An Accurate Model for Estimating H₂ Solubility in Pure Water and Aqueous NaCl Solutions

Abstract

By employing a specific particle interaction theory and a high-precision equation of states for the liquid and vapor phases of H₂, respectively, a new H₂ solubility model in pure water and aqueous NaCl solutions is proposed in this study. The model established by fitting the experimental data of H₂ solubility can be used to estimate H₂ 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 H₂ 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 H₂ solubilities was less than 1.14%.

1. Introduction

Hydrogen (H₂) is an important natural gas because it is light, storable, and reactive [1]. H₂ 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 H₂ does not emit pollutants and greenhouse gases; the only combustion product is H₂O; on the other hand, H₂ has a high calorific value of ~140 MJ/kg [3]. H₂ 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 H₂ in gas−liquid systems for the migration of fluids and the alteration of minerals induced during storage [7]. Moreover, H₂ 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, H₂ 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 H₂ solubility data at various temperatures and pressures. Additionally, H₂ solubility data have been accumulating since 1855. Using the pure physical absorption method, Bunsen [19] measured the absorption coefficients of H₂ in pure water at various temperatures (277.15–296.75 K) and atmospheric pressure. However, because H₂ 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 (273.15–373.15 K) 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 H₂ solubility in pure water and aqueous NaCl solutions (1, 3, and 5 mol/kg) at different temperatures (323.18–372.76 K) and pressures (28.623–229.720 bar).

A model based on experimental data can be used to predict H₂ solubility in an unmeasured system. Jauregui-Haza et al. [22] studied H₂ 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 H₂ at temperatures of 353, 363, and 373 K. The H₂ solubility error in the aforementioned solvents was ~2.6%; however, the model was only applicable to pure aqueous solutions and the Henry constant of H₂ was not determined in aqueous NaCl solutions. Li et al. [7] considered the system pressure, temperature, and formation fluid salinity in an H₂ solubility model. Their model reproduced all available experimental data and accurately predicted H₂ 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, H₂ 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 H₂ solubility using a fast method based on a Setschenow-type relation [24], which predicted H₂ 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 273.15–373.15 K and 1–203 bar, respectively, in pure water, and 323.15–373.15 K 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 H₂ solubility in nature. Torín-Ollarves and Trusler [25] proposed a simple model based on an analysis method for predicting H₂ solubility in aqueous solutions at temperatures and pressures of 273.15–423.15 K 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 high-precision 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 273.15–523.15 K 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., CaCl₂, KCl, and seawater) using the approximation principle. The calculated results were consistent with the experimental data. Later, the solubilities of N₂, CO₂, C₂H₆, and O₂ 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 H₂ solubility models for the H₂ + H₂O system and the H₂ + H₂O + 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.

2. H₂ Solubility Model

H₂ solubility in aqueous solutions was determined based on the balance between the chemical potentials of H₂ 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)):

$$\begin{array}{cc}{\mu }_{H_2}^l\left(T,P,m\right)& ={\mu }_{H_2}^{l\left(0\right)}\left(T\right)+RT\ln a_{H_2}\left(T,P,m\right)\\ & ={\mu }_{H_2}^{l\left(0\right)}\left(T,P\right)+RT\ln m_{H_2}+RT\ln {\gamma }_{H_2}\left(T,P,m\right)\end{array}$$

(1)

$$\begin{array}{cc}{\mu }_{H_2}^{\nu }\left(T,P,y\right)& ={\mu }_{H_2}^{\nu \left(0\right)}\left(T\right)+RT\ln f_{H_2}\left(T,P,y\right)\\ & ={\mu }_{H_2}^{\nu \left(0\right)}\left(T\right)+RT\ln y_{H_2}P+RT\ln {\phi }_{H_2}\left(T,P,y\right)\end{array}$$

(2)

where ${\mu }_{H_2}^{l\left(0\right)}$ and ${\mu }_{H_2}^{v\left(0\right)}$ represent the standard chemical potentials of H₂ in liquid and vapor phases, respectively. Here, ${\mu }_{H_2}^{l\left(0\right)}$ denotes the chemical potential in a hypothetical ideal solution of unit molality [34], and ${\mu }_{H_2}^{v\left(0\right)}$ denotes the chemical potential when the pressure of a hypothetical ideal gas is set to 1 bar.

At phase equilibrium ${\mu }_{H_2}^l={\mu }_{H_2}^v$, subsequently, we obtain Equation (3).

$$\begin{array}{cc}\ln \frac{y_{H_2}P}{m_{H_2}}& =\frac{{\mu }_{H_2}^{l\left(0\right)}\left(T,P\right)-{\mu }_{H_2}^{v\left(0\right)}\left(T\right)}{RT}-\ln {\phi }_{H_2}\left(T,P,y\right)\\ & +\ln {\gamma }_{H_2}\left(T,P,m\right)\end{array}$$

(3)

In parameterization, reference value ${\mu }_{H_2}^{v\left(0\right)}$ can be set to 0 for convenience as only the difference between ${\mu }_{H_2}^{l\left(0\right)}$ and ${\mu }_{H_2}^{v\left(0\right)}$ is important. Since the vapor phase has low water content, the fugacity coefficient of H₂ in gaseous mixtures is approximate to that of pure H₂ in the studied region. Therefore, $\ln {\phi }_{H_2}$ can be approximated from the equation of state of pure H₂ (refer to Appendix A) [32]. The mole fraction $y_{H_2}$. of H₂ in the gas is calculated as follows.

$$y_{H_2}=1-y_{H_{2}O}=\frac{P-P_{H_{2}O}}{P}$$

(4)

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], $\frac{{\mu }_{H_2}^{l\left(0\right)}}{RT}$ and $\ln {\gamma }_{H_2}^l$ 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:

$$y_{H_{2}O}=\frac{x_{H_{2}O}{P}_{H_{2}O}^S}{{\phi }_{H_{2}O}P}{e}^{\frac{v_{H_{2}O}^l\left(P-P_{H_{2}O}^S\right)}{RT}}$$

(5)

where $x_{H_{2}O}$ represents the mole fraction of H₂O in the liquid phase, which is approximated as 1 and 1–2$x_{NaCl}$ in the H₂ + H₂O and H₂ + H₂O + NaCl systems, respectively, when dissolved hydrogen is neglected. The saturation pressure $P_{H_{2}O}^S$ (in bar) of water was calculated using an empirical equation (refer to Appendix B). The molar volume $v_{H_{2}O}^l$ 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 ${\phi }_{H_{2}O}$ of water was calculated using the following equation, which is obtained by fitting the methane–water experimental data [30].

$${\phi }_{H_{2}O}=e^{a_1+a_{2}P+a_3{P}^2+a_{4}PT+\frac{a_{5}P}{T}+\frac{a_6{P}^2}{T}}$$

(6)

The values of $a_1-a_6$ are listed in

Table 1. Parameters of Equation (6) [30].

Parameters	Values
$a_1$	−1.42006707 × 10−2
$a_2$	1.08369910 × 10−2
$a_3$	−1.59213160 × 10−6
$a_4$	−1.10804676 × 10−5
$a_5$	−3.14287155 × 10
$a_6$	1.06338095 × 10−3

. 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.

$\ln {\gamma }_{H_2}$ is expressed as a virial expansion of excess Gibbs energy [33]:

$$\ln {\gamma }_{H_2}={\displaystyle \sum }_{c}2{\lambda }_{H_2-c}{m}_c+{\displaystyle \sum }_{a}2{\lambda }_{H_2-a}{m}_a+{\displaystyle \sum }_c{\displaystyle \sum }_a{\zeta }_{H_2-c-a}{m}_c{m}_a$$

(7)

where $\lambda$ and $\zeta$ 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.

$$\begin{array}{cc}\ln \frac{y_{H_2}P}{m_{H_2}}& =\frac{{\mu }_{H_2}^{l\left(0\right)}\left(T,P\right)}{RT}-\ln {\phi }_{H_2}\left(T,P,y\right)\\ & +{\displaystyle \sum }_{c}2{\lambda }_{H_2-c}{m}_c+{\displaystyle \sum }_{a}2{\lambda }_{H_2-a}{m}_a+{\displaystyle \sum }_c{\displaystyle \sum }_a{\zeta }_{H_2-c-a}{m}_c{m}_a\end{array}$$

(8)

Following Pitzer et al. [36], we selected the following equation for the T–P dependences of $\lambda$, $\zeta$, and $\frac{{\mu }_{H_2}^{l\left(0\right)}}{RT}$.

$$Par\left(T,P\right)=c_1+c_{2}T+\frac{c_3}{T}+c_4{T}^2+c_{5}P+\frac{c_{6}P}{T^2}+\frac{c_7}{P}+c_8\frac{T}{P}+c_9\frac{T^2}{P}+c_{10}\frac{T^3}{P}$$

(9)

The basis of our model parameterization consists of Equations (8) and (9).

