Artificially high stability of the sodium hydromolybdate ion pair in aqueous solution. Comment on Guan et al. (2023) “Different metal coordination in sub- and super-critical fluids: Do molybdenum (IV) chloride complexes contribute to mass transfer in magmatic systems?”, Geochim. Cosmochim. Acta 354, 240–251

Gleb S Pokrovski, Max Wilke, Maria A Kokh

► To cite this version:

Gleb S Pokrovski, Max Wilke, Maria A Kokh. Artificially high stability of the sodium hydromolybdate ion pair in aqueous solution. Comment on Guan et al. (2023) “Different metal coordination in sub- and super-critical fluids: Do molybdenum (IV) chloride complexes contribute to mass transfer in magmatic systems?”, Geochim. Cosmochim. Acta 354, 240–251. Geochimica et Cosmochimica Acta, In press, 10.1016/j.gca.2024.07.012. hal-04742326

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Geochimica et Cosmochimica Acta

Artificially high stability of the sodium hydromolybdate ion pair in aqueous solution.

Comment on Guan et al. (2023) “Different metal coordination in sub- and super-critical fluids: Do molybdenum(IV) chloride complexes contribute to mass transfer in magmatic systems?”, Geochim. Cosmochim. Acta 354, 240–251.

--Manuscript Draft--

<table><tr><td>Manuscript Number:</td><td>GCA-D-24-00210R2</td></tr><tr><td>Article Type:</td><td>Comment</td></tr><tr><td>Keywords:</td><td>hydrothermal fluid; molybdenum; HKF model; solubility; equilibrium constant; molybdate; ion pair; thermodynamic properties; hydration energy; uncertainty; Born coefficient</td></tr><tr><td>Corresponding Author:</td><td>Gleb S. Pokrovski, PhD, HDR<br>Centre National de la Recherche Scientifique (CNRS), Laboratoire Géosciences Environnement Toulouse<br>Toulouse, FRANCE</td></tr><tr><td>Corresponding Author's Institution:</td><td>Centre National de la Recherche Scientifique (CNRS), Laboratoire Géosciences Environnement Toulouse</td></tr><tr><td>First Author:</td><td>Gleb S. Pokrovski, PhD, HDR</td></tr><tr><td rowspan="3">Order of Authors:</td><td>Gleb S. Pokrovski, PhD, HDR</td></tr><tr><td>Max Wilke, PhD, HDR</td></tr><tr><td>Maria A. Kokh, PhD</td></tr><tr><td>Abstract:</td><td>Guan et al. (2023) reported a new set of thermodynamic parameters for the HKF equation of state for several aqueous species of hexavalent Mo(VI) (misprinted as Mo(IV) in the original title), based on selected published experimental studies or molecular dynamics simulations from their own work. The thermodynamic coefficients reported in Table 5 of Guan et al. (2023) imply that the NaHMoO40(aq) ion pair would be by far the dominant molybdate species in aqueous solution at ambient conditions, in a wide range of pH values (3–8) and salinities of >0.01 wt% NaCl. With increasing temperature to 300 °C in saline hydrothermal fluids, this ion pair becomes less stable, with its relative abundance decreasing in favor of its dissociated ion counterparts, HMoO4– and MoO42–. Those ions are, in turn, superseded by an oxychloride species, MoO2(OH)ClO(aq), with further temperature increase at acidic pH. Here we show that such speciation behavior with temperature for molybdate species is unphysical, being very likely an artifact of the thermodynamic analysis, and we caution the use of HKF model parameter sets derived from a few selected experimental data points within a narrow T-range.</td></tr></table>
Toulouse, June 30, 2024

Jeff Catalano, Executive Editor
Geochimica et Cosmochimica Acta

Dear Jeff,

Enclosed please find an electronic version of revised manuscript GCA-D-24-00210
"Artificially high stability of the sodium hydromolybdate ion pair in aqueous solution. Comment
on Guan et al. (2023) "Different metal coordination in sub- and super-critical fluids: Do
molybdenum(IV) chloride complexes contribute to mass transfer in magmatic systems?",
Geochim. Cosmochim. Acta 354, 240-251.", prepared by Maria Kokh, Max Wilke, and myself.

We have revised this submission as per your comment about rewording the use of some
citations. In a separate file entitled "Response to comments" you will find our more detailed
response. Thank you for your continued assistance and support.

We look forward to publication of our comment in GCA.

With best regards,

Gleb Pokrovski, correspondence author
**Response to Comments**

Our response is highlighted in blue. The original comment is copied in gray color, for
your convenience.

Executive Editor, Jeff Catalano:

Please revise the text to remove all uses of citations like "Guan et al.'s (2023)" and "Shang et al.'s (2020)".
Reword the sentences where such citation occur to remove the need to insert an " 's " in the middle of a citation.

In Revision 2, we have carefully reworded the use of citations to avoid the need to insert an
''s' in the middle of a citation. These corrections are highlighted in yellow color in the
manuscript version with tracked changes.

End of response.

Gleb Pokrovski
ms GCA-D-24-00210, Comment on Guan et al. (2023) to GCA, Revision_2, 30/06/2024_clean

1

2 Artificially high stability of the sodium hydromolybdate ion pair in aqueous

3 **solution**

4 Comment on Guan et al. (2023) “Different metal coordination in sub- and super-
5 critical fluids: Do molybdenum(IV) chloride complexes contribute to mass transfer in
6 magmatic systems?”, Geochim. Cosmochim. Acta 354, 240–251.

7

8 Gleb S. Pokrovski¹*, Max Wilke², Maria A. Kokh²

9

10 ¹ Experimental Geosciences Team (GeoExp), Géosciences Environnement Toulouse (GET), UMR
11 5563 of the Centre National de la Recherche Scientifique (CNRS), Université Paul Sabatier Toulouse
12 III, Institut de Recherche pour le Développement (IRD), Centre National d'Etudes Spatiales (CNES),
13 Observatoire Midi-Pyrénées, 14 av. Edouard Belin, 31400 Toulouse, France

14

15 ² Universität Potsdam, Institut für Geowissenschaften, 14476 Potsdam, Germany
16

17 * Corresponding author, e-mail: gleb.pokrovski@get.omp.eu; glebounet@gmail.com

18

19 **Keywords**: hydrothermal fluid; molybdenum; HKF model; solubility; equilibrium constant

20

21 Geochimica and Cosmochimica Acta

22 ms # GCA-D-24-00210

23 Revision 2

24 30 June 2024

25
Guan et al. (2023) reported a new set of thermodynamic parameters for the HKF equation of state for several aqueous species of hexavalent Mo(VI) (apparently misprinted as Mo(IV) in the original title), based on few selected published experimental studies or molecular dynamics simulations from their own work. The thermodynamic coefficients reported in Table 5 of Guan et al. (2023) imply that the NaHMoO₄⁰(aq) ion pair would be by far the dominant molybdate species in aqueous solution at ambient conditions, in a wide range of pH values (3–8) and salinities of >0.01 wt% NaCl. With increasing temperature to 300 °C in saline hydrothermal fluids, this ion pair becomes less stable, with its relative abundance decreasing in favor of its dissociated ion counterparts, HMoO₄⁻ and MoO₄²⁻. Those ions are, in turn, superseded by an oxychloride species, MoO₂(OH)Cl⁰(aq), with further temperature increase at acidic pH. Here we show that such speciation behavior with temperature for molybdate species is unphysical, being very likely an artifact of the thermodynamic analysis. We caution the use of HKF model parameter sets derived from a few selected experimental data points within a narrow T-range. Figure 1 summarizes available experimental values of the thermodynamic constant for reaction:

$$Na⁺ + HMoO₄⁻ = NaHMoO₄⁰(aq) \qquad K₁ \qquad (1)$$

We note the following flaws in the selection and analysis of some of those data by Guan et al. (2023).

(1) Guan et al. (2023) exclusively based the derivation of their HKF parameters on the $K₁$ values from measurements of MoO₃(s) solubility as a function of NaCl and NaCF₃SO₃ molality (to ~3 mol Na per kg H₂O, hereafter m) at 250, 300 and 350 °C, at saturated vapor pressure of the system ($P_{sat}$), reported by Shang et al. (2020). The use of these data was, however, cautioned by Plyasunov (2020) who showed that Mo concentrations in pure water measured by Shang et al. (2020) were 10 to 100 times lower than most available MoO₃(s) solubility data, thereby implying a potential lack of solid-solution equilibrium. Kudrin (1989) reported $K₁$ values from MoO₂(s) solubility measurements in NaCl-KCl-HCl solutions (to 1 m salt) from 300 to 450 °C at 500 bar. His values scatter by 2 log₁₀ units of the HKF predictions of Guan et al. (2023) (Fig. 1). We note that it is impossible to
unambiguously distinguish the effect of Na versus Cl in Mo complexing from solubility data obtained
in NaCl-HCl solutions, without constraints on the fluid pH. We also note that a formation constant
value for analogous KHMoO₄⁰(aq) reported by Kokh et al. (2017) from measurements of MoS₂(s)
solubility in pH-fO₂-fS₂ buffered H₂O-CO₂ fluids at 450 °C and 680 bar is, respectively, about 2 and 4
log₁₀ units lower than the values of Kudrin (1989) and the extrapolations of Guan et al. (2023) (Fig.
1). Recently, Dadze et al. (2017) and Dadze et al. (2018), not considered in Guan et al. (2023),
reported results of systematic solubility measurements of MoO₃(s) and CaMoO₄(s) at 300 °C and Pₐₙ
over a wide range of NaCl and NaClO₄ concentrations (to 3.8 m Na). They demonstrated that there
was no need to invoke the NaHMoO₄⁰(aq) or NaMoO₄⁻ ion pairs to describe the measured solubility,
which is largely dominated by the HMoO₄⁻ and MoO₄²⁻ anions and may simply be interpreted by
variations of their activity coefficients. Dadze et al. (2017, 2018) estimated a maximum value of
log₁₀K₁ < 1 that may still be possible within the solubility data points' scatter. Their K₁ value is thus
at least one log₁₀ unit lower than those of Shang et al. (2020) and Kudrin (1989) (Fig. 1). Under this
light, we must question the use of the selected 3 temperature data points to extract 9 HKF parameters
(G, S, c₁,₂, a₁,₂,₃,₄, and ω, even though some of them might roughly be constrained through existing
model correlations).

(2) In fact, the HKF parameters proposed by Guan et al. (2023) yield anomalously high values
of K₁ at 25 °C and 1 bar – the reference conditions at which data are primarily required for an accurate
HKF model parameterization. Their K₁ values are at least 7 to 8 orders of magnitude higher than all
other known so far constants for ion pairs of Na⁺ or K⁺ with –1 and –2 charged anions (e.g., Cl⁻,
HSO₄⁻, SO₄²⁻, B(OH)₄⁻, or HCO₃⁻ to name a few, as exemplified in Fig. 1). Such artificially high
reaction (1) constants are equally at odds with those known for the molybdate ion pairs, KMoO₄⁻ and
NaMoO₄⁻ (Essington, 1990; Fig. 1). Note that metal complexes with double-charged anions
(molybdate) should no doubt be stronger than with their single-charged counterparts
(hydromolybdate), as per basic rules of ion pair formation driven by electrostatic forces. The K₁ values
reported by Guan et al. (2023) would yield to an overestimation of the solubility of Mo-bearing solid phases by a factor of a million at ambient conditions in surficial moderately saline (<few wt% NaCl) waters. The proposed predominance of NaHMoO₄⁰(aq) over molybdic acid and molybdate ions in near-ambient solutions containing >1 millimol Na is in contradiction with the well-established mutually consistent dissociation constants of molybdic acid from the existing experimental studies that used solutions of variable salinity (e.g., see Dadze et al., 2018 for a recent compilation and references therein).

(3) In contrast to all known ion pairs, the log₁₀K₁ values of Guan et al. (2023) drop by 6–7 log₁₀ units with increasing temperature from 25 °C to 300 °C (Fig. 1). Such a trend contradicts to fundamentals of thermodynamics of ion association in aqueous solution driven by the well-known thermo-physical properties of water as a solvent. Ion pairing, largely driven by electrostatic interactions, always becomes stronger due to the decrease of the water dielectric constant (i.e., increase in temperature), as demonstrated by data for all ion pairs in Fig. 1. Furthermore, below 150 °C, the K₁ values of Guan et al. (2023) show an increase with increasing pressure (i.e., increase of the dielectric constant). A positive pressure effect on the stability of an ion pair is equally at odds with the trends for all other ion pairs for which there exist more robust data (Fig. 1). While the pressure dependency of the high-temperature (>300 °C) K₁ values of Guan et al. (2023) does roughly align with those of other species (Fig. 1), we are concerned that this prediction was generated from unreliable thermodynamic values (Gibbs energy, entropy, and heat capacity) that hold for ambient conditions.

