An Accurate Model for Estimating H 2 Solubility in Pure Water and Aqueous NaCl Solutions

: By employing a speciﬁc particle interaction theory and a high-precision equation of states for the liquid and vapor phases of H 2 , respectively, a new H 2 solubility model in pure water and aqueous NaCl solutions is proposed in this study. The model established by ﬁtting the experimental data of H 2 solubility can be used to estimate H 2 solubility in pure water at temperatures and pressures of 273.15–423.15 K and 0–1100 bar, respectively, and in salt solutions (NaCl concentration = 0–5 mol/kg) at temperatures and pressures of 273.15–373.15 K and 0–230 bar, respectively. By adopting the theory of liquid electrolyte solutions, the model can also be used to predict H 2 solubility in seawater without ﬁtting the experimental data of a seawater system. Within or close to experimental data uncertainty, the mean absolute percentage error between the model-predicted and experimentally obtained H 2 solubilities was less than 1.14%.


Introduction
Hydrogen (H 2 ) is an important natural gas because it is light, storable, and reactive [1].H 2 is considered the best energy carrier for the efficient storage of renewable primary energy sources such as solar and wind energy [2].On the one hand, the combustion of H 2 does not emit pollutants and greenhouse gases; the only combustion product is H 2 O; on the other hand, H 2 has a high calorific value of ~140 MJ/kg [3].H 2 is potentially suitable for large-scale geological storage in porous formations, saline aquifers, caverns, or depleted oil and gas reservoirs, all of which can provide significant storage capacity [4][5][6].To assess the stability and safety of the long-term operation of hydrogen storage reservoirs and the efficiency of energy storage, one should study the solubility and volumetric properties of H 2 in gas−liquid systems for the migration of fluids and the alteration of minerals induced during storage [7].Moreover, H 2 is abundantly present in nature.Hydrogen production can be divided into inorganic and organic geneses.Inorganic hydrogen is usually produced via earth degassing, water-rock reactions, and water radiolysis [8][9][10], whereas organic hydrogen is primarily produced via biogenesis and the thermal decomposition of organic matter [11,12].Natural hydrogen is abundant in the formation areas of terrestrial volcanic rocks, large fault basins, marine serpentinized areas, and hydrothermal vents [9,13,14].Hydrogen can be utilized as an electron donor in the reactions of photoautotrophic, photoheterotrophic, chemoautotrophic, and chemoheterotrophic organisms [15].Most typically, autotrophic hydrogen bacteria consume hydrogen to produce life-sustaining methane, which explains the abundance of hydrogen-consuming organisms in submarine hydrothermal vents [16][17][18].In studies on hydrogen-related biological activities or physicochemical processes, the hydrogen supply rate and hydrogen concentration in fluids must be determined.Moreover, H 2 solubility in a fluid largely determines the hydrogen transport rate and hydrogen concentration in the dissolved state.Experiments using various solutions have yielded a large amount of H 2 solubility data at various temperatures and pressures.Additionally, H 2 solubility data have been accumulating since 1855.Using the pure physical absorption method, Bunsen [19] measured the absorption coefficients of H 2 in pure water at various temperatures (277.15-296.75K) and atmospheric pressure.However, because H 2 shows low solubility in water at atmospheric pressure and the experimental conditions were limited, the measurement results were very similar.Using the same method, Wiebe and Gaddy [20] measured the absorption coefficients of hydrogen in pure water over a wide temperature range (273.15-373.15K) and different pressures (25-1000 atm).They treated nitrogen impurities in the hydrogen so that gas composition was close to pure hydrogen and the influence of the water-vapor partial pressure on solubility was corrected.Chabab et al. [21] employed the static analysis method to measure H 2 solubility in pure water and aqueous NaCl solutions (1, 3, and 5 mol/kg) at different temperatures (323.18-372.76K) and pressures (28.623-229.720 bar).
A model based on experimental data can be used to predict H 2 solubility in an unmeasured system.Jauregui-Haza et al. [22] studied H 2 solubility in water and organic solvents such as octene, toluene, and nonanal.They applied regular solution theory using the polar solvent factor correction method reported by Lemcoff [23].Moreover, they derived the Henry constant of H 2 at temperatures of 353, 363, and 373 K.The H 2 solubility error in the aforementioned solvents was ~2.6%; however, the model was only applicable to pure aqueous solutions and the Henry constant of H 2 was not determined in aqueous NaCl solutions.Li et al. [7] considered the system pressure, temperature, and formation fluid salinity in an H 2 solubility model.Their model reproduced all available experimental data and accurately predicted H 2 solubility in formation fluids under a range of typical geological hydrogen storage conditions (273-373 K, 1-500 bar, and 0-5 mol/kg NaCl).Within or close to the experimental data uncertainty, H 2 solubility was predicted with a maximum relative error of 5% in pure water; however, the error increased to 15% in brine.Chabab et al. [21] estimated the H 2 solubility using a fast method based on a Setschenow-type relation [24], which predicted H 2 solubility in pure water and aqueous NaCl solutions with average deviations of 0.5% and 2%, respectively.This model was adapted to the temperature and pressure ranges of 273.15-373.15K and 1-203 bar, respectively, in pure water, and 323.15-373.15K and 10-230 bar, respectively, in aqueous NaCl solutions.However, in aqueous NaCl, the lower bound of this model was 323.15 K, which is unsuitable for studying H 2 solubility in nature.Torín-Ollarves and Trusler [25] proposed a simple model based on an analysis method for predicting H 2 solubility in aqueous solutions at temperatures and pressures of 273.15-423.15K and 1-1010 bar, respectively.For reasons that defy a logical explanation, the prediction results of this model quite differed from those reported by Chabab et al. [21].
Duan et al. [26] established a solubility model of methane gas in aqueous solutions.Their model applies a specific theory of particle interactions for the liquid phase and a highprecision equation of state for the vapor phase.The methane solubility in both pure water and aqueous NaCl solutions was predicted for the temperature range of 273.15-523.15K and the pressure range of 0-1600 bar.The error between the calculated and experimental data was ~7%.The parameters in this model were fitted to the experimental data and represented the interactions between substances.The values of different parameters were closely related, suggesting that the model is applicable to complex brines (e.g., CaCl 2 , KCl, and seawater) using the approximation principle.The calculated results were consistent with the experimental data.Later, the solubilities of N 2 , CO 2 , C 2 H 6 , and O 2 in pure water to aqueous NaCl solutions were calculated using this model [27][28][29][30][31] and were also consistent with the experimental data.In conclusion, this model was widely applicable and can accurately calculate gas solubility in pure water and was easily employed in multiple ionic systems.Herein, we establish H 2 solubility models for the H 2 + H 2 O system and the H 2 + H 2 O + NaCl system, as well as for other ionized water systems that are applicable to a wide range of temperatures, pressures, and salinities.The gas-phase chemical potential of hydrogen was computed using the equation of state proposed by Peng and Robinson [32], whereas the liquid-phase chemical potential of hydrogen was defined using the theory of liquid electrolyte solutions proposed by Pitzer [33].The relevant parameters of this model were fitted to as many experimental data as possible.During comparison with experimental data, the model achieved high accuracy, thus providing a foundation for related marine geochemistry research.

