Ii. Linear Combinations, Lagrange Multipliers, and Legendre Transforms—A Look at Three Methods to Write the Gibbs Energy of Reactive Mixtures

A comparison of three methods to deduce a property of element potentials that allows to write the Gibbs energy expression for systems at chemical equilibrium is presented. These methods are: (1) the algebraic procedure widely used in reactive distillation modelling to obtain the , (2) the classical non-stoichiometric formulation of chemical equilibrium problems, and (3) the complete Legendre transformation of the Gibbs energy with respect to element composition variables. These three techniques lead to the same result in most cases, i.e. when elements are chosen among the system components. Because of its consistency with the rest of the Thermodynamics formalization, the latter gives deeper insight into the transformed composition variables and the auxiliary function of the Lagrange multipliers method. The element composition variables are the conjugate variables of the element potentials. The auxiliary function of the Lagrange multipliers method is the complete Legendre transform of the Gibbs energy with respect to element composition variables, and its value is zero for equilibrium states