A method is developed to complete an incomplete set of equations of state of a thermodynamic system. Once the complete set of equations is found, in order to verify the thermodynamic validity of a system, the Hessian and entropy methods are exposed. An original approach called the completeness method in order to complete all the information about the thermodynamic system is exposed. The Hessian method is improved by developing a procedure to calculate the Hessian when it is not possible to have an expression of the internal energy as a fundamental equation. The entropy method is improved by showing how to prove the first-degree homogeneous property of the entropy without having a fundamental expression of it. The completeness method is developed giving a total study of the thermodynamic system by obtaining the set of independent T d S equations and a recipe to obtain all the thermodynamics identities. In order to show the viability of the methods, they are applied to a typical thermodynamic system as the ideal gas. Some well-known and unknown thermodynamic identities are deduced. We also analyze a set of nonphysical equations of state showing that they can represent a thermodynamic system, but in an unstable manner. The rubber band, the paramagnetic solid and the Kelly equation of state for a plasma are corrected using our methods. In each case, a comparison is made between the three methods, showing that the three of them are complementary to the understanding of a thermodynamic system.
Keywords: Hessian; entropy; equations of state; fundamental equations.