where the upper sign corresponds to Fermi-Dirac statistics and the lower sign corresponds to Bose-Einstein statistics. The parameter is determined via

Finally, the partition function of the gas is given by

Let us investigate the magnitude of in some important limiting
cases. Consider, first of all, the case of a gas at a given temperature
when its concentration is made sufficiently low: *i.e.*, when
is made sufficiently small. The relation (618) can only
be satisfied if each term in the sum over states is made
sufficiently small; *i.e.*, if
or
for all states .

Consider, next, the case of a gas made up of a fixed number of particles
when its temperature is made sufficiently large: *i.e.*, when is
made sufficiently small. In the sum in Eq. (618), the terms of
appreciable magnitude are those for which
.
Thus, it follows that as
an increasing number of
terms with large values of contribute substantially to this
sum. In order to prevent the sum from exceeding , the parameter
must become large enough that each term is made sufficiently small: *i.e.*,
it is again necessary that
or
for all states .

The above discussion suggests that if the concentration of an ideal
gas is made sufficiently low, or the temperature is made sufficiently high,
then must become so large that

for all . It is conventional to refer to the limit of sufficiently low concentration, or sufficiently high temperature, in which Eqs. (620) and Eqs. (621) are satisfied, as the

According to Eqs. (617) and (620), both
the Fermi-Dirac and Bose-Einstein
distributions reduce to

(622) |

The above expressions can be combined to give

(624) |

Let us now consider the behaviour of the partition function (619)
in the classical limit. We can expand the logarithm to
give

(625) |

(626) |

(627) |

(628) |

(629) |

(630) |

In the classical limit, a full quantum mechanical analysis of an ideal gas reproduces the results obtained in Sects. 7.6 and 7.7, except that the arbitrary parameter is replaced by Planck's constant .

A gas in the classical limit, where the typical de Broglie wavelength of the
constituent particles is much smaller than the typical inter-particle
spacing, is said to be *non-degenerate*. In the opposite limit,
where the concentration and temperature are such that the typical
de Broglie wavelength
becomes comparable with the typical inter-particle spacing, and the actual
Fermi-Dirac or Bose-Einstein distributions must be employed, the
gas is said to be *degenerate*.