_3^{2-} + H^+ \qquad (4.28)$$

The bisulfite ion, $HSO_3^-$, in aqueous solution exists as two isomeric species, $(HS)O_3^-$ and $(HO)SO_2^-$ . The first form, sometimes called sulfonate or SH isomer, has a tetrahedral geometry with the proton bound to the sulfur atom. The second form, hydrogen sulfite or OH isomer, has a pyramidal geometry in which the proton is bound to one of the oxygen atoms. Due to their different geometries and bonding, these forms have distinguishable spectral signatures (Table 4.3) as seen by Raman (Littlejohn et al., 1992; Eldridge et al. 2018), S K-edge XANES (Risberg et al. 2007) or $^{17}$O-NMR (Horner and Connick 1986) spectroscopies that allowed determination of the isomerization constant:

$$(HS)O_3^- = (HO)SO_2^- \qquad Q_{29} \qquad (4.29)$$

The value of $Q_{29}$ (equilibrium quotient uncorrected for ionic strength, in contrast to K) at 25 °C is 2.7±0.5 (Risberg et al. 2007; Eldridge et al. 2018), implying an average ratio of 1:2.7 between $(HS)O_3^-$ and $(HO)SO_2^-$ in solution. According to the $Q_{29}$ variation with temperature, the $(HS)O_3^-$ isomer comprises from ~20 to 40% of the total bisulfite in the T-range 5–85 °C (Eldridge et al. 2018). The evolution of the isomerization phenomenon at higher temperatures would be important to know for interpreting S isotope signatures because S isotope fractionation between those species was calculated to be unusually large, despite the same redox state of $S^{4+}$ (1000×ln$^{34}$α ~25‰ at ambient conditions; Eldridge et al. 2016). The two isomers, when acting as ligands, might also have different affinities for metals in hydrothermal fluids.

In contrast to bond- and coordination-specific spectroscopy methods (Raman, XANES), most other methods such as potentiometry, calorimetry, conductivity or UV-Vis spectroscopy, which have been used to derive reaction constants (4.27) and (4.28), are not directly sensitive to isomerization reaction (4.29). Therefore, the thermodynamic properties of both isomers are treated collectively as the bisulfite ion. The properties of both bisulfite and sulfite ions are bound to reactions constants (4.27) and (4.28), which are compiled in Fig. 4.14. The thermodynamic values (enthalpy, Gibbs energy and entropy) of these reactions at ambient conditions have been reported in 1970s-1980s compilations based on critical revision of multiple experimental measurements that will not be discussed here (see Cobble et al. 1972; Goldberg and Parker 1985 and references therein). Using these data and some assumptions about the reaction (4.27) and (4.28) heat capacity changes, the constant was predicted with good confidence to at least 150–200 °C using van Hoff's types of equations (e.g., eqn. 4.11; Barbero et al. 1983; Goldberg and Parker 1985). These predictions have been confirmed by the more elaborate HKF and AD models, allowing extrapolations to higher
temperatures. Both models predict weakening of dissociation with increasing *T* that makes SO₂ and HSO₃⁻ the dominant SIV species in hydrothermal and metamorphic fluids. Both models also predict a moderate increase of dissociation with pressure (Fig. 4.14), consistent with the rare available data (Ellis and Anderson 1961) and general trends of protonation reactions (e.g., see sulfide and sulfate above). Alkali bisulfite and sulfite ion pairs, similar to those formed by bisulfate and sulfate (see above) are expected to further contribute to S⁴⁺ speciation at elevated temperatures in saline fluids, but their formation constants have not been thoroughly measured or estimated, and are therefore limited by the use of analogies with chloride or sulfate ion pairs.

*Disulfite ions.* Another complication of the sulfite system is formation of dimers, the pyrosulfite or disulfite ion, S₂O₅²⁻, and its protonated counterpart, HS₂O₅⁻:

$$2 \mathrm{HSO_3^-} = \mathrm{S_2O_5^{2-}} + \mathrm{H_2O} \qquad (4.30)$$

$$\mathrm{S_2O_5^{2-}} + \mathrm{H^+} = \mathrm{HS_2O_5^-} \qquad (4.31)$$

Their structure in solution likely involves S-S bonds, as shown by Raman and UV spectroscopy and is similar to the structure known in the solid phase (Connick et al. 1982; Littlejohn et al. 1992; references therein). Williamson and Rimstidt (1992), however, questioned such a structure in favor of a hypothetical S-O-S linkage that would be more coherent with Gibbs energy versus structure correlations for other oxyanion sulfur species suggested by these authors. However, their analysis was based on poorly constrained thermodynamic properties of S₂O₅²⁻ from old references subjected to large uncertainties. The dimerization constant of reaction (4.30) is well constrained to date at near-ambient conditions using a variety of methods. Remarkably, 1980s-1990s data and predictions (Goldberg and Parker 1985; Connick et al. 1982; Littlejohn et al. 1992; references therein) are well confirmed by more recent spectroscopic and calorimetric measurements (Ermatchkov et al. 2005; Beyard et al. 2014) allowing reaction (4.30) thermodynamic parameters to be experimentally constrained to at least 80 °C. The value of K₃₀ of ~0.05 at 25 °C, implies that the dimer may only be significant at too high total sulfite concentrations to be relevant to most natural low-temperature fluids (e.g., >10% of dimer at 0.5 m of sulfite). In addition, the constant decreases with increasing temperature, making S₂O₅²⁻ even less important in hydrothermal fluids. As a note, HKF parameters of S₂O₅²⁻ were also published by Shock et al. (1997) using inter-parameter correlations and old Gibbs energy and entropy values from Latimer (1952), which yields K₃₀ values 3 orders of magnitude lower than the more recent data above, and further decreasing with the temperature rise. We therefore do not recommend using those HKF parameters for S₂O₅²⁻ for modeling S-rich fluids. A protonated dimer, HS₂O₅⁻, has been inferred to form in acidic solutions according to reaction (4.31) whose constant at 25°C was reported in a single work of Beyad et al. (2014), logK₃₁ of ~3, meaning that HS₂O₅⁻ takes over S₂O₅²⁻ at pH below ~3.
Thiosulfates

