Thermodynamic properties of dilute hydrogen in supercritical water

Accepted Manuscript Thermodynamic properties of dilute hydrogen in supercritical water Andrey V. Plyasunov, Elena F. Bazarkina PII:

S0378-3812(17)30426-0

DOI:

10.1016/j.fluid.2017.11.004

Reference:

FLUID 11639

To appear in:

Fluid Phase Equilibria

Received Date: 30 August 2017 Revised Date:

30 October 2017

Accepted Date: 2 November 2017

Please cite this article as: A.V. Plyasunov, E.F. Bazarkina, Thermodynamic properties of dilute hydrogen in supercritical water, Fluid Phase Equilibria (2017), doi: 10.1016/j.fluid.2017.11.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Thermodynamic properties of dilute hydrogen in supercritical water Andrey V. Plyasunova,*, Elena F. Bazarkinab,c Institute of Experimental Mineralogy, Russian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia; b

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a

Institute of the Geology of Ore Deposits, Mineralogy, and Geochemistry (IGEM), Russian Academy of Sciences, Staromonetnyi per. 35, Moscow 119017, Russia;

Université Grenoble Alpes, CNRS, ISTerre, UMR 5275, BP 53, 38041 Grenoble, France

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c

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E-mail address: [email protected]

Abstract

A thermodynamic model is developed to calculate the fugacity coefficients and partial molar volumes of hydrogen at infinite dilution in water at 647.1-2000 K and pure water densities between 0 and 1500 kg m-3. The model is based on the predicted values of DCFI (the

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dimensionless integral of the infinite dilution hydrogen - water direct correlation). Values of DCFI at low water densities are calculated from accurately known second cross virial

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coefficients; at high water densities predictions are based on the relations from the theory of a mixture of hard spheres; DCFI values at intermediate water densities are interpolated using a

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variant of corresponding-states correlation. Predicted values of the hydrogen fugacity coefficients at infinite dilution in water are compared with experimental data and results of the literature equations of state. The included Excel spreadsheet allows calculations provided * that values of T, P, ρ1 , and κ T for pure water are entered by a user.

Keywords: hydrogen; infinite dilution partial molar volume; fugacity coefficient; thermodynamic model

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ACCEPTED MANUSCRIPT 1. Introduction The ever-increasing interest in hydrogen energy requires knowledge of thermodynamic properties of hydrogen in its pure state and in mixtures, including a very important H2O-H2 binary system. Technological needs cover a wide temperature range at relatively low

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pressures, while the studies of processes in deep Earth ask for thermodynamic data at extreme pressures, extending to GPa ranges.

It must be noted that both experimental studies and correlation methods, when applied

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to hydrogen-containing systems, meet specific difficulties, not characteristic for most other compounds. First, experimental investigations of hydrogen and its mixtures face severe

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technical problems. A review [1] emphasizes that “hydrogen at high temperatures and/or pressures can have a disastrous effect on the structural soundness of metal alloys” due to “hydrogen permeation and loss”, “hydrogen embrittlement” of materials, “hydrogen attacks” on steel, etc. On the other hand, correlating equations of state (EoS), usually successful in

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predicting/describing data for most gases, often perform significantly worse for H2-bearing fluids. The likely reason is the very low critical temperature for hydrogen, located in the region where quantum effects are important, therefore, the ordinary scaling methods to

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estimate EoS’ parameters may not be applicable for H2. As an illustration, Fig. 1 shows the values of logarithms of fugacity coefficients of H2,

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log10 ϕ 2* (here and below the index 2 refers to H2 (a solute), the index 1 – to H2O (a solvent), the superscript * denotes the property of a pure compound, and the superscript ∞ denotes the property at infinite dilution), predicted by various EoS’s at 1000 K and pressures up to 2000 MPa: by SUPERFLUID of Belonoshko et al. [2], based on an experimental and MDgenerated [3] data set; by LONER10 of Bakker [4], who coded and offered for download at http://fluids.unileoben.ac.at/Computer.html EoS by Duan et al. [5-7], also based on a combination of experimental and MD-generated PVT values; and the four-parameter EoS by 2

ACCEPTED MANUSCRIPT Churakov and Gottschalk [8-9], available at http://fluid-eos.web.psi.ch/, which is based on the thermodynamic perturbation theory. At the highest pressure of 2000 MPa, the difference between log10 ϕ 2* of various EoS reaches 0.45 log10 units. For comparison, for H2O the

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difference amounts to only 0.07 log10 units. The line in Fig. 1 shows the values calculated online at http://webbook.nist.gov/chemistry/fluid/ using the fundamental EoS for pure H2 [10]. As this EoS was published later, it serves as a stringent test of the quality of other model

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equations of state. As seen, only SUPERFLUID EoS [2] is in close agreement with reference values. Recently, the SAFT EOS was published for the H2O-H2 system [11]. Judging by

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comparison of predicted [11] and the NIST [10] recommended PVT properties of pure H2, see Fig. 2 in that paper, the SAFT model systematically overpredicts molar volumes of pure

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hydrogen, likely overpredicting the fugacity coefficients as well.

