# How to derive Gibbs Theorem?

How to derive Gibbs Theorem which states that

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.

• I've never seen a derivation of this mixing rule for an ideal gas. Also, I think you meant "except for entropy." Of course, other exceptions are functions derived from entropy, such as Helmholtz free energy and Gibbs free energy. I suppose it boils down to the fact that the various species act as if they are present independent of all the other species. – Chet Miller Jan 21 '17 at 13:32
• No, it is for volume. Mathematically if M is a molar property – Saketh Bharadwaj Jan 21 '17 at 18:32
• You're right. For an ideal gas, the partial molar volume of a species is equal to the volume of the pure species at the same temperature and pressure as the mixture. But, Gibbs theorem does not address the partial molar properties directly. It states that "A total thermodynamic property of an ideal gas mixture is the sum of the total properties of the individual species, each evaluated at the mixture temperature but at its own partial pressure.? – Chet Miller Jan 21 '17 at 18:49
• Thanks so much for pointing that out about volume being the exception. I guess intuitively I already knew that, but never formally assimilated it into my thinking. I learned something new. – Chet Miller Jan 22 '17 at 12:12
• You're welcome. And thanks for giving me an intuitive explanation. It's somewhat making sense now that for ideal gas, the total property would be the sum of pure component properties at mixture temperature but at its partial pressure. – Saketh Bharadwaj Jan 23 '17 at 19:42