A Thermodynamic Model for the Solubility of SO₂ in Multi-Ion Electrolyte Solutions and Its Applications

Abstract

A solubility model of SO₂ in multi-ion electrolyte solutions has been developed by the activity-fugacity relation at vapor-liquid equilibria. The fugacity coefficient of SO₂ in the vapor phase is calculated by the equation of state (EOS) of pure SO₂, and the activity coefficient of SO₂ in the liquid phase is calculated by the Pitzer activity coefficient theory. The model can reproduce the reliable solubility data of SO₂ in pure water and multi-ion electrolyte solutions (Na⁺, K⁺, Cl⁻, ${\mathrm{SO}}_4^{2-}$) within or close to experimental uncertainties. Although the second-order and third-order interaction parameters between SO₂ and Mg²⁺ and Ca²⁺ have been adopted by an approximation, the solubility model can also be extended to predict the SO₂ solubility in seawater. In addition, combining with the EOS of a CO₂-SO₂ fluid mixture, the model can be used to predict the solubility of a CO₂-SO₂ mixture in aqueous electrolyte solutions. The calculated results are consistent with experimental data, which indicates that the solubility model has certain predictive ability.

1. Introduction

SO₂ in the atmosphere mainly comes from volcanic eruptions [1], burning fossil fuels in power plants [2], metal processing and smelting [3]. In the process of CO₂ capture and sequestration (CCS), SO₂ is also an impurity, and to sequester it with CO₂ in the deep saline aquifers can reduce the cost. Deep saline aquifers contain a variety of electrolyte ions (e.g., Na⁺, K⁺, Ca⁺, Mg²⁺, Cl⁻, ${\mathrm{SO}}_4^{2-}$). Therefore, it is essential to study the solubility of single gases and mixed gases in multi-ion electrolyte solutions. The solubility model of CO₂ in multi-ion electrolyte solutions has been reported in our previous studies [4,5], so this work focuses on the solubility model of SO₂ in multi-ion electrolyte solutions under different temperature-pressure (T-P) conditions. This research is of great practical significance for the co-storage of CO₂-SO₂ gas mixturlles in deep saline aquifers.

Due to the strong corrosive potential of SO₂, experimental solubility data for SO₂ in multi-ion electrolyte solutions are difficult to collect, especially at high temperatures and pressures. Therefore, predictive SO₂ solubility models are desired. In recent decades, many scholars have developed thermodynamic models to predict SO₂ solubility in multi-ion electrolyte solutions (

Table 1. Thermodynamic models for calculating SO₂ solubilities in pure water, aqueous salt solutions or seawater.

References	Salt	T (K)	P (bar)	${\mathit{m}}_{\mathbf{salt}}$ (mol/kg)
Hunger et al. [6]	H2SO4, Na2SO4	298–323	10−4–0.1	0-saturation
Xia et al. [7]	NaCl, NH4Cl	313–393	0–37	0–6
Rodríguez-Sevilla et al. [8]	seawater	278.15–318.15	0.976	unstated
Mondal [9]	water	293–333	0–0.01 *	0
Zimmermann et al. [10]	HCl, NaCl, HBr, NaBr	298	1.015	0–4.613
Shaw et al. [11]	water	298–313	0.2–3.6	0
Tan et al. [12]	water, NaCl	313–393	0–40	0–6
Cox et al. [13]	seawater, brine	298.15–323.15	unstated	0–1
Zhou et al. [14]	water	263.15–393.15	10–300	0
Saadallah et al. [15]	water	293–393	0–25	0

Notes: T, P and $m_{\mathrm{salt}}$ denote the temperature, pressure and molality of salt, respectively; * is the partial pressure of SO₂.

).

Hunger et al. [6] applied the methods from Meissner [16,17] and Bromley [18] to estimate activity coefficients, which can predict the solubility of SO₂ in Na₂SO₄ or H₂SO₄ solutions from 10⁻⁴ to 0.1 bar at 298 K and 323 K. Xia et al. [7] developed a phase equilibrium model to calculate SO₂ solubility in NaCl and NH₄Cl aqueous solutions, from 313 to 393 K at total pressures up to 37 bar, which is capable of representing most of the experimental data within experimental uncertainties. Rodríguez-Sevilla et al. [8] adopted the extended Debye-Hückel theory [19] and the Pitzer [20] ion-interaction model to evaluate the activity coefficients of ionic species, which can be used to predict the absorption equilibria of dilute SO₂ in seawater in the temperature range from 278.15 to 318.15 K with a pressure of up to 0.976 bar. Mondal [9] proposed a model for the solubility of SO₂ using Henry’s Law in pure water with temperatures ranging between 293 and 333 K, with SO₂ partially pressurized up to 0.01 bar. Zimmermann et al. [10] developed a thermodynamic model with the Pitzer activity coefficient theory and the Sechenov’s approach from Weisenberger and Schumpe [21], which can predict the SO₂ solubility in HCl, NaCl, HBr and NaBr electrolyte solutions. The applicable P-T-m range is 298 K, 1.015 bar, 0–4.61 kg/mol. The developed thermodynamic model by Zimmermann et al. [10] is in good agreement with experimental data. Shaw et al. [11] developed a solubility model of SO₂ in pure water with a valid T-P range of 298–313 K and 0.2–3.6 bar. The model is in good agreement with experimental data at lower pressures, but improvements are needed at higher pressures (p > 1.5 bar). Tan et al. [12] adopted the PC-SAFT/PMSA and CPA EOS to evaluate SO₂ solubility in pure water and NaCl electrolyte solutions up to 393 K and 40 bar and 6 mol/kg. Miri et al. [22] applied the SAFT1 model to estimate CO₂-SO₂ solubility in pure water, covering a valid T-P range of 303.15–373.15 K and 60–300 bar. Cox et al. [13] developed an equilibrium model by the Pitzer and Sechenov approach, which provides a reliable prediction of SO₂ solubility in seawater and brine at 298 K and 323 K with molality of electrolytes up to 1 mol/kg. Zhou et al. [14] proposed a PC-SAFT model to predict the SO₂ solubility in water, covering a valid T-P range of 263.15–393.15 K and 10–300 bar, which can accurately predict the SO₂ solubility in water above 10 bar and 273.15 K, but at lower temperatures and pressures, it cannot accurately calculate the SO₂ solubility. Saadallah et al. [15] used the Monte Carlo (MC) molecular simulation technique to predict the vapor-liquid and liquid-liquid equilibria including the solubility data of SO₂-H₂O mixture in the temperature range of 293–393 K and pressures up to 25 bar, and the results from the MC simulation are in agreement with experimental data at liquid-vapor and liquid-liquid equilibria.

