DOC.
5
CONTRIBUTIONS TO
QUANTUM
THEORY
23
On the other
hand,
this derivation is also useful for
a
better
understanding
of
Nernst's
theorem, as can
be concluded from the fact that the derivation
of Planck's
formula
required hypothesis 1.
In order to
get a
better
understanding
of this
connection,
we
shall
try
to
extend
our
analysis
to structures
of
more
than
one degree
[p. 824]
of
freedom. How would he
reason
if
the resonator of
energy
ea
would have two
degrees
of
freedom?
In the derivation
of
1),
it
was
completely
immaterial
how
the
structure
of
energy
ea
was constituted; therefore,
this
equation
still is to be
kept.
In
a
similar
manner one
could
keep hypothesis
2.
If
we
also base ourselves
upon
hypothesis
1, we
again get equation 2)
for the
mean energy,
that is to
say, only
half
of what is correct for the two-dimensional
resonator.
In
order
to obtain
the correct
result
here,
the
entropy
constants
of
mixture
components
characterized
by
different
ea
can no longer
be considered
equal.
This is
immediately
understood
if
we
replace
the monochromatic resonator of
two
degrees
of
freedom
by
two resonators of
one degree
of
freedom
each.
The resonator
energy
em
is then
to be
taken
as
eat
=
(a
+
T)
hv.
We obtain the correct value for
mean energy
if
we
always
assume
the two kinds
of
molecules
a,r
and
a',r'
as separable
unless
a
=
a',
r
=
r',
and the
components
of
the mixture
satisfy hypothesis 1.
We then
get
»at-
v
not
__
-
BT
BPdT
n0Q
,1
t
(g +
*)
d
lgSSe
**
I
at
=
2N
hv.
2a)
e**
-
1
That
hypothesis
1
as
an
equivalent
to
Nernst's
theorem must not be used
as a
basis
when the
energy
carrier
ea
has two
degrees
of
freedom
and,
if
the
state
of the
molecule is characterized
only by
the
energy
ea
(without
regard
to
how this
energy
is distributed between the
degrees
of
freedom),
is
probably
connected
with
the
following: Hypothesis 1
is
permissible
if
and
only
if
the
state of
the mole-
cule-denoted
by
index
a
in
3)-is
completely
characterized
by
quantum-theoretic
[14]
standards
such
that it
can
be realized in
only one
way.
In this
case
the correct
[15]
distribution
law is
%
- £o
JO*
_
e
ät1
t
1a)
n0
If
we
limit ourselves
to
the
case
of
only
discrete
possibilities
of realization of
the
"inner state"
of
the
molecule,
to which
ea
refers,
then
we
have
to
stay
with
1a);
assuming
one
chooses for each
possibility
of realization
a
separate
index
(resp. a
[p. 825]