This type of sulfur species includes the well-known thiosulfate ion, S₂O₃²⁻, and its poorly known protonated counterparts, HS₂O₃⁻ and H₂S₂O₃. The S₂O₃²⁻ ion and its soluble alkali and alkali-earth salts are widely used chemical reagents and are kinetically stable solutes in water at neutral-to-basic pH. In geoscience-related fields, thiosulfate is used for extraction of gold and some other metals from ore because it forms very stable complexes with Au (e.g., Adams 2005; Marsden and House 2009). It is also a very convenient reagent in experimental mineralogy and petrology and isotope geochemistry, because, being nontoxic, water soluble and easy to handle at ambient conditions, it breaks down irreversibly to sulfate and sulfide, along with polysulfide species such as the radical ions (see below), when heated above 150 °C according to the formal reaction

$$S_2O_3^{2-} + H_2O = SO_4^{2-} + H_2S \quad (4.32)$$

This overall reaction may be decomposed into the three following steps (Kokh et al. 2020)

$$S_2O_3^{2-} = S^0 + SO_3^{2-} \quad (4.33)$$

This reaction is the most straightforward and kinetically easiest one, since it requires only breaking the S-S bond of the thiosulfate ion with minimal electron transfer between the two S atoms. With time, both S⁰ and SO₃²⁻ undergo transformation to H₂S and SO₄²⁻, which are thermodynamically stable at hydrothermal conditions:

$$4 SO_3^{2-} + 2 H^+ = H_2S + 3 SO_4^{2-} \quad (4.34)$$

$$4 S^0 + 4 H_2O = 3 H_2S + SO_4^{2-} + 2 H^+ \quad (4.35)$$

As a result, thiosulfate provides an excellent chemical proxy for many hydrothermal fluids in which sulfate and sulfide coexist (e.g., Ohmoto and Lasaga 1982 and references therein; Pokrovski and Dubessy 2015; Colin et al. 2020; Kokh et al. 2020). In particular, its use in S-rich hydrothermal experiments allows efficient fO₂ buffering by the sulfate-sulfide equilibrium

$$H_2S + 2 O_2 = SO_4^{2-} + 2 H^+ \quad (4.36)$$

The S₂O₃²⁻ ion contains 2 different sulfur atoms bound together in a S-S chemical bond, one sulfanyl (S-H) sulfur in a formal oxidation state of -1, and the other sulfonate (S-O₃) sulfur in +5 oxidation state which seem to be better compatible with S XANES spectra than the nominal -2 and +6 values (Vairavamurthy et al. 1993).

The thermodynamic properties of the S₂O₃²⁻ ion have been determined using heat of oxidation by Br₂ and solubility of Ag₂S and CaSO₄, as detailed and critically reviewed by Mel et al. (1956) and Cobble et al. (1972). The standard Gibbs energy and entropy values of the ion at reference T-P conditions of 25 °C and 1 bar from those studies are quite discrepant, spanning over almost 30 kJ/mol and 30 J/mol K, respectively; we are not aware of more recent experimental
measurements. As a result, the reference thermodynamic properties of S₂O₃²⁻ have not been updated since the 1970s and travel from one database to another with variations within those ranges. The high-temperature properties are probably best approximated by the HKF model that used in addition some limited Cp and V calorimetric data for thiosulfate salts (Shock et al. 1997). An uncertainty of >30 kJ/mol would be a realistic error when predicting the G⁰(S₂O₃²⁻) values at common hydrothermal conditions (200–400 °C).

Upon acidification to pH<5 at ambient conditions, thiosulfate solutions become unstable, rapidly producing SO₂, H₂S and elemental sulfur, according to mechanisms that would require simultaneous presence of S₂O₃²⁻ and its protonated forms, HS₂O₃⁻ and H₂S₂O₃ (e.g., Steudel and Steudel 2009b). However, direct spectroscopic evidence of these species in solution is yet lacking, even though the hydrothiosulfate ion, HS₂O₃⁻, has been identified in solid phase by its distinct Raman S-H and S-S frequencies (Table 4.3; Steudel and Prenzel 1989). The thiosulfuric acid, H₂S₂O₃, yet remains rather hypothetical from an analytical point of view. From a theoretical point of view, both protonated molecules have been extensively studied by DFT simulations that show two tautomers, HS-SO₃⁻ and S-SO₂(OH)⁻ (Steudel and Steudel 2009b), similar to the sulfite dimers (see above). Potentiometric measurements of the first half of 20th century performed in dilute acidic thiosulfate solutions are indicative of the reactions:

$$S_2O_3^{2-} + H^+ = HS_2O_3^- \qquad (4.37)$$

$$HS_2O_3^- + H^+ = H_2S_2O_3 \qquad (4.38)$$

The instability of thiosulfate upon acidification (see above) makes these reaction constants notoriously difficult to accurately measure. The work of Page (1953) reports logK values of 1.7 and 0.6 at 25 °C and 1 bar for reaction (37) and (38), respectively. These values mean that S₂O₃²⁻ significantly protonates only below pH ~2, and hypothetical thiosulfuric acid would dominate the thiosulfate speciation only in highly acidic solutions (pH < 0.6). These K values were the basis for estimating high-temperature reaction constants and thermodynamic properties of the protonated forms, using approximations for ΔS and ΔCp of these reactions (e.g., Naumov et al. 1971; Murray et Cubicciotti 1983), and HKF correlations (Shock et al. 1997). However, those theoretical estimations ignore a more recent accurate pH-metry work of Nikolaeva and Gusyakova (1989) who elegantly determined K values of reaction (4.38) in water and ethanol-water solutions to 90 °C. Their logK₃₈ values at 25°C (1.97±0.05) agree within 0.3 log units with the Page (1953) work and a few older studies not discussed here. Their K₃₈ values increase by an order of magnitude to 100 °C, which is within the same magnitude as the theoretical estimations above. All estimations predict a few orders of magnitude increase in both constants to 300 °C, suggesting a significant shift in the
thiosulfate speciation toward protonated species at near-neutral pH; however, the absolute *K* values among those estimations are discrepant over at least 2 log units at *T* ≥ 300 °C.