3. Results and Discussion

3.1. Review of H₂ Solubility Data

H₂ 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 H₂ + H₂O system in this model were determined to be 273.15–423.15 K and 0–1100 bar, respectively.

Alternatively, the experimental data of H₂ solubility in aqueous NaCl solutions showed poorer continuity and a narrower range than those in pure water. The measured values of H₂ 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 H₂ + H₂O + NaCl system yielded temperature, pressure, and salinity ranges of 273.15–373.15 K, 0–230 bar, and 0–5 mol/kg, respectively.

The H₂ solubility has been measured in solutions other than aqueous NaCl solutions. For example, Braun [37] measured the H₂ solubility in 0.16–0.34 mol/kg BaCl₂ solution. Thomas et al. [32] and Gordon et al. [39] measured H₂ 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.

Table 2. Aqueous H₂ solubility measurements in the literature.

References	System	T (K)	P (bar)	Na
Bunsen [19]	Water	277.15–296.75	1+	7
Timofejew [40]	Water	274.55–298.85	1+	5
Bohr and Bock [41]	Water	273.15–373.15	1+	48
Winkler [42]	Water	273.65–323.58	1+	6
Braun [37]	Water	278.15–298.15	1+	5
	0.21–4.03 m NaCl	278.15–298.15	1+	5
	0.16–0.34 m BaCl2	278.15–298.15	1+	5
Ipatiew jun et al. [43]	Water	273.65–318.15	20.265–141.855	17
Wiebe and Gaddy [20]	Water	273.65–373.15	25.331–1013.250	40
Morrison and Billett. [44]	Water	285.65–345.65	1+	12
Pray et al. [45]	Water	324.82–588.71	6.9–24.150	9
Ruetschi and Amlie [46]	Water	303.15	1+	1
	0.0011–15.2% H2SO4	303.15	1+	10
	0.0091–10.23% KOH	303.15	1+	9
Shoor et al. [47]	Water	298.15-333.15	1+	3
Longo et al. [48]	Water	310.15	1+	1
Power and Stegall [49]	Water	310.15	1+	1
Croizer and Yamamoto [38]	Water	274.60–302.47	1+	42
	27.665–39.927‰ Seawater	274.65–303.49	1+	180
	10.950–27.376‰ NaCl	274.03–301.51	1+	10
Gordon et al. [39]	Water	273.29–302.40	1+	7
	4.919–39.096‰ Seawater	272.80–302.41	1+	32
Chou D Hary et al. [50]	Water	323.15–373.15	25.331–101.325	4
Alvarez et al. [51]	Water	318.90–636.10	6.78–284.5	26
Kling and Maurer [52]	Water	323.15–423.15	31.8–153.7	10
Jauregui-Haza et al. [22]	Water	353.15–373.15	1+	3
Chabab et al. [21]	Water	323.18–372.73	29.272–121.706	6
	1–5 m NaCl	323.19–372.78	28.623–229.720	31
Torín-Ollarves and Trusler [25]	2.5 m NaCl	323.15–423.15	116.4–458.1	10

Note. 1⁺ denotes that the partial pressure of hydrogen is 1 atm. N^a, number of measurements.

Table 2. Aqueous H₂ solubility measurements in the literature.

References	System	T (K)	P (bar)	Na
Bunsen [19]	Water	277.15–296.75	1+	7
Timofejew [40]	Water	274.55–298.85	1+	5
Bohr and Bock [41]	Water	273.15–373.15	1+	48
Winkler [42]	Water	273.65–323.58	1+	6
Braun [37]	Water	278.15–298.15	1+	5
	0.21–4.03 m NaCl	278.15–298.15	1+	5
	0.16–0.34 m BaCl2	278.15–298.15	1+	5
Ipatiew jun et al. [43]	Water	273.65–318.15	20.265–141.855	17
Wiebe and Gaddy [20]	Water	273.65–373.15	25.331–1013.250	40
Morrison and Billett. [44]	Water	285.65–345.65	1+	12
Pray et al. [45]	Water	324.82–588.71	6.9–24.150	9
Ruetschi and Amlie [46]	Water	303.15	1+	1
	0.0011–15.2% H2SO4	303.15	1+	10
	0.0091–10.23% KOH	303.15	1+	9
Shoor et al. [47]	Water	298.15-333.15	1+	3
Longo et al. [48]	Water	310.15	1+	1
Power and Stegall [49]	Water	310.15	1+	1
Croizer and Yamamoto [38]	Water	274.60–302.47	1+	42
	27.665–39.927‰ Seawater	274.65–303.49	1+	180
	10.950–27.376‰ NaCl	274.03–301.51	1+	10
Gordon et al. [39]	Water	273.29–302.40	1+	7
	4.919–39.096‰ Seawater	272.80–302.41	1+	32
Chou D Hary et al. [50]	Water	323.15–373.15	25.331–101.325	4
Alvarez et al. [51]	Water	318.90–636.10	6.78–284.5	26
Kling and Maurer [52]	Water	323.15–423.15	31.8–153.7	10
Jauregui-Haza et al. [22]	Water	353.15–373.15	1+	3
Chabab et al. [21]	Water	323.18–372.73	29.272–121.706	6
	1–5 m NaCl	323.19–372.78	28.623–229.720	31
Torín-Ollarves and Trusler [25]	2.5 m NaCl	323.15–423.15	116.4–458.1	10

