Expressions for gas phase  using EOS
In this section we derive the expressions for computing the fugacity coefficients for individual species in a real gas mixture. As in the case of pure real gases (section 6.8) we consider the truncated virial EOS and the various cubic EOSs. From eqn. (6.90):
                                                                                                             
Further by eqn. (6.137):
                                                                                  
From these two relations we obtain respectively:
(6.140)
 (6.141)
Now, 
Hence: 

Species Fugacity Coefficients (in a gas mixture) from the Virial EOS:

The expression for computing the mixture virial coefficient corresponds to eqn. 2.25. One may rewrite the expression as follows:
(6.143)
Where,  (6.144)
Using (6.140 / 6.141): 
Or: (6.145)
Therefore:
 
Using (6.148) in (6.142):
(6.147)
Now putting (6.133) it may be shown that:
(6.148)
Therefore putting (6.148) in (6.147), one obtains:
(6.149)
In a similar manner, one may show that:

Fugacity coefficient expression for species in a mixture by pressure explicit EOS:
The eqn. 6.130 is useful for application to volume explicit EOSs only. For pressure explicit EOS (cubic EOSs) the expression for obtaining the species fugacity coefficient in a mixture is relatively complex. For the present purpose we provide below the generic equation used for pressure explicit EOS, while its proof is provided at the end of the present chapter.
(6.153)
We enlist below the resultant final expressions that obtain for species fugacity in a gaseous mixture corresponding to various forms of cubic EOSs. 
For example, for RK-EOS: 
Next one writes the parameters ‘a’ and ‘b’ for the mixture using typical mixing rules described by eqns. 2.29 to 2.31:

(6.155)
For SRK EOS:
(6.156)
For PR EOS:
(6.157)
(6.154)
(6.150)
(6.146)
(6.142)