Fig. 1. Values of the fugacity coefficients of pure hydrogen at 1000 K from various EoS (symbols), compared with the reference data (line, based on [10]). 3

ACCEPTED MANUSCRIPT Our primary interest lies not in the thermodynamic properties of pure hydrogen, but in another limit – of hydrogen at infinite dilution in water. The goal of the current study is to propose a model specifically for hydrogen at infinite dilution in water at temperatures above

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the critical temperature of water (647.096 K [12]) valid over wide density/pressure ranges.

2. The correlating model. 2.1. The general outline

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As practice shows, the success of any correlating model depends very strongly on the theoretical validity of relations in its basis. Over the years, Prof. J.P. O’Connell [13-22] has

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advocated the use of relations following from the Fluctuation Solution Theory (FST) [23] to build simple and robust models for correlating and predicting thermodynamic properties. FST provides rigorous relations connecting the thermodynamic properties of a system, such as the partial molar volume, the concentration dependence of the chemical potential, etc., with the

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structural characteristics of the system in terms of integrals of pair distribution function (solvent-solvent, solute-solute, solvent-solute). The original formulation [23] of FST in terms of total correlation function integrals (TCFI) faced difficulties in near-critical regions because

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TCFI diverged at such conditions. O’Connell [13] reformulated the theory in terms of short-

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ranged (not-diverging) direct correlation function integrals (DCFI), making it suitable for near-critical conditions as well. Rather lengthy and complex relations of the FST are strongly simplified for the case

of an infinitely dilute solute, where only the solvent-solute interactions need to be taken into account, and molar volumes and the coefficient of isothermal compressibility of a solvent are known with high precision. This particular case was thoroughly investigated by O’Connell and coworkers for correlating the infinite dilution partial molar volumes of many “gases” in

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ACCEPTED MANUSCRIPT various solvents [14], and of a number of electrolytes [15, 22] and “gases” [19, 21, 22] in water over wide temperature and pressure ranges. The property to develop correlations in the approach proposed by O’Connell is the ∞ dimensionless A12 function, defined as

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V2∞ A = 1− C = , κ T RT ∞ 12

∞ 12

(1)

∞ where C12 is the dimensionless integral of the infinite dilution solute–solvent direct

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∞ * * correlation function; V2 the partial molar volume in cm3 mol-1, κT = 1 / V1 (∂V1 / ∂P)T is the

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coefficient of the isothermal compressibility of water, MPa-1; R = 8.31446 J  K-1 mol-1 is the ∞ molar gas constant; T is the temperature in Kelvin. The partial molar volume V2 is a

“difficult” property to develop correlations for, as it experiences immense variations in the near-critical region, diverges in the critical point of a solvent, and approaches infinitely large

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∞ values at a zero density of solvent. For A12 the range of variations is far smaller, it does not ∞ diverge in the critical range and at zero density A12 is equal to 1. ∞ The virial expansion can be applied to A12 , giving [20]

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A12∞ = 1 + 2ρ1* ⋅ B12 + 3( ρ1* ) 2 ⋅ C112 + ...

(2)

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where B12 is the cross virial coefficient for the binary water-solute interactions, and C112 is the cross virial coefficient for the ternary water-water-solute interactions. As shown by ∞ O’Connell with coauthors [19, 20], the infinite dilution fugacity coefficient of the solute, φ2 , ∞ is calculated from the density dependence of A12 with the relation

ln φ = ∞ 2

ρ1*

∫ (A 0

∞ 12

− 1)

dρ1*

ρ1*

5

PV1* − ln , RT

(3)

ACCEPTED MANUSCRIPT ∞ It is important to understand the density and temperature dependence of the A12 function. As ∞ * an example, we show in Fig. 2 the values of A12 for aqueous CO2 (blue lines) and of the A11

function (black lines) at 1000 and 2000 K. The latter is defined for water as

V1* , κ T RT

(4)

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A11* =

∞ * and can be considered as “ A12 for water dissolved in water”. Values of A11 are calculated

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∞ from the high-quality Wagner-Pruβ EoS for water [12], and A12 for aqueous CO2 have been

computed previously [24], and shown to be in good agreement with the prediction of EoS for

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the H2O-CO2 system [9, 25]. As seen, the density dependence is expressed very strongly, reflecting the rise of repulsion as the density increases (and the interparticle distance ∞ decreases). However, over wide temperature ranges the changes of A12 along isochores must ∞ also be taken into account. The temperature dependence of A12 is two-fold: at low densities

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A12∞ rises with temperatures, reflecting the typical temperature course of the second virial ∞ coefficients; and at high densities A12 decrease with temperature, following the decline of the

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relative contribution of repulsive interactions at the growing thermal energy. ∞ The density dependence of A12 is more pronounced than its temperature dependence.

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∞ An earlier two-parameter correlation [19] ignored the temperature dependence of A12 and ∞ was proven to be successful in reproducing V2 values of aqueous CH4, CO2, H2S, NH3 and

B(OH)3 up to 725 K and 40 MPa (as a matter of fact, it was found that the quality of data description was better than when using the five-parameter relation for the HKF-model [26]). ∞ Even better description of V2 was achieved in later models [21, 22], which introduced on an ∞ empirical basis the temperature dependence of A12 to be able to reproduce not only the

volumetric, but also the caloric (the partial molar heat capacity, enthalpy, Gibbs energy) 6

ACCEPTED MANUSCRIPT properties of dilute solutes as well. However, while the description of data over the temperature range 273-725 K, including the near-critical region, was successful, a later analysis [27, 24] showed that the long temperature extrapolation cannot be recommended, ∞ ∞ because predicted V2 and φ2 started deviating from the values predicted by the successful

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literature EoS or by a “hard-spheres” theory.

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* ∞ Fig. 2. Values of the functions A11 (H2O, black lines) and A12 for aqueous CO2 (blue lines) at

1000 and 2000 K and various water densities.

In our opinion, attempts to find empirically the “magical” relations for successful

∞ extrapolations of A12 to higher temperatures and densities are not productive. The solution ∞ would be to develop ways to predict A12 at various temperatures and densities, based,

wherever possible, on sound theory-based relations [28, 27, 24].

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ACCEPTED MANUSCRIPT ∞ 2.2. Prediction of A12 values ∞ First, we note that the knowledge of B12 , see Eq. (2), allows to evaluate A12 values at

water densities up to ~100 kg m-3. For many water-solute interactions, B12 are known from

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the experimental measurements (see review [29]), high-quality ab initio calculations, for example, [30] for H2 in water, or can be sufficiently reliably estimated based on the corresponding-state correlations developed specifically for aqueous mixtures [31-33]. Thus,

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∞ the A12 values at low water densities can be in many cases estimated with acceptable

precision.

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An additional anchor exists at the critical point of water, where the so-called Krichevskii parameter [34], AKr , governs the thermodynamic properties of an infinitely diluted solute [35-37]. Values of AKr are known for several dozens of aqueous solutes [38]. Even in the absence of data, the knowledge of the chemical composition and properties of

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solutes may be used to roughly predict the approximate value of the Krichevskii parameter of ∞ aqueous neutral species [39]. The values of A12 in the critical point of water and AKr are

connected by the relation [40]

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Vc*1 A (Tc1 , ρ c1 ) = AKr , RTc1 ∞ 12

(5)

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where Vc*1 = 55.95 cm3 mol-1 and Tc1 = 647.096 are the critical volume and temperature of ∞ * water [12]. Although Eq. (5) calculates the value of A12 at the critical point only, at ρ1 = 322 ∞ * kg m-3 and T = 647.096 K, in practice it helps constrain the values of A12 at ρ1 =322 kg m-3

at quite a range of supercritical temperatures, because available examples suggest a weak ∞ positive temperature dependence of A12 at this water density.

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ACCEPTED MANUSCRIPT ∞ Another method of A12 evaluation comes from the equation of state for a mixture of

hard spheres. Lee [41] derived from the Mansoori-Carnahan-Starling-Leland [42] (MCSL) equation of state an expression for the partial molar volume of the solute in infinitely dilute

solvent a rather lengthy equation collapses into a simple relation:

V2∞ =N, V1*

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hard-sphere binary mixtures. For the case of a constant average packing density of a pure

(6)

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where N is a constant that has neither temperature nor density dependence. The hard-sphere model is a rough approximation of interactions in real systems. While Eq. (6) is the simplest

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∞ variant of this model for V2 , it appears more practical in application to real data compared

with more elaborate variants of the MCSL theory, which require the knowledge of the hardsphere radii of water and solutes and of the average packing densities of water at different T and P. Those are not directly measured properties, and more complicated MCSL-based

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∞ approaches lack the simple and direct connection with the experimentally measured V2 ∞ results offered by Eq. (6). Our analysis of available V2 data at ambient temperatures (278-

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348 K) at high water densities [28, 27, 24], up to 1100 kg m-3, showed that Eq. (6) offers an accurate representation of the partial molar volumes of dilute solutes at water densities above

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∞ * 970 kg m-3 (or at ρ1* > 3ρ c1 ). Experimental V2 results at ρ1 > 1000 kg m-3 and T > 350 K

are nonexistent. Still, predictions of EoS [25] for the H2O-CO2 system, together with a few experimental data at high T and P, suggest that the density range of applicability of Eq. (6) * * expands down to ρ1 > 850 kg m-3 (or at ρ1* > 2.5ρ c1 ) at T > 1000 K and to ρ1 > 450 kg m-3

(or at ρ1* > 1.5ρc1 ) at T > 1500 K, see [24] for detail. In all tested cases, the parameter N was ∞ close to constant, and its value could be computed from the V2 datum at 298.15 K and 0.1

MPa. 9

ACCEPTED MANUSCRIPT The region of water densities between 300 and 900 kg m-3 is most difficult for ∞ prediction of A12 . For this region, a variant of the corresponding-states correlation was ∞ proposed based on DCFIs [28] or the A12 function [24]. The following reduced parameter

∞ 12, r

A

A12∞ = ∞ , A12 , ref

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A12∞, r was introduced [24], defined as: (7)

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∞ ∞ where A12, ref is the value of the A12 function at the standard conditions, i.e. at 298.15 K and

0.1 MPa. It was found that the values of A12∞, r over extensive density ranges (>500 kg m-3) at

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temperatures 293-725 K are nearly identical within families of similar compounds (monohydric alcohols, diols, aliphatic amines and NH3, monocarboxylic acids, etc) and within several classes of solutes (“simple fluids”, “normal fluids”, and “hydroxides” like B(OH)3, As(OH)3, Si(OH)4 which form hydrogen bonds with water). At the same

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temperature and water density, A12∞, r regularly decrease in the following order: “simple fluids” → “normal fluids” → “hydroxides”. The polar solutes occupy space between “normal fluids”

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and “hydroxides”. At water densities above ~950 kg m-3, A12∞, r of all classes of solutes are close to each other, reflecting the dominance of repulsive interactions at high water densities,

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governed by excluded volume effects. It is useful to introduce a function A11* ,r , defined for pure water similar to Eq. (7) as

A11* ,r =

A11* A11* , ref

(8)

* * where A11, ref is the value of the function of A11 (see Eq. 4) at the standard conditions, i.e. at

298.15 K and 0.1 MPa. The theory of a mixture of hard spheres [41] predicts the equality of

A12∞,r and A11* ,r , see [24]. Therefore, the high-density limit of A12∞,r is well established. We

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ACCEPTED MANUSCRIPT remind the reader that density ranges of validity of a “hard-sphere” relation (6) for aqueous * * solutions are ρ1 > 970 kg m-3 at ambient temperatures, ρ1 > 850 kg m-3 at T > 1000 K and to

ρ1* > 450 kg m-3 at T > 1500 K, and at these conditions A12∞ ,r − A11* ,r will approach zero. Such

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considerations allow proposing [24] the following relations to interpolate A12∞ ,r − A11* ,r between experimental results, available mostly at temperatures below 673 K, and their hightemperature limiting values equal to zero along several isochores:

a1 a2 , + T2 T6

w2 = b0 +

(9) (10)

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w1 =

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A12∞ ,r − A11* ,r = w1 {exp[w2 ⋅ ∆ h G ∞ ] − exp[w2 ⋅ ∆ h G ∞ (H 2 O)]},

b1 b2 , + T3 T6

(11)

where ∆h G∞ is the Gibbs energy of hydration (proportional to Henry’s constant) of a solute at 298.15 K and 0.1 MPa. For all “simple fluids” to which H2 belongs to as well, it is

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recommended to use a “universal” value ∆h G∞ = 16.3 kJ mol-1, i.e. the value for methane [43]); ∆h G ∞ (H2O) = -18.55 kJ·mol-1 is the Gibbs energy of hydration of water at reference

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conditions [44]. The idea behind this correlation is to express A12∞ ,r − A11* ,r in terms of the

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difference between the water-water and water-solute interactions, although in an empirical way. Numerical values of parameters a1 , a2 , b0 , b1 , b2 are given in Table 1.

Table 1. Parameters of Eqs. (9) – (11) for several isochores.

ρ1* ,

-3 a1 ·10 ,

-15 a 2 ·10 ,

b0 ·102,

-7 b1 ·10 ,

-15 b2 ·10 ,

kg m-3

K2

K6

kJ-1 mol

kJ-1K-3mol

kJ-1K-6 mol

500

0.19581

9.6004

6.5922

-1.8381

1.4362

11

ACCEPTED MANUSCRIPT 0.54868

6.7761

9.4198

-2.5715

2.2568

700

-0.83090

5.0586

13.769

-4.3669

4.5074

800

-0.88961

3.5455

12.2802

-2.0309

0.78907

900

-0.51388

1.8954

12.2745

-2.2777

1.1942

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600

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2.3. Application to dilute aqueous H2 solutions 2.3.1. B12 for H2O-H2 interactions

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The study [30] used the scaled perturbation theory to construct a potential energy surface of H2O-H2 dimer. The second virial coefficient, B12, was calculated using an analytic fit of the potential-energy surface, including quantum effects, at 100 to 2000 K. A careful analysis of available experimental data leading to B12 and their temperature derivatives was performed as

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well. It was shown that the calculated and experimental B12 are mutually consistent, although calculated values have significantly smaller uncertainties. The temperature dependence of B12 over 200-2000 K in cm3 mol-1 is given by the expression [30]  T  B12 = ∑ a i    100  i

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4

bi

(12)

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and the parameters of Eq.(12) are presented in Table 2.

Table 2. Parameters of Eq. (12) [30]; B12 and ai are in cm3 mol-1. i

ai

bi

1

3.3047·101

-0.21

2

-2.5041·102

-1.50

12

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2.8542·102

-2.26

4

-1.8678·102

-3.21

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2.3.2. The Krichevskii parameter, AKr , of hydrogen in water The value of AKr of hydrogen in water can be evaluated from several types of experimental information: a). from the initial slopes of the critical line of the H2O-H2 system [45], b). and

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from the recommended constants of distribution of hydrogen between the vapor and liquid phases of water [43]. Details are given in [38], which recommended AKr = 170 ± 8 MPa for

ρ1*

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∞ hydrogen in water. This value, in accordance with Eq. (5), translates into A12 = 1.77 ± 0.08 at

= 322 kg m-3 and T = 647.096 K. At higher temperatures and

∞ of A12 are expected to increase slowly.

ρ1*

= 322 kg m-3 the values

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∞ 2.3.3. Evaluation of the parameter N (see Eq. 6), or V2 at 298.15 K and 0.1 MPa ∞ In order to evaluate A12 at high water densities, where the hard-sphere relation is valid, one

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needs the value of the parameter N, see Eq. (6). In all previously [24] tested cases (CH4, CO2, ∞ ∞ H2S, SO2, NH3) the parameter N was evaluated from V2 at 298.15 K and 0.1 MPa. This V2

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value is also necessary to convert A12∞,r , see Eqs. (9)-(11), estimated at water densities 500∞ ∞ 900 kg m-3, into A12 . The literature values of V2 for hydrogen dissolved in water at standard

conditions are given in Table 2. An important issue here is the uncertainty of reported data. For the case of poorly soluble gases, a major difficulty in experimental studies is the necessity to measure with high precision miniscule differences between densities of pure * water, ρ1 , and a gas-bearing solution, ρ . The solubility of H2 in water at 298.15 K under

partial pressure of gas of 0.1013 MPa is x2 = 1.413·10-5 or m = 7.843·10-4 [46]. Using the 13

ACCEPTED MANUSCRIPT ∞ definition of an apparent molar volume of a solute, φV , which is a close approximation of V2

at very low solute’s concentrations: φV =106 ( ρ1* − ρ ) /(mρ1* ρ ) + 103 M 2 / ρ , where M 2 is the molar weight of a solute, it is easy to show that the error of just ±1·10-3 kg m-3, or 1 ppm, in (

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ρ1* - ρ ) will result in the error ± 1.3 cm3·mol-1 in V2∞ of a solute, and a 2 percent error in the ∞ molality of a dissolved gas will change V2 by 0.4 cm3·mol-1. Therefore, we expect

uncertainties of reported values to be no less than 2 cm3·mol-1. Within these imprecision

∞

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∞ ranges, V2 data are consistent.

Reference

V2∞ , cm3 mol-1

[47]

26

[48]

25.2

[49]

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Table 2. Literature values of V2 at 298.15 K and 0.1 MPa for hydrogen dissolved in water

[50]

Comments

Dilatometer; 24 at 273 K and 323 K, 26 at 298 K Dilatometer

26.7

Vibrating tube

23.1

Vibrating tube

26.1

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[45]

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Indirect evaluations from experimental data at T > 647 K, see text

[51]

[52]

Vex ~ 0 at 673.15 K and 250 MPa Measurements

at

660-759

K

of

the

27.1 to 28.4

(>27)

fugacity/concentration ratios for H2 in water Fluid inclusions study at 1123 K and 2300 MPa

Nevertheless, the range of values from 23.1 to 26.7 cm3 mol-1 leads to the 15 percent ∞ * uncertainty of the N parameter and of A12 at ρ1 > 500 kg m-3. The resulting uncertainties of

* the logarithm of fugacity coefficient of H2 increase from 0.1 log10 units at ρ1 = 800 kg m-3 to

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ACCEPTED MANUSCRIPT * 0.3 log10 at ρ1 = 1000, quickly raising to 0.7 and 1.5 log10 units at water densities of 1300

and 1500 kg m-3, respectively. Existing experimental data at T > 647 K may provide additional constraints on the

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∞ value of the N parameter (or V2 at 298.15 K and 0.1 MPa).

As it appears, the most reliable datum comes from the PVTX studies of the system H2O-H2 up to 713 K and 250 MPa [45]. Regrettably, no primary data are reported, and the

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∞ ∞ direct evaluation of V2 and A12 is not possible. Nevertheless, when analyzing the excess

volumes of the mixture, Vex, at 673 K and various pressures, the authors noted that “at 2500

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bar, H2-H2O mixtures exhibit small positive or negative departures from ideality”, or in other words, at 673 K and 250 MPa at all compositions, Vex ~ 0. As it is known (see a formal derivation in [27], the Appendix), V2∞ = V2* + (∂V ex / ∂x2 )T ,P , x2 =0 . For this particular case it ∞ * means that the partial molar volume, V2 , is equal to the molar volume of pure hydrogen, V2 ,

at

this

T-P

point

V2*

=

37.5

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and

cm3

mol-1

([10],

calculated

at

∞ * http://webbook.nist.gov/chemistry/fluid/), or A12 (673.15, ρ1 =825.1 kg m-3) = 9.35. This

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∞ ∞ value of A12 can be predicted if V2 = 26.1 cm3 mol-1 at 298.15 K and 0.1 MPa.

Kishima and Sakai [51] reported at supercritical temperatures the concentrations of

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hydrogen in water in equilibrium with the mineral association magnetite (Fe3O4) + hematite ∞ (Fe2O3). The treatment of data, see Supplementary Materials, gives for V2 at 298.15 K and

0.1 MPa values at the range of 27.1 to 28.4 cm3 mol-1 with the expected uncertainty of 10 percent. These values are considered less reliable compared with the one evaluated from PVTX data [45], as they are obtained from data at lower water densities. An additional source of information is the study of Bali et al. [52]. They investigated the H2O-H2 mixture at 1000-1500 K at pressures from 1500 to 2600 MPa, with the mole

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ACCEPTED MANUSCRIPT fraction of hydrogen close to 50 mol%, by the method of synthetic fluid inclusions, see Supplementary Materials for detail. The approximate value of log 10 ϕ 2∞ >2.57 can be ∞ evaluated from data at 1123 K and 2300 MPa. This datum can be predicted if V2 > 27.0 cm3

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mol-1 at 298.15 K and 0.1 MPa. Still, this value was not accepted for reasons presented in Supplementary Materials, and it is shown in parentheses in Table 2.