(4) In the last 30 years, the HKF model has provided an unprecedented advance in our capability of predicting stabilities of aqueous complexes and solubilities of minerals at high temperatures and pressures. With modern software integrating HKF inter-parameter correlations, the ease of producing a new set of HKF parameters is tempting. However, care should be taken to respect the HKF model formalism and basics in such exercises with a restricted set of available Gibbs energy
data points. We note that some of HKF parameters reported in Table 5 of Guan et al. (2023) are at odds with HKF model fundamentals. Unfortunately, Guan et al. (2023) do not provide any detail about their derivation procedure nor error assessment for the new parameters. For example, the values of the electrostatic Born parameter (ω), which is responsible for the solvation contribution of the Gibbs energy of a dissolved species, for both NaHMoO₄⁰(aq) and MoO₂(OH)Cl⁰(aq) are highly negative (ω ≈ -1.7×10⁵ and -0.9×10⁵ cal/mol, respectively). In the HKF model, negative ω values are intrinsic to dissolved gases and volatile inorganic and organic compounds (e.g., CO₂, H₂, H₂S, CH₄, C₆H₁₄, and noble gases) that repulse water molecules, implying positive hydration energy in solution. In contrast, for all ionic species, both cationic and anionic, ω values are positive, reflecting species' strong solvation by water molecules (i.e. negative hydration energy). These different signs for ω are fundamentally expressed in the characteristic, either positive or negative, divergence of the solute partial Cₚ and V functions close to the water critical point, depending, respectively, on species gas-like (i.e. volatile, poorly soluble) or water-like (i.e. nonvolatile, soluble) nature (e.g., Plyasunov and Shock, 2001; references therein). Even though for uncharged nonvolatile metal complexes and ion pairs ω values are less constrained, they are systematically positive for species for which there are direct Cₚ and V measurements (e.g., boric, arsenic and many organic acids) or very detailed and extended solubility data sets (e.g., such as for silica and germanic acid) in the proximity of the water critical point (e.g., Perfetti et al., 2008). In the absence of such, a value of zero is commonly adopted as a first approximation for a neutral nongaseous species (e.g., Kokh et al., 2017). The ω values of Guan et al. (2023) are several times more negative than those for any gas-type species (ω ≈ -0.5 to -0.1 × 10⁵ cal/mol; e.g., Plyasunov and Shock, 2001; Schulte et al., 2001), including noble gases which are notoriously most volatile and most poorly soluble in water. Such highly negative ω values are definitely incompatible with the nonvolatile properties of molybdate species, clearly attested by direct measurements of Mo vapor-liquid partition coefficients that are largely in favor of the aqueous saline liquid (e.g., Kokh et al., 2016 and references therein). Thus, the unphysically negative ω values
reported by Guan et al. (2023) are simple artefacts of the HKF parameter derivation procedure
adopted in their work. They lead to fundamentally incorrect description of the two key derivatives of
the Gibbs energy function of the species (C<sub>p</sub> and V). While the error induced by an incorrect ω value
on the resulting Gibbs energy is small below 300 °C, it becomes impressively large at higher
temperatures (>350 °C). For example, setting the ω parameter of NaHMoO<sub>4</sub><sup>0</sup>(aq) to +0.1×10<sup>5</sup> cal/mol,
which is a typical value for nonvolatile uncharged species for which there are robust data (e.g., Perfetti
et al., 2008), would result in a Gibbs energy at 450 °C and 500 bar of >80 kJ/mol more positive than
the corresponding value of Guan et al. (2023). This difference translates to a K<sub>1</sub> value of almost 6
orders of magnitude lower than the predictions by Guan et al. (2023). Note that such a lower value
would be in far better agreement with the experimental data as well as the established ion association
trends shown in Fig. 1. Thus, a careful assessment of uncertainties is necessary to support any report
of a new set of thermodynamic parameters intended for geochemical modeling of natural fluids at
high temperatures and pressures.

As a conclusion, the work of Guan et al. (2023) demonstrated that much is yet to be done to
quantitatively comprehend molybdenum speciation in geological fluids. We emphasize that filling
this knowledge gap could only be achieved by a combination of critically evaluated and inter-
compared independent experimental measurements with robust thermodynamic models accompanied
by an explicit error assessment.

## Acknowledgments

We thank Editors J. Catalano and J.-F. Boily for handling this submission and two anonymous
reviewers for supportive comments. G.S.P. acknowledges support from the CNRS through the MITI
interdisciplinary program "Conditions Extremes 2024" (grant ExtremeS). M.A.K and M.W. thank
the Deutsche Forschungsgemeinschaft for funding (grants WI 2000/25-1 and WI 2000/30-1).

150
151 **Research Data**

152 Research Data associated with this article can be accessed at https://doi.org/10.17632/jk3rwxv3hh.2

153

154

References

155 Dadze, T.P., Kashirtseva, G.A., Novikov, M.P., Plyasunov, A.V., 2017. Solubility of MoO₃ in NaClO₄ solutions at 573 K. J. Chem. Eng. Data. 62, 3848–3853.

157 Dadze, T.P., Kashirtseva, G.A., Novikov, M.P., Plyasunov, A.V., 2018. Solubility of calcium molybdate in aqueous solutions at 573 K and thermodynamics of monomer hydrolysis of Mo(VI) at elevated temperatures. Monatsh. Chem. 149, 261–282.

160 Essington, M.E, 1990. Calcium molybdate solubility in spent oil shale and a preliminary evaluation of the association constants for the formation of CaMoO₄⁰ (aq), KMoO₄⁻ (aq), and NaMoO₄⁻ (aq). Environ. Sci. Technol. 24, 214–220.

163 Guan, Q., Mei, Y., Liu, W., Brugger, J., 2023. Different metal coordination in sub- and super-critical fluids: Do molybdenum(IV) chloride complexes contribute to mass transfer in magmatic systems? Geochim. Cosmochim. Acta 354, 240–251.

166 Kokh, M.A., Lopez, M., Gisquet, P., Lanzanova, A., Candaudap, F., Besson, P., Pokrovski, G.S., 2016. Combined effect of carbon dioxide and sulfur on vapor-liquid partitioning of metals in hydrothermal systems. Geochim. Cosmochim. Acta 187, 311–333.

169 Kokh, M.A., Akinfiev, N.N., Pokrovski, G.S., Salvi, S., Guillaume, D., 2017. The role of carbon dioxide in the transport and fractionation of metals by geological fluids. Geochim. Cosmochim. Acta 197, 433–466.

172 Kudrin, A.V., 1989, Behavior of Mo in aqueous NaCl and KCl solutions at 300–450 °C. Geochem. Intl. 26(8), 87–99.

174 Perfetti, E., Pokrovski, G.S., Ballerat-Busserolles, K., Majer, V., Gibert, F., 2008. Densities and heat capacities of aqueous arsenious and arsenic acid solutions to 350°C and 300 bar, and revised thermodynamic properties of As(OH)₃⁰(aq), AsO(OH)₃⁰(aq) and iron sulfarsenide minerals. Geochim. Cosmochim. Acta 72, 713–731.

178 Plyasunov, A.V., Shock, E.L., 2001. Correlation strategy for determining the parameters of the revised Helgeson-Kirkham-Flowers model for aqueous nonelectrolytes. Geochim. Cosmochim. Acta 65, 3879–3900.

181 Plyasunov, A.V., 2020. An experimental study of the solubility and speciation of MoO₃(s) in hydrothermal fluids at temperatures up to 350°C – a discussion. Econ. Geol. 115, 1871–1873.

183 Schulte, M.D., Shock, E.L., Wood, R.H., 2001. The temperature dependence of the standard-state thermodynamic properties of aqueous nonelectrolytes. Geochim. Cosmochim. Acta 65, 3919–3930.

185 Shang, L., Williams-Jones, A.E., Wang, X., Timofeev, A., Hu, R., Bi, X., 2020. An experimental study of the solubility and speciation of MoO₃(s) in hydrothermal fluids at temperatures up to 350°C. Econ. Geol. 115, 661–669.

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189

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Values of the formation constant of NaHMoO₄⁰⁻(aq) (reaction 1), and of analogous KHMoO₄⁰⁻(aq), NaMoO₄⁻ and KMoO₄⁻, as a function of temperature at pressures of *P*<sub>sat</sub> and 500 bar, from the indicated references (for experimental datapoints, errors bars as reported in original works, are smaller than symbol size). There data are compared with the known constants for the indicated Na⁺ and K⁺ ion pairs with common anions L⁻ (and L²⁻ for SO₄²⁻) shown by black curves as calculated using published HKF model parameters. Literature sources and all plotted data in digital format may be found at https://doi.org/10.17632/jk3rwxv3hh.2</figcaption></figure>

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195

196

197

198
1

2 Artificially high stability of the sodium hydromolybdate ion pair in aqueous
3 solution

4 Comment on Guan et al. (2023) “Different metal coordination in sub- and super-
5 critical fluids: Do molybdenum(IV) chloride complexes contribute to mass transfer in
6 magmatic systems?”, Geochim. Cosmochim. Acta 354, 240–251.