As mentioned above, the models of SO₂ solubility in multi-ion electrolyte solutions are still lacking, and most of them either have lower accuracy or have a limited temperature-pressure-salinity range. Therefore, it is necessary to develop an accurate model for the solubility of SO₂ in multi-ion electrolyte solutions. In this work, we first review experimental data of SO₂ solubility in pure water and multi-ion electrolyte solutions. Then an accurate thermodynamic model is developed to predict the SO₂ solubility in water and multi-ion electrolyte solutions by the activity-fugacity relationship at vapor-liquid equilibria. Finally, the applications of the model are briefly discussed.

2. Review of Experimental Data for SO₂ Solubility in Pure Water and Aqueous Electrolyte Solutions

In this article, the solubility data of four different systems (SO₂-H₂O, SO₂-NaCl-H₂O, SO₂-KCl-H₂O and SO₂-Na₂SO₄-H₂O) have been found and compiled as shown in

Table 2. SO₂ solubility measurements in pure water and aqueous single-salt solutions.

References	Solution	T (K)	P (bar)	Nd
Bunsen [23]	water	273.00–313.00	1+	41
Sims [24]	water	281.20–323.20	1+	22
Fox [25]	water	298.20–308.20	1+	2
Simth and Parkhurst [26]	water	278.20–333.20	0.20–1.60	8
Hudson [27]	water	283.15–363.15	1+	42
Sherwood [28]	water	273.15–323.15	0.00–1.00	109
Maass and Maass [29]	water	283.20–300.20	0.00–3.50	29
Campbell and Maass [30]	water	298.15–383.15	0.14–4.50	55
Morgan and Maass [31]	water	273.00–298.00	0.00–1.20	38
Conrad and Beuschlein [32]	water	298.20	0.40–1.00	6
Johnstone and Leppla [33]	water	298.20–323.20	0.00–0.20	16
Young [34]	water	373.00–423.00	1.70–7.20	22
Beuschlein and Simenson [35]	water	296.40–386.15	0.12–2.11	53
Parkinson [36]	water	263.00–294.00	0.00–0.06	42
Rabe and Harris [37]	water	303.15–353.15	0.09–1.20	43
Vosolsobe et al. [38]	water	293.20–333.20	0.04–0.34	35
Tokunaga [39]	water	283.15–313.15	1+	4
Lavrova and Tudorovskaya [40]	water	299.15–363.15	1+	5
Douabul and Riley [41]	water	278.97–303.25	1+	6
Byerley et al. [42]	water	298.15–323.15	1+	2
Rumpf and Maurer [43]	water	293.14–393.33	0.35–25.09	66
Siddiqi et al. [44]	water	291.05–294.45	0.02–0.03	50
Mondal [9]	water	293.00–333.00	0.02–0.21	20
Shaw et al. [11]	water	297.05–312.25	0.25–1.96	3
Cox et al. [13]	water	298.15–323.15	0.00–0.12	19
Zhou et al. [14]	water	273.15–373.15	10.00–300.00	45
Saadallah et al. [15]	water	293.15–353.15	0.00–12.00	45
	$m_{\mathrm{NaCl}}$ (mol/kg)
Fox [25]	0.50–3.22	298.00–308.00	1+	6
Millero et al. [45]	0.01–6.00	278.15–298.15	1+	17
Xia et al. [7]	2.90–6.00	313.15–393.15	0.30–37.00	90
Zimmermann et al. [10]	0.49–4.28	298.15	0.00–0.30	37
Cox et al. [13]	1.00	298.15–323.15	0.03–0.12	19
	$m_{\mathrm{KCl}}$ (mol/kg)
Fox [25]	0.00–0.05	298.00–308.00	1+	12
Hudson [27]	0.50–4.40	298.00–308.00	1+	43
Cox et al. [13]	0.05–1.00	298.15–323.15	1.00	19
	$m_{{\mathrm{Na}}_2{\mathrm{SO}}_4}$ (mol/kg)
Fox [25]	0.50–3.22	298.00–308.00	1	12
Hudson [27]	0.13–1.41	293.15–313.15	1+	23
Rumpf and Maurer [46]	0.50–1.00	313.15–393.15	1.00–28.00	65

Notes: T, P and $m_{\mathrm{salt}}$ denote the temperature, pressure and molality of salt, respectively; Salt stands for NaCl, KCl and Na₂SO₄, respectively; Nd is the number of experimental data. 1⁺ is the partial pressure of SO₂.

