Waals One-Fluid Theory

The most successful corresponding-states theory for mixtures is called the van der Waals one-fluid theory. This theory was developed on a molecular basis by Leland and co-workers and follows from an expansion of the properties of a system about those of a hard sphere system. A hard-sphere system is one whose molecules only have repulsive intermolecular potentials with no attractive contributions. The starting equation for the development of the van der Waals one-fluid (known by the acronym VDW-1) theory is a rigorous statistical-mechanical result for the equation of state of a mixture of pair wise-additive, spherically symmetric molecules  

In eq 6.54 equation Z is the compressibility factor pVjRT, gy is the radial distribution function which gives the probability of finding a molecule of type i at a distance r from a central molecule of type j, Uy is the intermolecular potential whose parameters are and Cy, the prime denotes differentiation with respect to distance r, k is Boltzmann s constant and p the number density. In the development of the VDW-1 theory the intermolecular potential is assumed to be composed of a hard-sphere term plus a long-range attraction, that is given by. 

F r) of eq 6.55 is a long-range attraction contribution to the potential, such as Cg/r . Before one can proceed, assumptions concerning the radial distribution 

The analogous result for a hypothetical pure fluid at the same reduced number 

we have at our disposal an infinite set of terms (coefficients of I ) from which we can choose two for the determination of the potential parameters of the hypothetical pure fluid. In the van der Waals one fluid model, the first two members of the series are chosen, giving 

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S. Murad, J. G. Powles, B. Holtz. Osmosis and reverse osmosis in solutions Monte-Carlo simulations and van der Waals one-fluid theory. Mol Phys 55 1473, 1995. 

The van der Waals one-fluid theory is quite successful in predicting the properties of mixtures of simple molecules. Unfortunately, the systems usually considered by chemists are considerably more complex, and often involve hydrogen bonding and other chemical interactions. Nevertheless, the material presented here outlines how one could proceed to develop models for more complex systems on the basis of the integral equation approach. 

The van der Waals one-fluid theory for mixtures assumes the properties of a mixture can be represented by a hypothetical pure fluid. Thus the thermodynamic behaviour of a mixture of constant composition is assumed to be isomorphic to that of a one-component fluid this assumption is not true near the critical point where the thermodynamic behaviour of a mixture at constant thermodynamic potential is isomorphic with that of a one-component fluid and this is discussed further in Chapter 10. 

The BWR equation has been used to calculate the thermodynamic properties of mixtures based on the idea that both the mixture and the pure-fluid equations should satisfy the same equation of state and provide the correct composition dependence of as many virial coefficients as possible. This was achieved with mixing rules similar to those obtained from the van der Waals one-fluid theory that are as follows ... 

Thus far we have only introduced the pure-fluid corresponding-states principle which, as mentioned above, has a rigorous basis in molecular theory. The extension of this theory to mixtures cannot, however, be made without further approximation and the problem of rigorous, yet tractable, prediction of mixture properties remains unsolved. These approximations take the form of mixing rules which are the topic of Chapter 5 in this volume. We will only discuss mixing rules from an illustrative basis to show problems that can arise in the implementation of a corresponding-states model. In that regard, we will focus our discussions on the one-fluid theories and primarily the van der Waals one-fluid theory proposed by Leland et The essence of this model... 

Equations 3.67 and 3.68 require a total of 42 constants, which are adjusted to fit experimental pure-component data of n-alkanes. This direct fitting to experimental data to some extent accounts for errors in the reference equation of state, the perturbing potential, and the radial distribution function, which appear in the integrals of Equations 3.63 and 3.64. The dispersion potential given by Equations 3.60 and 3.61 is readily extended to mixtures using the van der Waals one-fluid theory. 

Sarashina, E., Arai, Y., and Saito, S. Application of the van der Waals one fluid theory to predict vapor-liquid-equilibria. J. chem. Engng. Japan 6 (1973) 120. 

Efforts have been made to develop EOS for detonation products based on direct Monte Carlo simulations instead of on analytical approaches.35-37 This approach is promising given recent increases in computational capabilities. One of the greatest advantages of direct simulation is the ability to go beyond van der Waals 1-fluid theory, which approximately maps the equation of state of a mixture onto that of a single component fluid.38... 

Table I, in the column headed HSE-VW, shows the results of using Equations 2 through 6 to define the diameters with the HSE method to calculate the properties of an equimolar mixture of LJ fluids. The reference is a pure LJ fluid. Other columns show comparison with the machine-calculated results of Singer and Singer (8) in column MC. The van der Waals (VDW) one-fluid theory (9) and the VDW two-fluid theory (10) are in columns VDW-1 and VDW-2. The GHBL column gives the Grundke, Henderson, Barker, Leonard (GHBL) (11) pertu-bation theory results with each diameter determined by Equation 3.

A one-fluid theory is generally superior to a two-fluid theory in generalizing the LHW equation of state for mixtures. In the one-fluid theory the same form of equation is used for the mixture as for the components, but the equation of state parameters are assumed to be composition dependent. Van der Waals (VDW) forms, quadratic in mole fractions, prove satisfactory to describe this composition dependence for simple species. 

This equation shows that the excess Gibbs free energy computed from a cubic EOS of the van der Waals type and the one-fluid mixing rules contains three contributions. The first, which is the Flory free-volume term, comes from the hard core repulsion terms and is completely entropic in nature. The second term is very similar to the excess free-energy term in the regular solution theory, and the third term is similar to a term that appears in augmented regular solution theory. Consequently, one is led... 

The first and second order perturbation contributions could be evaluated by using rigorous expressions for and gjj as obtained from an integral equation theory. Such an approach has been recently undertaken with good results [301, 303], Unfortunately, the expressions are quite lengthy and somewhat inconvenient for further differentiation. For this reason, we will evaluate the perturbative contributions of the free energy by using a Van der Waals like one fluid approximation. In this approximation, one considers that the radial distribution function of the Lennard-Jones fluid mixture may be expressed in terms of the radial distribution function of a pure effective Lennard-Jones fluid with composition dependent parameters, try and ey, yet to be determined. More specifically, one assumes that gij (r) may be expressed in terms of the radial distribution function of a pure fluid as follows. 

Nearly all experimental eoexistenee eurves, whether from liquid-gas equilibrium, liquid mixtures, order-disorder in alloys, or in ferromagnetie materials, are far from parabolie, and more nearly eubie, even far below the eritieal temperature. This was known for fluid systems, at least to some experimentalists, more than one hundred years ago. Versehaflfelt (1900), from a eareflil analysis of data (pressure-volume and densities) on isopentane, eoneluded that the best fit was with p = 0.34 and 8 = 4.26, far from the elassieal values. Van Laar apparently rejeeted this eonelusion, believing that, at least very elose to the eritieal temperature, the eoexistenee eurve must beeome parabolie. Even earlier, van der Waals, who had derived a elassieal theory of eapillarity with a surfaee-tension exponent of 3/2, found (1893)... 

We should mention here one of the important limitations of the singlet level theory, regardless of the closure applied. This approach may not be used when the interaction potential between a pair of fluid molecules depends on their location with respect to the surface. Several experiments and theoretical studies have pointed out the importance of surface-mediated [1,87] three-body forces between fluid particles for fluid properties at a solid surface. It is known that the depth of the van der Waals potential is significantly lower for a pair of particles located in the first adsorbed layer. In... 

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

It should be emphasized that the comparatively large change obtained in more recent work is mainly caused by the application of finite-size scaling. Under these circumstances, one certainly needs to reconsider how far the results of analytical theories, which are basically mean-field theories, should be compared with data that encompass long-range fluctuations. For the van der Waals fluid the mean-field and Ising critical temperatures differ markedly [249]. In fact, an overestimate of Tc is expected for theories that neglect nonclassical critical fluctuations. Because of the asymmetry of the coexistence curve this overestimate may be correlated with a substantial underestimate of the critical density. 

We turn now to theories of ionic criticality that encompass nonclassical phenomena. Mean-field-like criticality of ionic fluids was debated in 1972 [30] and according to a remark by Friedman in this discussion [69], this subject seems to have attracted attention in 1963. Arguments in favor of a mean-field criticality of ionic systems, at least in part, seem to go back to the work of Kac et al. [288], who showed in 1962 that in D = 1 classical van der Waals behavior is obtained for a potential of the form ionic fluids with attractive and repulsive Coulombic interactions have little in common with the simple Kac fluid. 

The forces leading to microstructure formation in complex fluids are relatively few Excluded-volume, van der Waals, and electrostatic forces are the main ones. In some fluids, hydrogen bonding, hydrophobic, or various solvation forces are also important. Simplified theories can account for the effects of these forces on fluid structure and, to some extent, on relaxation rates. 

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