Generalized van der Waals Equations of State

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. 

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. 

The virial equations are unsuitable forhquids and dense gases. The simplest expressions appropriate (in principle) for such fluids are equations cubic in molar volume. These equations, inspired by the van der Waals equation of state, may be represented by the following general formula, where parameters b, 9 5, S, and Tj each can depend on temperature and composition ... 

Pressure is a manifestation of the kinetic energy of gas molecules. According to the van der Waals equation of state (see Eq. (36)), the pressure of 1 mole of gas must be increased by an amount a/v2 due to intermolecular attractions that decrease the pressure from what it would be if ideal. Use the term a/v2 as a general expression for the attraction between a pair of molecules and, based on this, reconstruct the argument leading to Equation (52). 

Bearing in mind that for a liquid which obeys the Van der Waals equation of state the specific cohesion U/V is ajV2, this formula can be written more generally as ... 

The first generalization of the ideal gas law was the van der Waals equation of state ... 

Such cubic equations of state as van der Waals correlate very satisfactorily the UCST-type behavior for polymers solutions, as shown by Harismiadis et al. ° A generalized correlation of the interaction parameter of the van der Waals equation of state for polymer blends based exclnsively on polystyrene blends has been presented. By nsing this equation, the van der Waals eqnation of state can be used as a predictive tool for investigating the compatibility of polymer blends. Predictive GC thermodynamic methods such as Entropic-FV, GC-Flory, UNIFAC, and UNIFAC-FV perform rather poorly, at least from a quantitative point of view. Entropic-FV performs best among these models, on a qualitative basis. For semiquantitative predictions in polymer blends, the approach proposed by Coleman et al. is recommended. 

Although the discussion of the previous section focused on the van der Waals equation and corresponding-states charts for both the compressibility factor Z and the thermodynamic departure functions, the modem application of the corresponding states idea is to use generalized equations of state. The concept is most easily demonstrated by again using the van der Waals equation of state. From Eqs. 6.2-38,... 

Modified van der Waals equations of state may be written in a general manner as ... 

Here is a useful leisure time exercise for a very attentive reader. The purpose is to understand the connection between virial expansion (8.7) and the well known van der Waals equation of state (i.e., the relationship between volume, pressure, and temperature) for an ordinary imperfect gas. You may have studied van der Waals equation in general physics and/or general chemistry class, it reads p- -a/V )(V — b) = NksT. Say, the volume is V, and the number of molecules in the gas is N. Then n = N/V. You can work out the pressure by differentiation p = — (9F/9V), where free energy F is defined by formula (7.19), F = U — TS = t/ + / , the internal energy U is given by (8.7), and... 

In a more general case one has a sum of different oscillators. The dispersion interactions are partially responsible for the well known attractive term a V between neutral molecules in the Van der Waals equation of state (p + ajV ) V — b)= RT. The corresponding energy is of the order of 0.1 kJ/M or 10 " eV/mol. By the way, the repulsive term (V — b) in the same equation that takes into account the excluded volume effect b is due to the steric interaction discussed next. 

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