Mixed phase Gibbs conditions

Let us start by giving a brief introduction into the general method of constructing mixed phases by imposing the Gibbs conditions of equilibrium [23, 18]. From the physical point of view, the Gibbs conditions enforce the mechanical as well as chemical equilibrium between different components of a mixed phase. This is achieved by requiring that the pressure of different components inside the mixed phase are equal, and that the chemical potentials (p and ne) are the same across the whole mixed phase. For example, in relation... 

It is easy to visualize these conditions by plotting the pressure as a function of chemical potentials (ji and Re) for both components of the mixed phase. This is shown in Figure 8. As should be clear, the above Gibbs conditions are automatically satisfied along the intersection line of two pressure surfaces (dark solid line in Figure 8). 

In order to fulfill the charge neutrality condition one can construct a homogeneous mixed phase of these states using the Gibbs conditions [33]. 

Chemists and physicists must always formulate correctly the constraints which crystal structure and symmetry impose on their thermodynamic derivations. Gibbs encountered this problem when he constructed the component chemical potentials of non-hydrostatically stressed crystals. He distinguished between mobile and immobile components of a solid. The conceptual difficulties became critical when, following the classical paper of Wagner and Schottky on ordered mixed phases as discussed in chapter 1, chemical potentials of statistically relevant SE s of the crystal lattice were introduced. As with the definition of chemical potentials of ions in electrolytes, it turned out that not all the mathematical operations (9G/9n.) could be performed for SE s of kind i without violating the structural conditions of the crystal lattice. The origin of this difficulty lies in the fact that lattice sites are not the analogue of chemical species (components). 

The influence of the chemical composition in (5) can be derived using a simplified versicai of Barker s lattice theory [55], The most important consequence of (5) is the fact that the segment-molar excess Gibbs free energy of mixing and, hence, the activity coefficients depend only oti the average value (yw) of the distribution function, but not on the distribution function itself. In continuous thermod5mamics, the phase equilibrium conditions read ... 

A region in which a one-phase blend is always stable for any composition, and a region in which two mixed phases are more stable than a homogeneous system, are divided by the critical point (CP). This is defined mathematically as the point at which the second and third derivatives of the Gibbs free energy with respect to the polymer volume fraction are zero. Applying these conditions to the Flory-Huggins equation (Eq. (3.14)), it is possible to define the... 

In this expression for the chemical potential, the first addendum, U(,(T), is a standard potential at a fixed pressure. The second addendum expresses the contribution from the fugacity of the pure component. The third addendum is due to mixing. The dependence/ (< >, T) may be found from the Gibbs adsorption equation (13b), where the integration is often carried out from zero pressure (and, correspondingly, the value of 0 is equal to zero). With such an expression for the chemical potentials of the adsorbed phase, equilibrium conditions (12), for the equilibrium with a nonideal gas phase, are reduced to the form... 

Physical Equilibria and Solvent Selection. In order for two separate Hquid phases to exist in equiHbrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy, G, of a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in two phases. Eor the binary system containing only components A and B, the condition (22) for the formation of two phases is... 

Huron and Vidal showed that equating the infinite pressure Gibbs energy of mixing to that of an activity model like the NRTL or UNIQUAC models provided a mixing rule that was sufficiently flexible to describe very complex phase behavior. With this modification, simple cubic equations like Soave s could be applied to nearly any kind of mixture at any conditions, including supercritical conditions. The Huron-Vidal mixing rule combined with NRTL activity model is illustrated below. 

Beside the condition (3.216) for the calculation of the molar excess Gibbs energy of mixing, the minimum necessary Gj coefficients for attaining thermodynamically consistent phase diagram and a reasonable standard deviation of approximation are required. The calculation is mostly performed assuming AfusH / T) and AG j / T). 

In the calculation of the phase diagram, the equilibrium molar fraction is inserted into LeChatelier-Shreder s equation and the temperature of primary crystallization of every constituent is then calculated. For the optimized phase diagram, the Gibbs energy of mixing of the system is calculated using the condition... 

At a given pressure and temperature, the total Gibbs free energy of mixing of a one-phase polymer-solvent system of composition 2 should be necessarily minimum, otherwise the system will separate into two phases of different composition, as it is represented in a typical AG versus cp phase diagram of a binary solution (Fig. 25.3). The volume fractions at the minima (dAGIdcp = 0), cp, and (p will vary with temperature (binodal) up to critical conditions (T and (p ) where cp = tp" (Fig. 25.3b). 

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