DOC.
5
CONTRIBUTIONS TO
QUANTUM
THEORY
21
corresponding
resonator
energies. Energy
U
and
entropy
S
of the mixture
are
then
given by
the
expressions
tJ
=
2
Wo {c
T
+11^
+
sa}
a
S
=
2
tia
{c
lg
Tf B
lg
F)
f
2
na
{s"

R
lg
«"•}.
[5]
The
specific
heat
c
(at
constant
volume)
per
molefollowing
the idea sketched
aboveis
to
be taken
at
constant resonator
energy
ea,
which
is
the
same
for all
components.
sa
is the
entropyconstant
of the
gas type
with resonator
energy
ea;
and
[6]
this constant
can
a priori
have different values for each
a.
Next
we
have
to
form the
free
energy
F
=
U

TS
and have to
state
the
condition that for
every
reaction to be
considered
SF
=
S
(UTS)
=
0.
We take into account the
totality
of
possible
resonator reactions
by
considering
for
each
v
the reaction
Sn0 = 1
Sn0
=
+1
.
In this
manner one
obtains the
system
of
equations
^So
H~
?p
 
0
[7]
or
M t
'
'v
eo~e0
no
__
(»a
»«)
YT~
n0
1)
This is the
equilibrium
distribution
we were
looking
for,
where
s'a
=
sa/R.
[p. 822]
Let the resonator under consideration
now
be
a
monochromatic
one
with
one
degree
of freedom and with
frequency v.
In order
to
arrive
at
Planck's
formula for
the
mean energy
of
such
a
structure,
we
have
to
introduce two
hypotheses:
1.
The
entropy
constants
of all
components
of
our
mixture
are
equal
even
though
the
components
differ in
resonator
energy;
i.e.,
for
all
a
S0
=
S0.
This
precondition corresponds
to
Nernst's
theorem.
[8]
2.
The
resonator
energy (per
mole)
is
an integral multiple
of
Nhv:
sa
=
asNhv.
This is the
quantum hypothesis
for
a
monochromatic structure.
Based
upon
these
hypotheses we get
ohv
na
=
n0e KT,
1a)
from which follows