Chapter 6 : Solution Thermodynamics and Principles of Phase Equilibria
Section 5 : Ideal Gas Mixtures and Liquid Solutions
 
(6.53)

Hence it follows:
(6.54)

The last result indicates that the molar volume for a species does not change between its pure state and in an ideal gas mixture at the same T & P.  It may then be concluded that for an ideal gas mixture the properties of each species are independent of that of the other ones. This may be easy to appreciate as the concept of an ideal gas is premised on the idea that the intermolecular interaction is non-existent in such a state. This conclusion leads to the well-known Gibbs theorem:

“Except for volume all other partial molar property of a species in an ideal-gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at a temperature same as that of the mixture, but at a pressure equal to its partial pressure in the mixture.”
(6.55)