Is the Gibbs-Duhem equation always valid?

Question

The derivation of the Gibbs-Duhem equation from Wikipedia uses:

As shown in the Gibbs free energy article, the chemical potential is just another name for the partial molar (or just partial, depending on the units of $N$) Gibbs free energy, thus $$G = \sum_{i=1}^I \mu_i N_i \,. $$

On the other hand, that article states:

Because some of the natural variables are intensive, $dG$ may not be integrated using Euler integrals as is the case with internal energy. However, simply substituting the Gibbs-Duhem relation result for $U$ into the definition of $G$ gives a standard expression for $G$:^[12]

$$\begin{align} G &= T S - p V + \sum_i \mu_i N_i + p V - T S\\ &= \sum_i \mu_i N_i \end{align}$$

This result applies to homogeneous, macroscopic systems, but not to all thermodynamic systems.^[13]

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[12] Salzman, William R. (2001-08-21). "Open Systems". Chemical Thermodynamics. University of Arizona. Archived from the original on 2007-07-07. Retrieved 2007-10-11.
[13] Brachman, M. K. (1954). "Fermi Level, Chemical Potential, and Gibbs Free Energy". The Journal of Chemical Physics 22 (6): 1152–1151. doi:10.1063/1.1740312.

This sounds like circular reasoning to me, or did I miss something? Anyway, my question is about the emphasized last sentence, in short:

When is the Gibbs-Duhem equation not valid? Is there a well-known example?

Answer 1

I think your question is really: Why is Gibbs-Duhem equation not valid for inhomogeneous and/or small systems? This is related to why Thermodynamics does not apply to these systems. Examples, discussion and a Thermodynamical theory for these systems can be found in Terrell Hill's works link.

But in short: both inhomogeneous and few-body systems have more degrees of freedom than their macroscopic homogeneous counterparts. So in the former ones, the Gibbs-Duhem relation is different and no longer restricts the changes in the intensive variables by allowing only 2 of them to be independent. Also Gibbs potential will need the addition of a term related to the new ways in which energy can be transformed.