To this uncertainty adds the one related to alkali ion pairs of thiosulfate anions in saline solutions whose stability has not been constrained so far. It is thus hard to accurately predict the equilibrium abundance of thiosulfate species in hydrothermal fluids above 200–300 °C. While $HS_2O_3^-$ might eventually be included with caution in sulfur speciation schemes to ~200 °C, the reported properties of its fully protonated hypothetical counterpart $H_2S_2O_3$ are not currently reliable enough to be used above ~100 °C. In situ Raman spectroscopy studies, performed on thiosulfate salts and sulfur aqueous solutions at acidic and neutral pH over a wide range of S concentrations (0.2–3.0 m) to at least 700 °C and 10 kbar, have not detected thiosulfate species within the inherently high detection limits of ~0.01–0.1 m for Raman spectroscopy at such conditions (Bondarenko and Gorbaty 1997; Yuan et al. 2013; Pokrovski and Dubessy 2015; Barré et al. 2017; Colin et al. 2020). Studies that used hydrothermal reactors, enabling fluid sampling and chemical separation of the different sulfur forms and their subsequent analyses, reported thiosulfate concentrations comparable with these detection limits. For example, Dadze and Sorokin (1993) analyzed, in acidic aqueous solutions in the presence of native sulfur at 250 to 450 °C, up to 0.01 m of thiosulfate that they tentatively attributed to the $H_2S_2O_3$ species. Kokh et al. (2020) measured, using S isotopes, the rate constants of decomposition to sulfide and sulfate of initial 0.2–0.5 m $K_2S_2O_3$ solutions at 300–450°C (see equations 4.1 and 4.32). The authors found rate values consistent with previously established ones from less concentrated solutions (Ohmoto and Lasaga 1982 and references therein; see also section 4.3.2).

## Polythionates

**Tetrathionate.** A highly characteristic reaction of $S_2O_3^{2-}$ is its facile oxidation to tetrathionate, $S_4O_6^{2-}$, which – when carried with iodine ($I_2$) – is used in quantitative analysis of both reagents (e.g., Kokh et al. 2020; Steudel 2020). Tetrathionate, kinetically stable in aqueous solution at neutral and acidic pH, is representative of a long series of water soluble polythionate anions, $[O_3S-S_n-SO_3]^{2-}$, in which two $SO_3$ terminal groups are linked through -S-S- bonds with *n* from 1 to 4. Such anions may be prepared in aqueous solution and solid phase by various routes (Steudel 2020). In addition, aqueous species with chain length up to 22 S atoms were identified by liquid chromatography (e.g., Steudel et al. 1987). The rich aqueous redox chemistry and reaction kinetic mechanisms of polythionates at ambient conditions have been thoroughly addressed in the chemical literature (see Steudel 2020 for the most recent review). This interest is because, these sulfur forms, along with thiosulfate and sulfite, are of concern in microbiological and sedimentary processes as
intermediates in sulfur transformation reactions (Eldridge et al. 2021; Turchyn et al., 2025 and references therein) as well as in hydrometallurgy of metals leaching from ore (e.g., Aylmore and Muir 2001; Molleman and Dreisinger 2002). Polythionates have been detected in natural near-surface aquatic environments which are out-of-equilibrium (Takano 1987; Zopfi et al. 2004; Leavitt et al. 2015; Findlay and Kamyshny 2017). However, the thermodynamic properties required for predicting polythionates abundances in hydrothermal fluids are either virtually non-existent or highly discrepant. Among them, the best thermodynamically known is the S<sub>4</sub>O<sub>6</sub><sup>2-</sup> ion. Its Gibbs energy, entropy (and enthalpy) and heat capacity values at 25 °C/1 bar have been reported in a number of thermodynamic compilations and databases (Latimer 1952; Wagman et al. 1982; Bard et al. 1985). However, the original data from which these values were established are hardly traceable and they appear highly discrepant. A rare example how the thermodynamic values have been derived in reality is given by Cobble et al. (1972) who critically revised the rare calorimetric and solubility data and estimated G<sup>0</sup><sub>298</sub> (and H<sup>0</sup>) and S<sup>0</sup> values for aqueous S<sub>4</sub>O<sub>6</sub><sup>2-</sup>. These values, compared together with those from the compilations cited above remain rather discrepant, with G<sup>0</sup><sub>298</sub> values spanning over >30 kJ/mol, and C<sub>p</sub> values over >100 J/(mol K), which corresponds to an uncertainty of at least 6 log units in the equilibrium constant value of a reaction involving S<sub>4</sub>O<sub>6</sub><sup>2-</sup> at ambient conditions plus additional 2 to 3 log units when extrapolating the K<sub>i</sub> value to >300 °C according to equations (4.6) and (4.11) and assuming Δ<sub>r</sub>C<sub>p</sub> to be constant.

*Other polythionate species.* Despite such potential uncertainties, the S<sub>4</sub>O<sub>6</sub><sup>2-</sup> ion is second “thermodynamically” best known one, after the thiosulfate ion, among the whole group of polysulfoxide species, including polythionates and dithionites, as well as peroxosulfates and associated radical oxysulfur ions (see below). Only scarce data on the thermodynamic properties have been reported for other polythionate ions and acids (S<sub>2</sub>O<sub>6</sub><sup>2-</sup>, H<sub>2</sub>S<sub>2</sub>O<sub>6</sub><sup>0</sup>, S<sub>3</sub>O<sub>6</sub><sup>2-</sup>, S<sub>5</sub>O<sub>6</sub><sup>2-</sup>), which were unsystematically listed in certain widely used compilations, some of those claiming to be based on critical revisions of available data, others just referring to previous compilations (e.g., Latimer 1952; Wagman et al. 1982; Bard et al. 1985). As a result, it is hard to trace and critically evaluate the original thermodynamic measurements from those compilations, which is beyond the scope of this chapter. A typical example of the degree of discrepancy is the value of standard enthalpy of formation (Δ<sub>r</sub>H<sup>0</sup><sub>298K, 1bar</sub>) of the S<sub>3</sub>O<sub>6</sub><sup>2-</sup> ion, which is a common intermediate in the thiosulfate-tetrathionate redox transformations (e.g., Steudel 2020). This value ranges from -1200 kJ/mol as reported by Wagman et al. (1982) and Bard et al. (1985) to -1167 kJ/mol as adopted by Latimer (1952) followed by Shock et Helgeson (1988), to -1083 kJ/mol as estimated by Williamson and Rimstidt (1992), thus spanning over >117 kJ/mol. Comparable variations are observed for other sulfoxide species such as dithionite (S<sub>2</sub>O<sub>4</sub><sup>2-</sup>) for which similarly discrepant G<sup>0</sup> (and H<sup>0</sup>) values can
be found in the above compilations. The corresponding $S^0$ values are more rarely reported because in generally less constrained. The protonated counterparts of the dithionite and analogous dianions are yet more hypothetical than real because have never been directly observed, while being extensively modeled using theoretical quantum-chemistry methods (e.g., Drozdova et al. 1998). Curiously, protonation constants at ambient conditions (analogous to reactions 4.37 and 4.38) for the $S_2O_4^{2-}$ ion can be calculated from $G^0$ values reported for $HS_2O_4^-$ and $H_2S_2O_4^{0}$ in some databases (logK values are between 0.3 and 2.5; Wagman et al. 1982; Bard et al. 1985). These data were adopted by Shock et al. (1997) who also used inter-parameter correlations of the HKF model to report a set of HKF coefficients for the three dithionite species. They suggest that $S_2O_4^{2-}$ remains dominantly deprotonated above pH~3 at ambient conditions and above pH~4–5 at 300 °C. We were unable, however, to identify the original source of these key data for dithionite protonation, and therefore do not recommend the inclusion of the protonated species in sulfur speciation models at elevated temperatures until their direct identification is provided.

## Persulfates

An amazing, but poorly known, group of sulfur oxidized species are persulfates, such as the peroxodisulfuric ($H_2S_2O_8^0$ and $S_2O_8^{2-}$) and peroxysulfuric ($HSO_5^-$ and $SO_5^{2-}$) acids and their anions, for which limited thermodynamic data have been reported. These species are used in industry as powerful oxidants and may be formed by oxidation of sulfuric acid in electrochemical processes or by concentrated $H_2O_2$ (Bard et al. 1985; Cotton et al. 1999). However, only $S_2O_8^{2-}$ and $HSO_5^-$ were observed in aqueous solution and are known to form alkali metal salts whose crystal structures have been investigated. Although the apparent formal oxidation state of S in these forms would be +7 and +8, respectively, the more correct structurally constrained state is +6, as in sulfate, because in persulfates the S atom is directly coordinated by 4 oxygens, one of which forming a peroxide group with a 5th oxygen, like $[O_3S-O-O-SO_3]$ and $[O_3S-O-OH]$. The values of $G^0(H^0)$ and $S^0$ of $S_2O_8^{2-}$ cited in the Wagman et al. (1982) and Bard et al. (1985) books are consistent within 5 kJ/mol. They very likely stem from the same data source, but no original references could be traced to enable more critical selection. Shock et al. (1997) adopted the Wagman et al. (1982) values and reported a set of HKF coefficients for the persulfate ion. Williamson and Rimsdidt (1992) estimated an uncertainty of >30 kJ/mol for the $G^0$ value using inter-species correlations, which is of the same order of magnitude as for other oxysulfur species discussed above. Neutral $H_2S_2O_8^0$ and $H_2SO_5^0$ are very strong acids and highly unstable in aqueous solution and their dissociation constants are not known. However, one might find $G^0$, $H^0$ and $S^0$ values of $H_2S_2O_8^0$ in the common thermodynamic books cited above. Paradoxically, these values appear to be identical to those of the $S_2O_8^{2-}$ ion,
which is devoid of any physical sense. A similar meaningless $G^0$ value, identical to that of the $S_2O_6^{2-}$ ion, has been attributed in those books to its protonated counterpart, $H_2S_2O_6^0$. As a result, such data for the protonated species should be disregarded. The $S_2O_8^{2-}$ ion, being stable in aqueous solution, forms ion pairs with alkalis and other common metals (e.g., Cu, Co), with stability constants similar to those with sulfate; e.g., see Smith et al. (2004) for a compilation and Chlebek and Lister (1971) and Aruga (1981) for representative original data sources. These are currently limited to near-ambient temperature solutions and cannot be used in modeling of hydrothermal fluids. The $HSO_5^-$ ion is seemingly a weak acid and would dissociate to $SO_5^{2-}$ with a reaction constant of $logK \sim 9.3$ at 25 °C (i.e. at pH above 9) according to Goodman and Robson (1963), which seems to be the only accessible original source of data. However, the description of the measurement of this constant in their article is limited by only 5 (!) lines of text. This is a typical example of the low degree of reliability of thermodynamic data for S species that are unstable at ambient conditions. There is no way of any reliable extrapolation of such data to higher T-P, even though tentative theoretical HKF coefficients for $HSO_5^-$ as generated by Shock et al. (1988) can be found in some widely used databases (e.g., SUPCRTBL); they should be used with extreme caution.

## Radical sulfoxide species

Persulfate ions decompose to sulfate via radical mechanisms involving $SO_4^{2-}$ (Cotton et al. 1999). Other oxysulfur radical ions such as $SO_2^{·-}$, $SO_3^{·-}$ and $SO_5^{·-}$ have also been identified as intermediates in photolysis and oxidation reactions in sulfoxide aqueous systems, relevant to industrial applications and atmospheric chemistry on Earth Venus and Io (e.g., Hayon et al. 1972; Lodders et al. 2025; Zolotov 2025). While the kinetics in such systems at near-ambient conditions have been thoroughly studied, thermodynamic properties of such radicals are yet inexistent. The concentrations of such radicals are rather low; for example in aerosol water in air $SO_4^{2-}$ concentrations were estimated to be $10^{-14}$ to $10^{-12}$ M (Cope et al. 2022). However, it does not mean that a species unstable in ambient solution would definitely be unstable at elevated T-P. The example of the trisulfur radical ion (see below) proves the contrary. By analogy, if sulfoxide radicals turn out to be more stable at high T-P conditions, they might contribute to a release of highly oxidized fluids during serpentinite breakdown in subduction zones. For example, recent data based on analyses of the $As^V/As^{III}$ ratio in serpentinites from Himalayas (Pokrovski et al. 2022a), indicate $O_2$ fugacity values in the fluid close to QFM+10 while equilibrium thermodynamic modeling in the fluid-rock system involving classical S species (sulfate, sulfide and sulfite) yields $f_{O_2}$ of QFM±2. Radical oxysulfur species might, therefore, be potentially important carriers of oxidation potential during such processes whose study would require in situ spectroscopy approaches.
## Polysulfide species

