Quasi-components

Swaminathan S, Beveridge DL (1977) A theoretical study of the structure of liquid water based on quasi-component distribution functions. J Am Chem Soc 99 8392-8398... 

Figure 2. (a) Geometrical parameters used in the hydrogen bond deflnition. (b) Quasi-component distribution functions for the parameter R ST2 water at lO C. (From Mezei and Beveridge 1981. )... 

P.K. Mehrotra and D.L. Beveridge. Structural analysis of molecular solutions based on quasi-component distribution functions. Application to [H2CO]aq at 25 oC. J. Am. Chem. Soc. 102 (1980) 4287-94. 

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). 

Let E be any extensive thermodynamic quantity expressed as a function of the variables T, P, and N (where N is the total number of molecules in the system). Viewing the same system as a mixture of quasi-components, we can express E as a function of the new set of variables T, P, and N. For correctness, consider a QCDF based on the concept of CN. The two possible functions mentioned above are then... 

The first turning point occurred in the beginning of 1972 when the Kirkwood-BufF theory was found useful in interpreting some properties of water and aqueous solutions. The main idea was to apply the Kirkwood-BufF theory of solutions, to pure one-component systems viewed as a mixture of various quasi-component systems. The KB theory was also applied in the analysis of various ideal solutions on a molecular level (Ben-Naim 1973b, 1974). 

Studying the MM approach based on a quasi-component distribution function had led to the formulation of what I shall refer to as the principal molecular property of water, or for short, the principle. While the importance of the structure of water and the underlying hydrogen bonds were recognized long ago, the new and more fundamental aspect of the intermolecular interactions which can explain both the structure and the outstanding properties of water were recognized much later. 

A mixture of quasi-components must be distinguished from a mixture of real components in essentially two respects. First, the quasi-components do not differ in their chemical composition ... 

As a second example, consider the quasi-component distribution function based on the concept of binding energy (BE). We recall that the vector (or the function) xbe gives the composition of the system when viewed as a mixture of molecules differing in their BE. Thus, XBE( )dv is the mole fraction of molecules with BE between v and v + dv. A possible TSM constructed from this function is... 

The above examples illustrate the general procedure by which we construct a TSM from any quasi-component distribution function. From now on, we assume that we have made a classification into two components, L and H, without referring to a specific example. The arguments we use will be independent of any specific classification procedure. We will see that in order for such a TSM to be useful in interpreting the properties of water, we must assume that each component in itself behaves normally (in the sense discussed below). The outstanding properties of water are then interpreted in terms of structural changes, i.e. redistribution of the molecules into various species that take place in the liquid. 

At the molecular level, the condition (2.3.58) can be formulated in terms of the quasi-component distribution function xbe,cn(v,K). For a normal liquid, we expect that the average binding energy of species having a fixed coordination number will be a decreasing function of K (see Fig. 2.5a). On the other hand, in water we expect that this function will increase... 

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 Sec. 2.3, we have seen that any quasi-component distribution function can be used as a basis for constructing an exact... 

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. 

In this representation, we identify the partial molar energies of the various quasi-components as... 

We now briefly mention a similar treatment of the partial molar volume of the solute. Consider the quasi-component distribution function based on the volume of the Voronoi polyhe-dra (VP) (Sec. 2.3). Let Nu)() and Ns((f>) be the corresponding singlet distribution functions. The total volume of the system is written as... 

Relation (3.7.27) is very general. First, it applies to any two-component system at chemical equilibrium, as well as to any classification procedure we have chosen for the two quasi-components. Second, because of the application of the Kirkwood-Buff theory of solutions, we do not have to restrict ourselves to any assumption of additivity on the total potential energy of the system. Furthermore, the quantities Gap appearing... 

Here, we refer to a system of two real components in chemical equilihrium, such as two isomeric forms of a molecule. Relation (3.7.27) was derived for a system of two quasi-components, but the derivative on the left-hand side refers to a system at equilibrium. 

Note that in defining the components L and H we count only water molecules in the neighborhood of the quasi-components. 

Finally, we reiterate that the explanation of the origin of the large negative enthalpy of solvation was based on the local density and the binding energy of the quasi-components L and H. The stabilization of the L component occurs because of its low local density. This shift towards more of the L component carries with it a decrease in energy or enthalpy Hi — Hh < 0). Because of the compensation effect, the same explanation applies to the entropy of solvation one does not need to invoke the concept of structure to explain the negative entropy of solvation. 

In this appendix, we present the generalized Euler theorem for homogeneous functions of order one. We first write the Euler theorem for a discrete quasi-component distribution function QCDF) and then generalize by analogy for a continuous QCDF. A more detailed proof is available. ... 

Finally, we derive some general relations between the spatial pair correlation functions of the various quasi-components. We begin with the simplest case of a two-structure model. We denote Fy gafi R) the pair correlation function for the pair of species a and p. Then, Paga iR) is the local density of an a molecule at a... 

P. K. Mehrota and D, L. Beveridge,/. Am. Chem. Soc., 102,4287 (1980). Structural Analysis of Molecular Solutions Based on Quasi-Component Distribution Functions. Application to [HjCO], at 25 °C. 

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