Pseudobinary systems

The experimental studies of three-component systems based on phase equilibria follow the same principles and methods discussed for two-component systems. The integral form of the equations remains the same. The added complexity is the additional composition variable the excess chemical potentials become functions of two composition variables, rather than one. Because of the similarity, only those topics that are pertinent to ternary systems are discussed in this section of the chapter. We introduce pseudobinary systems, discuss methods of determining the excess chemical potentials of two of the components from the experimental determination of the excess chemical potential of the third component, apply the set of Gibbs-Duhem equations to only one type of phase equilibria in order to illustrate additional problems that occur in the use of these equations, and finally discuss one additional type of phase equilibria. 

In some cases we are concerned with systems in which the mole ratio of two of the components is kept constant. Such a mixture can be considered as a solvent of fixed composition with the third component acting as a solute and is, therefore, a pseudobinary system. 

The thermodynamic relations for such pseudobinary systems are readily developed from those of a ternary system. The molar Gibbs energy of a 

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Springer, G. (1972) The pseudobinary system Cu2PeSnS4-Cu2ZnSnS4 and its mineralogical significance. Can. Mineral, 11, 535-541. 

Figure 3. Mixed overall adsorption of pseudobinary systems.

A typical sequence followed in this test series consists in injecting (1) a micellar slug of one pore volume of aqueous solution of 4% of the preceding pseudobinary system (2% sulfonate/2% Genapol) (2) a slug of desorbent solution corresponding to a fixed amount of additive (e.g. equal to 1 PV at a concentration of 0.5 %) (3) at least 1.5 PV of brine with no additive. 

Figure 7.3 depicts phase stability relations in the pseudobinary system CaMgSi206-CaAl2Si208 (diopside-anorthite). The original study of Bowen (1915) described crystallization behavior identical to the previously discussed case a mechanical mixture (Di-An) in equilibrium with a completely miscible melt. A later investigation (Osborn, 1942) showed that the system is not strictly binary... 

Figure 7,6 Phase stability relations in pseudobinary system NaAlSi30g-KAlSi30g, after Waldbaum and Thompson (1969). (A) High-T relations loop of metastable persistency of sanidine like a Roozeboom type III. (B) Expanded T-range with decreasing T, a solvus held opens downward.

The metal-H or alloy-H system can be regarded as a binary or pseudobinary system, i.e. c = 2 in the phase rule. Figure 3.17 shows schematically the relation between H2 and solid phase composition H/M for various values of T, assuming two solid phases a and p. These phases correspond to the... 

In this study, a thermodynamic framework has been presented for the calculation of vapor-liquid equilibria for binary solvents containing nonvolatile salts. From an appropriate definition of a pseudobinary system, infinite dilution activity coefficients for the salt-containing system may be estimated from a knowledge of vapor pressure lowering, salt-free infinite dilution activity coefficients, and a single system-dependent constant. Parameters for the Wilson equation may be determined from the infinite dilution activity coefficients. 

The phases form part of pseudobinary systems between the copper(I) halide and selenium or tellurium, respectively. 

Equations (10.205) and (10.207), together with Equation (10.206), comprise the basic equations to use in considering a ternary system that has a constant ratio of x2/x3 as a pseudobinary system. 

Darken [27] has made use of the concept of pseudobinary systems and has developed equations by which the molar excess Gibbs energy can be... 

Thus, when A/iE is determined as a function of x3 at a fixed k, AGE can be obtained for the pseudobinary systems by integration. The most convenient reference state of component 1 is the pure component at the chosen temperature and pressure, because the reference state chosen as the infinitely dilute solution of the component in a pseudobinary solvent is different for every ratio of x2/x3. 

With the appropriate choice of the reference state, values of the molar excess Gibbs energy can be determined over the entire range of composition of the ternary system by making studies on a set of pseudobinary systems. Values of A/if and A/iE can then be calculated by means of the usual methods. 

The methods of Darken and Gokcen discussed here can be applied to quaternary systems. New composition variables are defined in order to convert the quaternary system into pseudobinary systems. The paths of integration can be illustrated by the use of an equilateral tetrahedron. 

