# Gibbs' theorem and partial molar volume

#### kayan

In the chemical engineering text of Smith, VanNess, and Abbott, there is a section on partial molar volume. It states that Gibbs theorem applies to any partial molar property with the exception of volume. Why is volume different? In other words, when evaluating the partial molar volume of a mixture, we evaluate it at a T, P (total mixture pressure), but for other partial molar properties (like entropy), we evaluate them at T, Pi (partial pressure of i in the mixture).

There's a very similar thread on physics stack exchange, but I don't find a completely satisfying answer there (an neither did the OP): https://physics.stackexchange.com/questions/502788/gibbs-theorem-and-partial-molar-volume.

#### Attachments

• 281.9 KB Views: 19
Related Materials and Chemical Engineering News on Phys.org

#### Chestermiller

Mentor
What kind of answer were you looking for?

In the case of U and H, for an ideal gas mixture, these are independent of pressure, so the distinction "at the same partial" pressure doesn't come in. So these are just thrown in with the other properties that involve entropy.

S, A, and G all involve entropy S in their definitions. You can show that the reversible heat required to isothermally separate an ideal gas mixture into its pure constituents at their partial pressure in the mixture is zero. So the entropy of the mixture must be equal to the sum of the molar entropies of the pure constituents at the same temperature and pressure equal to their partial pressures in the mixture multiplied by the number of moles of each species. This means that S satisfies Gibbs theorem. And, since A and G involve U and H, respectively, and TS, if S satisfies Gibbs theorem, so must they.

So the real question should be "if V does not satisfy Gibbs theorem, how come U, H, S, A, and G do?"

#### kayan

Framing the question is definitely part of the problem, because since V, U, & H for an IG can all be evaluated at the total P, then Gibbs theorem seems to be the exception instead of the rule, with the exception being S (and anything related to S). I don't know what kind of answer I was looking for, just perhaps a better explanation of the rule instead of just accepting it by fiat, which is how any source that I've found has justified it.

It seemed like an issue that deserves more space in the textbooks than has been given to it. Your entropy explanation is something that makes sense, so I'm going to think about it for a bit and see if that resolves my concerns.

#### Chestermiller

Mentor
Don't be too hard on Smith and Van Ness. I think it's a really good book. Another great thermo book is Fundamentals of Engineering Thermodynamics by Moran et al. But the emphasis in not so much on ChE interests, such as solution thermodynamics.

#### kayan

Not meaning to bash Smith and VN at all. In fact, it's one of the only thermo books I've read that discusses this topic in any degree. I just wished they would've explained more since they are my only source on the subject!