Ternary phase rule

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... 

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

If we consider three-component (ternary) systems, for instance, we have C = 3 and, according to the phase rule, the variance is given byV=C — P + 2 = 5 Porin... 

A schematic illustration of the method, and of the correlation between binary phase diagram and the one-phase layers formed in a diffusion couple, is shown in Fig. 2.42 adapted from Rhines (1956). The one-phase layers are separated by parallel straight interfaces, with fixed composition gaps, in a sequence dictated by the phase diagram. The absence, in a binary diffusion couple, of two-phase layers follows directly from the phase rule. In a ternary system, on the other hand (preparing for instance a diffusion couple between a block of a binary alloy and a piece of a third... 

Identify the number of components present, the number of phases present, the composition of each phase, and the quantity of each phase from unary, binary, and ternary phase diagrams—that is, apply the Gibbs Phase Rule. 

Three-component systems, or ternary systems, are fundamentally no different from two-component systems in terms of their thermodynamics. Phases in eqnilibrium must still meet the equilibrium criteria [Eqs. (2.14)-(2.16)], except that there may now be as many as five coexisting phases in eqnilibrinm with each other. The phase rule still... 

Application of the phase rule to the binary (M-O) and ternary oxide system (M1-M2-O) in a closed system ... 

To explore the ramifications of the phase rule (7.6), we shall first consider the phase equilibria of pure chemical substances (c = 1). Subsequent sections will examine the more complex behavior of binary (c = 2) and ternary (c = 3) multiphase systems. 

Ternary A/B/C systems (c = 3) present further challenges to thermodynamic description. According to the phase rule, the number of degrees of freedom... 

Since both the oxide reactant and the spinel are ternary (nonstoichiometric) compounds when equilibrated with each other, at a given P and T, the boundaries (A, B)0/spinel and spinel/(B, A)203 are not invariant. They become invariant (and thus provide unique boundary conditions for the reaction) only if an additional intensive thermodynamic variable can be predetermined. This is a consequence of Gibbs phase rule. 

The equilibrium interfaces of fluid systems possess one variant chemical potential less than isolated bulk phases with the same number of components. This is due to the additional condition of heterogeneous equilibrium and follows from Gibbs phase rule. As a result, the equilibrium interface of a binary system is invariant at any given P and T, whereas the interface between the phases a and /3 of a ternary system is (mono-) variant. However, we will see later that for multiphase crystals with coherent boundaries, the situation is more complicated. 

In phase rule systems are categorized according to the number of components unary systems with only one component, binary systems with two components and (in this book) finally ternary systems with three components. The behaviour of the components in a system is determined by variables pressure, temperature and composition. 

Four distinct phases were observed in the microstructure of a ternary alloy at room temperature. Discuss this observation briefly in terms of the phase rule. What is the most likely explanation ... 

Consider the freezing of a ternary eutectic. The pressure is constant. The liquid simultaneously freezes to three solid phases, so there are four phases present during the freezing. One student applies the phase rule and concludes that there are zero degrees of freedom. Another student says that this is wrong because the amounts of the phases are not constant. Who is right Discuss briefly. 

Ternary phases with structures different from those of the phases of the binary boundary systems are more the exception than the rule. Such phases have been reported in the systems Nb-Mo-N, Ta-Mo-N, Nb-Ta-N, Zr-V-N, Nb-Cr-N, and Ta-Cr-N. Information about ternary transition metal-nitrogen systems is often available for specific temperatmes only. This is even more the case for quaternary nitride systems, which play a role in the production of carbonitride cermets where quaternary compounds of the types (Ti,Mo)(C,N) and (Ti,W)(C,N) are of interest (see Carbides Transition Metal Solid-state Chemistry), as well as in layer technology where titanium nitride-based coatings of the type Ti(C,B,N) are prepared by magnetron sputtering. Layers consisting of ternary compounds of the type (Ti,Al)N and (Ti,V)N also have favorable properties with respect to abrasion resistance. 

Simulations of ternary systems were performed using the pure component parameters in Table I and the cross parameters for the systems acetone/ CO2 and water/C02 determined previously (fi j - 1 and 0.81 respectively). Because of expected difficulties similar to the ones mentioned for the water/C02 system, no attempt was made to simulate the system acetone/water near room temperature. Thus, we set the acetone/water interaction parameters to the values from the Lorenz-Berthelot rules with fi j-l. Direct simulations of ternary phase equilibria have not been previously reported to the best of our knowledge. 

---