Conditions of phase equilibrium

The liquid sample must be large enough that all does not evaporate before equilibrium is achieved, The condition of phase equilibrium requires that two (or more) phases be present. 

A condition of phase equilibrium is the equality of the chemical potentials in the two phases. Therefore, at all points along the two-phase line, //(g) = p( ). But, as we have noted above, the approach to the critical point brings the liquid and gas closer and closer together in density until they become indistinguishable, At the critical point, all of the thermodynamic properties of the liquid become equal to those of the gas. That is, Hm(g) = Um(g) - /m(l),... 

In Chapter 5, we showed that the condition of phase equilibrium for multicomponent phases is that the chemical potential of each component must be the same in all the phases. That is... 

Conditions of phase equilibrium require that the chemical potential of polymer in each phase and that of solvent in each phase be equal ... 

Another general type of behavior that occurs in polymer manufacture is shown in Figure 3. In many polymer processing operations, it is necessary to remove one or more solvents from the concentrated polymer at moderately low pressures. In such an instance, the phase equilibrium computation can be carried out if the chemical potential of the solvent in the polymer phase can be computed. Conditions of phase equilibrium require that the chemical potential of the solvent in the vapor phase be equal to that of the solvent in the liquid (polymer) phase. Note that the polymer is essentially involatile and is not present in the vapor phase. 

We assume that the micelle is a pure surfactant phase with a chemical potential given by fi%. If the chemical potential of the monomer is m, then the condition of phase equilibrium is given by... 

We may, with appropriate attention to the reference state, develop the relations in terms of components between the intensive variables pertinent to multiphase systems that contain species other than the components. Such relations would be rather complex, because no account would be taken of the effect of the chemical reactions that occur in the system. All deviations from ideality would appear either in the activity coefficients for substances in condensed phases or in the coefficients used in some equations of state for the gas phase. Simpler relations are obtained when the conditions of phase equilibrium are based on species rather than components, once the species have been identified. 

Figure 8. Hypothetical concentration profiles in gas and aqueous phases indicating gradient in reagent concentration due to flux of material into and within drop. Concentration scales of aqueous-phase (r < a) left ordinate and gas-phase (r > a) right ordinate are chosen so that the same coordinate on each scale represents the condition of phase equilibrium. Departure from the uniform profile at the "bulk" (r =oo) value represents the inability of mass transport to maintain the reagent concentration as the reagent is consumed by aqueous-phase reaction. (Reproduced with permission from Ref. 28. Copyright 1986 Lewis Publishers, Inc.)...

In most cases of interest, VtAP is negligible in comparison with the other terms, so that the condition of phase equilibrium across a membrane becomes... 

Thus, we have proved the assertion that in the case of combined chemical and phase equilibria the conditions of phase equilibrium must be satisfied for all species in each of the phases and, furthermore, that chemical equilibrium must exist for each reaction... 

Summary. The last Section illustrating the basics of rational thermodynamics shows how phase equilibria can be treated by this methodology. Constitutive equations should be modified to describe the effects of different phases, see (2.107)-(2.109). Traditional condition of phase equilibrium in terms of chemical potentials was derived, (2.116) or (2.129). 

With aqueous solutions of electrolytes we have two types of equilibrium to consider phase equilibrium and chemical or ionic reaction equilibrium. Phase equilibrium of interest are primarily vapor-liquid and liquid-solid, though vapor-llquid-solid is often of great importance as, for example, in carbonate systems. The necessary condition of phase equilibrium is that the chemical potential of any species i in phase a is equal to the chemical potential of that same species 1 in phase b or... 

The conditions of phase equilibrium for the oxide and carbonate between their solid phases and their vapours are... 

The phase diagrams in Figure 13.1 only take into account fluid phase behaviour. In the case of solid reactants equations of state allow the evaluation of solute solubility in the solvent. The conditions of phase equilibrium between a supercritical fluid (1) and a solid component (2) are formulated on the basis of the isofugacity criterion. If the solid phase is assumed to be a pure component, the solubility in the gas phase can be directly obtained as ... 

The enthalpy change on adsorption is commonly referred to as the isosteric heat of adsorption. It was derived from the condition of phase equilibrium (i.e., the equality of chemical potentials between the adsorbed phase and the ambient gas phase) ... 

At any composition, the chanical potentials of the pure components are given by the tangent line, as shown in Hgure 15.3a. The condition of phase equilibrium in a danixed binary systan, which can also be used to determine the binodal, is given by... 

If we designate solvent properties by subscript 1 and solute properties by subscript 2, then the following relations hold at thermodynamic equilibrium, using the condition of phase equilibrium ... 

Once the interaction parameter in the form of Eq. (9.3.50) has been determined, the entire temperature-composition phase diagram or the binodal curve can be calculated using the conditions of phase equilibrium. At a chosen temperature, let the two polymer compositions in equilibrium with each other be and. Let... 

---