Equilibrium among coexisting phases

Equilibrium among coexisting phases the phase rule... 

CRITERIA FOR EQUILIBRIUM AMONG COEXISTING PHOSPHATES AND OTHER PHASES... 

Fugacity is a key concept in phase equilibria. The phase equilibrium condition consists of the equality of fugacities of a component among coexistent phases. The computation of fugacities implies two routes equation of states, for both pure components and mixtures, and liquid activity coefficients for non-ideal liquid mixtures. The methods based on equations of state are more general. 

Although these chemical effects are important in deciphering the genesis of ore minerals, it must be emphasized that the differences in the 8 8 values among coexisting condensed phases (hence the fractionation factor) at equilibrium are constant in each case because they depend only on T. 

Equilibrium—both physical and chemical, and including the equilibrium between a liquid and its vapor—is of enormous importance in chemistry, and it is important to acquire an understanding of the processes involved. Chemical equilibrium, which is examined in Chapter 9, deals with the equilibrium among the reactants and products of a chemical reaction. In this chapter, we deal with physical equilibrium, the state in which two or more phases of a substance coexist without a tendency to change. 

When the stability planes for three minerals are considered simultaneously (say, gibbsite, kaolinite, and beidellite), the three planes must intersect at a point (unless they happen to be parallel to one another). This intersection point represents the single solution composition consistent with all three minerals coexisting in equilibrium. In other words, equilibrium among these three mineral phases would fix the pH and the activities of AV and Si(OH)4 in solution at one unique invariant point. Since the gibbsite solubility expression (equation 6.33) can be written as... 

An interesting example of a one-component systems is SiOa, which can exist in five different crystalline forms or as a liquid or a vapor. As C = 1, the maximum number of phases that can coexist at equilibrium is three. Each phase occupies an area on the T P diagram the two-phase equilibria are represented by curves and the three-phase equilibria by points. Figure 13.1 (2, p. 123), which displays the equUi-brium relationships among the sohd forms of Si02, was obtained from calculations of the temperature and pressure dependence of AG (as described in Section 7.3) and from experimental determination of equUibrium temperature as a function of equilibrium pressure. 

It is also possible, as indicated previously, for a microemulsion containing spherical droplets to separate into a more concentrated microemulsion and excess continuous phase (e.g., an oil-in-water microemulsion in equilibrium with excess water), provided that attractive interaction among the droplets is sufficiently large. In other words, a microemulsion can have a limited capability to solubilize its continuous phase as well as its dispersed phase. Interfadal tension between the phases should be very low since both are continuous in the same component. Such a phase separation is similar to that which takes place at the cloud point of nonionic surfactants discussed previously. A simple theory of how it could occur for microemulsions was proposed by Miller et al. (1977). If excess continuous phase separates in this way and, at the same time, there is more dispersed phase present than can be solubilized, the microemulsion can coexist with both oil and water phases. While this situation of three-phase coexistence involving a microemulsion containing droplets probably exists for some compositions in some systems, in most situations the microemulsion in a three-phase region is bicon-tinuous. The above discussion emphasizes the early theoretical work on microemulsions with droplets, but numerous other developments have been reported since then. 

There is a paucity of information on fluid metal mixtures under extreme conditions. This is perhaps not surprising given the remarkable variety and complexity of the phase diagrams of mixtures of even simple molecular liquids (Rowlinson and Swinton, 1982). In a singlecomponent system, equilibriiun between two fluid phases is normally limited to liquid-gas equilibrium, and above the critical temperature the system exhibits the unique feature that its density can be varied in a continuous manner without the occurrence of an abrupt liquid-gas phase transition. We have discussed in detail the continuous metal-nonmetal transitions that occur in pure fluid metals under these conditions. The situation is quite different, however, for a two-component system. The equilibrium region where two fluid phases coexist is not necessarily limited to temperatures below the critical temperature of the less volatile component. On the contrary, among the more remarkable features of binary mixtures is the continuous existence of phase separation above the critical points of the pure components. One now speaks ot gas-gas or fluid-fluid equilibria. We consider the latter term to be the more appropriate since the phases are more accurately characterized as dense fluids than as gases under the conditions at which these separations occur. 

A first attempt to combine thermodynamic information on the Fe - Ti - O and the Fe - Si - O systems has been made by Taylor et al. (1972). According to a schematic plot (see Figure 10) the equilibrium curves for various monovariant reactions may be superimposed upon each other. As long as the equilibria are mutually independent - i.e. no titanium dissolves in the silicates and no silicon in the oxide phases - equilibrium relationships may be directly derived from this diagram. Based on the new data provided above a quantitative determination of equilibrium states among silicates and oxides coexisting with metallic iron in the system Fe - Si - Ti - O is presented in Figure 14. From analyti-... 

Remarkably, P and T are unique among the thermodynamic properties in that they both exist only in the intensive form and that they are equal across the different phases that coexist at equilibrium. Other thermodynamic properties (such as volume) can be written in both extensive and intensive forms, and most of these properties differ between two phases that coexist at equilibrium. 

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