## 3 The Applications of the ZMS EOS
### Calculating Excess Enthalpies
Figure 7: _Cont._  
Calculating the Solubility of CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) Mixtures in Aqueous Electrolyte Solution  
The solubility of the CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) gas mixtures in the electrolyte solutions can provide the quantitative assessment for the storage of CO\({}_{2}\) in deep saline aquifers. According to the thermodynamic principle, when the CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) gas mixtures reach dissolution equilibrium in the electrolyte solution, the chemical potential of component \(i\) in the liquid phase (\(\mu_{i}^{L}\)) and its chemical potential in the vapor phase (\(\mu_{i}^{V}\)) are equal. The chemical potential can be written in terms of fugacity in the vapor phase and activity in the liquid phase  
\[\mu_{i}^{L}=\mu_{i}^{L(0)}+RTlnm_{i}\gamma_{i} \tag{12}\]  
\[\mu_{i}^{V}=\mu_{i}^{V(0)}+RTln\varphi_{i}y_{i}P \tag{13}\]  
where \(i\) is the gas component in vapor mixtures, and \(P\) is the pressure in bar. \(y_{i}\) is the molar fraction of \(i\) in the vapor phase, \(\varphi_{i}\) is the fugacity coefficient of \(i\) in the vapor phase and \(m_{i}\) is the solubility of \(i\) in the liquid phase in mol/kg. \(\gamma_{i}\) is the activity coefficient of component \(i\) in the liquid phase. \(\mu_{i}^{L(0)}\) and \(\mu_{i}^{V(0)}\) are the standard chemical potential of \(i\) in the liquid and vapor phase, respectively. At the dissolution phase equilibrium, \(\mu_{i}^{V}=\mu_{i}^{L}\). From Equations (12) and (13), we can obtain  
\[lnm_{i}=lny_{i}P+ln\varphi_{i}-\frac{\mu_{i}^{L(0)}-\mu_{i}^{V(0)}}{RT}-ln\gamma _{i} \tag{14}\]  
\[H_{m}=RT\Big{(}1+\tau\Big{(}\alpha_{\tau}^{0}+\alpha_{\tau}^{r}\Big{)}+\delta \alpha_{\delta}^{r}\Big{)} \tag{8}\]0 Comparisons between experimental and calculated mixing enthalpies for the CH\({}_{4}\)-N\({}_{2}\) mixture at 195.15 K and 253.15 K. (**a**) \(T\) = 195.15 K; (**b**) \(T\) = 253.15 K. Experimental data (squares [77]), this work (solid curves); the Enhanced-Predictive-PR78 (E-PPR78) EOS (dash curves). \[H_{m}=RT\Big{(}1+\tau\Big{(}\alpha_{\tau}^{0}+\alpha_{\tau}^{r}\Big{)}+\delta \alpha_{\delta}^{r}\Big{)} \tag{8}\]1 Comparisons between experimental and calculated mixing enthalpies for the CO\({}_{2}\)-N\({}_{2}\) mixture, (**a**) \(T\) = 313.15 K, \(P\) = 10.13 bar; (**b**) \(T\) = 313.15 K, \(P\) = 30.40 bar; (**c**) \(T\) = 313.15 K, \(P\) = 81.06 bar; (**d**) \(T\) = 313.15 K, \(P\) = 121.59 bar. Experimental data (black squares [76]). PR (\(k_{\bar{\mathrm{ij}}}\) = 0) (red dash curves); PR (optimal \(k_{\bar{\mathrm{ij}}}\)) (green dash curves); PR + EOS/\(a_{\mathrm{res}}^{E,Wilson}\) (blue dash curves); This work (black solid curves). Since the water content in the vapor phase at equilibrium is generally small, it has a negligible effect on the fugacity coefficient. When calculating the fugacity coefficient, the water content in the vapor phase can be ignored, the vapor phase can be approximated as the CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) system and the mole fraction of gas component \(i\) in the vapor phase is expressed as \(y_{i}\). Consequently, \(\varphi_{i}\) can be calculated from the ZMS EOS developed in this work. Pitzer activity coefficient model [78] is chosen to calculate \(\gamma_{i}\)  
\[H_{m}=RT\Big{(}1+\tau\Big{(}\alpha_{\tau}^{0}+\alpha_{\tau}^{r}\Big{)}+\delta \alpha_{\delta}^{r}\Big{)} \tag{8}\]2  
where \(\lambda\) and \(\zeta\) are second-order and third-order interaction parameters, respectively; c and \(a\) refer to cation and anion, respectively. Substituting Equation (15) into Equation (14) yields  
\[ \begin{array}{rl}lnm_{i}=&lny_{i}P+ln\varphi_{i}-\frac{\mu_{i}^{L(0)}-\mu_{i }^{V(0)}}{RT}-\sum\limits_{c}2\lambda_{i-c}m_{c}\\ &&-\sum\limits_{a}2\lambda_{i-a}m_{a}-\sum\limits_{c}\zeta_{i-c-a}m_{c}m_{a}a \end{array}  \tag{16}\]  
As can be seen from Equation (17), \(m_{i}\) is related to the difference between \(\mu_{i}^{L(0)}\) and \(\mu_{i}^{V(0)}\), and is not related to the specific value of \(\mu_{i}^{L(0)}\) or \(\mu_{i}^{V(0)}\).