7

8

Gleb S. Pokrovski¹*, Max Wilke², Maria A. Kokh²

9

10 ¹ Experimental Geosciences Team (GeoExp), Géosciences Environnement Toulouse (GET), UMR
11 5563 of the Centre National de la Recherche Scientifique (CNRS), Université Paul Sabatier Toulouse
12 III, Institut de Recherche pour le Développement (IRD), Centre National d'Etudes Spatiales (CNES),
13 Observatoire Midi-Pyrénées, 14 av. Edouard Belin, 31400 Toulouse, France

14

15 ² Universität Potsdam, Institut für Geowissenschaften, 14476 Potsdam, Germany
16

17 * Corresponding author, e-mail: gleb.pokrovski@get.omp.eu; glebounet@gmail.com

18

19 **Keywords:** hydrothermal fluid; molybdenum; HKF model; solubility; equilibrium constant

20

21

Geochimica and Cosmochimica Acta

22

ms # GCA-D-24-00210

23

Revision 2

24

30 June 2024

25
Guan et al. (2023) reported a new set of thermodynamic parameters for the HKF equation of state for several aqueous species of hexavalent Mo(VI) (apparently misprinted as Mo(IV) in the original title), based on few selected published experimental studies or molecular dynamics simulations from their own work. The thermodynamic coefficients reported in Table 5 of Guan et al. (2023) imply that the NaHMoO₄⁰(aq) ion pair would be by far the dominant molybdate species in aqueous solution at ambient conditions, in a wide range of pH values (3–8) and salinities of >0.01 wt% NaCl. With increasing temperature to 300 °C in saline hydrothermal fluids, this ion pair becomes less stable, with its relative abundance decreasing in favor of its dissociated ion counterparts, HMoO₄⁻ and MoO₄²⁻. Those ions are, in turn, superseded by an oxychloride species, MoO₂(OH)Cl⁰(aq), with further temperature increase at acidic pH. Here we show that such speciation behavior with temperature for molybdate species is unphysical, being very likely an artifact of the thermodynamic analysis. We caution the use of HKF model parameter sets derived from a few selected experimental data points within a narrow T-range. Figure 1 summarizes available experimental values of the thermodynamic constant for reaction:

$$Na⁺ + HMoO₄⁻ = NaHMoO₄⁰(aq) \qquad K₁ \qquad (1)$$

We note the following flaws in the selection and analysis of some of those data by Guan et al. (2023).

(1) Guan et al. (2023) exclusively based the derivation of their HKF parameters on the $K₁$ values from measurements of MoO₃(s) solubility as a function of NaCl and NaCF₃SO₃ molality (to ~3 mol Na per kg H₂O, hereafter m) at 250, 300 and 350 °C, at saturated vapor pressure of the system ($P_{sat}$), reported by Shang et al. (2020). The use of these data was, however, cautioned by Plyasunov (2020) who showed that Mo concentrations in pure water measured by Shang et al. (2020) were 10 to 100 times lower than most available MoO₃(s) solubility data, thereby implying a potential lack of solid-solution equilibrium. Kudrin (1989) reported $K₁$ values from MoO₂(s) solubility measurements in NaCl-KCl-HCl solutions (to 1 m salt) from 300 to 450 °C at 500 bar. His values scatter by 2 log₁₀ units of the HKF predictions of Guan et al. (2023) (Fig. 1). We note that it is impossible to
unambiguously distinguish the effect of Na versus Cl in Mo complexing from solubility data obtained
in NaCl-HCl solutions, without constraints on the fluid pH. We also note that a formation constant
value for analogous KHMoO₄⁰(aq) reported by Kokh et al. (2017) from measurements of MoS₂(s)
solubility in pH-fO₂-fS₂ buffered H₂O-CO₂ fluids at 450 °C and 680 bar is, respectively, about 2 and 4
log₁₀ units lower than the values of Kudrin (1989) and the extrapolations of Guan et al. (2023) (Fig.
1). Recently, Dadze et al. (2017) and Dadze et al. (2018), not considered in Guan et al. (2023),
reported results of systematic solubility measurements of MoO₃(s) and CaMoO₄(s) at 300 °C and Pₐₙ
over a wide range of NaCl and NaClO₄ concentrations (to 3.8 m Na). They demonstrated that there
was no need to invoke the NaHMoO₄⁰(aq) or NaMoO₄⁻ ion pairs to describe the measured solubility,
which is largely dominated by the HMoO₄⁻ and MoO₄²⁻ anions and may simply be interpreted by
variations of their activity coefficients. Dadze et al. (2017, 2018) estimated a maximum value of
log₁₀K₁ < 1 that may still be possible within the solubility data points' scatter. Their K₁ value is thus
at least one log₁₀ unit lower than those of Shang et al. (2020) and Kudrin (1989) (Fig. 1). Under this
light, we must question the use of the selected 3 temperature data points to extract 9 HKF parameters
(G, S, c₁,₂, a₁,₂,₃,₄, and ω, even though some of them might roughly be constrained through existing
model correlations).

(2) In fact, the HKF parameters proposed by Guan et al. (2023) yield anomalously high values
of K₁ at 25 °C and 1 bar – the reference conditions at which data are primarily required for an accurate
HKF model parameterization. Their K₁ values are at least 7 to 8 orders of magnitude higher than all
other known so far constants for ion pairs of Na⁺ or K⁺ with –1 and –2 charged anions (e.g., Cl⁻,
HSO₄⁻, SO₄²⁻, B(OH)₄⁻, or HCO₃⁻ to name a few, as exemplified in Fig. 1). Such artificially high
reaction (1) constants are equally at odds with those known for the molybdate ion pairs, KMoO₄⁻ and
NaMoO₄⁻ (Essington, 1990; Fig. 1). Note that metal complexes with double-charged anions
(molybdate) should no doubt be stronger than with their single-charged counterparts
(hydromolybdate), as per basic rules of ion pair formation driven by electrostatic forces. The K₁ values
reported by Guan et al. (2023) would yield to an overestimation of the solubility of Mo-bearing solid phases by a factor of a million at ambient conditions in surficial moderately saline (<few wt% NaCl) waters. The proposed predominance of NaHMoO₄⁰(aq) over molybdic acid and molybdate ions in near-ambient solutions containing >1 millimol Na is in contradiction with the well-established mutually consistent dissociation constants of molybdic acid from the existing experimental studies that used solutions of variable salinity (e.g., see Dadze et al., 2018 for a recent compilation and references therein).

(3) In contrast to all known ion pairs, the log₁₀K₁ values of Guan et al. (2023) drop by 6–7 log₁₀ units with increasing temperature from 25 °C to 300 °C (Fig. 1). Such a trend contradicts to fundamentals of thermodynamics of ion association in aqueous solution driven by the well-known thermo-physical properties of water as a solvent. Ion pairing, largely driven by electrostatic interactions, always becomes stronger due to the decrease of the water dielectric constant (i.e., increase in temperature), as demonstrated by data for all ion pairs in Fig. 1. Furthermore, below 150 °C, the K₁ values of Guan et al. (2023) show an increase with increasing pressure (i.e., increase of the dielectric constant). A positive pressure effect on the stability of an ion pair is equally at odds with the trends for all other ion pairs for which there exist more robust data (Fig. 1). While the pressure dependency of the high-temperature (>300 °C) K₁ values of Guan et al. (2023) does roughly align with those of other species (Fig. 1), we are concerned that this prediction was generated from unreliable thermodynamic values (Gibbs energy, entropy, and heat capacity) that hold for ambient conditions.

(4) In the last 30 years, the HKF model has provided an unprecedented advance in our capability of predicting stabilities of aqueous complexes and solubilities of minerals at high temperatures and pressures. With modern software integrating HKF inter-parameter correlations, the ease of producing a new set of HKF parameters is tempting. However, care should be taken to respect the HKF model formalism and basics in such exercises with a restricted set of available Gibbs energy
data points. We note that some of HKF parameters reported in Table 5 of Guan et al. (2023) are at odds with HKF model fundamentals. Unfortunately, Guan et al. (2023) do not provide any detail about their derivation procedure nor error assessment for the new parameters. For example, the values of the electrostatic Born parameter (ω), which is responsible for the solvation contribution of the Gibbs energy of a dissolved species, for both NaHMoO₄⁰(aq) and MoO₂(OH)Cl⁰(ah) are highly negative (ω ≈ -1.7×10⁵ and -0.9×10⁵ cal/mol, respectively). In the HKF model, negative ω values are intrinsic to dissolved gases and volatile inorganic and organic compounds (e.g., CO₂, H₂, H₂S, CH₄, C₆H₁₄, and noble gases) that repulse water molecules, implying positive hydration energy in solution. In contrast, for all ionic species, both cationic and anionic, ω values are positive, reflecting species' strong solvation by water molecules (i.e. negative hydration energy). These different signs for ω are fundamentally expressed in the characteristic, either positive or negative, divergence of the solute partial Cₚ and V functions close to the water critical point, depending, respectively, on species gas-like (i.e. volatile, poorly soluble) or water-like (i.e. nonvolatile, soluble) nature (e.g., Plyasunov and Shock, 2001; references therein). Even though for uncharged nonvolatile metal complexes and ion pairs ω values are less constrained, they are systematically positive for species for which there are direct Cₚ and V measurements (e.g., boric, arsenic and many organic acids) or very detailed and extended solubility data sets (e.g., such as for silica and germanic acid) in the proximity of the water critical point (e.g., Perfetti et al., 2008). In the absence of such, a value of zero is commonly adopted as a first approximation for a neutral nongaseous species (e.g., Kokh et al., 2017). The ω values of Guan et al. (2023) are several times more negative than those for any gas-type species (ω ≈ -0.5 to -0.1 × 10⁵ cal/mol; e.g., Plyasunov and Shock, 2001; Schulte et al., 2001), including noble gases which are notoriously most volatile and most poorly soluble in water. Such highly negative ω values are definitely incompatible with the nonvolatile properties of molybdate species, clearly attested by direct measurements of Mo vapor-liquid partition coefficients that are largely in favor of the aqueous saline liquid (e.g., Kokh et al., 2016 and references therein). Thus, the unphysically negative ω values
reported by Guan et al. (2023) are simple artefacts of the HKF parameter derivation procedure adopted in their work. They lead to fundamentally incorrect description of the two key derivatives of the Gibbs energy function of the species ($C_p$ and $V$). While the error induced by an incorrect $\omega$ value on the resulting Gibbs energy is small below 300 °C, it becomes impressively large at higher temperatures (>350 °C). For example, setting the $\omega$ parameter of $NaHMoO_4^{0}(aq)$ to +0.1×10⁵ cal/mol, which is a typical value for nonvolatile uncharged species for which there are robust data (e.g., Perfetti et al., 2008), would result in a Gibbs energy at 450 °C and 500 bar of >80 kJ/mol more positive than the corresponding value of Guan et al. (2023). This difference translates to a $K_1$ value of almost 6 orders of magnitude lower than the predictions by Guan et al. (2023). Note that such a lower value would be in far better agreement with the experimental data as well as the established ion association trends shown in Fig. 1. Thus, a careful assessment of uncertainties is necessary to support any report of a new set of thermodynamic parameters intended for geochemical modeling of natural fluids at high temperatures and pressures.

As a conclusion, the work of Guan et al. (2023) demonstrated that much is yet to be done to quantitatively comprehend molybdenum speciation in geological fluids. We emphasize that filling this knowledge gap could only be achieved by a combination of critically evaluated and inter-compared independent experimental measurements with robust thermodynamic models accompanied by an explicit error assessment.

144

## Acknowledgments

We thank Editor J.-F. Boily for handling this submission and two anonymous reviewers for supportive comments. G.S.P. acknowledges support from the CNRS through the MITI interdisciplinary program "Conditions Extremes 2024" (grant ExtremeS). M.A.K and M.W. thank the Deutsche Forschungsgemeinschaft for funding (grants WI 2000/25-1 and WI 2000/30-1).

150
151 **Research Data**

152 Research Data associated with this article can be accessed at https://doi.org/10.17632/jk3rwxv3hh.2

153

154

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157 Dadze, T.P., Kashirtseva, G.A., Novikov, M.P., Plyasunov, A.V., 2018. Solubility of calcium molybdate in aqueous solutions at 573 K and thermodynamics of monomer hydrolysis of Mo(VI) at elevated temperatures. Monatsh. Chem. 149, 261–282.

160 Essington, M.E, 1990. Calcium molybdate solubility in spent oil shale and a preliminary evaluation of the association constants for the formation of CaMoO₄⁰ (aq), KMoO₄⁻ (aq), and NaMoO₄⁻ (aq). Environ. Sci. Technol. 24, 214–220.

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