H 2 Solubility Model
H 2 solubility in aqueous solutions was determined based on the balance between the chemical potentials of H 2 in the liquid and vapor phases.The potential can be expressed in terms of fugacity in the vapor phase (Equation ( 1)) and activity in the liquid phase (Equation ( 2)): where µ H 2 represent the standard chemical potentials of H 2 in liquid and vapor phases, respectively.Here, µ l(0) H 2 denotes the chemical potential in a hypothetical ideal solution of unit molality [34], and µ v(0) H 2 denotes the chemical potential when the pressure of a hypothetical ideal gas is set to 1 bar.
At phase equilibrium µ l H 2 = µ v H 2 , subsequently, we obtain Equation (3). ln In parameterization, reference value µ v(0) H 2 can be set to 0 for convenience as only the difference between µ l(0) H 2 and µ v(0) H 2 is important.Since the vapor phase has low water content, the fugacity coefficient of H 2 in gaseous mixtures is approximate to that of pure H 2 in the studied region.Therefore, ln ϕ H 2 can be approximated from the equation of state of pure H 2 (refer to Appendix A) [32].The mole fraction y H 2 . of H 2 in the gas is calculated as follows.
If the partial pressure P H 2 O of water in the vapor phase is approximated as the saturated pressure of pure water [26,[28][29][30][31], RT and ln γ l H 2 will contain errors of up to 5%.However, these errors can be largely canceled by parameterization.Herein, the mole fraction y H 2 O of water in the vapor phase is estimated using the following semiempirical equation: where x H 2 O represents the mole fraction of H 2 O in the liquid phase, which is approximated as 1 and 1-2x NaCl in the H 2 + H 2 O and H 2 + H 2 O + NaCl systems, respectively, when dissolved hydrogen is neglected.The saturation pressure P S H 2 O (in bar) of water was calculated using an empirical equation (refer to Appendix B).The molar volume v l H 2 O of water in the liquid phase was approximated to the saturated liquid-phase volume of water and was calculated using the equation proposed by Sun et al. [35].The fugacity coefficient Energies 2022, 15, 5021 4 of 15 ϕ H 2 O of water was calculated using the following equation, which is obtained by fitting the methane-water experimental data [30].
The values of a 1 − a 6 are listed in Table 1.The water content in the vapor phase can be calculated accurately using Equations ( 5) and (6).The results for different temperatures are plotted in Figure 1.
The values of 1 a -6 a are listed in Table 1.The water content in the vapor phase can be calculated accurately using Equations ( 5) and ( 6).The results for different temperatures are plotted in Figure 1.