The methods discussed in this section can be extended to systems that have more than three components. The problem is to convert each system to a pseudobinary system. For a quaternary system, the properties of an equilateral tetrahedron may be used to depict the composition of the system. The composition axes would be four lines drawn from the four apexes perpendicular to the opposite faces. Planes cutting the tetrahedron parallel to the bases would represent pseudoternary systems for which one composition variable would be constant. Pseudobinary systems would be depicted by the intersections of two of the pseudoternary planes. Indeed, the experimental measurements and calculations would be extensive. 

Dorfler HD. Relationships between miscibility behavior and chemical structure of phospholipids in pseudobinary systems. Colloid Polym. Sci. 2000 278 130-136. 

Safonov et al. (331) determined the liquidus surface of the ternary In-Te-Cl system by DTA, X-ray diffraction, and crystal optical methods. Only one ternary compound, InTeCl, exists. The crystallization field of InTeCl occupies 6% of the diagram, which demonstrates the considerable thermodynamic stability of this compound. InTeCl melts congruently at 453°C. It forms part of the two pseudobinary systems In Tes-InCls and InCl-Te (82). Whereas the first consists of the two eutectic parts In Tes-InTeCl and InTeCl InCls, the latter is more complicated. It is composed of the monotectic system InTeCl-InCl and the eutectic system Te-InTeCl, where tellurium forms a solid solution with InTeCl containing from 100 to 82 atom% of Te at the eutectic temperature (82). 

Multicomponent Systems. Most efficiency models and test data are based on binary systems. For multicomponent systems it is often possible to use the key components as a pseudobinary system. Chan and Fair (1984b) found that selection of the two components should be based on the dominant pair (if not the key components). Rigorous treatments of multicomponent separations are given by Taylor and Krishna (1993). 

FIG. 39. Pseudobinary systems in a composition FIG. 40. The temperature dependence of tetrahedron. nucleation rate (i) and crystal growth... 

V.K. Pecharsky and K.A. Gschneidner, Jr., Phase relationships and crystallography in the pseudobinary system Gd5Si4-GdsGe4, J. Alloys Comp. 260,98 (1997). 

Finally, the dissociation model of electrical conductivity was also successfully used in pseudobinary systems containing a trivalent cation MX-AIX3 (M = Li, Na, K ... 

The net diffusive mass flux for each phase still vanishes for binary systems as A s using Pick s law, whereas for dilute pseudobinary systems the latter relationship is only approximate. 

When more than two components are present, the efficiencies of each are not necessarily the same. The rigorous approach to handling multicomponent mixtures, outlined by Taylor and Krishna, uses the Maxwell-Stefan diffusional equations. Chan and Fair used the rigorous approach to compare multicomponent system separations with those predicted by the use of the equivalent pseudobinary systems. They found that if the dominant pair of components present in the mixture is used to determine efficiency for all of the components, the separation determined is quite close to that resulting from rigorous multicomponent procedures. 

But there are not a few systems in which the number of molecular species is greater than the number of components that is, substances which have the same chemical composition (but which may be isomeric forms) may give rise to different molecular species, between which, in the liquid or vapour state, a condition of equilibrium can exist. This fact may alter very markedly the behaviour of a system. Although, therefore, a system may appear to be unary, so far as chemical composition is concerned, it may, as a matter of fact, behave in some respects as a binary system. It forms a pseudo-binary system. The behaviour of these systems, as we shall see, depends largely on the rate at which the internal equilibrium between the different molecular species in the liquid or vapour phase is established. In the present chapter some of the more important aspects of these pseudobinary systems will be considered. 

While the IR dichroism and NMR studies can give information about the conformation at individual bonds, the small-angle neutron scattering (SANS) measures the global conformation of the polymers (Cotton et al., 1974). SANS has been successfully utilized, for example by Hardouin and coworkers (Hardouin et al, 1991 Leroux et al, 1993), to establish the evolution with the temperature of the global shape of some mesogen-jacketed liquid crystalline polymers from isotropic to the nematic phase. A pseudobinary system of fully protonated polymer (P4,4,4) with the same polymer... 

For a multicomponent mixture, it is common to neglect ternary and higher interactions and assume a pseudobinary system. The resulting van Laar expression for the activity coefficient depends only on composition and the binary constants. The following form given by NulF is preferred. 

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