Page 78 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

64 5 Perfect gases

m α /M. Thus it appears that we might require

2π¯h V 1/3 m α
2 1/2
λ = (5.20)
Mk B T N M
where m α is the lightest particle in the molecule. This condition would be very
stringent for electrons on molecules! However, when we consider the integrals on
r α , r α which enter the relevant term in the partition function, we see that the term

which must be neglected is proportional to

2 2
−2π(m α /M)| r α − r α | λ ∗ ∗
d r α d r α d {r} i d {r} e φ n i ( r α , {r} )φ ( r α , {r} )φ ( r α , {r} )φ n ( r α , {r} )

n
i
i
i
i
i
n i
i i
(5.21)
Here {r} means all the coordinates on i except r α or r α and {r} means all the

i i
coordinates on i except r α or r α . The local wave functions φ will overlap very

little in dilute gas and so, in particular, the terms involving exchange of electrons
will be small long before the condition (5.20) is satisfied. On the other hand, it is
true that both effects, associated with the momentum averaging and with spatial
averaging, indicate that other things (such as the strength of the binding of the
particle to the molecule) being equal, the lightest particles in the molecule will be
the easiest to exchange because the wave function overlaps will shrink exponentially
√
with m (as one can see from the WKB approximation, for example).
α
On the basis of these arguments we neglect terms in the partition function in
which one permutation of the particle labels occurs on one side of the matrix ele-
ment and another permutation, in which exchange of particles between molecules
has occurred relative to the permutation on the left hand side, occurs on the right
hand side. In this way one is grouping the terms involving different permutations of
the coordinates in (5.15) as follows. Start with a given assignment of particle num-
bers to molecules and add all permutations of labels within each molecule. Now
add all terms in which all the labels associated with one molecule are interchanged
with all the labels associated with another. Finally add all permutations resulting
in the assignment of different coordinate labels to the molecules and similarly
permute the coordinates in each assignment, first within the molecules, and then
interchange the labels of all the coordinates of each molecule for each such assign-
ment. Now the approximation to be made consists of two aspects. The overlaps of
terms involving different particle assignments to a given molecule can be ignored
as long as the range of the local wave functions φ is much less than the mean
distance between molecules. This criterion involves the temperature, because the
relevant molecular wave functions will have larger size for larger energies (and at
high enough energies the molecule will not be bound at all). Thus this aspect of
the approximation requires that the temperature be much less than the molecular