Criteria phase equilibrium

The criterion for phase equilibrium is given by Eq. (8.14) to be the equality of chemical potential in the phases in question for each of the components in the mixture. In Sec. 8.8 we shall use this idea to discuss the osmotic pressure of a... 

Donnan equilibrium arises from applying the phase equilibrium criterion to the indifferent electrolyte . From Eq. (8.13) this is /ia = +... 

This is the criterion of two-phase equilibrium. It is readily generahzed... 

The formulation of the residuals to be used in the objective function is based on the phase equilibrium criterion... 

In bulk matter the quark-hadron mixed phase begins at the static transition point defined according to the Gibbs criterion for phase equilibrium... 

Figure 1. Chemical potentials of the three phases of matter (H, Q, and Q ), as defined by Eq. (2) as a function of the total pressure (left panel) and energy density of the H- and Q-phase as a function of the baryon number density (right panel). The hadronic phase is described with the GM3 model whereas for the Q and Q phases is employed the MIT-like bag model with ms = 150 MeV, B = 152.45 MeV/fm3 and as = 0. The vertical lines arrows on the right panel indicate the beginning and the end of the mixed hadron-quark phase defined according to the Gibbs criterion for phase equilibrium. On the left panel P0 denotes the static transition point.

The following criterion of phase equilibrium can be developed from the first and second laws of thermodynamics the equilibrium state for a closed multiphase system of constant, uniform temperature and pressure is the state for which the total Gibbs energy is a minimum, whence... 

It is apparent that CMC values can be expressed in a variety of different concentration units. The measured value of cCMC and hence of AG c for a particular system depends on the units chosen, so some uniformity must be established. The issue is ultimately a question of defining the standard state to which the superscript on AG C refers. When mole fractions are used for concentrations, AG c directly measures the free energy difference per mole between surfactant molecules in micelles and in water. To see how this comes about, it is instructive to examine Reaction (A) —this focuses attention on the surfactant and ignores bound counterions — from the point of view of a phase equilibrium. The thermodynamic criterion for a phase equilibrium is that the chemical potential of the surfactant (subscript 5) be the same in the micelle (superscript mic) and in water (superscript W) n = n. In general, pt, = + RTIn ah in which... 

In this manner, the criterion, Pn(p) exp[At / Ate -1] = U represents the liquidus curve in the composition-temperature phase equilibrium diagram. Clearly, the shape of the L "( p, T ) = 1 curve must be parallel with the configuration probability, P"(p). The correspondence between these physical arguments and an actual phase equilibrium diagram (the Li-Hg system) for Ti.Tm, and T2 is illustrated in Fig. 12. 

The most important chemical thermodynamic property is the chemical potential of a substance, denoted /x.18 The chemical potential is the intensive property that is the criterion for equilibrium with respect to the transfer or transformation of matter. Each component in a soil has a chemical potential that determines the relative propensity of the component to be transferred from one phase to another, or to be transformed into an entirely different chemical compound in the soil. Just as thermal energy is transferred from regions of high temperature to regions of low temperature, so matter is transferred from phases or substances of high chemical potential to phases or substances of low chemical potential. Chemical potential is measured in units of joules per mole (J mol 1) or joules per kilogram (J kg 1). 

This criterion of equilibrium provides a general method for determination of equilibrium states. One writes an expression for G as a function of the numbers of moles (mole numbers) of the species in the several phases, and then finds the set of values for the mole numbers that minimizes G subject to the constraints of mass conservation. This procedure can be applied to problems of phase, chemical-reaction, or combined phase and chemical-reaction equilibrium it is most useful for complex equilibrium problems, and is illustrated for chemical-reaction equilibrium in Sec. 15.9. 

To apply this criterion, one develops an expression for dG as a function of the mole numbers of the species in the various phases, and sets it equal to zero. The resulting equation along with those representing the conservation of mass provide working equations for the solution of equilibrium problems. Equation (13.53) leads directly to Eq. (10.3) for phase equilibrium and it is applied to chemical-reaction equilibrium in Chap. 15. 

When liquid and gas phases are both present in an equilibrium mixture of reacting species, Eq. (11.30), a criterion of vapor/liquid equilibrium, must be satisfied along with the equation of chemical-reaction equilibrium. There is considerable choice in the method of treatment of such cases. For example, consider a reaction of gas A and water B to form an aqueous solution C. The reaction may be assumed to occur entirely in the gas phase with simultaneous transfer of material between phases to maintain phase equilibrium. In this case, the equilibrium constant is evaluated from AG° data based on standard states for the species as gases, i.e., the ideal-gas states at 1 bar and the reaction temperature. On the other hand, the reaction may be assumed to occur in the liquid phase, in which case AG° is based on standard states for the species as liquids. Alternatively, the reaction may be written... 

Up to the present we have dealt with fundamental principles applicable to general cases. In this chapter we specialize to ideal systems in which several components coexist in one or more phases. We proceed gradually from the simple equilibrium conditions to more complex cases the underlying thread is that equilibrium requirements involve constraints, all of which are ultimately based on Gibb s criterion At equilibrium the... 

The primary criterion for equilibrium between two multicomponent phases is that the chemical potential of each component, /Jt, must be equal in both phases I and II. 

Tlris criterion of equilibrium is tire one usually applied by chemical engineers in the solution of phase-equilibriunrproblenrs. 

Chemical and phase equilibria are attained when the chemical potential of the reactants equals that of the products. This criterion for equilibrium can be expressed by the mass action law as... 

Because the dn[ are independent and arbitrary, it follows that, -This is the criterion of two-phase equilibrium. It is readily generalized to multiple phases by successive application to pairs of phases. The general result is... 

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