Quasicomponent

The results are summarized in Fig. 8.16, which is comparable to Fig. 7.2, p. 152. This simple theory predicts the predominance of the HO [H20] quasicomponent, and that prediction is independent of the level of sophistication of electronic structure theory calculations with conventional basis sets give the same trends as calculations with larger basis sets (Asthagiri et al, 2003<7). 

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

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... 

Of course, N and Nb in (5.75) are not independent variables, i.e., we cannot prepare a system with any chosen values of N and Nb, as in the case of a real mixture of two components. Hence, we refer to the A-cules and the -cules as quasicomponents. One may envisage a device which prevents the conversion of molecules between A and B, Such a device may be called an inhibitor (or an anticatalyst) for the conversion reaction A B. A system in the presence of this inhibitor is referred to as being frozen in with respect to the conversion A B, Clearly, the partition function of our system in the frozen-in state is (5.75) and not (5.74). 

In the previous section, we presented two very simple classification procedures for splitting a one-component system into a mixture of, say two, quasicomponents. Here, we extend that idea to include more subtle procedures which will be particularly useful in the study of aqueous fluids. Most of the ground work for this extension has been carried out in Section 5.2, where various singlet GMDF s were introduced. 

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 Xq cannot be chosen at will [even when they satisfy the normalization condition (5.92)]. One consequence of this restriction is that quasicomponents have no existence in the pure state. ... 

There exists a certain analogy between a mixture of quasicomponents and a mixture of chemically reacting species. For the sake of simplicity, consider a dimerization reaction... 

Of course, for some special cases, one may envisage a system of pure quasicomponents. For instance, a solid in the regular close packing state is composed of particles having CN equal to A" == 12. This, in general, cannot be realized in a liquid. 

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 quasicomponents, we can express P as a function of the new set of variables P, P, and N. For concreteness, consider the QCDF based on the concept of CN. The two possible functions mentioned above are then ... 

Clearly, is the mole fraction of particles having a CN smaller than or equal to K. These may be referred to as particles with low local density (for more details, see Section 5.2). Similarly, Xjj is referred to as the mole fraction of high-local-density particles (i.e., particles for which K > K ). In this way, the system is viewed as a mixture of two quasicomponents, L and H. This point of view is called a two-structure model (TSM). In a similar fashion, one can construct TSM s from any other discrete or continuous QCDF. Therefore, the following treatment applies for any TSM, not necessarily the one defined in (5.122) and (5.123). 

Finally, we derive some general relations between the pair correlation functions of the various quasicomponents. We begin with the simplest case of a TSM, and denote by L and H the two species, and by W any molecule in the system. We denote by ga R) the pair correlation function for the pair of species a and Then, Qo,goip R) is the local density of a-cules at a distance R from a jS-cule. Conservation of the total number of W molecules around an L-cule gives... 

Relations similar to (5.137) can be generalized to any number of quasicomponents. In the case of a discrete QCDF, we get the relation... 

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... 

Such a splitting into four quasicomponents can serve as a rigorous basis for a mixture-model approach for this fluid. This has a direct relevance to the theory of real liquid water. Suppose we know the function x v) for water then the validity of various ad hoc models can be assessed according to the form of this function. 

In this section, we discuss a more general problem. Suppose we classify molecules into quasicomponents by any one of the classification procedures. We then select one of these species and inquire about the change in its concentration upon the addition of a solute. As a particular example, we may choose one of the species to be the fully hydrogen-bonded molecules hence, its concentration can serve as a measure of the structure of the solvent. 

Let L and H be two quasicomponents obtained by any classification procedure. The corresponding average numbers of L- and //-cules are N and Nji, respectively. We assume that the temperature and the pressure are always constant. The quantity of interest is, then, the derivative... 

Relation (7.113) is very general. First, it applies to any two-component system at chemical equilibrium, and to any classification procedure we have chosen to identify the two quasicomponents. Second, because of the application of the Kirkwood-Buff theory of solutions, we do not have to restrict ourselves to any assumptions on the total potential energy of the system. Furthermore, the quantities appearing here depend on the spatial pair correlation functions go, R), even though we may be dealing with nonspherical particles. 

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 ... 

In this representation, we identify the partial molar energies of the various quasicomponents as... 

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... 

There exists a certain analogy between a mixture of quasicomponents and a mixture of chemically reacting species. For the sake of simplicity, we recall the case of isomerization reaction, treated in section 2.4. We have stressed there that the distinction between the two species A and B was based on an (arbitrary) classification of all the states of the molecules into two groups. If the classification is such that transitions between the two groups of states is very slow compared with transitions within each group, then it might be possible to isolate the species A and B as pure components. This normally involves the introduction of an inhibitor to the reaction A B. However, whether such an inhibitor actually exists or not, it is of no importance for the formal theory of chemical equilibrium. Therefore, we can use the classification into quasicomponents to view the one-component system as if it were a mixture of species in chemical equihbrium. 

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... 

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