Mixtures molecular theories

Hirschfelder J C, Curtiss C F and Bird R B 1954 Molecular Theory of Gases and Liquids (New York Wiley) Rowlinson J and Swinton J 1983 Liquids and Liquid Mixtures 3rd edn (London Butterworth)... 

All binding processes in real-life systems occur in some solvent. The solvent is, in general, a mixture of many components, including water electrolytes and nonelectrolytes. At present, it is impossible to account for all possible solvent effects, even when the solvent is pure water. Yet, the solvent, whether a single or multi-component, cannot be ignored. Any serious molecular theory of cooperativity must deal with solvent effects. We shall see in this chapter that this is not an easy task even when the solvent is inert, such as argon, or a simple hydrocarbon liquid. ... 

During the period 1945-1960, Prigogine worked on an intensive research program on Mixtures and Solutions. It can be framed into what can be called Classical physical chemistry. It is clearly inspired by the professor he succeeded at the ULB and to whom many references are made Jean Timmermans, a remarkable experimental physico-chemist. The results of these research efforts were published in a monograph written by Ilya Prigogine, Victor Mathot, and Andre Bellemans The Molecular Theory of Solutions (LS.7), published in 1957 today this is still considered to be an important reference. 

An overarching goal of modem physical chemical studies is the molecular-level understanding of chemical and phase thermodynamics—a molecular theory of gases, liquids, and polymorphic solid phases of chemically reactive mixtures. Whereas thermodynamic sciences originated in an era when understanding of atomic and molecular phenomena... 

A well-known approximate molecular theory of a fluid at a planar interface is originally due to Helfand, Frisch and Lebowitz [76] and later to Henderson, Abraham and Barker [77] and Perram and White [78]. Consider a binary mixture (A,B) in which one of the species (A) becomes extremely dilute and infinitely large. S.E. [52] show that if the size of species A tends to infinity while the concentration of A tends toward zero, then a consequence of the OZ equation, coupled with the PY equation, is the relation... 

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

Understanding the relationship between the composition of a mixture and its properties is fundamental to the development of formulated products. In the pesticide industry, formulation chemists seek to translate such an understanding into products that meet criteria established for properties such as suspensibility, emulsibility, storage stability, compatibility, and most importantly, biological activity. The preferable way to acquire the necessary knowledge is to deduce the properties of mixtures in terms of mechanisms that are operative at the microscopic level. However, mixtures are extremely complex systems and the available theory is usually insufficient for developing useful theoretical models. For example, we are unable quantitatively to predict, on the basis of molecular theory, the suspensibility of a wettable powder from a knowledge of its composition. 

Section 12.1 introduces the concept of pressure and describes a simple way of measuring gas pressures, as well as the customary units used for pressure. Section 12.2 discusses Boyle s law, which describes the effect of the pressure of a gas on its volume. Section 12.3 examines the effect of temperature on volume and introduces a new temperature scale that makes the effect easy to understand. Section 12.4 covers the combined gas law, which describes the effect of changes in both temperature and pressure on the volume of a gas. The ideal gas law, introduced in Section 12.5, describes how to calculate the number of moles in a sample of gas from its temperature, volume, and pressure. Dalton s law, presented in Section 12.6, enables the calculation of the pressure of an individual gas—for example, water vapor— in a mixture of gases. The number of moles present in any gas can be used in related calculations—for example, to obtain the molar mass of the gas (Section 12.7). Section 12.8 extends the concept of the number of moles of a gas to the stoichiometry of reactions in which at least one gas is involved. Section 12.9 enables us to calculate the volume of any gas in a chemical reaction from the volume of any other separate gas (not in a mixture of gases) in the reaction if their temperatures as well as their pressures are the same. Section 12.10 presents the kinetic molecular theory of gases, the accepted explanation of why gases behave as they do, which is based on the behavior of their individual molecules. 

The corrections made by van der Waals to the kinetic molecular theory make physical sense, which makes us confident that we understand the fundamentals of gas behavior at the particle level. This is significant because so much important chemistry takes place in the gas phase. In fact, the mixture of gases called the atmosphere is vital to our existence. In the next section we consider some of the important reactions that occur in the atmosphere. 

I have a degree in chemistry and another in physics. My first scientific publications were mostly on mixtures. In 1957, I published a book on the molecular theory of solutions. Also, my early work in thermodynamics was closer to chemistry than to physics. This has changed in the later part of my life when I became mainly involved with the microscopic roots of the arrow of time associated to entropy. This part is closer to physics. Anyway, the distinction between chemistry and physics is somewhat arbitrary. My later work is dominated by my belief that the direction of time is of fundamental importance in nature. This was the main conclusion of my studies of non-equilibrium thermodynamics. 

Most substances can exist in three states depending on the temperature and pressure. A few substances, such as water, exist in all three states under ordinary conditions. States of a substance are referred to as phases when they coexist as physically distinct parts of a mixture. Ice water is a heterogeneous mixture with two phases, solid ice and liquid water. When energy is added or removed from a system, one phase can change into another. As you read this section, use what you know about the kinetic-molecular theory to help explain the phase changes summarized in Figure 13-22. 

