Relation between residual property and species fugacity coefficients in mixtures

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:
(6.131)
(6.132)
For the specific case of molar Gibbs free energy:
(6.133)
From eqns. 6.122 and 6.123 

On comparing eqns. 6.131 and 6.122:
(6.135)
But by definition: (6.136)

Thus, on inspecting 6.133 and 6.135: 

(6.137)
It follows that: 
(6.138) 
Since  is a partial molar property, it follows that the Gibbs-Duhem relation (eqn. 6.14) applies, that is:
(6.139)
(6.134)

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