Page 189 - Chemical equilibria Volume 4

Depending on the nature of the molecule-object, three branches of
statistics can be applied. Appendix 2 165
A2.1.1. Maxwell–Boltzmann statistics

Maxwell–Boltzmann statistics is applicable to objects for which there is
no need to draw on quantum mechanics – i.e. objects which are relatively
large and discernible. This type of statistics is also applicable for discernible
or localized quantum objects, such as the molecules distributed at the nodes
of a crystalline lattice.

The coefficient α in relation [A2.5] is given, in this case, by:

N
exp ( α− ) = [A2.7]
∑ g i exp − i )
( βε
i
The distribution law becomes:

Ng exp ( βε− )
n = ∑ g i exp − i ) [A2.8]
i
( βε
i i i

β is still defined by relation [A2.6].

A2.1.2. Bose–Einstein quantum statistics

Bose–Einstein quantum statistics applies for non-localized quantum
objects – that is, objects which are indiscernible and have integer spin (such
as most molecules and ions, atoms). The distribution of the objects obeys the
expression:

() = g i exp ( α βε−− i ) [A2.9]
n
( α βε
1exp − − i )
−
i BE
The value of the coefficient α is difficult to determine. We shall come
back to this later on (relation [A2.11]).