## 3 The Applications of the ZMS EOS  
### Calculating Excess Enthalpies  
The excess enthalpy (\(H^{E}\)) is an important thermodynamic property of the mixture for the mixing and separation processes, which is defined by  
\[H^{E}=\left\{H_{m}-\sum_{i}x_{i}H_{i}\right\}_{P,T} \tag{7}\]  
where \(H_{m}\) is the enthalpy of the mixture, \(H_{i}\) is the enthalpy of pure component \(i\), and \(x_{i}\) is the mole fraction of component \(i\) in the mixture. According to the thermodynamic relationship between Helmholtz free energy and enthalpy, \(H_{m}\) and \(H_{i}\) are given by  
\[H_{m}=RT\Big{(}1+\tau\Big{(}\alpha_{\tau}^{0}+\alpha_{\tau}^{r}\Big{)}+\delta \alpha_{\delta}^{r}\Big{)} \tag{8}\]  
\[H_{i}=RT\Big{(}1+\tau_{i}\Big{(}\alpha_{\tau i}^{0}+\alpha_{\tau i}^{r}\Big{)}+ \delta_{i}\alpha_{\delta i}^{r}\Big{)} \tag{9}\]  
where \(\alpha_{\tau}^{0}=\left(\frac{\partial\alpha^{0}}{\partial\tau}\right)_{g}\), \(\alpha_{\tau}^{r}=\left(\frac{\partial\alpha^{r}}{\partial\tau}\right)_{g}\), \(\alpha_{\delta}^{r}=\left(\frac{\partial\alpha^{r}}{\partial\delta}\right)_{ \tau}\), \(\alpha_{\tau i}^{0}=\left(\frac{\partial\alpha_{i}^{0}}{\partial\tau_{i}}\right) _{\delta i}\), \(\alpha_{\tau i}^{r}=\left(\frac{\partial\alpha_{i}^{r}}{\partial\tau_{i}} \right)_{\delta i}\) and \(\alpha_{\delta i}^{r}=\left(\frac{\partial\alpha_{i}^{r}}{\partial\delta_{i}^ {r}}\right)_{\tau i}\). \(\delta_{i}\) and \(\tau_{i}\) are the reduced density and inverse reduced temperature of pure component i, which are defined by  
\[\delta_{i}=\rho_{i}/\rho_{ci} \tag{10}\]  
\[\tau_{i}=T_{ci}/T \tag{11}\]  
where \(\rho_{i}\) is the density of component \(i\). Based on Equations (8)-(11), \(H_{m}\) can be calculated by this EOS of the CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) fluid mixtures and \(H_{i}\) can be calculated by the above-mentioned equations of pure CO\({}_{2}\), CH\({}_{4}\) and N\({}_{2}\) fluids. The calculated excess enthalpy curves of the CH\({}_{4}\)-CO\({}_{2}\) and CO\({}_{2}\)-N\({}_{2}\) mixtures are, respectively, shown in Figures 6 and 7, where the experimental data [75; 76] are included for comparison. The following EOSs are also included for comparison: the standard Peng-Robinson (PR) EOS, either optimized (optimal \(k_{ij}\)) or not (\(k_{ij}=0\)); the PR EOS with the residual part of the [PERSON] excess Helmholtz energy model (PR + EOS/\(a_{res}^{E,Wilson}\)) [19]. It can be seen from Figures 6 and 7 that this EOS is more accurate than the standard PR EOS, either optimized (optimal \(k_{ij}\)) or not (\(k_{ij}=0\)) at most cases. In general, the PR + EOS/\(a_{res}^{E,Wilson}\) and this EOS shows a similar predictive capability for the excess enthalpies of the CH\({}_{4}\)-CO\({}_{2}\) mixture. However, this EOS is slightly more accurate than the PR + EOS/\(a_{res}^{E,Wilson}\) for the CO\({}_{2}\)-N\({}_{2}\) mixture at low pressures. Figure 8 compares the excess enthalpy curves of the CH\({}_{4}\)-N\({}_{2}\) mixture calculated from the ZMS EOS with experimental data [77], and enthalpies calculated from the Enhanced-Predictive-PR78 (E-PPR78) EOS [20] are also added for comparison. Figure 8a shows the ZMSEOS are significantly more accurate than the (E-PPR78) EOS at high and middle pressures. Figure 8b shows the enthalpies of mixing calculated from the E-PPR78 EOS are slightly better than those of the ZMS EOS at low and middle pressures, but the enthalpies of mixing calculated from the ZMS EOS are better than those from the E-PPR78 EOS at \(P=101.33\) bar. Figure 6: Comparisons between experimental and calculated mixing enthalpies for the CH\({}_{4}\)-CO\({}_{2}\) mixture, (**a**) \(T\) = 283.15 K, \(P\) = 10.13 bar; (**b**) \(T\) = 283.15 K, \(P\) = 30.40 bar; (**c**) \(T\) = 313.15 K, \(P\) = 10.13 bar; (**d**) \(T\) = 313.15 K, \(P\) = 30.40 bar. Experimental data (squares [75]). PR(\(k_{ij}\) = 0) (red dash curves); PR(optimal \(k_{ij}\)) (green dash curves); PR + EOS/\(a_{res}^{E,Wilson}\) (blue dash curves); This work (black solid curves): \(h^{E}\) is the mixing enthalpy.