Generalized van der Waals Equation

The thermodynamic equilibrium of two phases of one component is governed by the equations 

The same is true for the reduced quantities. The pressure is calculated by resolving Eq. (8.7) with respect to the pressure and inserting either the reduced volume of the liquid phase I or the volume of the gas phase g, to get 

The numerical solution of the coexistence curve is calculated from Eqs. (8.8) and (8.9). In these two equations are three unknown variables, i.e., g, I, t. In the first step, a certain value for the reduced volume of the gas phase is prescribed. Next, the reduced volume of the liquid phase and the reduced temperature are calculated from these two equations. Finally, by inserting into Eq. (8.7), the reduced pressure is found. The results are given in Table 8.1 and plotted in Fig. 8.1, marked as theory. 

A graphical method for calculating the equilibrium values consists of the direct calculation of the Legendre transformation of the molar-free Gibbs enthalpy, which is essentially the chemical potential. 

In Fig. 8.2, leftdown the ordinary plot of the reduced pressure p against the reduced volume p is shown. Left up of the integral shows the molar reduced Helmholtz energy or some kind of chemical potential. The Legendre transformation of this function is the molar reduced Gibbs enthalpy, or the reduced chemical potential as function of temperature and pressure. Recalling that 

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One can effectively reduce the tliree components to two with quasibinary mixtures in which the second component is a mixture of very similar higher hydrocarbons. Figure A2.5.31 shows a phase diagram [40] calculated from a generalized van der Waals equation for mixtures of ethane n = 2) with nomial hydrocarbons of different carbon number n.2 (treated as continuous). It is evident that, for some values of the parameter n, those to the left of the tricritical point at = 16.48, all that will be observed with increasing... 

Application of a Generalized van der Waals Equation of State to Several Nonpolar Mixtures at High Pressures... 

Exactly as In lattice models, the walks are assumed to take place In a (self-consistent) field Ulz), which depends on the concentration profile

relation between U[z) and (p[z) one may use the Floiy-Hugglns theory usually in an expanded form, but other models, such as a generalized Van der Waals equation of state ) can also be taken. The most general expression for the self-consistent mean field U z) has been given by Hong and Noolandl K It has been shown ) that this expression is the continuum analogue of the lattice version of Scheutjens and Fleer, to be discussed in sec. 5.5. 

Table 8.1 Coexistence curves from the generalized van der Waals equation, Eq. (8.7)...

The representation of pVTx properties of mixtures by using the cubic EOS is still a subject of active research. Kiselev (1998), Kiselev and Friend (1999), and Kiselev and Ely (2003) developed a cubic crossover equation of state for fluids and fluid mixtures, which incorporates the scaling laws asymptotically close to the critical point and is transformed into the original classical cubic equation of state far away from the critical point. Anderko (2000) and Wei and Sadus (2000) reported comprehensive review of the cubic and generalized van der Waals equations of state and their applicability for modeling of the properties of multicomponent mixtures. 

A. Anderko, Cubic and Generalized Van der Waals Equations, Ch. 4 in Equations of State for Fluids and Fluid Mixtures. Part I. Experimental Thermodynamics Volume V, Part I, (ed.), J. V. Sengers, R. F. Kayser, C. J. Peters and H. J. White Jr., for International Union of Pure and Applied Chemistry, Elsevier, Amsterdam, 1991 p. 95. 

The generalized Van der Waals equation with relation [7.1] can be used for a gas mixture by defining coefficients a" and b" of the mixture rrsing two relatiorts, [8.21] and [8.22], which we call mixing laws. 

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