The Chemical Potential
In this section we focus on the properties of partial molar Gibbs free energy, which as we observed at the beginning of the chapter, is used for the description of phase and chemical reaction equlibria. The application of eqn. 6.3 to the molar Gibbs free energy of a mixture gives:
(6.38)
By definition the partial molar Gibbs free energy is termed the chemical potential of species in the mixture, i.e.,:  
(6.39)
Or:
Using the result in eqn. 5.7 the first two partial derivatives in eqn. 6.28, may be replaced by .  Eqn. 6.38 then becomes:
(6.41)
With eqn. 6.41 becomes:  
(6.42)
We next consider the use of chemical potential for obtaining a general criterion of thermodynamic equilibrium. Consider a closed system consisting of two phases which are in equilibrium with each other. These phases could be vapour and liquid, solid and liquid, solid and vapour etc. Each phase in the system may be treated as an open system, the interface between the two phases acting as the boundary across which material may be transferred. Thus we can apply 6.41 to both phases individually:
(6.43)

In general we should denote the temperature and pressure of each phase also with the superscript in order to distinguish them. However, for the present purpose we assume thermal and mechanical equilibrium to prevail, i.e.            

The total Gibbs free energy of the system changes with mass transfer between the two phases. The change in the total Gibbs energy of the two-phase system is the sum of the changes in each phase. The total volume and entropy of each phase is expressed by the following equations. 
(6.45)
And   (6.46)
Summing eqns. 6.43 and 6.44 and using eqns. 6.55 and 6.46 we get: 
= 0
Since the mass transfer of each species takes place between the two phases in question, their change of mass in each phase must be equal and opposite: 
In which case eqn. 6.47 becomes:
(6.48)
If the system is considered to be already under thermal and mechanical equilibrium no changes in temperature and pressure may occur, then the last equation simplifies to: 
(6.49)
We note that in the above sequence of equations no explicit constraint has been placed on the individual  which then are independent of each other. Thus for eqn. 6.49 to have general validity the coefficient of each  has to be identically zero. Thus: 
        (i = 1, 2, …N).
The above proof has been simplified by assuming identical temperature and pressure for each phase. However, a more rigorous mathematical derivation of the phase equilibrium criterion leads to the result that if a system is under thermodynamic equilibrium, the temperature and pressures of all the phases are the same. If there is third phase  in the system we have considered, a second equation of the type (6.50) obtains:
(6.51)
Thus, if there are a total of  phases in the system, one can generalize the result as follows:
By considering successive pairs of phases, we may readily generalize to more than two phases the equality of chemical potentials; thus for  phases:
      (i = 1, 2, …N)   (6.52)
The above result allows us to advance a more general statement of the phase equilibrium criterion: 
For a system under thermodynamic equilibrium, along with equality of the temperature and pressures of all phases, the chemical potential of each species is identical across all the phases.
The Ideal Solution:

We have already seen that owing to the fact that pure ideal gases and mixtures are not subject to intermolecular interactions the partial molar properties (apart from volume) of each species is the same as that of the pure species at the same temperature and pressure. In other words each species “sees” no difference in their environment in pure or mixed state. One can conceptually extend this idea to posit an ideal solution behaviour which may serve as a model to which real-solution behavior can be compared. Consider a solution of two liquids, say A and B. If the intermolecular interaction in the pure species, (i.e., A-A and B-B) is equal to the cross-species interaction A-B, neither A nor B type molecules will “see” any difference in their environment before and after mixing. This is in a sense the same condition as one obtains with idea gas mixtures. Hence an identical set of ideal solution property relations may be constructed based on the model of ideal gas mixture. By convention while describing properties of liquid solutions mole fractions yi are replaced by xi. The following relations therefore, ideal (liquid) solution properties (denoted by a superscript ‘id’):
 and,    (6.74)
(6.75)
Hence  (6.76)
Lastly,  (6.77)
As we will see later the ideal solution model can also serve to describe the behaviour of mixtures of real gases or solids
(6.50)
(6.47)
(6.44)
(6.40)