The Mixture-Model Approach to Liquid Water

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. 

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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). 

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... 

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). 

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. 

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. 

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). 

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. 

For many years, the only approach to the molecular theory of water was the so-called mixture-model (MM) approach. We discuss further the origin and the historical development of the MM to liquid water in Sec. 2.3. Here, we note that in choosing the MM approach, one has to make a decision about the type of... 

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. 

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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