, where the information of each data set is provided, including the temperature, pressure, salinity and total number of data points. SO₂ solubility in water represents the total amount of SO₂ species dissolved in water, including SO₂(aq), ${\mathrm{HSO}}_3^{-}$ and ${\mathrm{SO}}_3^{2-}$.

2.1. SO₂-H₂O Systems

Experimental data of SO₂ solubility in pure water are relatively extensive (Table 2). However, some of the solubility data are unreliable. Sherwood [28] measured the SO₂ solubility in water under relatively low pressure, but the data are quite different from the reliable data of others under the same conditions. Experimental data of SO₂ solubility from Johnstone and Leppla [33], Parkinson [36] and Vosolsobe et al. [38] were evaluated in the solubility data series book by Young [34]. Because their pressure is too low to be judged reasonably, experimental data of solubility have a large deviation. Experimental SO₂ solubility data of Siddiqi et al. [44], Mondal [9], Shaw et al. [11] and Cox et al. [13] are measured for the SO₂-N₂-H₂O system under extremely low pressures. The data yield big deviations when they are used in parameterization. Most of the experimental SO₂ solubility data of Morgan and Maass [31], Beuschlein and Simenson [35], Rabe and Harris [37], Zhou et al. [14] and Saadallah et al. [15] are consistent with one another. Figure 1 shows that the experimental SO₂ solubility data of Bunsen [23], Sims [24], Fox [25], Tokunaga [39], Lavrova and Tudorovskaya [40] and Byerley et al. [42] are consistent with each other when the partial pressure of SO₂ is 1 atm, but the data of Douabul and Riley [41] at 280 K yield a slight deviation. Therefore, all the data except the unreliable data mentioned above are used in the parameterization of this model, covering a wide T-P range of 273.15–393.15 K and 0.00–300.00 bar.

2.2. SO₂-NaCl-H₂O Systems

There are only five experimental data sets reported for SO₂-NaCl-H₂O systems (Table 2). The experimental SO₂ solubility data of Zimmermann et al. [10] and Cox et al. [13] are measured for SO₂-N₂-NaCl-H₂O systems under low pressures, and they are not used in the parameterization. The partially experimental SO₂ solubility data of Xia et al. [7] yield big deviations when they are used in parameterization. Therefore, the remaining reliable data are used to optimize the model parameters, covering a T-P-$m_{\mathrm{NaCl}}$ range of 278.15–393.15 K, 0.00–35.07 bar and 0–6 mol/kg.

2.3. SO₂-KCl-H₂O Systems

Experimental data of SO₂ solubility in aqueous KCl solutions are relatively scarce. Only 74 data points have been found for this type of system. Data from Cox et al. [13] for the SO₂-KCl-H₂O systems have the same problem as the SO₂-NaCl-H₂O systems. Due to the small amount of experimental data, the reliability of the experimental data cannot be determined. Therefore, the remaining two data sets are retained, covering a T-P-$m_{\mathrm{KCl}}$ range of 298.00–308.00 K, 1 bar and 0.00–4.40 mol/kg.

2.4. SO₂-Na₂SO₄-H₂O Systems

Only three data sets of SO₂ solubility have been reported for SO₂-Na₂SO₄-H₂O systems. When some SO₂ solubility data from Hudson [27] and Rumpf and Maurer [46] are added to optimize the model parameters, this yields big deviations. Therefore, the other reliable solubility data are used to optimize the model parameters, covering a T-P-$m_{{\mathrm{Na}}_2{\mathrm{SO}}_4}$ range of 298.00–393.15 K, 0.63–27.69 bar and 0.13–3.22 mol/kg.

3. The Thermodynamic Model for SO₂ Solubility in Multi-Ion Solutions and Parameterization

3.1. The Thermodynamic Model for SO₂ Solubility in Multi-Ion Solutions

SO₂ solubility in aqueous electrolyte solutions depends on the balance between the chemical potential in the liquid phase ${\mu }_{{\mathrm{SO}}_2}^l$ and that in the vapor phase ${\mu }_{{\mathrm{SO}}_2}^v$. The chemical potential can be written in terms of fugacity in the vapor phase and activity in the liquid phase:

$$\begin{array}{ll}{\mu }_{{\mathrm{SO}}_2}^l\left(T,P,m_{\mathrm{salt}}\right)& ={\mu }_{{\mathrm{SO}}_2}^{l\left(0\right)}\left(T,P\right)+RT\ln {\alpha }_{{\mathrm{SO}}_2}\left(T,P,m_{\mathrm{salt}}\right)\\ & ={\mu }_{{\mathrm{SO}}_2}^{l\left(0\right)}\left(T,P\right)+RT\ln m_{{\mathrm{SO}}_2}+RT\ln {\gamma }_{{\mathrm{SO}}_2}\left(T,P,m_{\mathrm{salt}}\right)\end{array}$$

(1)

$$\begin{array}{ll}{\mu }_{{\mathrm{SO}}_2}^v\left(T,P,y\right)& ={\mu }_{{\mathrm{SO}}_2}^{v\left(0\right)}\left(T\right)+RT\ln f_{{\mathrm{SO}}_2}\left(T,P,y_{{\mathrm{SO}}_2}\right)\\ & ={\mu }_{{\mathrm{SO}}_2}^{v\left(0\right)}\left(T\right)+RT\ln y_{{\mathrm{SO}}_2}P+RT\ln {\phi }_{{\mathrm{SO}}_2}\left(T,P,y_{{\mathrm{SO}}_2}\right)\end{array}$$

(2)

where ${\mu }_{{\mathrm{SO}}_2}^{l\left(0\right)}$, the standard chemical potential of SO₂ in liquid, is defined as the chemical potential in a hypothetically ideal solution of unit molarity [47], and ${\mu }_{{\mathrm{SO}}_2}^{v\left(0\right)}$, the standard chemical potential of SO₂ in vapor, is the hypothetical ideal gas chemical potential when the pressure is equal to 1 bar. T and P denote temperature and pressure, respectively. R is the molar gas constant. $f_{{\mathrm{SO}}_2}$ is the fugacity of SO₂ in the vapor phase, $y_{{\mathrm{SO}}_2}$ is the mole fraction of SO₂ in the vapor phase, ${\phi }_{{\mathrm{SO}}_2}$ is the fugacity coefficient of SO₂. ${\alpha }_{{\mathrm{SO}}_2}$ is the activity of SO₂, and $m_{{\mathrm{SO}}_2}$ and $m_{\mathrm{salt}}$ are the molality of SO₂ and salt, respectively. ${\gamma }_{{\mathrm{SO}}_2}$ is the activity coefficient of SO₂ in the liquid phase.

At phase equilibrium ${\mu }_{{\mathrm{SO}}_2}^l$ = ${\mu }_{{\mathrm{SO}}_2}^v$, we obtain

$$\begin{array}{ll}\ln m_{{\mathrm{SO}}_2}=& \ln y_{{\mathrm{SO}}_2}P+\ln {\phi }_{{\mathrm{SO}}_2}\left(T,P,y_{{\mathrm{SO}}_2}\right)\\ & -{\displaystyle \frac{{\mu }_{{\mathrm{SO}}_2}^{l\left(0\right)}\left(T,P\right)-{\mu }_{{\mathrm{SO}}_2}^{v\left(0\right)}\left(T\right)}{RT}}-\ln {\gamma }_{{\mathrm{SO}}_2}\left(T,P,m_{\mathrm{salt}}\right)\end{array}$$

(3)

where ${\mu }_{{\mathrm{SO}}_2}^{v\left(0\right)}$ can be assigned as zero for convenience in the parameterization because only the difference between these two chemical potentials is important. In this work, $\ln {\phi }_{{\mathrm{SO}}_2}$ can be calculated from EOS of pure SO₂ by Li [48] (see Appendix A), which will give rise to deviations, which can be canceled out in the later parameterization. $y_{{\mathrm{SO}}_2}$ is calculated by the following equation:

$$y_{{\mathrm{SO}}_2}={\displaystyle \frac{P-P_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{salt}}}{P}}$$

(4)

where $P_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{salt}}$ is the partial pressure of water in the vapor phase, which is approximated by Raoult’s law, defined as

$$P_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{salt}}=P_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{s}}{x}_{{\mathrm{H}}_2\mathrm{O}}$$

(5)

where $P_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{s}}$ is the saturated pressure of pure H₂O, which is calculated from the model of Wagner and Kretzschmar [49], and $x_{{\mathrm{H}}_2\mathrm{O}}$ is the gas-free mole fraction of H₂O in the liquid phase.

$\ln {\gamma }_{{\mathrm{SO}}_2}$ is expressed as a virial expansion of excess Gibbs energy [20].

$$\ln {\gamma }_{{\mathrm{SO}}_2}={\displaystyle \sum _{\mathrm{c}}2{\lambda }_{{\mathrm{SO}}_2-\mathrm{c}}{m}_{\mathrm{c}}+{\displaystyle \sum _{\mathrm{a}}2{\lambda }_{{\mathrm{SO}}_2-\mathrm{a}}{m}_{\mathrm{a}}}}+{\displaystyle \sum _{\mathrm{c}}{\displaystyle \sum _{\mathrm{a}}{\xi }_{{\mathrm{SO}}_{2^{-\mathrm{a}-\mathrm{c}}}}{m}_{\mathrm{c}}}}{m}_{\mathrm{a}}$$

(6)

where $\lambda$ and $\xi$ are second-order and third-order interaction parameters, respectively; c and a refer to cation and anion, respectively. ${\gamma }_{{\mathrm{SO}}_2}$ is not the real activity coefficient in aqueous salt solution. Substituting Equation (6) into Equation (3) yields

$$\begin{array}{ll}\ln m_{{\mathrm{SO}}_2}=& \ln y_{{\mathrm{SO}}_2}P+\ln {\phi }_{{\mathrm{SO}}_2}-{\displaystyle \frac{{\mu }_{{\mathrm{SO}}_2}^{l\left(0\right)}}{RT}}\\ & -{\displaystyle \sum _{\mathrm{c}}2{\lambda }_{{\mathrm{SO}}_2-\mathrm{c}}{m}_{\mathrm{c}}-{\displaystyle \sum _{\mathrm{a}}2{\lambda }_{{\mathrm{SO}}_2-\mathrm{a}}{m}_{\mathrm{a}}}}-{\displaystyle \sum _{\mathrm{c}}{\displaystyle \sum _{\mathrm{a}}{\xi }_{{\mathrm{SO}}_{2^{-\mathrm{a}-\mathrm{c}}}}{m}_{\mathrm{c}}}}{m}_{\mathrm{a}}\end{array}$$

(7)

Following Pitzer et al. [50], we choose the following equation for the P-T dependence of $\lambda ’\mathrm{s}$, $\xi ’\mathrm{s}$ and ${\mu }_{{\mathrm{SO}}_2}^{l\left(0\right)}/RT$:

$$Par(T,P)=c_1+{\mathrm{c}}_{2}T+{\displaystyle \frac{c_3}{T}}+c_{4}PT$$

(8)

Equations (7) and (8) form the basis of the solubility model.

3.2. Parameterization

Since measurements can only be made in electronically neutral solutions, one of the parameters must be assigned arbitrarily [51,52,53]. ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{Cl}}^{-}}$ is set to zero for convenience and remaining parameters are fit to the experimental solubility data. ${\mu }_{{\mathrm{SO}}_2}^{l\left(0\right)}/RT$ is directly evaluated from the solubility data of SO₂ in pure water. ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{Na}}^{+}}$ and ${\xi }_{{\mathrm{SO}}_2-{\mathrm{Na}}^{+}-{\mathrm{Cl}}^{-}}$ are obtained by fitting the data to the reliable solubility data of SO₂ in aqueous NaCl solution. ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{K}}^{+}}$ is determined by a linear regression to the reliable solubility data of SO₂ in aqueous KCl solutions reviewed above. ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{SO}}_4^{2-}}$ is determined from the reliable solubility data of SO₂ in aqueous Na₂SO₄ solutions. Up to now, experimental data of SO₂ solubility in aqueous MgCl₂, CaCl₂ solutions have not been reported. Because SO₂ and CO₂ are the acid gases, ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{Ca}}^{2+}}$, ${\xi }_{{\mathrm{SO}}_2-{\mathrm{Ca}}^{2+}-{\mathrm{Cl}}^{-}}$, ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{Mg}}^{2+}}$ and ${\xi }_{{\mathrm{SO}}_2-{\mathrm{Mg}}^{2+}-{\mathrm{Cl}}^{-}}$ are approximated as ${\lambda }_{{\mathrm{CO}}_2-{\mathrm{Ca}}^{2+}}$, ${\xi }_{{\mathrm{CO}}_2-{\mathrm{Ca}}^{2+}-{\mathrm{Cl}}^{-}}$, ${\lambda }_{{\mathrm{CO}}_2-{\mathrm{Mg}}^{2+}}$ and ${\xi }_{{\mathrm{CO}}_2-{\mathrm{Mg}}^{2+}-{\mathrm{Cl}}^{-}}$ of Shi and Mao [5], respectively. Because ${\xi }_{{\mathrm{SO}}_2}$-K⁺-Cl⁻ and ${\xi }_{{\mathrm{SO}}_2}$-Na⁺-${\mathrm{SO}}_4^{2-}$ have little effect on the SO₂ solubility, they are set as zero in this work.

Table 3. Interaction parameters for SO₂ solubility.

T-P Coefficient	${\mathit{\mu }}_{\mathbf{S}{\mathbf{O}}_2}^{\mathbf{l}\left(0\right)}/\mathit{R}\mathit{T}$	${\mathit{\lambda }}_{\mathbf{S}{\mathbf{O}}_2\text{-}\mathbf{N}{\mathbf{a}}^{+}}$	${\mathit{\xi }}_{\mathbf{S}{\mathbf{O}}_2\text{-}\mathbf{N}{\mathbf{a}}^{+}\text{-}\mathbf{C}{\mathbf{l}}^{-}}$
c1	0.13245076 × 102	−0.11148943 × 101	0.2296582 × 10−4
c2	−0.69839713 × 10−4	0.18138955 × 10−4
c3	−0.34267139 × 104	0.17449283 × 103
c4	0.48022032 × 10−7	−0.69223494 × 10−7
	${\lambda }_{{\mathrm{SO}}_2\text{-}{\mathrm{K}}^{+}}$	${\lambda }_{{\mathrm{SO}}_2\text{-}\mathrm{S}{\mathrm{O}}_4^{2-}}$
c1	0.14460352 × 101	0.46735219 × 101
c2	−0.019673290 × 10−5	−0.079666020 × 10−5
c3	−0.27249399 × 103	−0.68220367 × 103
	* ${\lambda }_{{\mathrm{SO}}_2\text{-}\mathrm{M}{\mathrm{g}}^{2+}}$	* ${\xi }_{{\mathrm{SO}}_2\text{-}\mathrm{M}{\mathrm{g}}^{2+}\text{-}\mathrm{C}{\mathrm{l}}^{-}}$
c1	1.6528428 × 10−1	−1.152724 × 10−2
c2	1.4572839 × 10−4
	* ${\lambda }_{{\mathrm{SO}}_2\text{-}\mathrm{C}{\mathrm{a}}^{2+}}$	* ${\xi }_{{\mathrm{SO}}_2\text{-}\mathrm{C}{\mathrm{a}}^{2+}\text{-}\mathrm{C}{\mathrm{l}}^{-}}$
c1	−5.513825 × 10−1	4.2222945 × 10−3
c2	1.0502705 × 10−3	−4.44659191 × 10−5
c3	1.3349619 × 102
c4	−1.9361945 × 10−4
c5	8.9529014 × 10−7
c6	−3.5780857 × 10−10

