The Mixture Model Approach to Liquids

The general idea of a mixture model (MM) approach is very simple and quite old. In particular, it has been applied extensively in the theory of water and aqueous solutions. In this section, we outline the basic procedure employed in the MM approach.  

Consider, for instance, the singlet generalized MDF, based on the coordination number (5.13.10)  

This has been interpreted in two ways. First, this is the mole fraction of particles having CN equal to K. Second, Xc K) is the probability that a specific particle, say 1 will be found with a CN equal to K. 

however, that since the particles exert strong repulsive forces when brought very close from each other, the occurrence of a large CN for a given particle must be a very improbable event. Therefore the sum over K in (5.13.37) effectively extends from A = 0 to, say, (depending of course, on the choice of Rc) Lq., the mole fractions Xc K), 

A mixture of quasicomponents must be distinguished from a mixture of real components in essentially two respects. First, the quasicomponents do not differ in their chemical composition or in their structure they are characterized by the nature of their local environment, which, in the above example, is the CN. A more important difference is that a system of quasicomponents cannot be prepared in any desired composition i.e., the components of the vector Xc cannot be chosen at will. One consequence of this restriction is that quasicomponents have no existence in the pure state. 

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We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

A simpler version of the same principle uses the language of the mixture-model approach to liquid water. Within this approach, the principle states that there exists a range of temperatures and pressures at which there are non-negligible concentrations of two species one characterized by large partial molar volume and large absolute value of the partial molar enthalpy, and a second characterized by a smaller partial molar volume and smaller absolute value of the partial molar enthalpy. In order to obtain the outstanding properties of water, one also needs to assume that the concentrations of these two species are of comparable magnitude (see Sec. 2.3 for details). 

Generalized Molecular Distribution Functions and the Mixture-Model Approach to Liquids... 

Z THE MIXTURE-MODEL APPROACH TO LIQUIDS CLASSIFICATIONS BASED ON LOCAL PROPERTIES OF THE MOLECULES... 

In section 2.7, we introduced the generalized molecular distribution functions GMDFs. Of particular importance are the singlet GMDF, which may be re-interpreted as the quasi-component distribution function (QCDF). These functions were deemed very useful in the study of liquid water. They provided a firm basis for the so-called mixture model approach to liquids in general, and for liquid water in particular (see Ben-Naim 1972a, 1973a, 1974). 

The term mixture model is somewhat misleading when applied to the exact mixture-model approach to liquids (see Sec. 2.3). 

This section presents various aspects of the mixture-model approach to the theory of water. We start with some historical notes in Sec. 2.3.1, and then proceed with one very simple representative of the MM approach due to Wada (1961). Although this model was successful in reproducing some of the anomalies of liquid water, it suffers from two serious drawbacks. One is the question of the validity of the specific MM approach used in the theory, and the second is the validity of the assumption of the ideality of the mixture. We shall discuss these problems in Sec. 2.3.3, where an exact MM is developed. We show that the assumption of ideality, though inconsistent with the requirement of a successful MM, is not essential to the interpretation of the properties of water. 

The procedure employed in this section will be generalized in the next section to obtain the basis of the so-called mixture-model approach to liquids. In fact, the treatment of a large number of problems in physical chemistry and especially in biophysics rests on arguments similar to those given above. 

In this section we generalize the concept of molecular distribution to include properties other than the locations and orientations of the particles. We shall mainly focus on the singlet generalized molecular distribution function (MDF), which provides a firm basis for the so-called mixture model approach to liquids. The latter has been used extensively for complex liquids such as water and aqueous solutions. 

These difficulties become more severe when we proceed to study aqueous solutions. Here again, various potential functions must be used to describe the interaction between pairs of molecules of different species. Because of these difficulties, it is no wonder that scientists have searched for other routes to study liquid water and aqueous solutions. Some of these routes are based on the mixture-model approach to these systems and will be described later in this chapter. 

We now use the definition of y/(X ) in (1.6.2) to construct an exact mixture model approach to liquid water. (This is exact within the definition of the primitive pair potential introduced in section 7.4.) In section 5.13 we showed that any quasicomponent distribution function can serve as the means for constructing a mixture model for any liquid. Specifically, for water, we construct the following mixture model. First we define the counting function... 

Lattice models for liquids are rarely used nowadays. The same is true of lattice models for water. Nevertheless, the model presented in this section is of interest for three reasons First, it presents a prototype of an interstitial model having features in common with many models proposed for water and used successfully to explain some of the outstanding properties of water and aqueous solutions. Second, this model demonstrates some general aspects of the mixture model approach to the theory of water, for which explicit expressions for all the thermodynamic quantities in terms of molecular properties may be obtained. Finally, the detailed study of this model has a didactic virtue, being an example of a simple and solvable model. 

Most of the recent theories of liquid solution behavior have been based on well-defined thermodynamic or statistical mechanical assumptions, so that the parameters that appear can be related to the molecular properties of the species in the mixture, and the resulting models have some predictive ability. Although a detailed study of the more fundamental approaches to liquid solution theory is beyond the scope of this book, we consider two examples here the theory of van Laar, which leads to regular solution theory and the UNIFAC group contribution model, which is based on the UNIQUAC model introduced in the previous section. Both regular solution theory and the UNIFAC model are useful for estimating solution behavior in the absence of experimental data. However, neither one is considered sufficiently accurate for the design of a chemical process. 