This geochemically important group of sulfur species includes polysulfide dianions, S<sub>n</sub><sup>2-</sup>, and their less well-known protonated counterparts, HS<sub>n</sub><sup>-</sup> and H<sub>2</sub>S<sub>n</sub><sup>0</sup>. The structurally correct formula of polysulfides is chain-like [S-S<sub>n</sub>-S]<sup>2-</sup> in which the two terminal S atoms are charged and the total chain length n is up to 11 atoms in aqueous solution and to 12 atoms in the solid state (e.g., Liebing et al. 2019). Comprehensive reviews of the chemistry of polysulfides, their reactions, synthesis, and detailed structural characterization and spectroscopic features in the solid state and both non-aqueous and aqueous solvents have been published (Steudel 2003, 2020; Chivers and Elders 2013; Sheldrick 2013; Steudel and Chivers 2019, to name a few). These species are important actors in sulfur-based batteries, which explains the continued interest of the chemical and material sciences communities. Polysulfide dianions have often been analyzed in sedimentary environments (e.g., Jorgensen 2021; references therein), geothermal springs (e.g., Stefánsson 2017), and more rarely in fluid inclusions (Barré et al. 2017; Hurai et al. 2017). Both in laboratory and natural systems, polysulfides commonly form by oxidation of hydrogen sulfide or reactions of native sulfur with hydrogen sulfide at pH >6 such as:

$$ (n-1)/8 S_{8(s)} + HS^{-} = S_{n}^{2-} + H^{+} \quad (4.39) $$

The reaction kinetics in the S<sup>0</sup>-HS<sup>-</sup>-S<sub>n</sub><sup>2-</sup> system being fast at most laboratory conditions (see section 4.3.2), equilibria among the above mentioned species in aqueous solution are rapidly and reversibly attained. This property allowed acquisition of thermodynamic parameters of the polysulfide anions based on reaction (4.39) in numerous experimental studies that employed solubility, UV-Vis spectroscopy, potentiometry, conductivity or chromatography methods (e.g., Giggenbach 1972, 1974; Berndt et al. 1994; Licht et al. 1986; Licht and Davis 1997; Kamyshny et al. 2004, 2007; and references therein for earlier work). For more detailed account of the polysulfide speciation, data analysis and stability constants from <1990s works, the reader is referred to Hamilton (1991) whose theoretical analysis represents a remarkable work for an unpublished MSc thesis. Among earlier published work, Giggenbach studies using high-temperature (up to 260 °C) UV-Vis spectroscopy should particularly be acknowledged; they are also most relevant to hydrothermal fluids. These and other lower-temperature works have reported reaction (4.39) constant values to 80–150°C, which served as the basis for thermodynamic compilations in which Δ<sub>f</sub>G<sup>0</sup>, Δ<sub>f</sub>H<sup>0</sup> and S<sup>0</sup> values at 25 °C and 1 bar for polysulfide dianions with S chain length up to 5 atoms were derived (Wagman et al. 1968, 1982, subsequently cited by Bard et al. 1985). These data were used to estimate the HKF-model coefficients for these ions with n ≤ 5 (Shock and Helgeson 1988). They were included in major databases of aqueous species such as SUPCRT92 and SUPCRTBL, allowing estimations of
polysulfide ions abundances in S speciation calculations at hydrothermal conditions. After those works, the acquisition of thermodynamic properties of polysulfide ions has almost stagnated for 20 years, until the early 2000s when Kamyshny et al. (2004, 2007) made a significant advance in polysulfide equilibrium speciation using a new approach, so called fast chemical derivation in a single phase. Their method is based on fast reactions of polysulfide ions with methyl triflate in aqueous solution with formation of dimethylpolysulfides and their subsequent analysis by HPLC coupled with UV detection. This approach allowed the initial distribution of inorganic polysulfide ions to be determined in experimental solutions corresponding to reaction (4.39) from 25 to 80 °C:

$$S_n^{2-} + 2 \text{CF}_3\text{SO}_3\text{CH}_3 = (\text{CH}_3)_2S_n + 2\text{CF}_3\text{SO}_3\text{-} \quad (4.40)$$

Kamyshny seminal studies allowed a revision of thermodynamic properties of the existing polysulfides to n = 5, and generation of Δf/G°, Δf/H° and S° values for longer-chain ions, up to n = 8. To enable the use of the new data at elevated temperatures, Barré et al. (2017) combined Kamyshny et al.'s values with the Shock and Helgeson (1988) and Sverjensky et al. (1997) correlations to generate HKF-model coefficients for the polysulfide ions from S₂²⁻ to S₈²⁻, as well as to integrate these data in the form of log K versus T polynomial equations compatible with the LLNL database of the PHREEQC code to 300 °C (Parkhurst and Appelo 1999). These data are actually the most complete set of thermodynamic properties of polysulfide ions for hydrothermal fluids. However, a very recent update was the study of Avetisian and Kamyshny (2022) that followed up the improved protocols of Kamyshny's work by specifically focusing on the disulfide ion, S₂²⁻, and its protonation constants (see below) to be measured from 10 to 70 °C. The re-determined Δf/G° value of S₂²⁻ at 25 °C and 1 bar is by 7 kJ/mol more positive and the S° value 50 J/mol K more negative than those previously reported by Kamishny et al. (2004, 2007) themselves. The new values make S₂²⁻ even less stable (i.e. less abundant) in aqueous solution compared to the previous studies. However, whatever the exact values for the S₂²⁻ and higher-order polysulfide ions, the majority of earlier and more recent studies remarkably agree about the predominance of tetra-, penta- and hexasulfide ions, and they yield quite similar total polysulfide concentrations in solution (ΣSₙ²⁻), whatever the exact set of data is chosen. Therefore, considering the relatively small overall differences in thermodynamic properties of the polysulfide ions among the different studies and the general robustness of the HKF model extrapolations for ionic species, these data may be used with confidence for modeling polysulfide dianions concentrations in liquid-like hydrothermal fluids.

