Gibbs-Duhem equation used to describe uncompensated and apparent heat transfer

Entropic analysis makes it possible to determine the equilibrium of a system with its environment, but heat exchange at boundaries is a difficulty in this exercise. In this article, we carry out the energy balance of a water liquid and air mixture: the Gibbs-Duhem equation is used to describe the balance. Diffusion terms are expressed taking into account for change of volume and their entropic contribution, leading to the fugacity expression. After mixing, the mixture spontaneously evolves combining mechanical, chemical and thermal changes towards equilibrium. By applying the balance equation to both an isenthalpic and an adiabatic paths, we highlight the energy exchange between the system and its environment. Some irreversibilities are included in transformations and mainly affect the chemical path. The comparison reveals heat transfer terms corresponding to uncompensated heat. This make it possible to illustrate the notion of uncompensated heat and propose its complement, the apparent heat.