DOC.
21
MOLECULAR MOTION IN SOLIDS
373
behavior of the
substance,
and
thus also
the
proper
frequency,
will
then
completely
be
determined
with
the addition of
a
further characteristic
quantity
of the
substance
that
is
not
determined
by
the
two
quantities
mentioned
above. As this
third
quantity
we
choose
the
melting point
Ts.
Of
course,
the latter
cannot
immediately
be
applied
in
the
dimensional
argument,
because
it cannot
be measured
directly
in
the
C.G.S.
system.
Instead of
Ts,
we
therefore choose the
energy quantity
x
=
RTs/N as
the
measure
of
temperature,
t
is
one-third of the
energy
that
an
atom
possesses
at
the
melting point
according
to
the kinetic
theory
of heat
(R
=
gas
constant,
N
=
number of
atoms in
a
gram-atom).
The dimensional
argument yields
immediately
v
=
C
N
-
=
C R^N1
md2 \
s
MvV3
=
C
•
0.77
•
1012
\| Mvm
The Lindemann formula reads
v
=
2.12
•
1012
N
T
Mvm
Thus,
the dimensionless
constant C
is
here
also
of the
order
of
magnitude
one.
The
investigations
by
Nernst and
his students8 show
that,
even though
it
is
based
on
a
very
daring
assumption,
this
formula
yields a surprisingly good
agreement
with
the
v
values
determined from the
specific
heat. From
this it
seems
to follow
that the
law
of
corresponding
states holds in
remarkably
good approximation
for
simple
bodies
in
the
solid
and
liquid
states.
It
even
appears
that Lindemann's
formula holds
much better than
my
formula, which rests
on a
less
daring
assumption.
This
is
all
the
more
remarkable
because
my
formula
also,
of
course,
can
be deduced
from
the
law
of
corresponding
states.
If both
formulas,
mine
as
well
as
Lindemann's,
are
correct,
then
it follows
from the
division
of the
two
formulas that
M/pTsk
must
be
independent
of the
nature
of the
substance;
in
fact,
this
relation
can
also
be deduced
directly
from
the
law
of
correspond-
ing
states.
However,
if
one uses
Gruneisen's
values9
for the
compressibility
of
metals,
one
obtains
values
for
this
quantity
that fluctuate
roughly
between
6.10-15
and
15.10-15!
In
view
of the
fact
that the
law
of
corresponding
states holds
up so
well
in the
case
of
Lindemann's
formula,
this
is quite
peculiar. Might
it not
be
possible
that
systematic
errors
still lie
hidden
in all
determinations of the
cubic
compressibility
of metals?
Compression
under
equal pressure
from
all sides has not
yet
been
applied
for the
purpose
of
measurement,
probably
because of the considerable
experimental
difficulties
[19]
[21]
[23]
8 Cf.
especially
W.
Nernst,
Sitzungsber.
d.
preuss.
Akad.
d. Wiss. 13
(1911):
311.
9
E.
Grüneisen,
Ann.
d.
Phys.
25
(1900):
848.
[20]
[22]