Expression for Fugacity of a Pure Liquid

Expression for Fugacity of a Pure Liquid
The representation of liquid state by EOS is generally difficult. Thus, the calculation of fugacity of a compressed (or sub-cooled) liquid is based on the saturated liquid state as a reference state. One starts with the two generic relations (that apply to any pure real fluid ‘i’) already introduced in the section 6.7, namely:

The two equations above may be combined to yield:
As shown (eqn. 6.96) at the saturation condition for co-existing vapour and liquid phases:
Thus for a compressed liquid state at a given pressure one can write for an isothermal change of pressure from :
On equating the last two equations:
Since Vi, the liquid–phase molar volume, is a relatively weak function of pressure at temperatures well below the critical temperature, one may approximate where,  is the molar volume of the saturated liquid at the temperature of interest.
Using in the integrated form of eqn. 6.108 gives:
The exponential term on the left side of the last equation is known as a Poynting factorIt may be noted that the calculation of the term  can be made using a suitable gaseous EOS suitable to the temperature T and pressure 
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: