# In the case of ionic liquids the

In the case of ionic liquids, the association scheme should be determined and considered for calculations. H. Soltani Panah (2017) suggested the 2B scheme for the imidazolium-based ionic liquids. His speculation has been used in this study for the modeling [22]. Considering 2B association scheme, the CPA pure component parameters for the ionic liquids were obtained using the liquid density data [[33], [34], [35]] through minimization of the following objective function in Eq. (17). The Fmincon toolbox in the MATLAB software was used for optimization of the objective function. Fig. 1 shows the algorithm of calculation for obtaining the CPA pure parameters. The obtained adjusted parameters are reported in Table 2. The calculation of vapor pressures of the six studied ionic liquids using the tuned pure parameters results in the low values as expected for ionic liquids. In addition, the results indicated that the more volatile ionic liquids have higher vapor pressures and the levels of vapor pressure increases with temperature rise reasonably. The results of vapor pressure calculation are shown in Fig. 2. Subsequently, the binary interaction parameters in the CPA EoS (kij) were tuned for ionic liquid-water and methane–ionic liquid systems using binary-mixture data [2,33,[36], [37], [38], [39], [40], [41], [42]]; However, the lack of experimental data for binary systems of CH4-[BMIM][Cl], CH4-[BMIM][Br], CH4-[EMIM][HSO4], and CH4-[OH-EMIM][BF4] prompted fitting their kij from gas hydrate data. The binary interaction parameters are obtained through minimization of the below objective function.Where, x denotes the β-Funaltrexamine hydrochloride sale fraction of ionic liquid in the liquid phase. Moreover, the binary interaction parameter available in the literature was used for water-methane system. The obtained binary coefficients and the percentages of error (amounts of OF) are reported in Table 3. To predict the equilibrium pressures of CH4 hydrate formation in the presence of ionic liquids, at first the VLE calculation was performed for ternary mixture of methane-water-ionic liquid. The fugacities of both liquid and vapor phases were calculated through CPA EoS. Then, the chen-guo method was used for calculating the gas hydrate equilibrium conditions. The algorithm of the calculations is shown in Fig. 3. It should be mentioned Septum in the thermodynamic modeling of gas hydrate through Chen-Guo theory, to calculate the f(a) in Eq. (5), the activity of water in the liquid phase should be obtained. So, the activity coefficient (γw) was calculated using Equation (19).Where, and are the fugacity coefficients of pure water and water in the liquid phase mixture, respectively. The CPA EoS was used to calculate these fugacity coefficients. The thermodynamic modeling of methane hydrate formation in the absence and presence of 10 weight percentage of imidazolium-based ionic liquids was performed using Chen-Guo method coupled with CPA EoS. The P-T equilibrium diagrams of hydrate formation are presented in Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9. According to the previous experimental investigations [2,43], the studied ionic liquids shift the P-T equilibrium diagrams of methane hydrate toward higher pressures and lower temperatures, suggesting the inhibitory effects of these materials on the equilibrium conditions of methane hydrate formation. As can be observed from Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, the proposed model can predict these shifts satisfactorily especially for [BMIM][BF4] and [EMIM][EtSO4], for which no hydrate data were used for adjusting the parameters. The accuracy of the model for predicting the hydrate formation pressures in the presence of ionic liquids was examined by calculating the average absolute deviation of P (AADP%) through Eq. (20). The calculated AADPs are reported in Table 4. According to the results, the overall average absolute deviation between the model and experimental data was obtained to be lower than 4.2%, suggesting good agreement between the model and experimental data. Among the studied ionic liquids, the minimum amounts of AADP% were obtained for [HO-EMIM][BF4] and [EMIM][EtSO4] (below 1.3%), while the maximum value belonged to [BMIM][Br] (4.2%). As already mentioned, the binary interaction parameters for [EMIM][EtSO4] and [BMIM][BF4] were completely tuned by binary mixture data, for which no hydrate data were used. Therefore, the AADPs calculated for hydrate equilibrium conditions for these ionic liquids (1.25% and 1.66%) can prove the prediction validity of the proposed model.