334
DOC.
13
ELASTIC BEHAVIOR AND SPECIFIC HEAT
(2)
1
26
N
v
=
a
•
2tt^ 3
M
and
[9]
(2a)
A.
=
2ite
3_M_
26
aN
Based
on
the
same
approximative assumptions,
we now
calculate
the
coefficient
of
compressibility
of the
substance.
To
this
end,
we express
in two
different
ways
the
work
A
that
must
be
applied
in
uniform
compression,
and
set
the
two
expressions
equal to
one
another.
The
work that
must
be
applied
to
reduce the
distance
between
two
neighboring
molecules
by
A is
(a/2)
A2.
Since
each molecule
has 26
neighboring
molecules,
the
work
to
be
applied
to
reduce
its distance from
the
neighboring
molecules is
26.
(a/2)
A2.
Since
there
are
N/v
molecules
in
a
unit
volume,
and
each
term
(a/2)A2 belongs
to two
molecules, we
obtain
[10]
.
26
N".2A2.
A
=
--a
4
v
On
the other
hand,
if
k
denotes the
compressibility,
and
B
the contraction of the
unit
volume,
then
A
=
1/2k
.
B2,
or,
since B
=
3
A/d,
A
=
9
A
2k.d2
Equating
these
two values
of
A,
we
obtain
(3)
18 v
K
=
26 a-(d2.N
Eliminating
a
and d from
equations
(1), (2a),
and
(3) we
obtain
[11]
/r\l/3
=
2n
M1/3
p1/6
=
l.08.l0M1/6.
s/6
N1/3
Of
course,
the formula
assumes
that
no
polymerization
takes
place.
In
what
follows,
I
used
this
formula
to calculate
the
proper
wavelengths (as
a measure
of
proper
frequencies)
of those metals
for which
the
cubic
compressibilities were
determined
by
Grüneisen.3
These
are
the
results:4
[12]
3 E. Grüneisen, Ann.
d.
Phys.
25
(1908):
848.
4
The
temperature dependence
of
cubic
compressibility
has
been
neglected.