Cubica

Chapter 3 Equations of State
The simplest way to derive the Helmholtz function of a fluid is to directly integrate the equation of state with respect to volume (Sadus, 1992a, 1994). An equation of state can be applied to either vapour -liquid or supercrit ical phenomena without any conceptual difficulties. Therefore, in addition to liquid-liquid and vapour -liquid properties, it is also possibleto determine transitions between these phenomena from the same inputs. All of the physical properties of the fluid except ideal gas are also simultaneously calculated. Many equations of state have been proposed in the literature with either an empirical, semiempirical or theoretical basis. Comprehensive reviews can be found in the works of Martin (1979), Gubbins (1983), Anderko (1990), Sandler(1994), Economou and Donohue (1996), Wei and Sadus (2000) and Sengers et al. (2000).

The van der Waals equation of state (1873) was the first equation to predict vapour-liquid coexistence. Later, the Redlich-Kwong equation of state (Redlich and Kwong, 1949) improved the accuracy of the van der Waals equation by proposing a temperature dependence for the attractive term. Soave (1972) and Peng andRobinson (1976) proposed additional modifications of the Redlich-Kwong equation to more accurately predict the vapour pressure, liquid density, and equilibria ratios. Guggenheim (1965) and Carnahan and Starling (1969) modified the repulsive term of van der Waals equation of state and obtained more accurate expressions for hard sphere sys tems. Christoforakos and Franck (1986) modified both theattractive and repulsive terms of van der Waals equation of state. Boublik (1981) extended the Carnahan-Starling hard sphere term to obtain an accurate equation for hard convex geometries.

50

In addition to modeling small and simple molecules, considerable emphasis has been placed on modeling long and convex molecules. Based on theory of Prigogine (1957) and Flory (1965), an equation formolecules treated as chains of segments, which is called PerturbedHard-Chain-Theory (PHCT) was constructed by Beret and Prausnitz (1975) and Donohue and Prausnitz (1978). To reduce the mathematical complexity of Perturbed-Hard-ChainTheory, Kim et al. (1986) developed a simplified version of the theory by replacing the complex attractive part by a simpler expression. At the almost same time, Vimalchandand Donohue (1985) obtained a fairly accurate multipolar mixture calculation by using the Perturbed Anisotropic Chain theory, and Ikonomou and Donohue (1986) extended the Perturbed Anisotropic Chain Theory to the Associated Perturbed Anisotropic Chain Theory by taking into account the existence of hydrogen bonding.

Wertheim

(1987)

proposed

a

thermodynamic

perturbation

theory(TPT),

which

accommodates hard-chain molecules. Chapman et al. (1988) generalized the TPT model to obtain the compressibility factor of a hard-chain of segments. Ghonasagi and Chapman (1994) and Chang and Sandler (1994) modified TPT for the hard-sphere chain by incorporating structural information for the diatomic fluid (TPT-D). Sadus (1995) derived the simplified thermodynamicperturbation theory–dimer (STPT-D) equation from TPT-D. Sadus (1999b) later developed STPT-D to the empirical simplified thermodynamic perturbation theory-dimer (ESTPT-D) equation, and tested the accuracy of the equation against simulation data for hard-sphere chains containing up to 201 hard-sphere segments. Sadus (1999 a) also derived an equation of state for hard convex body chains from the TPT of hardsphere chains (Wertheim, 1987; Chapman et al., 1988). 51

Jin et al. (1993), Povodyrev et al. (1996) and Kiselev (1997) developed theoretical crossover equations of state for pure fluids and binary mixtures which incorporate the scaling laws asymptotically close to the critical point and which are transformed into the regular classical expansion far away from the critical point. Kiselev (1998)...