Soubiran and Militzer [53] performed ab initio simulations of water–hydrogen

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mixtures in the range from 2 to 70 GPa and from 1000 to 6000 K, and calculated the Gibbs energy of this mixture, which showed the complete miscibility of water and hydrogen at all

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studied conditions. It was found that the model of an ideal mixing describes all the obtained ∞ * data, or, in other words, at all conditions ϕ2 ≈ ϕ2 . This conclusion appears unlikely for a

mixture of these two very different compounds, “simple fluid” H2 and hydrogen-bonded H2O. * ∞ * According to [10], at 1000 K and 2000 MPa ϕ 2 ≈ 1.33. In order to get ϕ2 ≈ ϕ2 at this T-P

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∞ point, the value of V2 at 298.15 K and 0.1 MPa should be less than 18 cm3 mol-1, which is

incompatible with the available experimental determinations. Results of [53] are not

desirable.

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accepted. A new, more detailed ab initio investigation of the H2O-H2 system would be

The best-constrained result, as appears, is from the PVTX study of Seward and Frank

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∞ [45] – it corresponds to V2 at 298.15 K and 0.1 MPa equal to 26.1 cm3 mol-1 with the

expected uncertainty of 5%. This value is consistent with the direct determinations of the partial molar volume of hydrogen in water at the standard T and P, and it is close to the mean value of all data in Table 2. Therefore, for the H2O-H2 system we selected

V2∞ (298.15 K, 0.1 MPa) = 26.1 cm3 mol-1; and N = 26.1/18.07 ≈ 1.444.

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ACCEPTED MANUSCRIPT 3. Results and discussion ∞ 3.1. Evaluation of the density dependence of A12 on isotherms. ∞ With all parameters in place, the evaluation of A12 for hydrogen in water at

values of the function

Y = ( A12∞ − 1) / ρ1*

(13)

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at various water densities, see Fig. 3.

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supercritical temperatures is straightforward. As before [27, 24], we calculate and plot the

∞ * Fig. 3. Values of the function Y = ( A12 − 1) / ρ1 at 647.1 (triangles) and 2000 (circles) K and

various water densities for aqueous H2. The star shows Y at 647.1 K and calculated from the value of the Krichevskii parameter.

17

ρ1*

= 322 kg m-3,

ACCEPTED MANUSCRIPT * ∞ * According to Eq. (2), at ρ1 =0 the function ( A12 − 1) / ρ1 is equal to 2B12 , the initial

departure of this function is linear in density, and the example of water shows that the linear part may extend up to 300 kg m-3. The value of B12 has to be converted from cm3 mol-1 to the m3 kg-1 units. At the critical temperature and density, 647.096 K and 322 kg m-3, a helpful

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constraint is given by the Krichevskii parameter, see a star on Fig. 3. In such a way, we evaluated many values of the Y function at 647.1 to 2000 K and between 0 and 1500 kg m-3.

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All results can be reproduced by the equations

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∞ ∞ 3.2. Parameters of the correlating model to calculate A12 and φ2 for aqueous H2

Y = a o + a1 ρ1* + a 2 (ρ1* ) + a3 (ρ1* ) + a 4 (ρ1* ) ,

(14)

ao = a00 + a01 / T + a02 / T 2 + a03 / T 3 + a04 / T 4 ,

(15)

a1 = a10 + a12 T 2 + a13 T 3 + a14 T 4 ,

(16)

a2 = a20 + a22 T 2 + a23 T 3 + a24 T 4 ,

(17)

a3 = a30 + a31 T + a32 T 2 ,

(18)

a 4 = a 40 + a42 T 2 + a44 T 4 .

(19)

4

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2

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Numerical values of parameters a00 - a 44 are given in Table 3. They should be used at temperatures between 647 and 2000 K and at water densities up to 1500 kg m-3. Values of

a00 ...a04 have been determined from the temperature dependence of B12 [30]. ∞ The values of A12 are calculated with

A12∞ = Y ⋅ ρ1* + 1

(20)

∞ and lnφ2 - according to

a1 * 2 a 2 * 3 a3 * 5 a 4 * 7 PV1* ln φ = ao ρ + (ρ1 ) + (ρ1 ) + (ρ1 ) + (ρ1 ) − ln . 2 3 5 7 RT ∞ 2

* 1

18

(21)

ACCEPTED MANUSCRIPT ∞ Calculated values of A12 and log10 φ2∞ at 647.1, 1000 and 2000 K are shown in Fig.