In contrast to the dianions, the data on their protonated counterparts in aqueous solution, HSₙ⁻ and H₂Sₙ°, described by the following reaction constants

$$S_n^{2-} + H^+ = HS_n^- \quad K_{0-1} \quad (4.41)$$
$$HS_n^- + H^+ = H_2S_n^0 \qquad K_{1-2} \qquad (4.42)$$

are far more scarce and discrepant. This is mostly because protonation of the dianions occurs at neutral and acidic pH where $S_8^{0}(s)$ solubility is much lower than in alkaline solutions in which $S_n^{2-}$ has commonly been measured. As a result, acidification causes precipitation of native sulfur inducing large and time-dependent losses of polysulfide concentration from solution making it hard to perform accurate measurements. The seminal experimental study of Schwarzenbach and Fischer (1960) employed potentiometric pH titrations in 0.1 M KCl solutions at 20 °C and derived values of reaction constants (4.41) and (4.42) for tetra- and pentasulfide ions and tentative values for disulfide and trisulfide. Remarkably, all theoretical estimations and extrapolations of protonation constants in the polysulfide series that followed are based on that seminal work. Using the Schwarzenbach and Fischer (1960) data, Schoonen and Barnes (1988) applied a simple concept of the linear dependence of association constants versus reciprocal of the chain length ($1/n$) to estimate the second dissociation constant of $H_2S$ (i.e. $n = 1$; see above for detailed assessment of the $H_2S$ dissociation). With increasing $n$, the charges of the two terminal S atoms become more and more localized and independent of one another, resulting in identical protonation constants for both atoms at $n$ tending to infinity (Fig. 4.15). This trend places an additional constraint on extrapolations to $n = 1$. Kamyshny et al. (2004) used the same concept and basis data for $S_4^{2-}$ and $S_5^{2-}$ to estimate the constants for higher ($n = 6$ to 8) and lower ($n = 2$ and 3) polysulfides. Chadwell et al. (1999), using electrochemical titrations, experimentally confirmed the $K_{0-1}$ value for pentasulfide within <0.3 log units of that of the Schwarzenbach and Fischer (1960) study. Finally, Liu et al. (2013a) used molecular dynamics approaches (FPMD) to provide a good independent confirmation of $K_{0-1}$ of sulfide and disulfide within <0.5 log unit, but the agreement with the experimental data and $1/n$ relationships for longer-chain polysulfides was less good, with differences of >1–2 log units (Fig. 4.15).

It follows from this overview that in the polysulfide group, disulfide and pentasulfide dianions and their protonated species are currently the best thermodynamically constrained ones. Their respective relative distributions as a function of pH at 25 °C is shown in Fig. 4.16e,f. There are currently no published thermodynamic coefficients for the protonated polysulfide species allowing extrapolations to elevated temperatures. Qualitatively, it would be expected that the pH range of their predominance domains shown in Fig. 4.16 would become larger with increasing $T$, compared to that of $S_n^{2-}$. The uncharged sulfanes $H_2S_n^0$, being volatile species similarly to $H_2S$ and $SO_2$, may further contribute to increasing partitioning of sulfur in the vapor phase during boiling and vapor-liquid immiscibility in hydrothermal systems.
## Molecular sulfur

Molecular sulfur, $S_n^0$, has an extremely rich solid-, liquid- and gas-phase chemistry thoroughly discussed in comprehensive reviews (e.g., Meyer 1976 and Steudel 2003, 2020 to name a few). It is known to form a large variety of allotropes composed of ring molecules, with $n$ from 6 to at least 20, isolated as individual compounds. In the molten state (typically >120 °C), it exists as polymeric chains or $S_n$ rings with $n$ typically from 8 to 12, as well as less abundant higher-$n$ cycles. In the gas phase, at least to 1000 °C, all $S_n$ molecules with $n$ ranging from 2 to 8 form in equilibrium (Rau et al. 1973). Remarkably, among all these S allotropes and molecules, cyclooctasulfur, $S_8(s)$, is the thermodynamically stable solid, and $S_8(l,g)$ is the most abundant molecule both in the molten and gas phase over a wide $T$ range, from ambient to at least 500 °C. Other shorter-chain sulfur allotropes, such as $S_2$, $S_3$ and $S_4$, are likely deposited from high-temperature magmatic plumes at the surface of Io, a famous Jupiter's satellite, yielding green and red colors, which rapidly transform to yellow $S_8$ (e.g., Davies and Vorburger 2023). Cyclooctasulfur and most other molecular sulfur allotropes exhibit an extremely large range of solubilities in different organic solvents, spanning from <0.1 wt% in ethanol, to 2–3 wt% in acetone and hexane, and to ~35 wt% S in carbon disulfide at ambient temperature for $\alpha$-$S_8(s)$ (see Meyer 1976 and Steudel 2020 for an extended list). Being hydrophobic, elemental sulfur easily forms colloids in water, so called sulfur or Weimarn sols, which are persistent for durations of days in aqueous solution at near ambient temperature, before being precipitated as $S_8(s)$ or further oxidized. This colloidal sulfur is rather common in sedimentary pore waters, geothermal springs and ocean where it forms by oxidation of $H_2S$, microbial activity or during fluid cooling (e.g., Jorgensen 2021). The solubility of $S_8(s)$ as the $S_8^0$ molecule in aqueous solution is very small at ambient conditions, ~$10^{-8}$ m of $S_8$, as measured in the few experimental studies (Boulëgue 1978; Kamyshny 2009; Wang and Tessier 2009). Like in most organic solvents, sulfur aqueous solubility increases strongly with temperature rise, but both speciation and thermodynamics of $S_n^0$ (aq) remain poorly constrained in hydrothermal fluids.

Only two experimental studies, to the best of our knowledge, have reported the stability of molecular sulfur in aqueous solution in equilibrium with solid or molten sulfur above 25 °C. Kamyshny (2009) conducted accurate measurements of $\alpha$-$S_8(s)$ solubility in water from 4 to 80 °C and interpreted the data in terms of equilibrium constant of the reaction:

$$ \alpha \cdot S_{8(s)} = S_{8(aq)}^0 \qquad (4.43) $$

Their data allowed derivation of $\Delta/G^0$, $\Delta/H^0$ and $S^0$ of $S_8^0$ (aq) at 25 °C, 1 bar by fitting the $K_{43}$ values to a van't Hoff equation. Dadze and Sorokin (1993) analyzed $S^0$ concentrations as hexane extracts, along with other sulfur species such as sulfate, sulfide and thiosulfate, in aqueous solutions in
equilibrium with molten sulfur from 200 to 440 °C and from $P_{sat}$ to 450 bar in a hydrothermal reactor. Their experiments produced 4 to 5 orders of magnitude higher $S^0$ concentrations than those of Kamyshny (2009), attesting increasing solubility of sulfur molecules whose exact stoichiometry could not be assessed from bulk solubility data. Pokrovski and Dubessy (2015) tentatively regressed the whole set of $S^0$ solubility data assuming reaction (4.43), and reported a set of HKF coefficients for a conventional $S_8^0(aq)$. However, applying the HKF model to Kamishny (2009) low-temperature data only results in much lower $S_8^0(aq)$ concentrations predicted above 250 °C (Fig. 4.17). This discrepancy points out to the predominance at elevated temperatures of molecular sulfur species other than $S_8^0(aq)$. Their evidence is indirectly supported by the temperature trend of Dadze and Sorokin (1993) data points, which is opposite to that of $S_8^0(aq)$. A more direct proof of such new $S_n^0$ species, with their maximum abundance around 300 °C, was provided by in situ Raman spectroscopic data in solutions in the presence of molten sulfur (Fig. 4.3), which requires, however, rather high dissolved sulfur concentrations (>0.1 wt% $S_{tot}$) atypical for natural fluids at such temperatures. With increasing temperature above 300 °C, these sulfur polymers break down to sulfur radical ions that are described as follows.