Note. 1⁺ denotes that the partial pressure of hydrogen is 1 atm. N^a, number of measurements.

3.2. Construction and Validation of the Model

3.2.1. Determination of Model and Calculation of H₂ Solubility

To estimate H₂ solubility as a function of temperature, pressure, and salinity, we must determine parameters $\lambda$ and $\zeta$ of Na⁺ and Cl⁻ in the liquid phase and the standard chemical potential ${\mu }_{H_2}^{l\left(0\right)}$ in Equation (8). Because measurements can only be performed in electronically neutral solutions, one of the parameters must be assigned arbitrarily [53]. We set ${\lambda }_{H_2-Cl}$ to zero and fitted the remaining parameters. First, $\frac{{\mu }_{H_2}^{l\left(0\right)}}{RT}$ was evaluated using the H₂ solubility data for pure water (92 related experimental data values), with a root mean square error of 1.62. Next, ${\lambda }_{H_2-Na}$ and ${\zeta }_{H_2-Na-Cl}$ 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. Values of the interaction parameters in Equation (9).

Parameters	$\frac{{\mathit{\mu }}_{{\mathit{H}}_{\mathbf{2}}}^{\mathit{l}\left(0\right)}\left(\mathit{T}\right)}{\mathit{R}\mathit{T}}$	${\mathit{\lambda }}_{{\mathit{H}}_{\mathbf{2}}-\mathit{c}}$	${\mathit{\lambda }}_{{\mathit{H}}_{\mathbf{2}}-\mathit{a}}$	${\mathit{\zeta }}_{{\mathit{H}}_{\mathbf{2}}-\mathit{c}-\mathit{a}}$
c1	4.18266086 × 101	−7.68559552 × 100	0	−1.44839161 × 10−2
c2	−8.24713967 × 10−2	1.91233146 × 10−2
c3	−4.60318630 × 103	1.04890475 × 103
c4	6.03537635 × 10−5	−1.52746819 × 10−5
c5	4.12979459 × 10−4	1.59803686 × 10−4
c6	1.82081207 × 101	−1.92667249 × 101
c7	3.73478602 × 101	−4.75822792 × 101
c8	−3.87633253 × 10−1	4.72712503 × 10−1
c9	1.34370747 × 10−3	−1.56750050 × 10−3
c10	−1.55621990 × 10−6	1.73272315 × 10−6

.

Substituting the parameters (Table 3) into Equation (8), we obtain the H₂ 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

Table 4. Calculated H₂ solubility (mol/kg) in pure water.

P (Bar)	T (K)
	273.15	303.15	333.15	363.15	393.15	423.15
1	0.00095	0.00075	0.00070	0.00046	0	0
50	0.04700	0.03719	0.03560	0.03828	0.04370	0.05039
100	0.09248	0.07334	0.07030	0.07565	0.08657	0.10094
150	0.13675	0.10864	0.10426	0.11225	0.12856	0.15039
200	0.18006	0.14325	0.13758	0.14818	0.16980	0.19892
300	0.26458	0.21086	0.20269	0.21838	0.25031	0.29359
400	0.34720	0.27691	0.26622	0.28677	0.32863	0.38553
500	0.42864	0.34188	0.32857	0.35373	0.40514	0.47517
600	0.50942	0.40612	0.39002	0.41954	0.48013	0.56282
700	0.58990	0.46987	0.45080	0.48441	0.55384	0.64876
800	0.67033	0.53333	0.51105	0.54849	0.62644	0.73319
900	0.75091	0.59664	0.57091	0.61191	0.69806	0.81626
1000	0.83180	0.65989	0.63046	0.67477	0.76880	0.89811
1100	0.91310	0.72318	0.68979	0.73713	0.83875	0.97881

,

Table 5. Calculated H₂ solubility (mol/kg) in a 1 mol/kg NaCl solution.

