Multicomponent systems - solution thermodynamics

The changes in properties when a solution is formed from its components are described by the mixing quantities. The partial molar mixing quantities for each component are given by 

In multicomponent systems the partial Gibbs free energy of mixing and chemical potential of a given component are related to composition via 

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The preceding material of this section has focused on the most important phenomenological equation that thermodynamics gives us for multicomponent systems—the Gibbs equation. Many other, formal thermodynamic relationships have been developed, of course. Many of these are summarized in Ref. 107. The topic is treated further in Section XVII-13, but is worthwhile to give here a few additional relationships especially applicable to solutions. 

Examples of ideal binary systems ate ben2ene—toluene and ethylben2ene—styrene the molecules ate similar and within the same chemical families. Thermodynamics texts should be consulted before making the assumption that a chosen binary or multicomponent system is ideal. When pressures ate low and temperatures ate at ambient or above, but the solutions ate not ideal, ie, there ate dissimilat molecules, corrections to equations 4 and 5 may be made ... 

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

Bertrand G. L., Acree W. E. Jr., and Burchfield T. (1983). Thermodynamical excess properties of multicomponent systems Representation and estimation from binary mixing data. J. Solution. Chem., 12 327-340. 

Wood B. I (1987). Thermodynamics of multicomponent systems containing several solid solutions. In Reviews in Mineralogy, vol. 17, P. H. Ribbe (series ed.), Mineralogical Society of America. 

A thermodynamic parameter (dV/dnB)T,F,n g which describes how the volume of component S in a multicomponent system depends on the change in its amount expressed in mol. Hpiland recently summarized the partial molar volumes of numerous biochemical compounds in aqueous solution. See Dalton s Law of Partial Pressures Concentrations Molecular Crowding... 

Two different methods have been presented in this contribution for correlation and/or prediction of phase equilibria in ternary or mul> ticomponent systems. The first method, the clinogonial projection, has one disadvantage it is not based on concrete concepts of the system but assumes, to a certain extent, additivity of the properties of individiial components and attempts to express deviations from additivity of the properties of individual components and attempts to express deviations from additivity by using geometrical constructions. Hence this method, although simple and quick, needs not necessarily yield correct results in all the cases. For this reason, the other method based on the thermodynamic description of phase equilibria, reliably describes the behaviour of the system. Of cource, the theory of concentrated ionic solutions does not permit a priori calculation of the behaviour of the system from the thermodynamic properties of pure components however, if a satisfactory equation is obtained from the theory and is modified to express concrete systems by using few adjustable parameters, the results thus obtained are still substantially more reliable than results correlated merely on the basis of geometric similarity. Both of the methods shown here can be easily adapted for the description of multicomponent systems. 

The thermodynamic equations for the Gibbs energy, enthalpy, entropy, and chemical potential of pure liquids and solids, and for liquid and solid solutions, are developed in this chapter. The methods used and the equations developed are identical for both pure liquids and solids, and for liquid and solid solutions therefore, no distinction between these two states of aggregation is made. The basic concepts are the same as those for gases, but somewhat different methods are used between no single or common equation of state that is applicable to most liquids and solids has so far been developed. The thermodynamic relations for both single-component and multicomponent systems are developed. 

The usual choice of a reference state other than the pure components is the infinitely dilute solution for which the mole fractions of all solutes are infinitesimally small and the mole fraction of the solvent approaches unity that is, the values of the thermodynamic properties of the system in the reference state are the limiting values as the mole fractions of all the solutes approach zero. However, this is not the only choice, and care must be taken in defining a reference state for multicomponent systems other than binary systems. We use a ternary system for purposes of illustration (Fig. 8.1). If we choose the component A to be the solvent, we may define the reference state to be the infinitely dilute solution of both B and C in A. Such a reference state would be useful for all possible compositions of the ternary systems. In other cases it may be advantageous to take a solution of A and B of fixed... 

We have developed the basic equations for the thermodynamic functions of the defined surface in the preceding paragraphs, but have not discussed the determination of the position of the boundary. Actually, the position is somewhat arbitrary, and as a result we must also discuss the dependence of the properties of the surface on the position. The position can be fixed by assigning the value of zero to one of Equations (13.25)—(13.27) that is, by making one of the nf equal to zero. For a one-component system there is only one such equation. For multicomponent systems we have to choose one of the components for which nf is made zero. The value of nf for the other components then would not be zero in general. The most appropriate choice for dilute solutions would be the solvent. The position of the surface for a one-component system is illustrated in Figure 13.2, where the line c is determined by making the areas of the two shaded portions equal. 

