Phase transition coexistence coefficients

A, A,B,C,... are phenomenological coefficients that are functions of the independent intensive variables P,T,... In order to meet the equilibrium condition at rj = 0, A must vanish. Furthermore, for stable phases, A (7 ,/ )>0 and C(T,P) is also >0, otherwise G would become excessively negative for larger values of tj. (G-G°) vs. 11 is depicted in Figure 12-5a. If B2 = 4 AC, the minima for (G-G°) at t] = 21/UC and -4 AC have the same value G. This means that here two phases would coexist in equilibrium, which is characteristic of a first-order phase transition. 

The presence of transition metal ions in mineral structures may significantly modify phase equilibria at high pressures as a result of increased CFSE acquired by certain cations in the dense oxide phases believed to constitute the Lower Mantle. The additional electronic stabilization can influence both the depth in the Mantle at which a phase transition occurs and the distribution coefficients of transition metals in coexisting dense phases in the Lower Mantle. 

Notice, that d = 1 corresponds to the equilibrium phase coexistence temperature. The values d = 9/8 and = 0 are the upper and lower limits of the metastable nematic and isotropic states, respectively. The quantity <5k = 1 — T /Tk which sets a scale for the relative difference of the temperature from the equilibrium phase transition is known from experiments to be of the order 0.1 to 0.001. On the other hand, it is related to the coefficients occurring in the potential function according to... 

The site preferences shown by cations in the spinel structure demonstrate that transition metal ions prefer coordination sites that bestow on them greatest electronic stability. In addition, certain cations deform their surrounding in order to attain enhanced stability by the Jahn-Teller effect. These two features suggest that similar factors may operate and cause enrichments of cations in specific sites in silicate structures, leading to cation ordering or intersite (intracrystalline ) partitioning within individual minerals which, in turn, may influence distribution coefficients of cations between coexisting phases. 

One of the most successful applications of crystal field theory to transition metal chemistry, and the one that heralded the re-discovery of the theory by Orgel in 1952, has been the rationalization of observed thermodynamic properties of transition metal ions. Examples include explanations of trends in heats of hydration and lattice energies of transition metal compounds. These and other thermodynamic properties which are influenced by crystal field stabilization energies, including ideal solid-solution behaviour and distribution coefficients of transition metals between coexisting phases, are described in this chapter. 

In liquid-liquid extraction, a solvent is added to a liquid matrix (feed) to remove selectively transition components by the formation of two coexisting, immiscible liquid phases. The selected solvent (receiver phase) must be capable of preferentially dissolving the solutes to be extracted and be either immiscible or only partly miscible with the carrier (release phase). This process is, therefore, based on the different affinities of the solute distributing between the two coexisting liquid phases. Of the two phases, the solvent-rich solution containing the extracted solute is the extract and the solvent-lean, residual feed mixture is the raffinate. In the case of a closed miscibility gap, the correlation of the solute mole fraction in the extract and the raffinate phase is called the distribution coefficient (partition coefficient) K ... 

Figure 1 represents the isotherms for two lipid components which are miscible in the condensed monolayer state. The major feature of the isotherms for the pure components (1 0, 0 1) is the transition region in which the surface pressure is independent of surface area here the limits of the transition region are at the low area end, Ac, and at the high area end, Ax. These areas are characteristic of each lipid and represent the area per molecule of the lipid in the condensed and vapor states (10). For an equimolar mixture of the two components (1 1), the surface pressure in the transition region depends on the surface area according to the phase rule (11, 12, 13, 14), two surface phases coexist here a condensed phase of lipids and the surface vapor phase. To obtain the activity coefficient of the ith component in the condensed phase the following relation may be used ... 

In second-order transitions, the first derivatives of a thermodynamic parameter are continuous, but the second derivatives with respect to the corresponding parameter change in steps. Such transitions are not attended by a heat effect and are characterized by a change in the heat capacity and in the expansion and pressure coefficients, which means that the coexisting phases differ not in volume and their store of energy but in the values of their derivatives. 

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