372
DOC.
21
MOLECULAR MOTION
IN
SOLIDS
v
=
C
d
^
m k,
where
C
is
again
a
dimensionless numerical
factor.
Substituting
the molecular
volume
v
for d
(d
=
\jv/N)
and the
so-called
atomic
weight
M
for
m (M
=
N
m),
one
obtains
1111
_
i
i i
[14]
v
=
CN\~6M
V5
=
C
1.9
•
107M
where
p
denotes the
density.
The formula
I
found
by means
of
a
molecular-kinetic
argument,
1 1 1
[15] A.
=1.08
•
lO'M'pV,
or
v
=
2.8
•
107M~VV5,
agrees
with this
formula
with
a
factor C
whose
order of
magnitude
is
one.
The
numerical factor obtained from
my
earlier
argument is
in
satisfactory
agreement
with
experiment.6 Thus,
the
value
for
copper
is
[16]
v
=
5.7
•
1012,
when
calculated
from
the
compressibility by
means
of
my
formula,
and
[17]
v
=
6.6
•
1012
when
calculated from
specific
heat
using
the formula of
Nernst,
discussed in
§2.
However,
this value
of
v
is not to be conceived
as
the
"true
proper
frequency."
We
only
know
about the latter that
it lies
between Nernst's
v
and half
of this value.
In the
absence
of
an
exact
theory,
the
most
logical
thing
to
do
is
to view
v
+v/2
as
the "true
proper
frequency,"
from which value
one
obtains,
for
copper,
according
to
Nernst
v
=
5.0
• 1012,
which
is
in close
agreement
with
the
value
calculated
from
the
compressibility.
Let
us now
turn
to Lindemann's
formula.7
We
assume again
that, above all
else,
the
mass
of
an
atom
and the
distance
d between
two
adjacent
atoms
influence the
proper
frequency.
Besides
that,
we assume
the
existence
of
a
law
of
corresponding
states
for the
solid
state,
whose
degree
of
accuracy
is
sufficiently good
for
our
present purposes.
The
6
Regarding
the
degree
of
approximation
with
which this
formula
holds,
cf.
the
last
paragraph
of
this
section.
[18]
7
F.
Lindemann,
Physik.
Zeitschr.
11
(1910):
609.