Water mixture model approach

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. 

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). 

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... 

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. 

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. 

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). 

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. 

We can now re-interpret the quantity Um)s — Um)o oti the right-hand side of (3.4.7) in terms of structural changes. A more appropriate term would be redistribution of quasi-components. We shall do it in two steps. First, we use the binding energy distribution function xbe introduced in Sec. 2.3. Second, we shall reformulate this quantity in terms of structure as defined in Sec. 2.7.4. Finally, we shall use the same quantity to apply to a two-structure mixture-model approach to water. 

Note that the temperature dependence of AG depends on the degree of structure of water, but it is not necessarily a monotonic dependence. We have seen that A5 depends on the structural changes induced by the solute, and the extent of the structural changes depends, in an ideal mixture model for water, on product XiXp. This means that in some regions, an increase in X can either increase or decrease the product Xi — xi). In a nonideal mixture-model approach, the dependence of the structural changes induced by the solute is not so simple as xi — Xi), but the general conclusion that A5 is non-linear or even monotonic in Xi is still valid. 

To the best of my knowledge, the formulation of the principle and its significance to the understanding of the anomalous properties of water was first published in 1972. This principle and its implications were repeatedly used later by many authors who rediscovered it. Similarly, the exact entropy-enthalpy compensation theorem and its implications for the theory of aqueous solutions were first discussed in terms of a mixture-model approach in 1965. Later, it was proved in a much more general form in Ben-Naim (1975b, 1978b). This theorem was reproved several times by several authors using different nomenclature and different notations. 

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. 

As is the case for all other models of water, it is difficult to assess the validity of the assumptions underlying the choice of the particular partition function (6.25). Similar works using the mixture-model approach have accomplished a similar agreement with experimental results [for instance, models worked out by Samoilov (1946), Grjotheim and Krogh-Moe (1956), Wada (1961), Marchi and Eyring (1964), Jhon et al. (1966), Hagler et aL (1972)]. We elaborate in detail on the merits of the simplest mixture-model approach in Section 6.8. 

In the next two sections, we undertake to describe the general theoretical reasoning underlying the mixture-model approach to water. In Section 6.7, we describe a special group of mixture models, which may be called interstitial models, and we outline the theory common to all of the models in this group. In Section 6.8, we discuss the general consequences of the two-structure models, without commitment to any specific model. 

A comment regarding nomenclature is now in order. We distinguish between a one-component approach and a mixture-model approach to water. Within the latter, we further distinguish between continuous and discrete mixture-model approaches, according to the nature of the parameter used for the classification procedure (for instance, v and K in Chapter 5). All of these are equivalent and formally exact. The various ad hoc mixture rnodels for water may be viewed as approximate versions of the general mixture-model approach. This will be discussed in Section 6.9. 

The purpose of this section is threefold. First, it presents a prototype of an interstitial model having features in common with the models proposed for water, notably, the Samoilov (1957) and Pauling (1960) models. These have been worked out in considerable detail by Frank and Quist (1961) and Mikhailov (1967). Second, this model demonstrates some general aspects of the mixture-model approach to the theory of water, for which explicit expressions for 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. 

Let us now consider some general features of the present model. At the very outset, we emphasize that the basic variables of our system are r, P, and i.e., we have a one-component system, and there is no trace of a mixture-model approach. The variables and in (6.32) play the role of convenient intermediary variables. Once we have carried out the summation in (6.32), the dependence of the partition function on disappears. Nevertheless, the nature of the model suggests a new way of looking at this one-component system namely, we decide to refer to a lattice molecule as an L-cule and to an interstitial molecule as an LT-cule (the letters L and H in the present context are chosen to remind us of water in lattice and water in holes. The same notation will be used in Section 6.8 to indicate the type of environment, that of low local density and high local density, respectively.) Once we have made this classifica-... 

Mikhailov (1967) worked out a detailed statistical mechanical theory of the Samoilov model. In particular, he made an estimate of the temperature dependence of the mole fraction of the lattice molecules. It is puzzling to find a comment in this article saying We do not consider Samoilov s ice-like model to be one of the so-called two-structure water models, since by the very fact that we choose to classify molecules into two or more species, we have already made the choice of the mixture-model approach (in the sense of Chapter 5). 

L and H are both water molecules. However, by their very definition, they differ markedly in local environment hence, it is unlikely that they obey the condition for symmetric ideal solutions. This point will be discussed further in Section 6.8, when we analyze a more general mixture-model approach to water. [See also the discussion following (6.52).]... 

The simplest version of the mixture-model approach and, in fact, the most useful one is the so-called two-structure model (TSM). We studied one specific example of a TSM in the previous section. Many ad hoc models of liquid water fall into the realm of the TSM [notable examples are those of Samoilov (1946), Hall (1948), Grjotheim and Krogh-Moe (1954), Pauling (1960), Wada (1961), Danford and Levy (1962), Krestov (1964), Marchi... 

Kell (1972), and Davis and Jarzynski (1972).] The major argument raised against the mixture-model approach is that no direct experimental evidence exists to support it. Therefore, one should view the whole approach as basically speculative. In the process of constructing the mixture-model approach in this section (based on Chapter 5), we have made no reference to experimental arguments. Therefore, experimental information cannot be used as evidence either to support or refute this approach. There is, however, a viewpoint from which to criticize specific ad hoc models for water. These can be viewed as approximate versions of the general and exact mixture-model formalism, and will be discussed in the next section. 

EMBEDDING AD HOC MODELS FOR WATER IN THE GENERAL FRAMEWORK OF THE MIXTURE-MODEL APPROACH... 

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. 

The second part consists of Chapter 5 alone, which comprises a bridge connecting the formal theory of fluids, on the one hand, and its application to water and aqueous solutions on the other. The construction of this bridge is rendered possible through the generalization of the ideas of molecular distribution functions, which lays the foundation for the so-called mixture-model approach to the theory of fluids. The latter may be viewed as the formal basis for various ad hoc mixture models for water and aqueous solutions that have been suggested by many authors. 

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). 

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. 

THE STRUCTURE OF WATER AND THE MIXTURE MODEL APPROACH TO THE THEORY OF WATER... 

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 5.13 we have seen that any quasicomponent distribution function can be used as a basis for constructing an exact mixture model for any liquid. We now develop a mixture model approach which is particularly suitable for liquid water. A variety of approximate versions of such mixture model approaches have been used in the development of theories of water and aqueous solutions. 

Having the PF of the system, we can derive all the required thermodynamic quantities by standard relations. As in any mixture model approach (section 5.13), we start with a one-component system. The independent variables are T, P and Nyy. We then choose to refer to a lattice water molecule as L and to an interstitial molecule as an H molecule. Once we have made this classification, we may adopt the point of view that our system is a mixture of two components, L and H molecules. 

In this section we formulate the general aspect of the application of the simplest mixture model approach to water. We shall use an exact two-structure model as introduced in section 7.9. In the following subsection, we shall illustrate the application to solutions of an interstitial model for water, and in section 7.13 we shall discuss the application of a more general MM approach to this problem. 

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