Quasicomponent distribution functions

From here on we make a specific choice of a quasicomponent distribution function QCDF) based on binding... 

Xjr may be referred to as the mole fraction of molecules with low (L) local density, and x as that of molecules with relatively high (H) local density. The new vector composed of two components (x, x ) is also a quasicomponent distribution function, and gives the composition of the system when viewed as a mixture of two components, which we may designate as L- and i7-cules. Starting with the same vector = ( c( )j c(1)j )> we may, of course, derive many other TSM s differing from the one in (6.73). A possibility which may be useful for liquid water is... 

As a second example, consider the quasicomponent distribution function, based on the concept of binding energy (BE) (Section 5.2). We recall that the vector (or the function) x gives the composition of the system when viewed as a mixture of molecules differing in their BE. Thus, X (v) dv... 

The above examples illustrate the general procedure by which we construct a TSM from any quasicomponent 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 anomalous properties of water are then interpreted in terms of structural changes that take place in the liquid. 

The property (in the sense of Chapter 5) that is now used to construct a quasicomponent distribution function is the number of hydrogen bonds connected to the /th molecule, defined by... 

It is instructive to demonstrate that with a specific choice of a quasicomponent distribution function (QCDF), we can express Es as a pure relaxation term. To do this, we specialize to the case of very dilute solutions of S in W, and also assume pairwise additivity of the total potential. We define the following two QCDF s for W and S molecules ... 

We now briefly mention a similar treatment of the partial molar volume of the solute. Consider the quasicomponent distribution function based on the volume of the Voronoi polyhedra (VP) (Chapter 5). Let and... 

We can extend the above argument to any structural change in the solvent. Let N be any vector that may be used as a quasicomponent distribution function (Chapter 5). For simplicity, we assume that N contains discrete components. Let N and N be the composition of the solvent when the two solutes are at R and at = oo, respectively. Then instead of (8.133) we have... 

General Relations between Thermodynamics and Quasicomponent Distribution Functions... 

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. 

On a molecular level, the condition (7.9.13) can be formulated in terms of the quasicomponent distribution function 5,c(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. On the other hand, in water we expect that this function will increase with K, at least for some range of P and T. This has indeed been demonstrated for two-dimensional water-like particles, and recently also by Monte Carlo simulation on water. 

A series of Monte Carlo computer simulation studies of the structure and properties of molecular liquids and solutions have recently been carried out in this Laboratory.The calculations employ the canonical ensemble Monte Carlo-Metropolis method based on analytical pairwise potential functions representative of ab initio quantum mechanical calculations of the intennolecular interactions. A number of thermodynamic properties including internal energies and radial distribution functions were determined and are reported herein. The results are analyzed for the structure of the statistical state of the systems by means of quasicomponent distribution functions for coordination number and binding energy. Significant molecular structures contributing to the statistical state of each system are identified and displayed in stereographic form. 

All calculated quantities reported, with the exception of the free energy for liquid water, are based on simple ensemble averages and are produced in a straightforward manner in the Metropolis procedure. The analysis of results is based particularly on quasicomponent distribution functions as Introduced by Ben-Naim. Quasicomponent distribution functions are defined on the statistical state of the system and give the distribution of... 

Figure 2. Calculated quasicomponent distribution function xc(Kj ds. coordination number K for liquid water...

Figure 3. Calculated quasicomponent distribution function xn(r) vs. binding energy v for liquid xcater...

An analysis of the structure of the dilute aqueous solution of methane was also developed in terms of quasicomponent distribution functions and stereographic views of significant molecular structures. The coordination number of methane in this system was calculated on the basis of 5.38, fixed at the first minimum In the methane-water radial distribution function. A plot of the mole fraction of methane molecules x (K) vs. their corresponding water coordination number is given in Figure 7. 

The calculated quasicomponent distribution function for binding energy, the mole fraction of methane molecules Xg(v) as a function of methane binding energyvis shown in Figure 8. 

Figure 7. Calculated quasicomponent distribution function xcfKj vs. methane coordination number K for the dilute aqueous solution of methane...

Figure 16. Calculated quasicomponent distribution functions Xr(K) vs. ion coordination number K for dilute aqueous solutions of alkali metal cations...

Figure 19. Calculated quasicomponent distribution functions Xn(v) vs. ion binding energy v for dilute aqueous solutions for halide anions...

The series of studies of molecular liquids presented herein collect results on a diverse set of chemically relevant systems from a uniform theoretical point of view ab initio classical statistical mechanics on the (T,V,N) ensemble with potential functions representative of ab initio quantum mechanical calculations of pairwise interactions and structural analysis carried out in terms of quasicomponent distribution functions. The level of agreement between calculated and observed quantities is quoted to indicate the capabilities and limitations to be expected of these calculations and in that perspective we find a number of structural features of the systems previously discussed on... 

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