Molecular distribution functions in mixtures

Molecular distribution functions (MDFs) in mixtures are defined in a similar way as in the case of the one-component system the only complication is 

Here we have assumed pairwise additivity of the total potential energy and adopted the convention that the order of arguments in the parentheses corresponds to the order of species as indicated by subscript of U. Thus X in the first sum on the rhs of (2.129) is the configuration of the ith molecule ( / = 1,2. NA) of species A, whereas X -, in the last term on the rhs of (2.129), stands for the configuration of the jth molecule (j= 1, 2. NB) of species B. 

The singlet distribution function for the species A is defined in complete analogy with the definition in the pure case (section 2.1), 

Clearly, p (X ) is the probability of finding any molecule of type A in a small region dX at X. Similar interpretation applies to pB x ). 

As in the case of a one-component system, (X ) is also the average density 

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Before embarking on the concepts of molecular distribution functions in mixtures, it is appropriate to digress to a brief discussion of the inter-molecular potential function for two particles of different species. We denote hy X") the work required to bring two molecules of species A... 

Cochran, H. D., and L. L. Lee. 1989. Solvation structure in supercritical fluid mixtures based on molecular distribution functions. In Supercritical fluid science and technology, ed. K. P. Johnston, J. M. L. Penninger, ACS Symposium Series 406 27. 

Cochran, H.D., Lee, L.L., "Solvation Structure in Supercritical Fluid Mixtures based on Molecular Distribution Functions," ACS Symp. Ser, 1989,406,28. 

We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxide-naphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (Ifl) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M. Lee, L. L. Cochran, H. D. Int. J. Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. 

Fluid microstructure may be characterized in terms of molecular distribution functions. The local number of molecules of type a at a distance between r and r-l-dr from a molecule of type P is Pa T 9afi(r)dr where Pa/j(r) is the spatial pair correlation function. In principle, flr (r) may be determined experimentally by scattering experiments however, results to date are limited to either pure fluids of small molecules or binary mixtures of monatomic species, and no mixture studies have been conducted near a CP. The molecular distribution functions may also be obtained, for molecules interacting by idealized potentials, from molecular simulations and from integral equation theories. 

In this chapter, we introduce the concepts of molecular distribution function (MDF), in one- and multicomponent systems. The MDFs are the fundamental ingredients in the modern molecular theories of liquids and liquid mixtures. As we shall see, these quantities convey local information on the densities, correlation between densities at two points (or more) in the system, etc. 

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. 

For systems that do not obey the assumption of pairwise additivity for the potential energy, equation (3.67) becomes invalid. In a formal way, one can derive an analogous relation involving higher order molecular distribution functions. This does not seem to be useful at present. However, in many applications for mixtures, one can retain the general expression (3.55) even... 

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). 

The Kirkwood-Buff theory of solutions was originally formulated to obtain thermodynamic quantities from molecular distribution functions. This formulation is useful whenever distribution functions are available either from analytical calculations or from computer simulations. The inversion procedure of the same theory reverses the role of the thermodynamic and molecular quantities, i.e., it allows the evaluation of integrals over the pair correlation functions from thermodynamic quantities. These integrals Gy, referred to as the Kirkwood-Buff integrals (KBIs), were found useful in the study of mixtures on... 

An indirect route has been developed mainly by Kirkwood, which involves molecular distribution functions (MDF) as an intermediate step. The molecular distribution function approach to liquids and liquid mixtures, founded in the early 1930s, gradually replaced the various lattice theories of liquids. Today, lattice theories have almost disappeared from the scene of the study of liquids and liquid mixtures. This new route can be symbolically written as... 

Routes II and III are identical in the sense that they use the same theoretical tools to achieve our goals. There is however one important conceptual difference. Clearly, molecular properties are microscopic properties. Additionally, all that has been learned about MDF has shown that in the liquid phase, and not too close to the critical point, molecular distribution functions have a local character in the sense that they depend upon and provide information on local behavior around a given molecule in the mixture. By local, we mean a few molecular diameters, many orders of magnitude smaller than the macroscopic, or global, dimensions of the thermodynamic system under consideration. We therefore rewrite, once again, route II in different words, but meaning the same as III, namely... 

This equation is valid for a homogeneous state of equilibrium. For slow reactions, it is reasonable to assume that the chemical process is proceeding through states that are in thermomechanical equilibrium at all times, although a chemical equilibrium is not established. This assumption is supported by the fact that, for example, for a gaseous mixture, collisions not causing reactions are far more frequent than collisions that cause reactions. At a specified composition, the molecular distribution functions describing temperature and pressure are effectively the same as those at equilibrium. 

Most of the properties of the molecular distribution functions discussed in Chapter 2 hold for mixtures as well. In this section, we dwell upon some new features that are specific to multicomponent systems. We shall be... 

One sees that an expression for AE (Ri, R2) involves molecular distribution functions of up to order four. Therefore, this approach to the problem is impractical at present. In the next section, we pursue a different, conceptually simpler method which involves the mixture-model approach to the solvent. 

Excluding the introductory chapter, the book is organized into three parts. The first. Chapters 2-4, presents the general molecular theory of fluids and mixtures. Here the notions of molecular distribution functions are developed with special attention to fluids consisting of nonspherical particles. We have included only those theories judged to be potentially useful in the study of aqueous fluids, so this part may not be considered as an introduction to the theory of fluids per se. 

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. 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

Liquid chromatography (LC) and, in particular, high performance liquid chromatography (HPLC), is at present the most popular and widely used separation procedure based on a quasi-equilibrium -type of molecular distribution between two phases. Officially, LC is defined as a physical method... in which the components to be separated are distributed between two phases, one of which is stationary (stationary phase) while the other (the mobile phase) moves in a definite direction [ 1 ]. In other words, all chromatographic methods have one thing in common and that is the dynamic separation of a substance mixture in a flow system. Since the interphase molecular distribution of the respective substances is the main condition of the separation layer functionality in this method, chromatography can be considered as an excellent model of other methods based on similar distributions and carried out at dynamic conditions. 

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