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.