Natural Mathematical Derivation of the Gibbs-Duhem Equation Related to ΔG and partial differential partial derivativeG/partial differential partial derivativeξ

A long-standing scientific problem concerning Delta G and partial differential G/ partial differential xi, involving the Gibbs-Duhem equation, demonstrates that wrong concepts are sometimes dominant in the literature even though corrections are readily available. This fact shows that the Gibbs-Duhem equation and especially its applications are among the most difficult topics in thermodynamics. Moreover, it demonstrates that there are often weak points in existing methods which necessitate new perspectives. There exist several different derivations of the Gibbs-Duhem equation. A novel mathematical derivation of the equation is developed here and is demonstrated to be more natural, direct, and intuitive than its thermodynamics counterpart. Different perspectives of mathematical and thermodynamic derivations are illustrated along with discussions concerning methods for solving specific scientific problems. The introduction of different self-consistent methods to solve a specific problem is vital in providing researchers with a variety of perspectives, thus, Legendre transformations are used here to introduce thermodynamic functions since this convenient and underused method is unfamiliar to many researchers in this field.

Automated summary

The study of the Gibbs-Duhem equation and its applications is one of the most difficult topics in thermodynamics. This paper presents a novel mathematical derivation that is shown to be more natural, direct, and intuitive than the traditional thermodynamic derivation. By introducing Legendre transformations, multiple perspectives for problem-solving are provided.