As well as pure gases, we can apply the kinetic molecular theory to mixtures of gases. In a mixture of gases, each gas contributes to the pressure in the same proportion as it contributes to the number of molecules of the gas. This makes sense, given the kinetic molecular theory, because molecules have no volume, no interactive forces other than collisions, and kinetic energy is conserved when they collide. Thus, each gas in a mixture essentially behaves as if it were in its container alone. The... 

In specific cases the gas mixture is almost isotherm and the chemical process are not altering the mixture molecular mass very much, thus the system and transport properties may be considered constant. Otherwise, the system and transport properties have to be considered temperature and composition dependent and calculated from approximate parameterizations or kinetic theory relations. 

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. 

As in the case of a one-component system, ideal-gas (IG) mixtures also enjoy having a simple and solvable molecular theory, in the sense that one can calculate all the thermodynamic properties of the system from molecular properties of single molecules. We also have a truly molecular theory of mixtures of slightly nonideal gases, in which case one needs in addition to molecular properties of single molecules, also interactions between two or more molecules. 

Another difficulty in developing a molecular theory of liquid mixtures is the relatively poor knowledge of the intermolecular interactions between molecules of different species. While the intermolecular forces between simple spherical particles are well-understood, the intermolecular forces between molecules of different kinds are usually constructed by the so-called combination rules, the most well-known being the Lorentz and the Berthelot rules. 

During the 1950s and the 1960s, two important theories of the liquid state were developed, initially for simple liquids and later applied to mixtures. These are the scaled-particle theory, and integral equation methods for the pair correlation function. These theories were described in many reviews and books. In this book, we shall only briefly discuss these theories in a few appendices. Except for these two theoretical approaches there has been no new molecular theory that was specifically designed and developed for mixtures and solutions. This leads to the natural question why a need for a new book with the same title as Prigogine s ... 

The kinetic molecular theory states that the average kinetic energy of the gas particles increases as the temperature increases. Kinetic energy is proportional to (velocity)2. Therefore, as the temperature increases the gas particle velocity increases and the rate of mixing increases as well. Dalton s law states that the total pressure of a mixture of gases is the sum of the partial pressures of the component gases. 

The great appeal of the virial equations derives from their interpretations in terms of molecular theory. Virial coefficients can he calculated from potential fonctions describing interactions among moleculas. More importantly, statistical mechanics provides rigorous expressions for the composition dependeace of ihe virial coefficients. Thus, the nth virial coefficient of a mixture is nth order in the mole fractions ... 

The application of the conformal solution method in industrial calculations requires the use of the approximation A = Ax to avoid the lengthy computation required to calculate the higher order terms in Equation 2. Thus, a practical strategy for choosing the exponents k, 1, m, p, q, r, u, v, and w in Equations 3, 4, and 5 would be through minimization of the difference A — Ax (actually, data for all available mixture thermodynamic properties can be used simultaneously to determine the exponents by regression). However, most applications of the conformal solution method have involved the use of exponents based on molecular theory and so this approach was used in the initial phases of the present work. 

In chemical kinetics the concept of the order of a reaction forms the basis of a kinematics which constitutes a frame for most of the molecular theories of chemical reactions. The fundamental magnitudes of this kinematics are the concentrations and the specific rate constants. In simple cases only the time enters as an independent variable, whereas in a diffusion process both time and space are involved. Diffusion processes are generally described in terms of diffusion coefficients, volume concentrations and thermodynamic potential or activity factors. Partial volume factors and friction coefficients associated with the components of the diffusing mixture are also essential in the description. A feature of the macro-dynamical theory is that it covers any region of concentration. Especially simple equations are connected with the differential diffusion process (diffusion with small concentration differences), for which the different coefficients or factors mentioned above are practically constant. 

The coefficients Bn(T) are functions of temperature and depend on the type of molecular interaction between species i and /. They can be determined from molecular theory or by experiments on gas mixtures. If only pure-gas data are available, we may assume B, to be of the form... 

As already emphasized, theoretical development in the area of aqueous binary mixtures has been comparatively slow and to date no satisfactory molecular theory exists that can describe the complex physical chemistiy of a binaiy solution. The reason is the complexity of the intermolecular potential. While binary mixtures have often been studied by using a cell or lattice theory (as we discussed in the description of a polymer solution in the Hydrophobic effects chapter), even such a description is hard here because of the amphiphilic nature of the solute. It is really hard to develop a quantitative theory that includes the two different types of local heterogeneity at two sides of a given solute molecule. 

The molecular theory considers a dipolar liquid where the two constituents are Lennard-Jones spheres each with an embedded dipole moment at the center. The Lennard-Jones parameters (sizes, interaction strength parameters) and also values of the dipole moments are different for the two species. The theoiy properly includes the differing inter- and intramolecular correlations that are present in a binary mixture. As a result, the theory can explain several important aspects of the nonideality of equilibrium solvation energy (broadly known as preferential solvation) observed in experiments. The non-ideality of solvation is found to depend on both the molecular sizes and the magnitude of the dipole moments of the solvent... 

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