Fugacity and Fugacity Coefficient of Species in Mixture

 Fugacity and Fugacity Coefficient of Species in Mixture
The definition of the fugacity of a species in a mixture is parallel to the definition of the pure species fugacity. For an ideal gas mixture the chemical potential is given by eqn. (6.63):
Or:     (at const. T)  (6.120)
 But   (at const. T) 
Thus: (6.121)
Or:      (6.122)
Using the same idea implicit in eqn. 6.86, we extend eqn. 6.122 to define a similar expression for fugacity of species in a real mixture:

Here,  is the fugacity of species i in the mixture, replacing the partial pressure As we will demonstrate later, is a partial molar property, and is therefore denoted by a circumflex rather than by an overbar, as are partial properties. On integrating eqn. 6.113 between the any two multi-component phases  in equilibrium with each other: 
Hence, (6.125)
But by eqn. 6.42, for equilibrium: 
It follows, therefore, 
That is:  (6.126)
Thus for an arbitrary number of phases in equilibrium with each other:
(i = 1,2…N)
Thus, eqn. 6.126 offers a criterion of equilibrium correspondent to that provided by eqn. 6.42. The great advantage this equivalence offers is that, the fugacity coefficients of species in a mixture can be related to the volumetric properties of the mixtures, which facilitates the solution of phase and chemical reaction equilibria problems. 
On comparing eqns. 6.122 and 6.123, the following limiting condition obtains:  
It follows that a fugacity coefficient  (dimensionless) of a species in a real gas mixture may be defined as:
Although the derivation above defines the fugacity coefficient with respect to gaseous mixture the definition may be extended to represent fugacity coefficient of a species in a real liquid solution. Accordingly, in that case it is defined in the following manner: