DOC.
12 THERMODYNAMIC DEDUCTION 149
SS
=
±
-
EH~•Ta
r
T
Equation
(2)
becomes
on
account
of
(a), (b),
(b')
and
(c):
[8]
(2a)
EH
|
-
T
-
R
log
n
+ j
"
yr
+
R
log
P
-
R
log
P'=
0.
e e
This
equation
should also hold in the
special
case
when
a
=
1.
In that
case
(a)
reduces
to
(1),
which is
to
say
that the
state
becomes
one
of
proper
thermodynamic
equilibrium.
Then
Ts'
=
T and p'
=
p.
From this it follows that the first
term
of
(2a)
must vanish;
one
thus obtains
EShA
a'X
-
^
-
R
log
n
=
0,
which
equation
is
none
other than
the familiar condition for thermochemical
equilibrium
between ideal
gases.
In
consequence, equation
(2a)
becomes
log
p'
=
JL
+
log
p.
RT'
RT
On
account
of
equation (2),
the second
term is
a
constant
for
a
given temperature
of
the
gas
and
a
given frequency.
Hence it is
independent
of
p'
and
T's.
The
same
must
be
true
for the first
term,
hence
e
[9]
(3)
p'Ae.
This
equation
shows that
according
to
the
hypotheses
we
have
formulated,
the
temperature dependence
of
monochromatic radiation
must
obey
Wien's
law,
according
to
which
Q
Z.
3
-HVN
(3a)
p
=
8rftv3
g
c3
We know
that,
for
a
given frequency,
Wien's formula holds
only
for
sufficiently
weak
radiation.
Thus,
we see
here that
our
hypotheses
are
not
valid for
arbitrary
radiation densities.
But the fact that
our
analyses yield, on
the
one
hand,
the familiar
formula
for the
thermochemical
equilibrium and,
on
the other
hand,
the Wien radiation
formula,
shows that
at
sufficiently
low radiation densities
they
lead
to
results that
are
in accord
with
observations.
A
comparison
of
equations
(3)
and
(3a)
shows that
(4)
e
=
Nhv,
where N
is
the number
of
molecules in
a
gram-molecule
and
h
the well-known Planck
constant.
This
equation expresses
the law of
photochemical equivalence,
which has
already
been
deduced earlier from the
quantum hypothesis.
[10]