## 1 Introduction
### The Thermodynamic Model for \(\text{SO}_{2}\) Solubility in Multi-Ion Solutions  
\(\text{SO}_{2}\) solubility in aqueous electrolyte solutions depends on the balance between the chemical potential in the liquid phase \(\mu_{\text{SO}_{2}}^{l}\) and that in the vapor phase \(\mu_{\text{SO}_{2}}^{v}\). The chemical potential can be written in terms of fugacity in the vapor phase and activity in the liquid phase:  
\[ \begin{array}{ll}\mu_{\text{SO}_{2}}^{l}(T,P,m_{\text{salt}})&=\mu_{\text{ SO}_{2}}^{l(0)}(T,P)+RT\ln\alpha_{\text{SO}_{2}}(T,P,m_{\text{salt}})\\ &=\mu_{\text{SO}_{2}}^{l(0)}(T,P)+RT\ln\eta_{\text{SO}_{2}}+RT\ln\gamma_{\text{ SO}_{2}}(T,P,m_{\text{salt}})\\ \mu_{\text{SO}_{2}}^{v}(T,P,y)&=\mu_{\text{SO}_{2}}^{v(0)}(T)+RT\ln\gamma_{ \text{SO}_{2}}\big{(}T,P,y_{\text{SO}_{2}}\big{)}\\ &=\mu_{\text{SO}_{2}}^{v(0)}(T)+RT\ln y_{\text{SO}_{2}}P+RT\ln\varphi_{\text{ SO}_{2}}\big{(}T,P,y_{\text{SO}_{2}}\big{)}\end{array}  \tag{1}\]  
where \(\mu_{\text{SO}_{2}}^{l(0)}\), the standard chemical potential of \(\text{SO}_{2}\) in liquid, is defined as the chemical potential in a hypothetically ideal solution of unit molarity [47], and \(\mu_{\text{SO}_{2}}^{v(0)}\), the standard chemical potential of \(\text{SO}_{2}\) 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. \(f_{\text{SO}_{2}}\) is the fugacity of \(\text{SO}_{2}\) in the vapor phase, \(y_{\text{SO}_{2}}\) is the mole fraction of \(\text{SO}_{2}\) in the vapor phase, \(\varphi_{\text{SO}_{2}}\) is the fugacity coefficient of \(\text{SO}_{2}\). \(\alpha_{\text{SO}_{2}}\) is the activity of \(\text{SO}_{2}\), and \(m_{\text{SO}_{2}}\) and \(m_{\text{salt}}\) are the molality of \(\text{SO}_{2}\) and salt, respectively. \(\gamma_{\text{SO}_{2}}\) is the activity coefficient of \(\text{SO}_{2}\) in the liquid phase. At phase equilibrium \(\mu_{\text{SO}_{2}}^{l}=\mu_{\text{SO}_{2}}^{v}\), we obtain  
\[ \begin{array}{ll}\ln m_{\text{SO}_{2}}=&\ln y_{\text{SO}_{2}}P+\ln\varphi_{ \text{SO}_{2}}\big{(}T,P,y_{\text{SO}_{2}}\big{)}\\ &-\frac{\mu_{\text{SO}_{2}}^{l(0)}(T,P)-\mu_{\text{SO}_{2}}^{v(0)}(T)}{RT}-\ln \gamma_{\text{SO}_{2}}(T,P,m_{\text{salt}})\end{array}  \tag{3}\]  
where \(\mu_{\text{SO}_{2}}^{v(0)}\) can be assigned as zero for convenience in the parameterization because only the difference between these two chemical potentials is important. In this work, \(\ln\varphi_{\text{SO}_{2}}\) can be calculated from EOS of pure \(\text{SO}_{2}\) by Li [48] (see Appendix A), which will give rise to deviations, which can be canceled out in the later parameterization. \(y_{\text{SO}_{2}}\) is calculated by the following equation:  
\[y_{\text{SO}_{2}}=\frac{P-P_{\text{H}_{2}\text{O}}^{\text{salt}}}{P} \tag{4}\]  
where \(P_{\text{H}_{2}\text{O}}^{\text{salt}}\) is the partial pressure of water in the vapor phase, which is approximated by [PERSON]'s law, defined as  
\[P_{\text{H}_{2}\text{O}}^{\text{salt}}=P_{\text{H}_{2}\text{O}}^{\text{s}}x_{ \text{H}_{2}\text{O}} \tag{5}\]  
where \(P_{\text{H}_{2}\text{O}}^{\text{s}}\) is the saturated pressure of pure H\({}_{2}\)O, which is calculated from the model of [PERSON] and [PERSON] [49], and \(x_{\text{H}_{2}\text{O}}\) is the gas-free mole fraction of H\({}_{2}\)O in the liquid phase. In \(\gamma_{\text{SO}_{2}}\) is expressed as a virial expansion of excess Gibbs energy [20]. \[\ln\gamma_{\text{SO}_{2}}=\sum_{\text{c}}2\lambda_{\text{SO}_{2}-\text{c}}m_{ \text{c}}+\sum_{\text{a}}2\lambda_{\text{SO}_{2}-\text{a}}m_{\text{a}}+\sum_{ \text{c}}\sum_{\text{a}}\xi_{\text{SO}_{2}-\text{a}-\text{c}}m_{\text{c}}m_{ \text{a}} \tag{6}\]  
where \(\lambda\) and \(\xi\) are second-order and third-order interaction parameters, respectively; c and a refer to cation and anion, respectively. \(\gamma_{\text{SO}_{2}}\) is not the real activity coefficient in aqueous salt solution. Substituting Equation (6) into Equation (3) yields  
\[