Parameters Values
where  and  represent the second-order and third-order interaction parameters, respectively.The subscripts c and a denote cations and anions, respectively.Substituting Equation (7) into Equation (3) yields the following.ln γ H 2 is expressed as a virial expansion of excess Gibbs energy [33]: where λ and ζ represent the second-order and third-order interaction parameters, respectively.The subscripts c and a denote cations and anions, respectively.Substituting Equation (7) into Equation (3) yields the following. ln Following Pitzer et al. [36], we selected the following equation for the The basis of our model parameterization consists of Equations ( 8) and (9).

Review of H 2 Solubility Data
H 2 solubility in pure water and aqueous NaCl solutions was measured in a wide range of temperatures, pressures, and ionic strengths (Table 2).The remaining experimental data in pure water show good continuity and correlation, with the exception of some obvious deviations and relative dispersions.After including most of the experimental pure water data in the parameterization, the optimal ranges of temperatures and pressures for the H 2 + H 2 O system in this model were determined to be 273.15-423.15K and 0-1100 bar, respectively.
Alternatively, the experimental data of H 2 solubility in aqueous NaCl solutions showed poorer continuity and a narrower range than those in pure water.The measured values of H 2 solubility reported by Torín-Ollarves and Trusler [25] are obviously inconsistent with those reported by Chabab et al. [21].These abnormal data were excluded, and the experimental data reported by Braun [37], Croizer and Yamamoto [38], and Chabab et al. [20] were selected for the parameterization.Finally, the solubility models for the H 2 + H 2 O + NaCl system yielded temperature, pressure, and salinity ranges of 273.15-373.15K, 0-230 bar, and 0-5 mol/kg, respectively.
The H 2 solubility has been measured in solutions other than aqueous NaCl solutions.For example, Braun [37] measured the H 2 solubility in 0.16-0.34mol/kg BaCl 2 solution.Thomas et al. [32] and Gordon et al. [39] measured H 2 solubility in seawater with different salinities.Although the temperature ranges in these experiments were wide, the pressure was kept constant (1 atm).Because the aforementioned parameterization requires the combined effect of temperature, pressure, and salinity, these data were excluded from parameterization.To estimate H 2 solubility as a function of temperature, pressure, and salinity, we must determine parameters λ and ζ of Na + and Cl − in the liquid phase and the standard chemical potential µ l(0) H 2 in Equation (8).Because measurements can only be performed in electronically neutral solutions, one of the parameters must be assigned arbitrarily [53].
We set λ H 2 −Cl to zero and fitted the remaining parameters.First, RT was evaluated using the H 2 solubility data for pure water (92 related experimental data values), with a root mean square error of 1.62.Next, λ H 2 −Na and ζ H 2 −Na−Cl were evaluated simultaneously by the least-squares fitting of the solubility data for the aqueous NaCl solutions (41 related experimental data values), with a root mean square error of 1.42.The temperature-and pressure-dependent coefficients are listed in Table 3.
Table 3.Values of the interaction parameters in Equation (9).Substituting the parameters (Table 3) into Equation ( 8), we obtain the H 2 solubilities in pure water and aqueous NaCl solutions.The calculated solubilities in pure water and in 1, 3, and 5 mol/kg NaCl solutions are displayed in Tables 4-7.