## Tri- and disulfur radical ions

Sulfur has a rich radical chemistry, with a variety of polysulfide, sulfoxide and organic thiol radicals as intermediates in different sulfur transformation reactions both in aqueous and non-aqueous solution and gas phase. Among them, the trisulfur radical ion, $S_3^{·-}$, is the most ubiquitous and well-studied radical species since the early 1970s. Compared to the trisulfide dianion ($S_3^{2-}$, and other polysulfide species discussed above), the trisulfur ion has an unpaired electron (that is why it is termed radical) and the resulting electrical charge of -1 (see Fig. 4.3 for its molecular structure). It should be noted that its formula is sometimes misunderstood or misspelled in modern literature including high-rating journals (e.g., Sverjensky 2025) by mistaking it for a triply charged $S^3-$ ion that does not exist in sulfur chemistry. The $S_3^{·-}$ ion is stable in a variety of non-aqueous materials such as S-bearing organic and inorganic solvents, alkali halide melts, borosilicate glasses, ultramarine pigments, and zeolite-type minerals whose blue color is due to this chromophore ion (see Chivers and Elder 2013; Steudel and Chivers 2019 for comprehensive reviews). It is also an important intermediate in organic synthesis reactions and alkali metal-sulfur battery cycles (Song et al. 2023). The $S_3^{·-}$ ion is easily detectable by a variety of spectroscopic techniques such as Raman, UV-Vis and EPR (see section 4.2), and has been the subject of theoretical molecular modeling studies (e.g., Tossell 2012; Blanchard et al. 2024). Although being much less stable at ambient conditions in water than in non-aqueous matrixes mentioned above, the blue chromophore $S_3^{·-}$ ion
has been known since long time to form on heating above 100 °C in aqueous solution of polysulfide ions and sulfur (e.g., Chivers 1974). However, its blue color and UV-Vis and Raman spectral patterns in aqueous solution in some earlier spectroscopic studies (Giggenbach 1971b; Uyama et al. 1985; Bondarenko and Gorbaty 1997) were erroneously attributed to S₂⁻ or other species (see Pokrovski and Dubessy 2015 for discussion). Note that hydrothermal batch-reactor experiments in S-rich high-T aqueous solutions (Dadze and Sorokin 1993; Kokh et al. 2020) could not directly detect S₃⁻, because of its rapid breakdown to sulfate, sulfide and molecular sulfur during fluid sampling or quench. The temperature-reversible formation of S₃⁻ in aqueous solutions of thiosulfate, sulfur and sulfate-sulfide (e.g., Fig. 4.3) was unambiguously demonstrated by recent in situ Raman spectroscopy work over a wide range of T-P (Pokrovski and Dubrovinsky 2011; Jacquemet et al. 2014; Pokrovski and Dubessy 2015). Using this method Pokrovski and Dubessy (2015) were able to quantify the equilibrium concentrations of S₃⁻ in sulfate-sulfide aqueous solutions to 500 °C and 2 kbar. Combined with previous spectroscopic work (Pokrovski and Dubrovinsky 2011; Giggenbach 1971b), they have generated the first set of equilibrium constants of the reaction of S₃⁻ formation from sulfide and sulfate (Fig. 4.18)

$$2 \mathrm{H}_2\mathrm{S} + \mathrm{SO}_4^{2-} + \mathrm{H}^+ = \mathrm{S}_3^{\cdot -} + 0.75 \mathrm{O}_{2(\mathrm{g})} + 2.5 \mathrm{H}_2\mathrm{O} \qquad \mathrm{K}_{44} \qquad (4.44)$$

These independent data that arise from Raman measurements that employed 3 different laser wavelengths (623, 514, and 457 cm⁻¹), two different types of optical cell (HDAC and silica capillaty), and covered 2 log units of S₃⁻ concentrations, along with a UV-Vis dataset (Giggenbach 1971b), are all in mutual agreement, confirming the K₄₄ value validity across the whole T-P range covered (200–500 °C, 15 bar–15 kbar), and with no evidence of potential artefacts related to Raman signal absorption in colored solutions as claimed in a later study (Schmidt and Seward 2017). Naturally, S₃⁻ thermodynamic stability obeys the same laws as for any other aqueous species. Note that the formal reaction (4.44) was exclusively intended to report equilibrium constants within a conventional thermodynamic framework to show the dependence on the major fluid thermochemical parameters (pH, fO₂ and sulfide and sulfate activity). In no way it reflects any chemical mechanism of S₃⁻ formation, as was misleadingly interpreted in some recent studies (e.g., Farsang and Zajacz 2025). The obtained thermodynamic data, both in the form of a van't Hoff-type equation and a set of HKF-model parameters compatible with the SUPCRT database, can be used for predicting S₃⁻ concentrations in different types of geological fluids across wide T and P ranges, up to 700 °C and 30 kbar. It may sound surprising, but S₃⁻ stands out now as one of the sulfur species that has been studied experimentally over the largest T-P range. Fig. 4.19 illustrates the distribution of S₃⁻, together with other major sulfur species discussed above, in representative hydrothermal fluids, at 450 °C as the middle temperature of the hydrothermal range, as a function
of fluid key parameters (P, S<sub>tot</sub> concentration, f<sub>O2</sub> and pH), but the pattern is similar in a wide T range, at least from 300 to 600 °C. Reaction (4.44) can be re-arranged in terms of the sulfide/sulfate ratio as

