The Condition for Equilibrium between Phases

The stability of the phases can be shown in a diagram like Fig. X VII-3, called a phase diagram. In this, temperature is plotted as ordinate, composition as abscissa, and lines separate the regions in which various phases are stable. Thus, at high temperature, the liquid is stable for all compositions. Running from 1702° to 1356° are two curve s, one called the liquidus (the upper one) and the other called the solidus. For GOO any T-x point lying between the liquidus and solidus, the stable goo 

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In this section we limit our discussion to the phase equilibria involved with pure substances. In this case, the condition for equilibrium between phases A, B, C,..., becomes... 

The chemical potential of a homogeneous material (a phase) is a function of two intensive variables, usually chosen as temperature and pressure. We say that such a material has two degrees of freedom (i.e., we are free to set two intensive variables). (Note that only intensive variables count as degrees of freedom.) In addition to being able to specify a number of intensive variables equal to the number of degrees of freedom of a system, we are also at liberty to specify the size of the phase with one extensive variable. The chemical potential can be represented as a surface on a plot of p versus P and T. The condition for equilibrium between phase a and phase p is, according to Eq. (24),... 

The conditions for equilibrium between phase A which is the gas mixture under consideration and phase B which consists of pure gas / are... 

Thus, the condition for equilibrium between two phases of a pure substance is given by... 

The fundamental condition for equilibrium between phases, the equality of the chemical potential of a component in every phase in which the component is present still applies. We then substitute for the chemical potential of a component the equivalent chemical potential of the same substance in terms of species or appropriate sums of chemical potentials of the species, as determined by the methods of Section 8.15 and used in the preceding sections. Several examples are discussed in the following paragraphs. 

The equilibrium state is reached when there is no change with time in any of the system s macroscopic properties. The phase rule by Gibbs gives the general conditions for equilibrium between phases in a system. It is assumed that the equUibrium is only influenced by temperature and pressure, that is, surface, magnetic, electrical, and magnetic... 

In this section, we derive the Clapeyron equation. This equation relates changes in the pressure to changes in the temperature along a two-phase coexistence curve (e.g., the vapor pressure curve or the melting curve). Note that the condition for equilibrium between two phases is given by... 

The condition for equilibrium between vapor and liquid phases, expressed by Equation 1.19 as the equality of each component fugacity in the two phases, applies to equilibrium between two liquid phases or any number of phases such as two liquids and a vapor. When the deviations from ideality are large enough, mixtures can form two immiscible liquids at equilibrium with each other. It is easy to see that an ideal solution cannot form two liquid phases at equilibrium. In order for this to occur, the condition = (p,%)P, where a and P designate each liquid phase, must be satis-... 

The conditions for equilibrium between two phases in a binary system are given by the equality of the chemical potentials in the two phases that is. 

The thermodynamic conditions for equilibrium between phases at constant pressure are immediately apparent in Fig. 12.1. Solid and liquid coexist in equilibrium when A soiid = A iiq j that is, at the intersection point of curves S and L. The corresponding temperature is, the melting point. Similarly, liquid and gas coexist in equilibrium at the tempera-... 

Thus, a condition for equilibrium between phases 1 and 2, when the volume of 1 cannot decrease, is that the pressure of phase 2 be larger than, or equal to, the pressure of phase 1. This condition is intuitively obvious. If the wall separating 1 and 2 is completely rigid,... 

Whichever view of the adsorbed phase is adopted, the condition for equilibrium between the adsorbed phase and the ambient gas phase is simply... 

The condition for equilibrium between the copper-ammine solution and the solid phase, Cu(N03)2 3Cu(0H)2, can be formulated as the expression, [Cu ][NHo], being a constant. This expression was shown to be quantitatively consistent with the step complexity constants found from die analysis of die vapor pressure data over a wide range of concentrations of free ammonia. 

According to thermodynamic principles, the condition for equilibrium between two phases requires that the partial molar free energy of each component be equal in each phase. This condition requires that the first and second derivatives of AGi in equation (3.23) with respect to V2 be zero, where V2 represents the volume fraction of polymer. The critical concentration at which phase separation occurs may be written... 

For equilibrium between phases, the fugacity of any component should be the same in each phase in which it is present. It therefore follows that if some means exists for calculating the fugacities of the components present in the phases of a system as functions of temperature, pressure and composition, it also becomes possible to calculate the conditions for equilibrium between the phases in the system. 

There is a long history of observations of phase equilibrium in polymer solutions. Phase separation in polymer solutions was first considered in detail from a statistical thermodynamic view by Flory [27,40,41). In his original papers on the statistical thermodynamics of these systems, the conditions for equilibrium between two separated phases are the classical conditions of when the partial molar Gibbs free energies are the same for each phase. The partial molar free energies are... 

The condition for equilibrium between the electrolyte and the bulk metal phase is obtained equating and ... 

A useful relationship between the temperature and pressure of phases in equilibrium can be derived from the condition for equilibrium. We start with equilibrium between phases A and B written as... 

The derivation of the quantitative relationship between this equilibrium temperature and the composition of the liquid phase may be carried out according to the well-known thermodynamic procedures for treating the depression of the melting point and for deriving solubility-temperature relations. The condition of equilibrium between crystalline polymer and the polymer unit in the solution may be restated as follows ... 

The condition of equilibrium between the adsorbed molecules and molecules in the gas state requires that the chemical potential for the adsorbed species be the same in both the gas phase and the adsorbed state ... 

Figure 13.14 shows the phase diagram for liquid 3He in the very low (millikelvin) temperature region. Line abc gives the experimental conditions for equilibrium between the normal liquid and a superfluid designated as liquid A, and between the normal liquid and a different superfluid (liquid B). The transition between the normal liquid and either superfluid is a continuous... 

The conditions where the last domain vanishes is found as follows. A condition for equilibrium between a two-phase structure and a homogeneous-melt structure is that the chemical potential of a repeat unit in the domain and in the monomolecular melt state pA are the same ... 

The process of crystallization from solution is generally understood in terms of three interdependent conditions supersaturation, nucleation and growth. For crystallization from the melt or the gas phase, supersaturation is simply interpreted as undercooling. Of the essence is a metastable phase, at chemical potential higher than that of the crystalline phase under the same conditions. The transition from the metastable state to the state of equilibrium between phases corresponds to nucleation. 

The condition of equilibrium between adsorbate and gas presumes the same temperature for tlie two phases and requires ... 

We conclude this chapter with some applications of the above formulae. Let us consider two phases both of which we will suppose, to fix our ideas, to be solutions. Introducing activity coefficients into the condition for equilibrium with respect to transport of matter between the two phases a-, we see that the equilibrium is governed by the relation (c/. 7.42 ISAS)... 

Finally the condition for equilibrium with respect to mass exchange between the gaseous and liquid phase also holds. This leads, as was illustrated, for instance, in [4.50], to... 

Now let s derive the mathematical conditions for equilibrium between two coexisting phases. We consider an isolated system that is separated into two phases, which we label A and B. The volume occupied by each phase can change in addition, the both phases can freely exchange energy and material with each other. Because the system is isolated, the total energy U, the total volume V,... 

We shall now show that the first eigenvector is in a direction parallel to the limiting tie line in the vicinity of P. To do this, we begin with the condition of equilibrium between two phases and " for which the following relation must be satisfied... 

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