Basic Thermodynamics of Phase Equilibrium

The starting point for a phase equilibrium calculation is the thermodynamic requirement that the temperature, pressure, and partial molar Gibbs free energy of each species be the same in all phases in which that species is present. That is, 

Here T and P are temperature and pressure, respectively, Xj represents all the mole fractions in phase J, is the molar Gibbs free energy of the phase, and the subscript indicates that the derivative is to be taken with respect to the number of moles of species i in phase J with all other mole numbers held constant. The partial molar Gibbs free energy of a species is equal to its chemical potential /r,-, and this is shown as the last equality in eqn, (2.1.2). 

Equation (2.1.1) is an exact relation from thermodynamics. However, in chemical engineering design and process simulation, what is needed is interrelations between the compositions of the phases in equilibrium rather than among the chemical potentials. Consequently, considerable effort in applied thermodynamics is devoted to converting the relation of eqn. (2.1.1), together with the definition of the chemical potential in eqn. (2.1.2), into interrelations between the compositions of the equilibrium phases. 

For the ideal solution the activity coefficients of the constituents are unity, and for the real solutions they are defined with respect to a suitable reference state with the limitation that the temperature of the reference state must be that of the solution. We will return to the activity coefficient concept later when we discuss models for the liquid mixtures. 

With equations of state, real mixture behavior is described by introducing the fugacity, /, The fugacity of a species in a real mixture is 

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