Canonical ensemble

Let us assume that the reservoir temperature does not change. The probability that the system has a given energy is given by[1] :

The above is called Bolzmann distribution. The quantity is called partition function and it's given by:

Thus, the mean value of the energy in such an ensemble is given by:

In such an ensemble the Helmoltz free energy takes the role of a thermodynamic potential:

We do present here some useful relations that involve and its differential form:

Using and it can be proved that

That is the equation of state for ideal gases. The partition function of a single particle can be written as[2]:

The value is called quantum density, i.e. the total number of accessible states per unit of volume. We do define the DeBroglie thermal wavelength as the of a free particle with kinetic energy given by : . Hence we can rewrite as: . Using we can find the expression of the Sackur-Tetrode entropy:

All these considerations hold for non-degenerate system, that is for systems where : meaning that we do assume that the number of particles is way smaller than the number of allowed energy states.

Gibbs free energy[edit | edit source]

This quantity is defined as the Legendre transformation of within respect to . Hence:

Some important relations involving and its differential form:

  1. Usually the term is called , and so
  2. In the derivation of this relation, is the dimension of the discrete element of space where the w.f. is constrained.