Model Validation
Figures 2 and 3 show a comparison of the experimental data with the results predicted using our model.The model adequately represented most of the experimental data, remaining within or close to the experimental uncertainty (~1.14%).
In the T-P-m range covered by our model, H 2 solubility increased with increasing pressure and decreased with increasing ionic strength.The temperature dependence of H 2 solubility was more drastic (Figure 4).H 2 solubility was slightly dependent on the temperature at low pressures (<200 bar) but decreased and then increased with increasing temperature at high pressures (>200 bar).The isobaric minimum solubility point was observed at ~320 K and 200 bar (Figure 4).Morrison [44], Longo et al. [48] and Power and Stegall [49]; (b-f) The point reported by Wiebe and Gaddy [20]; All the curves calculated using the proposed model).In the T-P-m range covered by our model, H2 solubility increased with increasing pressure and decreased with increasing ionic strength.The temperature dependence of H2 solubility was more drastic (Figure 4).H2 solubility was slightly dependent on the temperature at low pressures (<200 bar) but decreased and then increased with increasing temperature at high pressures (>200 bar).The isobaric minimum solubility point was observed at ~320 K and 200 bar (Figure 4).for the calculation of Henry's constant.The functions are expressed using Equations ( 10)-( 14). ) Tables 8 and 9 list the molar heat of the solution and Henry's constants of H2 in water, respectively, obtained experimentally and calculated using our model.When the temperature was high, Henry's constants were closer only at higher pressures.There will still be small errors, but this is an acceptable range.Both sets of results demonstrated good agreement, affirming the reliability of the model from another perspective.The aforementioned solubility model may also be used to calculate the partial molar volume V H 2 (l) , Henry's constant k H , and the heat of solution H s m of H 2 in aqueous NaCl solutions.At a given temperature, we can set P to 20 times of P S H 2 O for the calculation of Henry's constant.The functions are expressed using Equations ( 10)- (14).
( ∂Par(T, P) ∂P ) T,m ( ∂Par(T, P) ∂T Tables 8 and 9 list the molar heat of the solution and Henry's constants of H 2 in water, respectively, obtained experimentally and calculated using our model.When the temperature was high, Henry's constants were closer only at higher pressures.There will still be small errors, but this is an acceptable range.Both sets of results demonstrated good agreement, affirming the reliability of the model from another perspective.A model developed using the specific interaction approach can be evaluated using binary and ternary data and then applied to more complicated systems [55].Seawater often contains NaCl, KCl, MgCl 2 , CaCl 2 , and sulfate, as well as carbonate salts, although NaCl commonly dominates.As an example, Table 10 lists the main components in seawater with a salinity of 34.7‰.As data were limited, the model could only be directly fitted to the experimental results for the H 2 -NaCl-H 2 O system.We incorporated Ca 2+ , K + , Mg 2+ , and SO 4 2− into this model to tackle more complicated systems, utilizing the approximation suggested by Duan et al. [26].The interaction parameters (λ and ζ) of ions with the same charge achieved approximately the same values.Within experimental accuracy, the interaction parameters of CH 4 -bivalent cations were approximately double those of CH 4 -monovalent cations at different temperatures and pressures.The interaction parameters of CH 4 -anions were relatively small and therefore contributed little to the calculations.Hence, Duan et al. approximated all interaction parameters of CH 4 -monovalent cations and CH 4 -bivalent cations using λ CH 4 −Na and 2λ CH 4 −Na , respectively [26].By adopting the same approach, we approximated the H 2 solubility in seawater-type brines by setting the interaction parameters of H  (15) where λ H 2 −SO 2− 4 = −3.572.To check the accuracy of the approximation, we compared the calculated results of Equation ( 15) with the experimental data on the solubility of H 2 in seawater (Figure 5).The modeled results (p = 1 atm and T < 290 K) demonstrated excellent agreement with the experimental measurements.

3 Figure 1 .
Figure1.Pressure-dependent water content in the vapor phase as predicted using the model (The point reported by Wiebe and Gaddy[20]; The curves calculated using the proposed model).

Figure 1 .
Figure1.Pressure-dependent water content in the vapor phase as predicted using the model (The point reported by Wiebe and Gaddy[20]; The curves calculated using the proposed model).

Figures 2
Figures 2 and 3 show a comparison of the experimental data with the results predicted using our model.The model adequately represented most of the experimental data, remaining within or close to the experimental uncertainty (~1.14%).

Figures 2
Figures 2 and 3 show a comparison of the experimental data with the results predicted using our model.The model adequately represented most of the experimental data, remaining within or close to the experimental uncertainty (~1.14%).

Figure 3 .
Figure 3. H 2 solubility versus pressure in aqueous NaCl solutions with different concentrations: model predictions (colored lines) and experimental data (colored circles).((a) The point reported by Crozier and Yamamoto [38]; (b-d) The point reported by Chabab et al. [21]; All the curves calculated using the proposed model).

Figure 4 .
Figure 4. Isobaric minimum solubilities of H2 in pure water.The aforementioned solubility model may also be used to calculate the partial molar volume 2 ( ) H l V , Henry's constant H k , and the heat of solution

Figure 4 .
Figure 4. Isobaric minimum solubilities of H 2 in pure water.

Table 2 .
Aqueous H 2 solubility measurements in the literature.
+ denotes that the partial pressure of hydrogen is 1 atm.N a , number of measurements.

Table 8 .
Molar heat of solution of H2 in water.

Table 8 .
Molar heat of solution of H 2 in water.

Table 10 .
Main components of seawater with a salinity of 34.7‰.