In Chapters 6, 7, and 8, the thermodynamic framework is successively apphed to phase transformations of single-component systems, chemical reactions, and ideal solutions. Included are discussions of the thermodynamics of open systems, the phase rule, and colligative properties. Chapter 9 gives the framework for discussing nonideal multicomponent systems and describes a... 

Here, we only present the simplest thermodynamic expressions used in the CALPHAD method for the major phase classes observed in multicomponent systems namely, disordered miscible and immiscible phases and ordered sublattice phases. The reader is referred to specialized textbooks for further discussion. The Gibbs energies for disordered two-component solid and liquid solution phases are most easily represented by the regular solution model (Eq. 2.10) or one of its variants ... 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

In the category of physical continuation methods are the thermodynamic homotopies of Vickery and Taylor [AIChE J., 32, 547 (1986)] and a related method due to Frantz and Van Brunt (AIChE National Meeting, Miami Beach, 1986). Thermodynamic continuation has also been used to find azeotropes in multicomponent systems by Fid-kowski et al. [Comput. Chem. Engng., 17, 1141 (1993)]. Parametric continuation methods may be considered to be physical continuation methods. The reflux ratio or bottoms flow rate has been used in parametric solutions of the MESH equations [Jelinek et al., Chem. Eng. Sd.,28, 1555(1973)]. 

Several formalisms have been developed leading to what may be called practical thermodynamics. These treatments include the analog of solution thermodynamics, where the adsorbent and the adsorbate are considered as components in a two-phase equilibrium [6]. Another way to study the system is to use the surface excess approach, whereby the properties of the adsorbed phase are determined in terms of the properties of the real two-phase multicomponent... 

Examples of synergistic effects are now very numerous in catalysis. We shall restrict ourselves to metallic oxide-type catalysts for selective (amm)oxidation and oxidative dehydrogenation of hydrocarbons, and to supported metals, in the case of the three-way catalysts for abatement of automotive pollutants. A complementary example can be found with Ziegler-Natta polymerization of ethylene on transition metal chlorides [1]. To our opinion, an actual synergistic effect can be claimed only when the following conditions are filled (i), when the catalytic system is, thermodynamically speaking, biphasic (or multiphasic), (ii), when the catalytic properties are drastically enhanced for a particular composition, while they are (comparatively) poor for each single component. Therefore, neither promotors in solid solution in the main phase nor solid solutions themselves are directly concerned. Multicomponent catalysts, as the well known multimetallic molybdates used in ammoxidation of propene to acrylonitrile [2, 3], and supported oxide-type catalysts [4-10], provide the most numerous cases to be considered. Supported monolayer catalysts now widely used in selective oxidation can be considered as the limit of a two-phase system. 

The chemical potential is the most important quantity in chemical thermodynamics and, in particular, in solution chemistry. There are several routes for obtaining a relationship between the chemical potential and the pair correlation function. Again we start with the expression for the chemical potential in a one-component system, and then generalized to multicomponent systems simply by inspection and analyzing the significance of the various terms. 

We see that the shear- (i.e., tangential-) stress components are discontinuous across the interface whenever gradv y is nonzero. Now, the interfacial tension for a two-fluid system, made up of two pure bulk fluids, is a function of the local thermodynamic state - namely, the temperature and pressure. However, it is much more sensitive to the temperature than to the pressure, and it is generally assumed to be a function of temperature only. If the two-fluid system is a multicomponent system, it is often the case that there may be a preferential concentration of one or more of the components at the interface (for example, we may consider a system of pure A and pure B, which are immiscible, with a third solute component C that is soluble in A and/or B but that is preferentially attracted to the interface), and then the interfacial tension will also be a function of the (surface-excess) concentration of these solute components. Both the temperature and the concentrations of adsorbed species can be functions of position on the interface, thus leading to spatial gradients of y. 

The equilibrium ratio [Equation (12.1)] involves physical equilibrium between phases. The system involved may be binary or multicomponent and ideal or nonideal, according to the terminology used in solution thermodynamics. Methods for correlating or predicting equilibria are based on an application of thermodynamics to each phase and to the solutions of the components within each phase. The techniques are described in standard reference works such as Walas and Reid et al. For multicomponent systems, the usual approach is to model the equilibria for each of the binary pairs and then combine the binary models in special ways to obtain the multicomponent model. 

Multicomponent Effects. Only limited experimental data are available for multicomponent diffusion in liquids. The binary correlations are sometimes employed for (he case of a solute diffilsing through a mixed solvent of uniform composition.3,33 It Is clanr that thermodynamic nonideslities In multicomponent systems can cause sigaiflcanl effects. The resder is referred to Cussler s book4 for a discussion of available experimental information on diffusion in multicomponent systems. 

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