Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.38.
The chemical potential is defined in terms of the entropy as
This definition leads to a general thermodynamic identity
For a mixture of ideal gases, each species constitutes a molar fraction of the total number of molecules, so each species has its own chemical potential defined as
where all with are held constant in the derivative.
Also, for an ideal gas, each species contributes its own portion of the overall entropy, independently of the other species. We can see this by noting that if we have a mixture of, say, 2 gases, then for each configuration of the gas molecules there is a multiplicity of of the gas molecules and since for an ideal gas, the molecules don’t interact, the total multiplicity of the mixture is , so the entropy is the sum of the entropies for the separate species: .
Since ideal gas molecules don’t interact, species contributes a fraction of the total pressure, or in other words, its partial pressure is
We can therefore write the thermodynamic identity for a mixture of ideal gases as
Since in 3 (because only the number of species is changing, and no properties of any of the other species are changing), we can write the chemical potential of species as
But this is the definition of chemical potential in a system containing only species at partial pressure in volume . Thus, for a mixture of ideal gases, the chemical potential of each species is independent of the other species. In a mixture of real gases, however, this is probably not the case, since interactions between the species means the total entropy isn’t a simple sum of the entropies of the individual species.