This paper explores the advantages of using relative free energy instead of exergyto build a mathematical theory of thermodynamic costs to diagnose malfunctions in thermalsystems. This theory is based on the definition of a linearized characteristic equation that representsthe physical behavior of each component. The physical structure of the system described by itsenergy interrelationships is called “primal”, and its derivatives are the costs and consumptions.The obtained costing structure is the mathematical “dual” of its primal. The theory explains whythe F and P cost assessment rules and any other suggestion may (or may not be) rational under a givendisaggregation scheme. A result of the theory is a new thermodynamic function, called the relative free energy, and a new parameter called deterioration temperature due to a component’s deteriorationcause, characterized by a h-s thermodynamic trajectory describing the effects on the exiting stream.The relative free energy function allows for an exact relationship between the amount of usedresources and the increase in entropy generation caused by the deterioration path of the component.This function allows the obtaining of, for the first time, an appropriate characteristic equation fora turbine and a new definition of efficiency that does not depend on the environment temperaturebut on its deterioration temperature. Also, costing with relative free energy instead of exergy mayopen a new path for more precise and straightforward assessments of component deteriorations.