Applications of the Mixture Model Approach

There is a long history of the theory of water and aqueous solutions based on various mixture model (MM) approaches. One of the earliest documented explanations of some anomalous properties of water is due to Rontgen, who in 1892 proposed to view liquid water as consisting of two kinds of molecules, one of which he referred to as ice-molecules. The general idea of explaining the properties of water by viewing it as a mixture of species probably originated much earlier. 

Beginning in the early 1960s, efficient simulation techniques developed for the study of simple liquids were also applied to aqueous fluids. This approach requires us to start from a model for water molecules defined in terms of their pair potential. (This method is sometimes referred to as ab initio, or a continuous, theory of water.) 

In this section we present an example of the application of a two-structure model, based on the exact MM approach to the theory of liquids (section 5.13). Then we extract a particular MM which can be viewed as an approximation of the general exact MM approach. The latter, because of its simplicity and solvability, is useful in the study of some thermodynamic aspects of both pure water and aqueous solutions of simple solutes. 

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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. 

APPLICATION OF THE MIXTURE MODEL APPROACH TO AQUEOUS SOLUTIONS OF SIMPLE SOLUTES... 

Component-based approaches are based on the assumption that the toxicity of a mixture can be assessed based on knowledge of the individual components in combination with the application of a mixture model. A component-based approach is generally applied to mixtures with relatively few chemical components that have all or partially been identified. If only partial information is available, the approach corresponds to the partially characterized boxes in Figure 5.11. 

Successful stoichiometric modeling is demonstrated for industrial pyrolysis of light hydrocarbon and their mixtures. A semikinetic approach is more appropriate for naphtha pyrolysis. Although the final form of such a model is simple, its development generally requires more innovations. Applicability of the naphtha model to olefins production is evidenced by the successful prediction of commercial plant performances. 

In ternary mixtures of oil, water, and surfactant the ordering properties of the system follow from the vectorial character of the interactions of the surfactant molecules with both the oil and the water molecules. The typical size of the ordered domains, much larger than the molecular size, justifies the application of the mesoscopic Landau-Ginzburg approach to the ordering. In the simplest approach of Gompper and Schick [3,12], which we call here the basic Landau-Ginzburg model, the orientational degrees of free-... 

Equilibrium constants calculated from the composition of saturated solutions are dependent on the accuracy of the thermodynamic model for the aqueous solution. The thermodynamics of single salt solutions of KC1 or KBr are very well known and have been modeled using the virial approach of Pitzer (13-15). The thermodynamics of aqueous mixtures of KC1 and KBr have also been well studied (16-17) and may be reliably modeled using the Pitzer equations. The Pitzer equations used here to calculate the solid phase equilibrium constants from the compositions of saturated aqueous solutions are given elsewhere (13-15, 18, 19). The Pitzer model parameters applicable to KCl-KBr-l O solutions are summarized in Table II. 

The object of this work was to extend the field of application of the equation-of-state method. The method was applied to aqueous systems in conjunction with a model that treats water as a mixture of a limited number of polymers, an approach similar to that previously adopted for the carboxylic acids (2). Association is calculated by the law of mass action corrections for non-ideal behaviour are made by means the equation of state. A major problem of the method is the large number of parameters needed to describe the properties and concentrations of the polymers together with their interaction with molecules of other substances. The Mecke-Kemptner model (15) (also known as the Kretschmer-Wiebe model (16) and experimental values for hydrogen-bond energies were usecT for guidance in fixing these parameters. 

The problem of the application of any TD model in ecotoxicology to mixtures is that there is an extreme lack of data on toxic effects of mixtures measured as a function of time. Almost all studies focus on the effects of the mixture after a fixed exposure period only, which is of limited use for the application of dynamic approaches. 

Whenever appropriate, and in line with the probabilistic concept of risk, probability distributions are used in ecological risk assessment of mixtures. This applies to the assessment of exposure (e.g., the probabilistic application of multimedia fate models see Hertwich et al. 1999 Ragas et al. 1999 MacLeod et al. 2002), as well as to the assessment of effects, especially the SSD approach. Recent developments (both conceptually and practically) suggest that joint probability assessments (looking at exposure and effects distributions simultaneously) are applied more frequently. This relates to the refined questions being posed, but also to theory development (e.g., Aldenberg et al. 2002) and technical facilitation by software (e.g., Van Vlaardingen et al. 2004). 

Which effect assessment method should be applied in a particular situation depends on the nature of the mixture problem at hand. Because the diversity in assessment methods is large, it is important to clearly describe the problem. For example, derivation of a safe level for a proposed industrial mixture emission requires a different approach than the prioritization of a number of sites contaminated with mixtures. The former problem requires the assessment of realistic risks, for example, by the application of a suite of fate, exposure, and effect models, whereas the application of a simple consistent method suffices to address the latter problem, for example, a toxic unit approach. A successful and efficient assessment procedure thus starts with an unambiguous definition of the mixture problem at hand. The problem definition consists of the assessment motive, the regulatory context, the aim of the assessment, and a structured or stepwise approach to realize the aim. Elaboration of the problem definition is an iterative process (Figure 5.1) that strongly depends on factors such as resources, methods, data availability, desired level of accuracy, and results of previous studies. 

Cerofolini and Rudzihski [43] have reviewed the theoretical principles of single gas and mixture adsorption on heterogeneous surfaces. Their review is chronologically arranged from the earliest to the latest approaches. In the same book, Tovbin [44] reported the application of lattice-gas models to explain mixed-gas adsorption equilibria on heterogeneous surfaces he also discussed [45] the kinetic aspects of adsorption-desorption on flat heterogeneous surfaces. The book [46] also contains other papers on different aspects of adsorption for the reader interested in surface diffusion processes. 

A few years ago, we presented a review of the thermodynamic models for the treatment of hydrogen bonding in fluids and their mixtures. In that work, we gave an account of the association models and reviewed the work that was done to that time with models adopting the combinatorial approach. The two approaches were compared and applied to the description of phase equilibria and mixture properties of systems of fluids. The key conclusion was that, in the systems where both approaches apply, they prove to be essentially equivalent. However, the combinatorial approach has a much broader field of applications as it can be applied even to systems forming three-dimensional hydrogen bonding networks. 

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