Page 190 - Chemical equilibria Volume 4

166 Chemical Equilibria
A2.1.3. Fermi–Dirac quantum statistics
Fermi–Dirac quantum statistics applies for non-localized quantum objects –
i.e. those which are indiscernible and have fraction spin (some molecules and
ions, electrons). The distribution of the objects obeys the expression:
( α βε−−
exp
g
() = 1exp − − i i ) ) [A2.10]
i
n
( α βε
+
i BE
The value of the coefficient α is just as difficult to calculate as in the
previous case.
For the two branches of statistics pertaining to non-discernible objects,
we content ourselves, as regards that coefficientα , with a limited expansion
of the following form:

N ⎛ N ⎞ 2 ⎛ N ⎞ 3
−
exp( α ) = a + a A + a 2 ⎜ A ⎟ + a 3 ⎜ A ⎟ + ... [A2.11]
0
A
1
z A ⎝ z A ⎠ ⎝ z A ⎠
By laborious calculations, it can be shown that the coefficients a i in that
expansion are:
1 1 1
a = 0 ; a = ; a =± ; a = − ; etc. [A2.12]
1
2
1
0
2 3/2 3 4 3 3/2
In the coefficient a 2, the + sign is applied for Fermi–Dirac statistics, and
the – sign for Bose–Einstein statistics.
A2.1.4. Classic limiting case

The three branches of quantum statistics (Maxwell-Boltzmann, Bose–
Einstein and Fermi–Dirac) meld into one, known as the classic limiting case,
if the following condition is met:

exp ( α− ) << 1 [A2.13]

Here, the value of α must be very high.

In these conditions, the three laws are combined in the form:
n = g exp ( α βε− − ) [A2.14]
i i i