138

DOC.

11

ARGUMENTS FOR MOLECULAR AGITATION

varied

arbitrarily, independently

of the

temperature.

The

dependence

of the

energy

on

the

frequency

at

constant

temperature

would

depend considerably

on

the existence

of

a

zero-point energy. Unfortunately,

we

have

no

experience

with

structures

of

this

kind. But

we are

familiar with

structures-namely,

rotating gas

molecules-whose

thermal motions exhibit

a far-reaching similarity1

to

those

of monochromatic

structures,

and for which the

mean frequency

varies with

temperature.

Thus,

the

justification

for

assuming

a

zero-point energy should,

first of

all,

be checked in the

case

of

these

structures.

In

what follows

we

will first

investigate

to

what extent

conclusions about the theoretical

behavior

of

such

structures

can

be drawn from

Planck's formula.

The

Specific

Heat of

Hydrogen

at Low

Temperatures

The

question

at issue

is how

the rotational

energy

of

a

diatomic molecule

depends

on

the

temperature. By analogy

with the

theory

of the

specific

heat of

solids,

we

are

justified

in

assuming

that the

mean

kinetic

energy

of rotation does

not

depend

on

whether the

molecule does

or

does

not

possess an

electric

moment

in the direction

[4]

of

its

symmetry

axis. If

the

molecule does

possess

such

a

moment,

it must

not

disturb

the

thermodynamic equilibrium

between

gas

molecules and radiation. From this

we

can

conclude that

the kinetic

energy

of rotation which the molecule

acquires

under

the influence of radiation alone

must

be the

same

as

that which it

acquires through

collisions with other molecules.

Thus,

the

question

is,

for what

mean

value of

rotational

energy

will

an

unreactive,

rigid dipole

be in

equilibrium

with radiation

of

[5] a

specific temperature.

Whatever the laws of radiation

might

be,

one

will still have

to

keep

the

assumption

that

a rotating dipole

radiates

twice

as

much

energy per

unit

time

as a

one-dimensional

resonator,

for which the

amplitude

of

the electric and

mechanical

moment

equals

the electric and mechanical

moment

of the

dipole.

An

analogous assumption

shall hold for the

mean

value

of

the absorbed

energy. If,

for

the sake

of

simplicity,

we

also

assume

that,

approximately,

all

dipoles

of

our gas

rotate

with the

same speed

at

a

given temperature,

we

will be led

to

the conclusion

that

at

equilibrium

the kinetic

energy

of

a

dipole

must

be twice

as

great

as

that

of

a

one-dimensional

resonator

of

the

same

frequency.

Based

on

the

assumptions

that

we

have

made,

we can

use

expressions

(1)

and

(2)

to

calculate

directly

the kinetic

energy

of

a

gas

molecule

rotating

with two

degrees

of

freedom,

where at each

temperature

E and

v are

related

by

the

equation

[3] 1The

first to draw attention to this

was Nernst,

cf.

Zeitschr.

f.

Elektroch.

17

(1911):

270,

825.