## 2 The ZMS EOS  
The EOS of the CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) mixtures is in terms of dimensionless Helmholtz free energy \(\alpha\), which is represented by  
\[\alpha=\alpha^{0}+\alpha^{r} \tag{1}\]  
where \(\alpha^{0}\) and \(\alpha^{r}\) are the ideal-gas part and the residual part of dimensionless Helmholtz free energy. \(\alpha^{0}\) and \(\alpha^{r}\) are defined by  
\[\alpha^{0}=\sum_{i=1}^{n}x_{i}\Big{[}\alpha_{i}^{0}+lnx_{i}\Big{]} \tag{2}\]  
\[\alpha^{r}=\sum_{i=1}^{n}x_{i}\alpha_{i}^{r}(\delta,\tau)+\alpha^{E}(\delta, \tau,x) \tag{3}\]  
where \(n\) is the number of components in the mixture, \(x_{i}\) is the mole fraction of component \(i\) in the mixture, the superscripts \"0\", \"r\" and \"E\" denote the ideal-gas part, the residual part, and the excess of dimensionless Helmholtz free energy, respectively. The subscript \(i\) denotes the component \(i\). \(\alpha_{i}^{r}\) can be calculated by the EOSs of pure CO\({}_{2}\), CH\({}_{4}\) and N\({}_{2}\) fluids [41; 42; 43], which are in terms of dimensionless Helmholtz free energy and are recommended as the standard EOSs of pure CO\({}_{2}\), CH\({}_{4}\) and N\({}_{2}\) fluids by the National Institute of Standards and Technology (NIST). \(\delta\) and \(\tau\) of Equation (3) are the reduced mixture density (\(\delta=\rho/\rho_{c}\)) and the inverse reduced mixture temperature (\(\tau=T_{c}/T\)), where \(\rho\) and \(T\) are the density and temperature of the mixture, and \(\rho_{c}\) and \(T_{c}\) are the pseudo-critical density and temperature of the mixture. \(\rho_{c}\) and \(T_{c}\) are defined by  
\[\rho_{c}=\left[\sum_{i=1}^{n}\frac{x_{i}}{\rho_{ci}}+\sum_{i=1}^{n-1}\sum_{j=i+ 1}^{n}x_{i}x_{j}\zeta_{ij}\right]^{-1} \tag{4}\]  
\[T_{c}=\sum_{i=1}^{n}x_{i}T_{ci}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}x_{i}^{\beta_{ ij}}x_{j}\zeta_{ij} \tag{5}\]  
where \(x_{j}\) is the mole fraction of component \(j\) in the mixture, \(\rho_{ci}\) and \(T_{ci}\) are the critical density and temperature of component \(i\), respectively. The critical temperatures and densities of the three components considered in this study are listed in Table 1. \(\zeta_{ij}\), \(\beta_{ij}\) and \(\zeta_{ij}\) are binary parameters associated with components \(i\) and \(j\). \(\alpha^{E}\) of Equation (3) takes the general form developed by [PERSON] (LJ-1999 EOS) [35]. \[\alpha^{E}(\delta,\tau,x)=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}x_{i}x_{j}F_{ij}\sum _{k=1}^{10}N_{k}\delta^{d_{k}}\tau^{t_{k}} \tag{6}\]  
where \(F_{ij}\) is a binary parameter associated with components \(i\) and \(j\), \(N_{k}\), \(d_{k}\) and \(t_{k}\) are the general parameters independent of fluids, which can be obtained from [PERSON] and [PERSON] [35]. There are four binary parameters (\(\zeta_{ij}\), \(\beta_{ij}\), \(\zeta_{ij}\) and \(F_{ij}\)) in the above equations. Binary interaction parameters of CH\({}_{4}\)-CO\({}_{2}\) and CO\({}_{2}\)-N\({}_{2}\) have been determined in previous work [34; 40]. In this work, the binary interaction parameters of CH\({}_{4}\)-N\({}_{2}\) systems are obtained by the nonlinear regression of selected experimental data. Since experimental _PVTx_ data available are much more than experimental VLE data, only part of new and accurate experimental _PVTx_ data are selected to fit the parameters to make the _PVTx_ and VLE data keep similar weights in the fitting. In this work, the Levenberg-Marquardt algorithm of the nonlinear least squares method is used for fitting, which is an efficient and widely used mathematical optimization technique. The fitting condition is to minimize the weighted sum of squares of the calculation errors of the equation on the selected experimental _PVTx_ and VLE data. Compared to [PERSON]'s method (e.g., The Newton-Raphson method used in REFPROP [44]), it combines the advantages of two numerical minimization algorithms: the gradient descent method and the Gauss-Newton method, and it has good convergence and robustness.