We consider a simple pure substance under hydrostatic conditions
described by the following fundamental equation:

where the extensive variables *U*, *V* and *N* are the internal
energy, the volume, and the number of particles respectively, and the
intensive variables *T*, *p* and are the temperature, the
pressure and the chemical potential respectively.

Equation () corresponds to the choice of the variables *U*,
*V* and *N* as independent variables of the entropy *S*(*U*,*V*,*N*). These
variables are precisely those which are fixed and determine the
macrostate of the members of the Microcanonical Ensemble and
consequently *S* is the relevant potential in this statistical
ensemble.

It is useful to define the following quantities: ,
and so that Eq. () can
then be written in the dimensionless form:

In general, for other thermodynamic systems with degrees of
freedom, one will have:

where are extensive variables, and the corresponding
entropic conjugate variables. Massieu-Planck functions
[] are entropic thermodynamic potentials defined as
Legendre transformations of the entropy. In the case of a pure
substance, the following (dimensionless) potentials can be formally
defined:

The function was first introduced by Massieu
[], and it is called Massieu's potential. The function
was introduced by Planck[] and is called Planck's.
potential.

Given the extensivity of , and using Euler's theorem for
homogeneous functions, it is easy to see that . Therefore
the Legendre transformation of all variables redefines the entropy,

Substituting Eq. () into the differentials of the potentials
defined above one gets:

From Eq. () one obtains:

The above equations allow a re-derivation of all the standard
thermodynamic equations in terms of , and . For
instance, Maxwell relations can be deduced, by imposing that the
equations ()-() are exact differentials (equality of
crossed derivatives). Moreover, Eq. () is the Gibbs-Duhem
equation which states that the complete set of intensive variables of
the system are not all independent. On the other hand, the extremal
condition of leads us to deduce that , and
are homogeneous at equilibrium [].

Tue Oct 24 11:27:45 CEST 2000