Non-applicability of the Gibbs-Duhem relation in nonextensive thermodynamics. Case of micellar solutions

This work is a discussion on the applicability of the Gibbs-Duhem relationship. We show that it does not exist in the nonextensive thermodynamic approaches. To illustrate this, we have chosen to present the properties of three surfactants (dodecyltrimethylammonium bromide (DTABr), sodium dodecyl sulfate (SDS), sodium decanoate (NaDec) in aqueous solutions at 298 K) whose behavior after the micellization threshold can be described by the rules of the nonextensive thermodynamics. We show that the Gibbs-Duhem relationship does not apply to these systems and we propose to formalize the deviation from this law. The consequences of this study are discussed for the different approaches that involve the use of Gibbs-Duhem relation in the case of micellized solutions. In particular, the applicability of the Gibbs relation which links surface tension and the chemical potentials of solutes is considered. More generally, this study warns against the blind application of the Gibbs-Duhem (or Gibbs) relation, to any system.