DOC.
11
ARGUMENTS FOR MOLECULAR AGITATION
141
increasing temperature;
thus, at 102°
abs.,
v
=
11.4
.
1012;
at
189°, v
=
12.3
.
1012;
at
323°, v
=
14.3
.
1012.
This
explains why, relatively speaking,
Eucken could
still
best
represent
his
measurements
by
the
simple
Einstein formula with
a
tempera-
ture-independent
v
(curve
III, Fig.
2).
Even
so,
one sees
that
even
this formula fails
[10]
at
higher temperatures,
not to
mention the fact that without the
assumption
of
a
zero-point
energy
the
constancy
of
v
remains
totally incomprehensible.
Thus
one sees
that the
specific
heat
of
hydrogen
makes the existence of
a
zero-point energy
probable,
and
it
only
remains
to
check
to
what
degree
the
particular
value hv/2 is
to
[11]
be considered
secure.
Since in the
following investigation
on
the
radiation law it
must
be assumed that the
zero-point energy
is
hv,
we
have
calculated
the
specific
heat of
hydrogen
for this
assumption
as
well
(p
=
5.6
.
10-40,
curve IV, Fig.
2).
It is
apparent
that the
curve
is
too
steep
and too
high
at
higher temperatures.
On the other
hand,
it should be noted
that,
in
any
case,
the
curve might
turn out to
be
somewhat
flatter if the
velocity
distribution
among
the molecules
were
taken into
account.
Therefore,
this
possibility
cannot
be
definitively
ruled
out,
even though
it is
improbable
that the
zero-point
energy equals
hv.4
Derivation of the
Radiation Law
In what follows it shall be shown
how,
on
the basis
of
the
assumption
of
a
zero-point
energy,
one can
derive Planck's radiation formula in
an
unforced,
though
not
quite
4If
one assumes
that the
entropy
of
rotating
structures at T
=
0
equals
zero,
like the
entropy
of
solids
according
to Nernst's
theorem,
one
obtains for the total fraction
of
the
entropy
in
one
mole that derives from the rotation of
diatomic molecules
Sr
=
f-dT
=
fin
V
+
V°
dv
=
+
fcin py_
-
1 lT
{,
v"vo T
h
For
high temperatures we
obtain:
S
=
RlnT
+
2
R
+
R
ln^^.
h2
According
to Sackur
(Nernst-Festschrift, 1912, 414),
the
entropy
constant for rotation:
r
+
r in
167r3jk
h2
[13]
[12]
is in the
main,
namely, as regards
the
expression
J
k/h2,
identical
with
the
expression given
above.
As
it
happens, one gets
the
same
result
if
one
substitutes formula
(6),
rather than
formula
(5),
for
cr.
[14]