P (Bar)	T (K)
	273.15	293.15	313.15	333.15	353.15	373.15
1	0.00073	0.00063	0.00060	0.00055	0.00036	0
50	0.03002	0.02767	0.02700	0.02743	0.02858	0.03014
100	0.05953	0.05481	0.05341	0.05423	0.05663	0.06014
150	0.08884	0.08164	0.07942	0.08053	0.08406	0.08940
200	0.11810	0.10828	0.10512	0.10642	0.11097	0.11800
250	0.14743	0.13482	0.13059	0.13195	0.13741	0.14601

,

Table 6. Calculated H₂ solubility (mol/kg) in a 3 mol/kg NaCl solution.

P (Bar)	T (K)
	273.15	293.15	313.15	333.15	353.15	373.15
1	0.00047	0.00043	0.00046	0.00036	0.00012	0
50	0.01336	0.01504	0.01654	0.01776	0.01857	0.01882
100	0.02691	0.03011	0.03288	0.03520	0.03702	0.03824
150	0.04091	0.04539	0.04921	0.05241	0.05502	0.05703
200	0.05542	0.06096	0.06559	0.06945	0.07264	0.07524
250	0.07053	0.07687	0.08206	0.08635	0.08992	0.09292

and

Table 7. Calculated H₂ solubility (mol/kg) in a 5 mol/kg NaCl solution.

P (Bar)	T (K)
	273.15	293.15	313.15	333.15	353.15	373.15
1	0.00034	0.00033	0.00039	0.00027	0.00004	0
50	0.00668	0.00918	0.01137	0.01291	0.01355	0.01319
100	0.01366	0.01857	0.02273	0.02566	0.02718	0.02731
150	0.02115	0.02834	0.03424	0.03830	0.04044	0.04086
200	0.02920	0.03853	0.04595	0.05089	0.05340	0.05388
250	0.03789	0.04921	0.05790	0.06345	0.06608	0.06640

.

3.2.2. Model Validation

Figure 2 and Figure 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, 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).

The aforementioned solubility model may also be used to calculate the partial molar volume ${\overline{V}}_{H_2\left(l\right)}$, Henry’s constant $k_H$, and the heat of solution $△H_m^s$ of H₂ in aqueous NaCl solutions. At a given temperature, we can set P to 20 times of $P_{H_{2}O}^S$ for the calculation of Henry’s constant. The functions are expressed using Equations (10)–(14).

$$\frac{{\overline{V}}_{H_2\left(l\right)}}{RT}=\frac{\partial }{\partial P}{(\frac{{\mu }^{l\left(0\right)}}{RT})}_{T,m}+{(\frac{\partial \ln {\gamma }_{H_2}}{\partial P})}_{T,m}$$

(10)

$$k_H\left(T\right)=\frac{y_{H_2}{\phi }_{H_2}P}{x_{H_2}}{e}^{\frac{-{\overline{V}}_{H_2\left(l\right)}\left(P-P_{H_{2}O}^s\right)}{RT}}$$

(11)

$$-\frac{△H_m^s}{R{T}^2}=\frac{\partial }{\partial T}{(\frac{{\mu }^{l\left(0\right)}}{RT})}_{P,m}+{(\frac{\partial \ln {\gamma }_{H_2}}{\partial T})}_{P,m}$$

(12)

$${(\frac{\partial Par\left(T,P\right)}{\partial P})}_{T,m}=c_5+\frac{c_6}{T^2}-\frac{c_7}{P^2}-\frac{c_{8}T}{P^2}-\frac{c_9{T}^2}{P^2}-\frac{c_{10}{T}^3}{P^2}$$

(13)

$${(\frac{\partial Par\left(T,P\right)}{\partial T})}_{P,m}=c_2-\frac{c_3}{T^2}+2c_{4}T-2\frac{c_{6}P}{T^3}+\frac{c_8}{P}+2\frac{c_{9}T}{P}+3\frac{c_{10}{T}^2}{P}$$

(14)

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

T (K)	P (Bar)	$\mathbf{-}\mathbf{\Delta }{\mathit{H}}_{\mathit{m}}^{\mathit{s}}$ (KJ/mol)
275.15	1	62.30
280.15	1	57.58
285.15	1	51.59
290.15	1	44.15
295.15	1	35.12
300.15	1	24.33
305.15	1	11.60

and

Table 9. Henry’s constant ($k_H$) of H₂ in pure water ($k_{H1}$ reported by Fernandez-Prini and Roberto [54]; $k_{H2}$ calculated using the proposed model).

