128
DOC. 26 THEORY OF TETRODE AND SACKUR
allow all N
systems
to
take all
possible positions
and orientations. How often do
I
have
a
realization of each distinct distribution of all the numbered
atoms
in the
gas?
I
now
think of
a
distinct distribution of coordinate
systems
in the first
one
of
the
models. When
I
now exchange
two
of
these
systems
(together
with their
atoms)
with
each
other,
I
get
a
different distribution of
atoms
insofar
as
these
atoms
are imagined
as
being
numbered.
However,
if I make
a
realization of this second distribution
system
in all
models,
I
will find
one
model
of
this second distribution with
exactly
the
same
spatial
distribution of
atoms
as
I
had in the first model of the first
system
distribution.
Consequently, every
distribution
of
atoms is
realized
N!
times in
our
"thought-experiment,"
corresponding
precisely
to
the
N!
permutations
of
the
coordinate
systems.
The number of realizations of
every
individual distribution of
(numbered)
atoms
in
space
need not be exhausted
by
this. One rather finds
more
frequent repetitions
of
distributions if the molecule has
symmetries.
If,
for
example,
there
are p positions
of
the
molecule's coordinate
system
such that fixed
points
of
atoms coincide in
pairs,
then
there
are
pN
configurations
for
p
coordinate
systems
that allow
for
the
same
distribution
of
atoms.
The
same
atomic distribution
occurs, therefore, pN
times
more
frequently
in
our thought-experiment
than if there
were
no symmetry properties
of
the molecule.
Consequently,
our
thought-experiment produces every configuration
of
atoms,
in
the
total, N!pN-fold.
Our result would be
too
large by
just
this factor if
we
could
multiply
the
phase integral (which corresponds
to
one single
distribution
of
atoms
over
all
systems) by
C =
II(Na!),
the latter
being
the individual distribution
of
atoms
over
N
systems.
The
correct
multiplier
is
therefore
II(Na!).
N!pN
[p. 11]
Instead
of
(2b)
we now
get
S
=
E_
©
lg
/
*dr
hnN\p
N*
(4)
where
the
phase integral
is
to
be taken "under
preservation
of
the molecule
compound."
This
formula does not allow for
an
immediate exact
evaluation,
because
we
have
only rough
clues about the laws
of
interaction
of
atoms within
a
molecule. But
we
can
use
it in those "normal"
areas
where the
temperature
is
so
low that the
degrees
of
freedom,
which
correspond
to the relative motion
of
atoms within the
molecule,
are
"asleep."
The
phase integral
is
equal
to the N-th
power
of
the
integral
extended
over
the
variables
of
state of
one
molecule