When developing an equation of state for molecular fluids one can follow two different strategies. First, it is possible to take an empirical equation of appropriate functional form and to find the values for its coefficients so as to reproduce available experimental data. The main disadvantage of this approach is that it is usually valid only in the restricted P-T range where experimental data are available.

The second possibility is to describe interaction of molecules in the fluid by an abstract physical model. Then it is necessary to construct an equation, describing the behavior of the hypothetical fluid with a chosen model of intermolecular interaction. In contrast to the empirical approach the experimental data are not used to fit coefficients of the equation but to choose a realistic model of intermolecular interaction. The basic idea is that interaction between molecules change only slightly with P and T. This means, that the model obtained using the experimental data in one P-T range will be also accurate at other P-T conditions. Last makes theoretical equations safe for extrapolations.

To develop an equation of state for a given model of intermolecular interaction, one can use molecular dynamic or/and Monte-Carlo simulations or thermodynamic perturbation theory.
Moreover, there are strong connection between both ideas.
Both methods benefits and suffer some limitations.
In this work perturbation theory have been implemented.
The physical pattern of the intermolecular interaction considered here is the model the *dipolar "Lennard-Jones"-spheres*.

In the next part the basic ideas of the perturbation theory are reported. We do not go far into detail, especially the ones relating to computational problems, since that can be found in the references given in the text.

The main contribution to the potential interaction energy between two polar molecules is given by the sum of Lennard-Jones, dipole-induced dipole, and dipole-dipole interaction:

where

with *r _{ij}* distance between centers of molecules

Figure 1.

Figure 2. Lennard-Jones interaction potential.

Gray and Gubbins, 1984 used the thermodynamic perturbation theory to develop equation of state for the fluid mixtures interacting with such potential. While thermodynamic properties of the system with Lennard-Jones interaction are well known, it convenient to take Lennard-Jones system as a reference one and consider contribution from other type of interaction(dipole-dipole, dipole-induced dipole) as perturbation (deviation from properties of Lennard-Jones system). Then Helmholtz free energy A is expressed in form of power expansion:

where A^{LJ} is Helmholtz free energy of Lennard-Jones fluid and A_{1}, A_{2}, A_{3} are first, second and third order perturbation contribution.

Explicit expressions for and A_{1}, A_{2}, A_{3} in the fluid mixtures are follow (Gubbins and Twu 1978a):

where *x _{i}* -mole fraction of compound

where U and Z are internal energy and compressibility of Lennard-Jones fluid.

In order to improve convergence of perturbation expansion Stell at al. (1972, 1974) recommended to use [0,1] Pade approximation for dipole-dipole interaction, which gives final expression for Helmholtz free energy of system as:

Thermodynamic properties of reference Lennard-Jones system were calculated with modified Weeks- Chandler-Andersen perturbation theory (Weeks at al., 1971; Andersen at al., 1972).:

there A_{HS} is free energy of hard sphere system ( Carnahan and Starling, 1969; Boublic 1970), A_{HTA} is so called high temperature approximation. In current work Shmulovich, at al.(1982) results were used for A_{HTA}. Simple analytical expression for A_{RPA} term (random phase approximation) proposed by Miyano and Masuoka (1984). For the details of practical implementation see discussion of Verlet and Weis (1972, 1973).

Perturbation expansion discussed above give Helmholtz free energy of the system *A=A(T, r)* as function of temperature and fluid density. Other thermodynamic quantities of interest are derived by differentiation:

where n number of moles; f-fugacity of pure fluid; f_{i} fugacity coefficient of compound *i* in the mixture. List of other important derivatives is given by Praushnitz at al (1986).

See: pdf-documentation file to GEMS-package

- Andersen H. C., Chandler D. and Weeks J. D. (1972) Role of repulsive forces in liquids: The optimized random phase approximation. J. Chem. Phys. 56, 3812-3823
- Carnahan N.F. and Starling K.E. (1969) Equation of state for non-attracting rigid spheres. J. Chem. Phys. 51, 635-636
- Gubbins K.E. and Twu C.H. (1978) Thermodynamics of polyatomic fluid mixtures. I Theory, Chem. Eng.. Sci. 33, 863-878
- Miyano Y. and Masuoka H. (1984) Equation of state based on Weeks-Chandler-Andersen perturbation theory. Fluid Phase Equilib. 18, 131-146
- Shmulovich K.I., Tereshchenko Ye.N. and Kalinichev A.G. (1982) An equation of state and isochors for nonpolar gases to 2000K and 10 GPa. Geochem. Internat. 19, 49-64
- Stell G., Rasaiah J.C. and Narang H. (1972) Thermodynamic perturbation theory for simple polar fluids, I. Mol. Phys. 23, 393-406
- Stell G., Rasaiah J.C. and Narang H. (1974) Thermodynamic perturbation theory for simple polar fluids, II. Mol. Phys. 27, 1393-1414
- Verlet L. and Weis J.J. (1972) Perturbation theory for the thermodynamic properties of simple liquids. Mol. Phys.24, 1013 - 1024
- Verlet L. and Weis J.J. (1972) Equilibrium theory of simple liquids. Phys Rev. 5, 939-952
- Weeks J. D. Chandler D. and Andersen H. C. (1971) Role of repulsive forces in determinig the equilibrium structure of simple liquids. J. Chem. Phys, 54, 5237-5247