T (K)	${\mathit{P}}_{{\mathit{H}}_{\mathbf{2}}\mathit{O}}^{\mathit{S}} \left(\mathbf{Bar}\right)$	${\mathit{k}}_{\mathit{H}\mathbf{1}}$	${\mathit{k}}_{\mathit{H}\mathbf{2}}$
279.15	0.009	60,739.000	60,742.2020
290.15	0.019	67,113.000	65,737.9586
300.15	0.036	71,721.108	69,872.8408
350.15	0.420	77,317.000	71,095.8440
400.15	2.475	63,369.653	58,877.2595
423.15	4.751	54,667.000	53,262.5159

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.

3.3. H₂ Solubility in Seawater: Extrapolation of the Model

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, MgCl₂, CaCl₂, and sulfate, as well as carbonate salts, although NaCl commonly dominates. As an example,

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

Components	Concentration (mol/kg)
Cl−	0.5405
Na+	0.4645
K+	0.01022
Ca2+	0.4096
Mg2+	0.0526
SO42−	0.0279

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 H₂–NaCl–H₂O system. We incorporated Ca²⁺, K⁺, Mg²⁺, and SO₄²⁻ into this model to tackle more complicated systems, utilizing the approximation suggested by Duan et al. [26].

The interaction parameters ($\lambda$ and $\zeta$) of ions with the same charge achieved approximately the same values. Within experimental accuracy, the interaction parameters of CH₄–bivalent cations were approximately double those of CH₄–monovalent cations at different temperatures and pressures. The interaction parameters of CH₄–anions were relatively small and therefore contributed little to the calculations. Hence, Duan et al. approximated all interaction parameters of CH₄–monovalent cations and CH₄–bivalent cations using ${\lambda }_{C{H}_4-Na}$ and 2${\lambda }_{C{H}_4-Na}$, respectively [26]. By adopting the same approach, we approximated the H₂ solubility in seawater-type brines by setting the interaction parameters of H₂–monovalent cations and H₂–bivalent cations as ${\lambda }_{H_2-Na}$ and 2${\lambda }_{H_2-Na}$, respectively. All ternary parameters were treated similarly. With this simplification, we achieved the following:

$$\begin{array}{cc}\ln m_{H_2}& =\ln y_{H_2}P+\ln {\phi }_{H_2}-\frac{{\mu }_{H_2}^{l\left(0\right)}\left(T\right)}{RT}\\ & -2{\lambda }_{H_2-N{a}^{+}}\left(m_{N{a}^{+}}+m_{K^{+}}+2m_{C{a}^{+}}+2m_{M{g}^{+}}\right)\\ & -{\zeta }_{H_2-N{a}^{+}-C{l}^{-}}\left(m_{N{a}^{+}}+m_{K^{+}}+2m_{C{a}^{+}}+2m_{M{g}^{+}}\right)\times \left(m_{C{l}^{-}}+2m_{S{O}_4^{2-}}\right)\\ & -2{\lambda }_{H_2-S{O}_4^{2-}}{m}_{S{O}_4^{2-}}\end{array}$$

(15)

where ${\lambda }_{H_2-S{O}_4^{2-}}$= −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 H₂ in seawater (Figure 5). The modeled results (p = 1 atm and T < 290 K) demonstrated excellent agreement with the experimental measurements.

4. Conclusions

By applying a high precision equation of state to the vapor phase and the theory of liquid electrolyte solutions proposed by Pitzer to the liquid phase, we developed an accurate model of H₂ solubility in pure water and aqueous NaCl solutions. The results were within or close to the experimental uncertainty in pure water (273.15–423.15 K and 0–1100 bar) and aqueous NaCl solutions (273.15–373.15 K, 0–230 bar, and 0–5 mol/kg). After a simple extrapolation, the model predicted H₂ solubility in complex aqueous solutions such as seawater containing Na⁺, K⁺, Mg²⁺, Ca²⁺, Cl⁻, and SO₄²⁻ with remarkable accuracy. The mean absolute percentage error between this model and experimentally obtained H₂ solubilities was less than 1.14%. The model can calculate hydrogen solubility not only in the subsea environments but also under several typical hydrogen geological storage conditions.