Equivalence of thermodynamical fundamental equations

The Gibbs function, which depends on the intensive variables T and P , is easier to obtain experimentally than any other thermodynamical potential. However, textbooks usually first introduce the internal energy, as a function of the extensive variables V and S, and then proceed, by Legendre transformations, to obtain the Gibbs function. Here, taking liquid water as an example, we show how to obtain the internal energy from the Gibbs function. The two fundamental equations (Gibbs function and internal energy) are examined and their output compared. In both cases complete thermodynamical information is obtained and shown to be practically the same, emphasizing the equivalence of the two equations. The formalism of the Gibbs function is entirely analytical, while that based on the internal energy, is, in this case, numerical. Although it is well known that all thermodynamic potentials contain the same information, usually only the ideal gas is given as an example. The study of real systems, such as liquid water, using numerical methods, may help students to obtain a deeper insight into thermodynamics.