According to this theory the chemical potential
According to this theory, the chemical potential of the basic hydrate in the equilibrium condition can be calculated as:
, and denote the chemical potentials of the basic hydrate, water and gas, respectively. Also, is the number of gas molecules per water molecules in the basic hydrate. After substituting the relations for each of the chemical potentials and to simplify the equations, the following equation is obtained :Where, denotes the number of linked cavities per water molecules and . In Equation (2), for gas hydrate with structure I and for structure II. is the fraction of linked cavities occupied by guest molecules which can be calculated using Langmuir theory, Further, f is the gas fugacity and denotes the product of three functions representing the contribution of temperature, pressure and activity of water (a) as
The pressure function is: is a structural parameter equal to 0.4242 K/bar for structure I and 1.0224 K/bar for structure II, where the activity function is:
And,Where , and are adjustable parameters which can be obtained by optimizing the experimental data for a particular hydrate structure and a special guest molecule .Where, i and j denote the guest gas component, zi is the betaxolol fraction of guest molecule i in the basic hydrate cavities, Cj represents the Langmuir constant which has been reported by Chen and Guo  for CH4 in the structure I. Also, denotes the fugacity of gas species in the basic hydrate which can be calculated from Eq. (3).
The CPA EoS proposed by Kontogeorgis et al., in 1996 was composed of two parts (SRK EoS + association term), which can be expressed in term of pressure as follows :
In this equation, denotes the molar density ( ), is the fraction of sites A on molecule i which do not form bonds with other active sites. Also, xi denotes the mole fraction of species i. can be calculated using association strength between two sites belonging to two various molecules, through Equations (11), (12):Where , and g and η denote the radial distribution function and reduced density, respectively, which are obtained as follows;Where, the parameter represents the association energy and is called the association volume. Also, the parameters b and a are the co-volume and energy parameter of CPA EoS, respectively. The energy parameter is given by:
Where, Tr is the reduced temperature. These parameters are pure component parameters and are adjusted using pure liquid density data for the six studied imidazolium-based ionic liquids.
Moreover, the conventional mixing rules are used in the SRK term for the energy and co-volume parameters.
Where, kij is the binary interaction parameter. In this study, the binary coefficients for methane-ionic liquids and water-ionic liquids have been obtained using binary VLE data except for the binary coefficients of CH4-[BMIM][Cl], CH4-[BMIM][Br], CH4-[EMIM][HSO4] and CH4-[OH EMIM][BF4] systems, which were obtained using the experimental data of gas hydrate phase equilibria.
In addition, Equation (16) is employed to calculate the fugacity coefficient of species i in a mixture using CPA EoS :Where, Z is the compressibility factor of the mixture.
Results and discussion In this study, the aim has been to calculate the P-T equilibrium conditions of methane gas hydrate formation and dissociation in the presence of six imidazolium-based ionic liquids including [EMIM][HSO4], [EMIM][EtSO4], ([BMIM][BF4], [OH-EMIM][BF4], [BMIM][Cl], and [BMIM][Br] using two-step hydrate formation theory of Chen and Guo coupled with the CPA EoS. To perform this, the pure component parameters of CPA EoS have been required for the methane, water and ionic liquids. In addition, the association schemes should be considered for the species. These parameters relating to CH4 and H2O along with the association scheme of water were taken from Refs.  and , as presented in Table 1.