Relation between residual property and species fugacity coefficients in mixtures
As we know, for any molar property (M), the definition of residual property (MR) is given by:

It follows that:
For the specific case of molar Gibbs free energy:
From eqns. 6.122 and 6.123 

On comparing eqns. 6.131 and 6.122:
But by definition: (6.136)

Thus, on inspecting 6.133 and 6.135: 

It follows that: 
Since  is a partial molar property, it follows that the Gibbs-Duhem relation (eqn. 6.14) applies, that is: