## 1 Introduction  
The CO\({}_{2}\), CH\({}_{4}\) and N\({}_{2}\) are important natural fluids, and non-aqueous CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) mixtures are often reported in the studies of fluid inclusions [1; 2; 3]. In recent decades, the amount of greenhouse gas CO\({}_{2}\) in the atmosphere has gradually increased because of the development of industry. Numerous studies have shown that CO\({}_{2}\) capture and storage (CCS) is an effective method to reduce the amount of CO\({}_{2}\) in the atmosphere [4; 5; 6]. However, CO\({}_{2}\) in industrial exhaust gases is usually impure, mixing with other components, such as CH\({}_{4}\) and N\({}_{2}\)[7; 8]. The impurities can affect the design of the CCS processes [9; 10; 11]. Therefore, predicting thermodynamic properties of the CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) fluid mixtures, especially the pressure-volume-temperature-composition (_PVTx_) and vapor-liquid equilibrium (VLE) properties at different temperatures and pressures, is of great significance for the related CCS and fluid inclusion studies [12; 13; 14; 15]. Although experimental thermodynamic data are the most reliable in the corresponding applications, they are costly and time-consuming to obtain. On the other hand, the predictive EOS is a better choice for us. In the past century, the cubic EOSs (e.g., Peng-Robinson (PR) EOS [16], Soave-Redlich-Kwong (SRK) EOS [17]) originated from the van der Waals EOS [18] are highly preferred for the CCS researchers because of their simplicity. Based on the standard PR EOS, some more accurate EOSs have been presented for the CCS mixtures, such as a model that integrates the classical PR EOS with advanced mixing rules, called the \"Peng-Robinson + residual part of excess Helmholtz energy model\" [19], and the Enhanced-Predictive-PR78 (E-PPR78) EOS [20]. However, the cubic EOSs cannot well reproduce the _PVTx_ properties of fluids under high pressure-temperature conditions, as can be seen in Section 2. Based on Heyen EOS [21], [PERSON] et al. [22] simulated the phase equilibria of the CH\({}_{4}\)-CO\({}_{2}\) system below 50 \({}^{\circ}\)C and 100 bar; [PERSON] and [PERSON] [23] predicted the phase equilibria of the CO\({}_{2}\)-N\({}_{2}\) system below 20 \({}^{\circ}\)C and 90 bar. Because cubic EOSs cannot well reproduce the _PVTx_ properties, [PERSON] et al. [24; 25] and [PERSON] and [PERSON] [26] modeled the phase equilibria of the CO\({}_{2}\)-CH\({}_{4}\)-N\({}_{2}\) system including liquid, vapor and solid by the cubic EOSs and the _PVTx_ properties by the Lee-Kesler correlation [27]. However, these models have the disadvantage of using two EOSs that may cause incoherency of various fluid parameters [28]. In recent years, the Statistical Associating Fluid Theory (SAFT) EOS [29] from [PERSON]'s thermodynamic perturbation theory (TPT) has become popular for CCS fluid mixtures. Because it is based on statistical mechanics and has an explicit physical meaning, it has more reliable extrapolation and more accurate predictive capability than traditional empirical models. Many successful modifications of the SAFT-based EOSs have been presented, such as the EOS of perturbed-chain SAFT (PC-SAFT) [30; 31], variable range SAFT (SAFT-VR) [32], and Lennard-Jones SAFT (LJ-SAFT) [33]. However, the SAFT EOSs cannot well reproduce experimental VLE data in the near-critical region of the CO\({}_{2}\)-N\({}_{2}\) fluid mixture, as can be seen in the previous work [34]. Up to now, the EOS in terms of Helmholtz free energy is also commonly used to calculate the thermodynamic properties of CCS fluid mixtures [35; 36; 37; 38; 39], because the EOS in the form of Helmholtz free energy has high precision, wide temperature-pressure range, and can calculate all thermodynamic properties of the fluids by the derivation from the Helmholtz free energy EOS, such as enthalpy, heat capacity, entropy, etc. The derivation from the Helmholtz free energy EOS is not complex, as can be seen later.