EN General thermodynamic relations for thermal variables formed by first-order partial derivatives and not mixed second-order derivatives

The theory of differential equations is a powerful tool necessary for constructing mathematical models of various applied problems and their solution. The study and analysis of the general differential relations between the derivatives of the thermodynamic functions of substances is a new direction in the theory of their thermodynamic properties. This research is the development of thermodynamics associated with the establishment and use of differential equations. This paper presents two new thermodynamic relationships between thermal variables pressure, absolute temperature, and specific volume. They relate first-order partial derivatives with unmixed second-order derivatives for these variables. Revealing new general relations of thermodynamics made it possible to formulate and solve a number of problems, among which we can note: creation of algorithms for calculating second-order mixed derivatives for a given thermal equation of state; analysis of the features of the second mixed derivatives at the critical point of pure substance; investigations of the general thermodynamic equations with entropy. The procedure for establishing new equations is based on the use of a rigorous mathematical apparatus and Maxwell's differential equations. Therefore, they are of a general nature, that is, they are valid for any pure substance or any mixture of fixed composition. They are also applicable to any model of the equation of state of these substances. With a formal approach to the concept of the term "equation of state", the new relations themselves can be considered as the equation of state of substance. The obtained new relations can be used to develop the theory of the thermodynamic properties of substances. In particular, in this work it is shown that the ratios of two partial derivatives of pressure over the volume, which are zero critical conditions, have no singularities at the critical point itself