Page 79 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

Molecular ideal gas 65

binding energy. On the other hand, if we wish to treat the centers of mass of the
molecules classically, then the thermal wavelength associated with the molecular
mass must be much less than the distance between molecules so that the effects
of permutations of entire sets of coordinates between molecules can be ignored.
This second requirement puts a lower bound on the temperatures where the ap-
proximations are valid, while the first requirement puts an upper bound on the
temperature.
If we have N particles, combined into M molecules so that there are n = N/M
coordinates associated with each molecule, then the three varieties of permutations
discussed above are (n!) M permutations of the internal coordinates, M! permuta-
tions of all the coordinates of each molecule with all the coordinates of each other
M
molecule and N!/(n!) M! assignments of coordinate labels to the molecules. We
have been saying that we can ignore cross terms associated with the last kind of per-
mutations as long as the temperature and density are low enough so that the range
of all the molecular wave functions is much less than the intermolecular distance.
We can ignore cross terms associated with the M! permutations of all the coordi-
nates of one molecule with all those of another as long as the temperature is high
enough so that the molecular thermal wavelength is much less than the intermolec-
ular distance. We cannot ignore the permutations associated with internal degrees
of freedom of the molecules. (Molecules whose internal dynamics is classical are
are already appropriately symmetrized
not known to exist.) Assuming that the φ n i
or antisymmetrized with respect to permutations of labels within a molecule, one
can work with one assignment of particle labels to molecules because each of
the (n!) M terms associated with different assignments will give the same result in
the partition function and cross terms are ignored. Thus one can work with the wave
function

M
1
= √ e i k i · R P (i) ({q} i ) (5.22)
{k i M/2
ψ },{n i } φ n i
M!V
P i=1
in which the sum on permutations only includes those in which all the coordinates
assocated with one molecule have been interchanged with all the coordinates asso-
ciated with another. The factor η in (5.15) has been replaced by M! assuming that,
in the semiclassical limit which we consider, no plane wave state associated with
the motion of the center of mass of the molecules is macroscopically occupied. We
may now calculate Z by changing sums into integrals as in the discussion of the
semiclassical limit, adding a factor M! to take account of the fact that a new state
is not produced by permuting the {k i }. Then we have

M
V M −β ¯h 2 k 2 i
V M
M
Z c = d k e i 2m +
n i = e −β
n i (5.23)
M! M!λ 3M
{n i } i=1 n i