Page 195 - Chemical equilibria Volume 4

Appendix 2 171
A2.3.1. Canonical set
A canonical set is a set composed of replicas of the system under study.
Each element is enclosed, so the number of molecules N is identical in all the
elements of the set. That number is constant, because there is no exchange of
matter between the elements and between the elements and the outside of the
set. The volume V is the same for all the elements. The elements are in
thermal contact with one another, and are therefore able to exchange energy.
Their temperature is identical T. Each element has an energy level E j. The
total energy of the canonical set will be E C. That energy is constant, because
the set is insulated from the outside world.

A2.3.2. Canonical partition functions

In the same way as for molecules, we define the partition function for the
canonical set by the sum:

C ∑
Z = exp ( β− E j ) [A2.32]
j

This sum is extended to all the elements of the set.

A2.3.3. Canonical partition function and molecular partition
functions

The canonical partition function is linked, firstly to the molecular
canonical functions, and secondly to the thermodynamic functions that
define the phase on the macroscopic level (U, F, G, S, etc.). These two types
of relation mean that the canonical partition function forms the link between
the microscopic definition of the phase and its macroscopic thermodynamic
properties.

In order to calculate the canonical partition function on the basis of the
molecular functions, we distinguish two cases, depending on whether the
molecules are discernible or indiscernible.

A2.3.3.1. Case of sets of discernible molecules
If the molecules are all identical and discernible, we can show that the
following expression can be used: