Gibbs triangles

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

If the property evaluated, for instance, the critical micelle concentration, can be approximated by a suitable plot, it is depicted in the ternary system as a concave area (e.g., cM area) located in the space above the Gibbs triangle as the basis for the distinct concentrations. The property axis describing the cM data stands vertically on the base triangle. 

Figure 2.24 The Gibbs triangle for the representation of the composition in a ternary system.

Fig. 31. Liquid isotherms and solid-solution isoconcentration lines in the Hg-rich corner of the Gibbs triangle.

Figure 9-2. Schematic A-B-0 phase diagram (Gibbs triangle) with tie lines between the following phases of complete solubility (A,B), (A,B)0, (A, B)304, (A,B)203. B-oxides are more stable than A-oxides. I, II, III denote two-phase fields.

We have discussed the oxidation kinetics of metal alloys and of oxide solutions. These reactions lead to dispersed internal products rather than to external product layers. In the present section, let us pose a different question can the reduction of (nonmetallic) solid solutions e.g., (A,B)2Oa to (A,B)304, (A,B)304, to (A,B)0, or (A, B)0 to (A, B)) similarly lead to internally precipitated particles of the reduced product If so, then do these reactions occur in field III, II, or I of the Gibbs triangle plotted in Figure 9-2 We further note that the reaction (A,B)0->(A,B) is the fundamental process of ore reduction. 

X2 = 0, X3 = X3) in the Gibbs triangle. This line connects the pure component 2 and the binary mixtures 1-3 with a mole fraction of component 1 equal to x°. Physically speaking, this line represents the locus of compositions of the ternary mixtures, formed by adding a solute to a binary mixture of a SC fluid and an entrainer. 

It is worth noting that with higher molar volume amphiphiles, such as C12E4, a significant amount of the amphiphile can be present in the oil phase, even at T. Here, too, the plait points CPa and CPp will be inside or not be inside the Gibbs triangle depending on the relative positions of T, Ta, and 7p. 

For T < T < T, the corner of the 3PT corresponding to the amphiphile phase remains close to the water side but moves clockwise in the Gibbs triangle. 

Winsor behavior is not the only characteristic of water-oil-nonionic amphiphile systems. The lyotropic mesophases appearing on the water-amphiphile binary phase diagrams expand to some extent in the Gibbs triangle (Figure 3.19). 

Foam is produced when air or some other gas in introduced beneath the surface of a liquid that expands to enclose the gas with a film of liquid. Foam has a more or less stable honeycomb structure of gas cells whose walls consist of thin liquid films with approximately plane parallel sides. These two-sided films are called the lamellae of the foam. Where three or more gas bubbles meet, the lamellae are curved, concave to the gas cells, forming what is called the Plateau border or Gibbs triangles (Figure 7-1). 

Figure 1.2 Isothermal Gibbs triangles of the system water (A)-oil (B)-non-ionic surfactant (C) at different temperatures. Increasing the temperature leads to the phase sequence 2-3-2. A large miscibility gap can be found both at low and high temperatures. While at low temperatures a surfactant-rich water phase (a) coexists with an oil-excess phase (b), a coexistence of a surfactant-rich oil phase (b) with a water-excess phase (a) is found at high temperatures. At intermediate temperatures the phase behaviour is dominated by an extended three-phase triangle with its adjacent three two-phase regions. The test tubes illustrate the relative change in phase volumes.

Stacking the isothermal Gibbs triangles on top of each other results in a phase prism (see Fig. 1.3(a)), which represents the temperature-dependent phase behaviour of ternary water-oil-non-ionic surfactant systems. As discussed above, non-ionic surfactants mainly dissolve in the aqueous phase at low temperatures (2). Increasing the temperature one observes that this surfactant-rich water phase splits into two phases (a) and (c) at the temperature T of the lower critical endpoint cepp, i.e. the three-phase body appears. Subsequently, the lower water-rich phase (a) moves towards the water corner, while the surfactant-rich middle phase (c) moves towards the oil corner of the phase prism. At the temperature Tu of the upper critical endpoint cepa a surfactant-rich oil phase is formed by the combination of the two phases (c) and (b) and the three-phase body disappears. Each point in such a phase prism is unambiguously defined by the temperature T and two composition variables. It has proved useful [6] to choose the mass fraction of the oil in the... 

Figure 10.9 Schematic Gibbs triangles for the system hhO/NaCI-natural fat-non-ionic surfactant, (a) T-y cut at = 0.50 (b) T-yb cut with varying water to oil plus water volume fraction.

Figure 10.12 Schematic of the variation of the phase behaviour during the degreasing process. In the short float the ultra-low interfacial tension between water and oil ensures efficient degreasing. Upon reducing the salt mass fraction the phase behaviour shifts to higher temperatures. At the degreasing temperature now an oil-in-water microemulsion coexists with an oil-excess phase. Shearing induces the formation of a stable macroemulsion that prevents the depositing of the fat on the skin and ensures the transport of the fat away from the skin. Note that only the Gibbs triangles correspond to the real experimental conditions. The T-y cuts are shown for clarity.

Figure 15.5 Effect of molecular weight dispersity ( )) (formerly known as polydispersity) using schematic Gibbs triangle diagrams for polymer-solvent system, generation of cloud point curve and shadow curve in temperature-composition diagram.

In chemistry, we use often the Gibbs triangle coordinates or tetrahedron coordinates. We emphasize here that the three-dimensional analogue of the triangle coordinates is the tetrahedra coordinates. Triangle coordinates can plot only the mole fractions. However, with the transformation... 

Such a set of axes (or triangle) is known as a mass balance triangle (MBT), or sometimes referred to as a Gibbs Triangle, and is typically used for ternary systems. The region enclosed by the triangle represents all the physically attainable compositions in a ternary system, that is, 0[Pg.18]

As discussed in Section 2.3, the MET or Gibbs triangle represents the region of physically achievable profiles in ternary systems. One may define this triangular region mathematically as 0 < X/ < I for all components /, with every point obeying the unity summation constraint of Equation 2.2. 

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