Barthel J, Bachhuber K, Buchner R, Hetzenbauer H (1990) Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols. Chem Phys Lett 165 369-373 Bates RG (1973) Determination of pH theory and practice. Wiley, New York Battino R, Clever HL (2007) The solubility of gases in water and seawater. In Letcher TM (ed) Developments and applications of solubihty, RSC Publishing, Cambridge, pp. 66-77 Ben-Naim A (1972) Mixture-model approach to the theory of classical fluids. II. Application to liquid water. J Chem Phys 57 3605-3612... 

In Sec. 2.3.5, we shall see that a central concept in the theories of water is that of the structure of water SOW). We shall devote Sec. 2.7.4 to discussing this concept. Here, we only point out that one way of defining the structure of water, is to use the mixture-model approach. The idea is very simple. We assume that liquid water may be viewed as a mixture of two components, say, an ice-like component and a close-packed one. We then identify the ice-like component with the more structured component, and hence, the degree of the structure can be measured simply by the concentration of this component. 

We have described two main lines of development in the theory of liquid water. The first, and the older, was founded on the mixture-model approach (Chapter 5) to liquids, which offers certain approximate or ad hoc models for the fluids as a whole. The second approach may be referred to as the ab initio method, based on first principles of statistical mechanics. In the past, these two lines of development were thought to be conflicting and a vigorous debate has taken place on this issue. As we have stressed throughout this and the previous chapter, both approaches can be developed from first principles, and, in fact, provide complementary information on this liquid. Once we attempt to pursue this theory along either route, we must introduce serious approximations. Therefore, it is very difficult to establish a clear-cut preference for one approach or the other. 

G. Nemethy and H. A. Scheraga, Structure of water and hydrophobic bonding in proteins. 1. a model for the thermodynamic properties of liquid water. J. Chem. Phys. 36, 3382-3400 (1962). A. Ben-Naim, Mixture-model approach to the theory of classical fluids. J. Chem. Phys. 56, 2864-2869 (1972). 

A. Ben-Naim, Mixture-model approach to the theory of classical fluids. IF Application to liquid water. J. Chem. Phys. 56, 3605-3612 (1972). 

Note that conditional averages appear in AEc, but not in A S- We shall now reinterpret this observation from a chemical equilibrium point of view i.e., the quantity Ep g - (Ep)o will be related to the shift in the chemical equilibrium L H induced by the adsorption of G. To do that we adopt a mixture-model point of view of the same system. This approach will be generalized in Chapter 5 to treat any liquid and in particular aqueous solutions. In our model we have two states of the adsorbent molecules. The mixture-model approach follows from the classification of molecules in state L as L-molecule, and likewise molecules in state H as an //-molecule. This is the same procedure we have discussed in section 2.4. We now assign partial molecular quantities to the species L and H, viewing Ml and Mh as independent variables. Theoretically these are defined as partial derivatives of the corresponding thermodynamic functions (see below). From the physical point of view, these can be defined only if we have a means of varying Ml and Mh independently, e.g., by placing an inhibitor that prevents the conversion between... 

Intermediate liquid 8 values are obtained by mixing liquids of known solubility parameter SPS makes use of this. The 8 value of the mixture is equal to the volume-weighted sum of the individual component liquid 8 values. Thus, the mass uptake of a miscible liquid mixture by an elastomer may be very much greater than the swelling which would occur in the presence of either one of the constituent liquids alone. The mixture could of course comprise more than two liquid components, and an analogous situation would apply MERL have applied this approach for the offshore oil-production industry to allow realistic hydrocarbon model oils to be developed,basically by mixing one simple aliphatic (paraffinic) hydrocarbon, one naphthenic, and one aromatic to proportions that meet two criteria, namely, that... 

Parameter Estimation from Phase Diagram Assessment. The second approach used to determine solid-solution behavior is to estimate solution model parameters from a fit to the measured phase diagram. Many of the solution models used to describe the liquid solution have been used to model the solid mixture. The simple-solution expression and its special cases have been used most extensively. 

This chapter is not organized according to the chronological sequence of the published theories, nor is it organized according to the extent of resemblance of the models to real liquid water, but rather in what I believe is the order of their interpretive power. Therefore, I will start with the mixture-model (MM) approach (which happens to be the older approach), then proceed with the 1-D models for water, which though very far from real water is the most useful one for understanding the properties of water. I will follow with the 2-D and 3-D models in subsequent sections. 

Such a splitting into four quasi-components can serve as a rigorous basis for a mixture-model approach for this liquid. This has direct relevance to the theory of real liquid water. 

In the next step, we focus our attention on one of the GMDF s, the singlet GMDF, which by a reinterpretation, can be used to view a one-component system as a mixture of various quasicomponents. This step provides a firm and rigorous basis for the so-called mixture-model (MM) approach to liquids. It now requires only a small step to reach the various... 

Because of the above difficulties, it is no wonder that many scientists have incessantly searched for other routes to studying liquid water and its solutions. The most successful approach has been the devising of various ad hoc models for water. In subsequent sections, we describe some of these theories and view them as approximate versions of the general mixture-model approach treated in Chapter 5. 

It is difficult to trace the historical development of the theory of water. One of the earliest documents attempting an explanation of some anomalous properties of water is Rontgen s (1892) article, which postulated that liquid water consists of two kinds of molecules, one of which is referred to as an ice-molecule. Rontgen himself admitted that his explanation of the properties of water, using the so-called mixture-model approach, had been known in the literature for some time, but he could not point out its originator. An interesting review of the theories of water until 1927 was presented by Chad well (1927). Most of the earlier theories were concerned with association complexes, or polymers of water molecules. There has been little discussion on the structural features of these polymers. 

In Chapter 5, we elaborated on the general aspects of the mixture-model (MM) approach to the theory of liquids. In this section, we present various ad hoc models for water as approximate versions of the general MM approach. We illustrate the point by a few examples. 

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