Note: * denotes that ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{Ca}}^{2+}}$, ${\xi }_{{\mathrm{SO}}_2-{\mathrm{Ca}}^{2+}-{\mathrm{Cl}}^{-}}$, ${\lambda }_{{\mathrm{SO}}_2-{\mathrm{Mg}}^{2+}}$ and ${\xi }_{{\mathrm{SO}}_2-{\mathrm{Mg}}^{2+}-{\mathrm{Cl}}^{-}}$ are approximated as ${\lambda }_{{\mathrm{CO}}_2-{\mathrm{Ca}}^{2+}}$, ${\xi }_{{\mathrm{CO}}_2-{\mathrm{Ca}}^{2+}-{\mathrm{Cl}}^{-}}$, ${\lambda }_{{\mathrm{CO}}_2-{\mathrm{Mg}}^{2+}}$, and ${\xi }_{{\mathrm{CO}}_2-{\mathrm{Mg}}^{2+}-{\mathrm{Cl}}^{-}}$ [5], respectively.

lists the optimized parameters.

4. Comparison with Experimental Data

Table 4. Calculated deviations of SO₂ solubilities in pure water and aqueous single-salt solutions.

References	T (K)	P (bar)	msalt (mol/kg)	Nd	AAD (%)	MAD (%)
Bunsen [23]	273.00–313.00	1+	0	41	3.58	16.39
Sims [24]	281.20–323.20	1+	0	22	2.65	−6.77
Fox [25]	298.20–308.20	1+	0	2	3.62	−5.76
Simth and Parkhurst [26]	278.20–333.20	0.20–1.60	0	8	4.72	−7.39
Hudson [27]	283.15–363.15	1+	0	42	2.91	−5.43
Maass and Maass [29]	283.20–300.20	0.00–3.50	0	29	2.53	−5.67
Campbell and Maass [30]	298.15–383.15	0.14–4.50	0	55	6.95	−15.82
Morgan and Maass [31]	273.00–298.00	0.00–1.20	0	22	6.75	−16.90
Conrad and Beuschlein [32]	298.20	0.40–1.00	0	6	2.03	−3.03
Young [34]	373.00–423.00	1.70–7.20	0	9	6.30	13.72
Beuschlein and Simenson [35]	296.40–386.15	0.12–2.11	0	51	5.44	−17.55
Rabe and Harris [37]	303.15–353.15	0.09–1.20	0	39	4.79	−15.76
Tokunaga [39]	283.15–313.15	1+	0	4	3.78	−6.76
Lavrova and Tudorovskaya [40]	299.15–363.15	1+	0	5	2.04	3.48
Douabul and Riley [41]	278.97–303.25	1+	0	6	5.53	−10.35
Byerley, et al. [42]	298.15–323.15	1+	0	2	11.37	15.30
Rumpf and Maurer [43]	293.14–393.33	0.35–25.09	0	55	6.04	−17.19
Zhou et al. [14]	273.15–373.15	10.00–300.00	0	34	8.11	−21.46
Saadallah et al. [15]	293.15–353.15	0.00–12.00	0	36	6.98	−19.08
$m_{\mathrm{NaCl}}$(mol/kg)
Fox [25]	298.00–308.00	1+	0.50–3.22	6	1.46	3.64
Millero et al. [45]	278.15–298.15	1+	0.01–6.00	17	4.52	−13.96
Xia, et al. [7]	313.15–393.15	0.30–37.00	2.90–6.00	57	5.33	−14.11
$m_{\mathrm{KCl}}$(mol/kg)
Fox [25]	298.00–308.00	1+	0.00–0.05	12	11.53	−21.56
Hudson [27]	298.00–308.00	1+	0.50–4.40	43	1.28	−4.6
$m_{{\mathrm{Na}}_2{\mathrm{SO}}_4}$(mol/kg)
Fox [25]	298.00–308.00	1	0.50–3.22	12	1.46	3.64
Hudson [27]	293.15–313.15	1+	0.13–1.41	18	3.45	−5.73
Rumpf and Maurer [46]	313.15–393.15	1.00–28.00	0.50–1.00	35	6.18	18.07

Notes: T, P and m_salt denote the temperature, pressure and molality of salt, respectively; Nd is the number of experimental data; AAD is the average absolute deviation calculated from this model; MAD is the maximal absolute deviation calculated from this model. 1⁺ is the partial pressure of SO₂.

shows the average and maximal absolute deviations of the model from each data set for the SO₂ solubility in pure water and aqueous NaCl, KCl and Na₂SO₄ solutions. Figure 2, Figure 3 and Figure 4 show the comparisons between the experimental results and model predictions.

4.1. SO₂-H₂O Systems

It can be seen in Table 4 and Figure 2 that the SO₂ solubility in pure water calculated from this model is in good agreement with most of the experimental data. Although there are large deviations for individual experimental data, such as Bunsen [23], Campbell and Maass [30], Morgan and Maass [31], Beuschlein and Simenson [35], Rabe and Harris [37], Byerley et al. [42], Rumpf and Maurer [43], Zhou et al. [14] and Saadallah et al. [15], the average relative deviation is 5.24%, which is within the experimental uncertainties of solubility (≈7%) [43].

Figure 2b–e shows the comparison between experimental data at 298.15–343.15 K and the calculated solubility curves from this model, Shaw et al. [11] and Saadallah et al. [15]. Figure 2b,c indicates that this solubility model is better than the model of Shaw et al. [11] at high pressures. This model can reproduce most of the experimental data at low pressures, but the model of Saadallah et al. [15] yields somewhat big deviations (Figure 2c–e). In all, the model can reproduce the solubility of SO₂ in water up to 423 K and 300 bar.

Figure 2. SO₂ solubilities in pure water (models vs. experimental data) [11,14,15,23,24,25,26,27,29,32,39,40,43]. (a) partial pressure of SO₂ = 1 atm; (b) T = 298.15 K; (c) T = 313.15 K at low pressures; (d) T = 333.15 K; (e) T = 343.15 K; (f) T = 313.15 K at high pressures. $m_{{\mathrm{SO}}_2}$, T and P denote the solubility of SO₂, temperature and pressure, respectively.

Figure 2. SO₂ solubilities in pure water (models vs. experimental data) [11,14,15,23,24,25,26,27,29,32,39,40,43]. (a) partial pressure of SO₂ = 1 atm; (b) T = 298.15 K; (c) T = 313.15 K at low pressures; (d) T = 333.15 K; (e) T = 343.15 K; (f) T = 313.15 K at high pressures. $m_{{\mathrm{SO}}_2}$, T and P denote the solubility of SO₂, temperature and pressure, respectively.

4.2. SO₂-NaCl-H₂O Systems

Figure 3a–c shows that the SO₂ solubilities in aqueous NaCl solutions calculated from this model are in good agreement with the experimental data. The average absolute deviation of SO₂ solubility in an aqueous NaCl solution is 4.84%, within the experimental uncertainties of solubility. Figure 3 shows that the SO₂ solubility curve calculated from this model is consistent with the model of Tan et al. [12] at 393.15 K, but this model is more accurate than the model of Tan et al. [12] at 353.215 K and 373.15 K. The model can be applied in the T-P-m_NaCl range of 273–400 K, 0–50 bar and 0–6 mol/kg.

Figure 3. SO₂ solubilities in aqueous NaCl or KCl solutions (models vs. experimental data) [7,12,27,45]. (a) molality of NaCl = 6 mol/kg; (b) molality of NaCl = 3 mol/kg; (c) T = 278.15 K and P = 1 atm; (d) partial pressure of SO₂ = 1 atm and molality of KCl = 4.05 mol/kg. $m_{{\mathrm{SO}}_2}$, T and P denote the solubility of SO₂, temperature and pressure, respectively. $m_{\mathrm{NaCl}}$ and $m_{\mathrm{KCl}}$ are the molality of NaCl and KCl.

Figure 3. SO₂ solubilities in aqueous NaCl or KCl solutions (models vs. experimental data) [7,12,27,45]. (a) molality of NaCl = 6 mol/kg; (b) molality of NaCl = 3 mol/kg; (c) T = 278.15 K and P = 1 atm; (d) partial pressure of SO₂ = 1 atm and molality of KCl = 4.05 mol/kg. $m_{{\mathrm{SO}}_2}$, T and P denote the solubility of SO₂, temperature and pressure, respectively. $m_{\mathrm{NaCl}}$ and $m_{\mathrm{KCl}}$ are the molality of NaCl and KCl.

4.3. SO₂-KCl-H₂O Systems

Figure 3d shows that the solubility curve of SO₂ in aqueous KCl solutions calculated from this model is in good agreement with the experimental data. The average absolute deviation of SO₂ solubility in an aqueous KCl solution is 3.51%, within the experimental uncertainties of solubility. The model can reproduce experimental solubility data from 298 to 308 K with a salinity of 0–4.5 mol/kg at one atmospheric pressure.

4.4. SO₂-Na₂SO₄-H₂O Systems

Figure 4 shows that the SO₂ solubilities in aqueous Na₂SO₄ solutions calculated from this model are in agreement with the experimental data. The average absolute deviation of SO₂ solubility in an aqueous Na₂SO₄ solution is 4.55%, which is within experimental uncertainties. On the whole, the model can approximately predict the solubility of SO₂ in aqueous Na₂SO₄ solutions.