$$19 H_2S + 5 SO_4^{2-} + 2 H^+ = 8 S_3^{2-} + 20 H_2O \quad (4.45)$$

According to the stoichiometry of reaction (4.45), the maximum of S<sub>3</sub><sup>-</sup> concentration (or activity) at a given pH is reached at a sulfide/sulfate ratio of 19/5, i.e. at ~80% of the total sulfur as sulfide. Thus, the key parameter for S<sub>3</sub><sup>-</sup> is redox conditions of the sulfide-sulfate (±SO<sub>2</sub>) coexistence, which are within the range of f<sub>O2</sub> between the hematite-magnetite (HM, at T<500°C) and nickel-nickel oxide (NNO, T≥600°C) buffers. Additional factors favorable for S<sub>3</sub><sup>-</sup> are i) temperature above 250 °C in a wide P range of liquid-like fluids (ρ>0.4–0.5 g/cm<sup>3</sup>), ii) elevated S<sub>tot</sub> concentrations (> 0.5 wt%), iii) moderately acidic-to-neutral pH (4<pH<6) and, additionally iv) the excess of alkali or alkali-earth cations over chloride, offering a favorable electrical balance for the negatively charged trisulfur ion. Such conditions commonly occur in two crustal settings: magmatic-hydrothermal porphyry-epithermal Cu(-Au-Mo) systems associated with volcanic arcs, and metamorphic belts hosting orogenic Au deposits. The S<sub>3</sub><sup>-</sup> abundance in typical fluids from such settings is illustrated in Fig. 4.20.

The thermodynamic predictions of S<sub>3</sub><sup>-</sup> abundance in geological fluids at moderate temperatures (<350 °C) of Pokrovski and Dubessy (2015) are in very good agreement with independent studies that have identified S<sub>3</sub><sup>-</sup> in thermochemical sulfate reduction experiments in the presence of organic matter (Truche et al. 2014) as well as natural fluid inclusions (Barré et al. 2017). In constast, the Raman analyses of polysulfide solutions at high pressures (Schmidt and Seward 2017) overestimate, by at least a factor of 10, the S<sub>3</sub><sup>-</sup> concentrations predicted for their conditions using thermochemical data of Pokrovski and Dubessy (2015). Predictions above 500 °C, beyond the range so far covered by direct measurements strongly depend on the source of the thermodynamic properties of H<sub>2</sub>S<sub>(aq)</sub> (see above subsection “Sulfide species” and detailed discussions in Pokrovski and Dubessy 2015). Realistic overall uncertainties for S<sub>3</sub><sup>-</sup> concentration estimations in most types of liquid-like hydrothermal fluids at pressures below 5 kbar, using currently available thermodynamic properties, are about ±0.5 log unit below 450 °C, ±1 log units around 500 °C and approaching ±1.5 log units toward 700 °C. Note that this is a fairly good precision compared to the other intermediate valence sulfur species discussed above. However, uncertainties of thermodynamic model extrapolations for all sulfur species increase dramatically above 700 and 5 kbars, but were often disregarded in recent modeling work (e.g., Baudry and Sverjensky 2024). In particular, the use of data sources for the major sulfur species, hydrogen sulfide, sulfate and its ion pairs, other that those that were employed in derivation of the S<sub>3</sub><sup>-</sup> thermodynamic properties,
introduces a fundamental inconsistency that resulted in misleading interpretations of S₃⁻ stability in magmatic fluids at shallow-crust conditions (>800 °C, 2 kbar; Farsang and Zajacz 2025; Pokrovski 2025b).

The discovery of sulfur radical species in hydrothermal fluids invigorated the interest to S₃⁻ in geosciences, having been until recently limited mostly to the mineralogy of ultramarines, by opening the door to the domain of deep geological fluids and magma-fluid transition. The potentially important role of S₃⁻ (and S₂⁻) in sulfur transfers from silicate melts to aqueous fluids under subduction zone conditions has recently been demonstrated by direct in situ Raman spectroscopy measurements in a diamond-anvil cell in fluid-felsic melt systems at 700 °C and 3–15 kbar by Colin et al. (2020). In the aqueous fluid phase along with sulfate and sulfide both radicals were found to be abundant, in good agreement with the HKF model predictions of Pokrovski and Dubessy (2015), but in the coexisting hydrous silicate melt phase, they were close to detection limit. Remarkably, the radicals partition 10 to 1000 times more than sulfide and sulfate into the fluid phase relative to the melt (Fig. 4.21). The study has direct implications for sulfur degassing from magmas in subduction zone settings. This degassing would be enhanced, together with accompanying metals, in the redox window of the sulfate-sulfide transition where the radical ions have the highest abundance. Due to its capability to form very strong complexes with chalcophile metals such as Au, PGE and, potentially, Mo and Re, the S₃⁻ ion could play a key role in the formation of economic metal resources in subduction zone settings (see section 4.4 below).

In contrast to its S₃⁻ counterpart, the disulfur radical ion (S₂⁻, which is a diatomic ion with an unpaired electron and a –1 charge, Fig. 4.3 and 4.5) remains much poorly constrained. Similarly, it is misspelled in some publications, by mistaking it for the sulfide doubly-charged ion (S²⁻). Pokrovski and Dubessy (2015) tentatively estimated concentrations of S₂⁻ in their Raman spectroscopy experiments at 450 and 500 °C and 700–1400 bar (e.g., Fig. 4.3), assuming that Raman scattering cross-sections of S₂⁻ and S₃⁻ are equal at 457 nm excitation (Ledé et al. 2007). The data at 450 °C were used by Pokrovski et al. (2019) to estimate, by thermodynamic speciation calculations, the equilibrium constant of the reaction of S₂⁻ formation from sulfate and sulfide analogous to that of S₃⁻ (4.45):

$$13 \text{ H}_2\text{S} + 3 \text{ SO}_4^{2-} = 8 \text{ S}_2^{\cdot -} + 2 \text{ H}^+ + 12 \text{ H}_2\text{O} \qquad (4.46)$$

To complete this small dataset, we used the same speciation calculations for the few points reported by Pokrovski and Dubessy (2015) to estimate the reaction constant value at 500 °C. Thus, the currently available logK₄₆ values are –16.2±0.5 at 450 °C and 700 bar and –18.4±3.7 at 500 °C and 1000(±300) bar (±1 s.d.). These first values were used to estimate S₂⁻ concentrations in the