Thermodynamic Potential - Euler Integrals

Euler Integrals

Again, define and to be conjugate pairs, and the to be the natural variables of the internal energy. Since all of the natural variables of the internal energy U are extensive quantities

it follows from Euler's homogeneous function theorem that the internal energy can be written as:

From the equations of state, we then have:

Substituting into the expressions for the other main potentials we have:

As in the above sections, this process can be carried out on all of the other thermodynamic potentials. Note that the Euler integrals are sometimes also referred to as fundamental equations.