Figure 4. SO₂ solubilities in aqueous Na₂SO₄ solutions (models vs. experimental data), [25,27,43]. (a) P = 1 bar; (b) partial pressure of SO₂ = 1 atm; (c) T = 363.15 K and molality of Na₂SO₄ = 0.546 mol/kg; (d) T = 393.15 K and molality of Na₂SO₄ = 0.493 mol/kg. $m_{{\mathrm{SO}}_2}$, T and P denote the solubility of SO₂, temperature and pressure, respectively. $m_{{\mathrm{Na}}_2{\mathrm{SO}}_4}$ is the molality of Na₂SO₄.

Figure 4. SO₂ solubilities in aqueous Na₂SO₄ solutions (models vs. experimental data), [25,27,43]. (a) P = 1 bar; (b) partial pressure of SO₂ = 1 atm; (c) T = 363.15 K and molality of Na₂SO₄ = 0.546 mol/kg; (d) T = 393.15 K and molality of Na₂SO₄ = 0.493 mol/kg. $m_{{\mathrm{SO}}_2}$, T and P denote the solubility of SO₂, temperature and pressure, respectively. $m_{{\mathrm{Na}}_2{\mathrm{SO}}_4}$ is the molality of Na₂SO₄.

5. Applications of the Solubility Model

5.1. Prediction of SO₂ Solubility in Seawater

The solubility model of SO₂ in multi-ion electrolyte solutions can be used to predict the solubility of SO₂ in seawater. The average absolute deviation of our model from experimental SO₂ solubility in seawater is about 7.48% (

Table 5. Calculated deviations of SO₂ solubilities in seawater.

System	Reference	T (K)	P (bar)	${\mathit{m}}_{\mathbf{Seawater}}$ (g/kg)	Nd	AAD (%)	MAD (%)
SO2-Seawater	Douabul and Riley [41]	278.97–303.25	1+	10.00–40.00	24	7.48	12.75

Notes: T, P and $m_{\mathrm{Seawater}}$ denote the temperature, pressure and salinity of seawater, respectively; Nd is the number of experimental data; AAD is the average absolute deviation calculated from this model; MAD is the maximal absolute deviation calculated from this model. 1⁺ is the partial pressure of SO₂.

). It can be seen in Figure 5 that the calculated results are a bit lower than the experimental data. It may be that the experimental solubility data of SO₂ in aqueous MgCl₂, CaCl₂ and K₂SO₄ solutions have not been found, so the second-order and third-order interaction parameters between SO₂ and Mg²⁺ and Ca²⁺ cannot be obtained, and they are only approximated as ${\lambda }_{{\mathrm{CO}}_2-{\mathrm{Ca}}^{2+}}$, ${\xi }_{{\mathrm{CO}}_2-{\mathrm{Ca}}^{2+}-{\mathrm{Cl}}^{-}}$, ${\lambda }_{{\mathrm{CO}}_2-{\mathrm{Mg}}^{2+}}$ and ${\xi }_{{\mathrm{CO}}_2-{\mathrm{Mg}}^{2+}-{\mathrm{Cl}}^{-}}$ from Shi and Mao [5].

5.2. Prediction of CO₂ Solubility of the CO₂-SO₂ Mixture in Pure Water

In this work, the fugacity-activity model can be used to predict the solubility of CO₂-SO₂ mixed gases in multi-ionic electrolyte solutions. If a fugacity coefficient of CO₂ or SO₂ is calculated from the EOS of CO₂-SO₂ of Li [48], combining the solubility model of CO₂ developed by Shi and Mao [5] and the solubility model of SO₂ here, we can calculate the solubility of CO₂ and SO₂ in multi-ion electrolyte solutions. However, only experimental CO₂-SO₂ solubility data in pure water have been reported [54]. Figure 6 shows the calculated solubilities of CO₂ of CO₂-SO₂ mixtures in pure water under different T-P conditions. It can be seen in Figure 6 that the CO₂ solubilities of CO₂-SO₂ mixtures in pure water increase with the increase of pressure at the same temperature and mole fraction of SO₂, but decrease with the increase of SO₂ at the same T-P conditions. The calculated solubilities of CO₂ with a 0.1 mole fraction of SO₂ are in agreement with the experimental data of Zhou [54] (Figure 6a,c,d).

6. Conclusions

A SO₂ solubility model has been developed based on the EOS of pure SO₂ and the Pitzer activity coefficient theory. This model can accurately reproduce most of the experimental data with or close to experimental accuracy. Although the second-order and third-order interaction parameters between SO₂ and Mg²⁺ and Ca²⁺ have been adopted by an approximation, based on the electrolyte solution theory of Pitzer [20], this model can also be used to predict SO₂ solubility in seawater. Combined with the EOS of CO₂-SO₂ fluid mixture, the developed model can approximately predict the solubility of CO₂-SO₂ mixed gases in more complex brines.

It should be noted that the experimental data on the solubility of SO₂ and CO₂-SO₂ mixtures in aqueous CaCl₂, K₂SO₄ and MgCl₂ solutions are still lacking. Experimental studies can be focused on the solubility of these systems in the future.