# Is the Gibbs-Duhem equation always valid?

The derivation of the Gibbs-Duhem equation from Wikpedia 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]

[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?

• (or, apparently equivalent, when is $G\neq\sum_i\mu_i N_i$?) – Tobias Kienzler Oct 8 '14 at 9:48