4(a,b). The unusual shape of the log10 φ2∞ density dependency at 550-800 kg m-3 along the isotherm 647.1 K, where log10 φ2∞ remain practically constant, arises from a combination of

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polynomial terms and the log10 ( PV1* / RT ) term in Eq. (21). Indeed, the - log10 ( PV1* / RT ) values at 674.1 K form a "hump-like" maximum, see Fig. 4(c). A similar, although less pronounced, shape persists up to 700-750 K, and then wanes farther away from the critical

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∞ ∞ isotherm. Calculated values of A12 , V2 , and log10 φ2∞ are given in the Excel file of

Supplementary Materials at temperatures 647.1, 673.15, 700, and then with a step of 50 K to

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1000 K, and then with a step of 100 K to 2000 K, at round water densities between 25 and 1500 kg m-3 with a step of 25 kg m-3. Tables are sufficiently detailed to allow interpolations. The included Excel spreadsheet allows additional calculations provided that values of T, P,

ρ1* , and κ T are entered by a user (the κ T is only necessary to convert A12∞ into V2∞ ).

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The use of Eqs. (14)-(19) outside of the fitted range of water temperatures and densities is not recommended. If one needs to perform calculations at higher temperatures or

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∞ water densities, then it is necessary first to evaluate A12 results at new conditions following

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the method, described in this work.

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Table 3. Numerical values of parameters a00 - a 44 of Eqs. (14)-(19).

ai1

ai 2

0

1.38107·10-3

1.30421

-1.75656·103

1

9.868772·10-6

-2.2191437·10-11

2

-1.756703·10-8

4.360627·10-14

3

4.588544·10-14

4

-5.133027·10-21

ai3

ai 4

7.49394·105

-1.33785·108

2.003317·10-14

-4.620274·10-18

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ai 0

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-4.108981·10-17

-3.449499·10-17

7.265531·10-21

1.098607·10-20

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2.152018·10-27

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-3.1376777·10-34

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∞ Fig. 4. Values of A12 (a) and log10 φ2∞ (b) at 647.1 (a black line), 1000 (a red line) and 2000

K (a blue line) and various water densities for aqueous H2 predicted with the proposed model,

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and the values of the - log10 ( PV1* / RT ) term for water as a solvent (c) at the same conditions.

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∞ 3.3. Comparison with experimental φ2 values

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As was already discussed, Kishima and Sakai [51] reported at supercritical temperatures 660-759 K and pressures up to 100 MPa the concentrations of hydrogen in water in equilibrium with the mineral association magnetite (Fe3O4) + hematite (Fe2O3). As ∞ follows from the study, one can calculate φ2 values with the relation

φ 2∞ =

1244 ⋅Y , P (bars )

(22)

where Y is the fugacity/concentration ratios for dilute hydrogen in water, reported by the authors, and 1244 is the numerical coefficient to convert the custom concentration units to the 22

ACCEPTED MANUSCRIPT mole fractions. The reproducibility of individual data points is within 5 percent. Nevertheless, the absolute values of Y may bear a small systematic shift due to selection by the authors of a specific value of the oxygen fugacity of the magnetite-hematite mineral O2 buffer at atmospheric pressure. In other words, the pressure dependence of Y may be more precise than

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individual Y values themselves.

Fig. 5. Values of log10 φ2∞ of H2 in water at 689.2 K measured in experiments of Kishima and Sakai [51] and predicted by various EoS.

For comparison, we selected the isotherm 689.2 K, see Fig. 5. Data at this isotherm cover the largest range of water densities, from 389 to 669 kg m-3. In addition, this is a sufficiently low supercritical temperature, where our model predicts a rather unusual density

23

ACCEPTED MANUSCRIPT dependence of log10 φ2∞ , see Fig. 4(b), which may raise questions. The experimental data are shown by stars connected by a solid line, and the predictions of literature EoS – by various symbols. Our model and the model of Akinfiev and Diamond [54] provide a close

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∞ reproduction of experimental φ2 , with an excellent agreement for the density increment of

log10 φ2∞ of aqueous hydrogen between ρ1* equal to 389 to 669 kg m-3: 0.27 and 0.30, correspondingly, in comparison with an experimental value of 0.28±0.03. The popular in

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geochemistry HKF-model in application to neutral species [26], with calculations made using the online tool at http://geopig3.la.asu.edu:8080/GEOPIG/index.html), and the EoS of

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Belonoshko et al. [2, 3] performs satisfactorily, although not within experimental accuracy of data: the density increment of log10 φ2∞ is 0.21 and 0.40, respectively, compared with 0.28±0.03 in the experiment. A failure of the model of Duan et al. [5-7] coded by Bakker [4] is likely due to a poor reproduction of PV properties of pure water at this temperature (for

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* example, the model predicts ρ1 = 169 kg m-3 at P = 36.5 MPa instead of the reference [12] * value of ρ1 = 389 kg m-3). The EoS [9] could not be tested, because the lowest temperature

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of its use is 750 K.

There are a few other works [55-57] reporting the chemical potentials of dilute

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hydrogen in water at supercritical temperatures: at 973 K and 80 MPa [55], at 673 and 723 K and 27.5 MPa [56], at 658-723 K and 25 MPa [57]. However, these studies appear to be less precise (the scatter of reported Henry’s constants may be up to 50% of the value [57]), they are performed at water densities below 300 kg m-3, and thus cannot provide a definite test of the quality of various models for predicting thermodynamic properties of H2 in its dilute aqueous solutions.

3.4. Comparison with literature EoS 24

ACCEPTED MANUSCRIPT Fig. 6(a,b) compares values of the fugacity coefficients of hydrogen at infinite dilution in water at 1000 K (a) and 1500 K (b) over a wide range of densities. Our values are given by a solid line, while those of other EoS are shown by symbols. At moderate densities, from 400 to 800 kg m-3, our results at 1000 K are in close agreement with the values predicted by

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SUPERFLUID [2], but systematically lower than the predictions of EoS of Duan et al. [4] and Churakov and Gottschalk [9]. For these equations of state, calculations are made for the mole fraction of H2 equal to 10-4. The model of Akinfiev and Diamond [54] is developed

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specifically for properties at infinite dilution. Its predictions, especially for 1500 K, are

observed earlier for other “gases” [24].

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systematically lower than values of all other models; similar behavior of this model was

A dashed line shows the predictions of the high-pressure variant of the HKF model, popular in geochemistry. Calculations are made with the DEW (Deep Earth Water) model [58], available for download at deepcarbon.net//feature/deep-earth-water-model-download-

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now. As was already noted [59, 19, 60, 54, 24], this Born-based model cannot be recommended for calculating thermodynamic properties of neutral species in hightemperature water at low and moderate water densities.

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At high water densities, our predictions are systematically larger than those of SUPERFLUID [2] and approach values calculated with EoS’ [4] and [9]. We expect that

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∞ * values of φ2 , calculated with our model at ρ1 > 1000 kg m-3, bear the largest uncertainties. * In order to properly constrain our model at ρ1 > 1000 kg m-3, we need reliable ∞ ∞ thermodynamic values ( V2 or φ2 ) at these high water densities. Available so far data are

limited by partial molar volumes of hydrogen at ambient temperatures, which bear significant ∞ uncertainties, see Table 2. Good candidates would be values of V2 at ambient temperatures

measured at high pressures of hydrogen (up to tens MPa), which would increase the

25

ACCEPTED MANUSCRIPT concentrations of H2 in solution by hundreds of times compared with measurements at ∞ atmospheric pressure, thus significantly reducing the errors in the V2 value. Another

possibility would be to measure spectroscopically concentrations of hydrogen in high-

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O2), although this represents a strong experimental challenge.

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pressure water in equilibrium with a mineral association, which buffers the fugacity of H2 (or

26

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Fig. 6. Values of log10 φ2∞ at 1000 K (a) and 1500 K (b) and various water densities for aqueous H2 predicted by various EoS.

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4. Conclusion

As shown by J.P. O’Connell and his coworkers [14, 15, 19-22], DCFI-based models

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(DCFI stands for direct correlation function integrals) offer a simple and robust way for correlating thermodynamic properties of dilute solutions over extremely wide ranges temperatures and solvent’s densities. An important feature of DCFI-based models is the simple shape of DCFI everywhere, including the near-critical region. In practice, it may be ∞ easier to work with the function A12 , which is linearly related to DCFI, see Eq. (1). The ∞ function A12 has simple and understandable density and temperature dependencies, as shown ∞ * in Fig. 2, and it is easy to describe A12 by polynomials of T and ρ1 .

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ACCEPTED MANUSCRIPT Our work is concerned with thermodynamic properties of hydrogen at infinite dilution in water at temperatures from 647.1 to 2000 K and densities up to 1500 kg m-3. Experimental data for H2 in supercritical water are fragmentary, and our model is heavily based on ∞ predicted values of A12 . Our predictions use theoretically sound methods and include

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precisely known [30] values of B12 , the second cross H2O-H2 virial coefficients, which ∞ ∞ provide A12 at low water densities. At high water densities, above 900-1000 kg m-3, A12 can

be predicted based on relations following from a theory of mixtures of hard spheres [41]. In

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∞ * order to use this theory, one needs to know the V2 of aqueous hydrogen at ρ1 > 1000 kg m-3,

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however, existing data bear significant uncertainty, primarily due to the low solubility of H2 ∞ in water. New measurements of V2 , preferably at pressures of hydrogen up to tens MPa, ∞ would be useful to significantly reduce the uncertainty of V2 . The most difficult for

predictions is the region of intermediate water densities, between 300 and 900 kg m-3,

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although the variant of the corresponding-states correlation [28, 24] offers help here. We have also performed a comparison of our predictions with experimental data [51] and some literature EoS for the H2O-H2 system, although a definite test has to wait until new

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experimental data, preferably at high water densities, become available for this system, which

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is so crucially important for industry and for Earth and Space sciences. The detailed Table of ∞ ∞ calculated values of A12 , V2 , and log10 φ2∞ is available as the Excel file in Supplementary

Materials, together with the spreadsheet, that allows additional calculations provided that * ∞ values of T, P, ρ1 , and κ T are entered by a user (the κ T is only necessary to convert A12 ∞ into V2 ).

28

ACCEPTED MANUSCRIPT Acknowledgements AVP is privileged to personally know Prof. J.P. O’Connell, which definitely had an impact on the development of the DCFI-based models for aqueous solutes over wide T and P ranges. The picture of the thermodynamics of mixtures in terms of DCFIs is simpler than those in

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terms of the partial molar properties themselves, promising advances in the development of robust predictive models for many industrially and scientifically important systems. This research was partially supported by Russian Foundation for Basic Research (grant 14-05-

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00488).

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