DOC.

26

THE PROBLEM OF SPECIFIC HEATS

403

ics.

This

one

result

already

shows

that molecular

mechanics

cannot

yield

correct

specific

heats

for solid

bodies-at

least not at low

temperatures. Further,

the

laws

of

dispersion

led to

the

conclusion

that instead of

consisting

of

only one

material

point,

the

atom

may

possess electrically

charged

material

points (polarization electrons)

that

move

indepen-

dently

of the

atom

as a

whole and

which-statistical

mechanics

notwithstanding-make

no

contribution

to

the

specific

heat.

[3]

We

were

not in

the

position

to

relate these

inconsistencies

of the

theory

to

other

physical

properties

of

matter

until

a

few

years ago,

when Planck's

investigations

on

thermal radiation

quite unexpectedly

shed

new light on

this area.1

Though

we

have not

yet come

to

the

point

where

we

need

to

supplant

classical mechanics with

a

mechanics

that

would

be

able to

yield

correct

results for

fast

thermal

oscillations

as

well,

still

we

have

found the

law from which

the

deviations from

the

Dulong-Petit

law

follow,

and

we

learned that these

deviations

are

related

by

law to

other

physical

properties

of the

substances.

In

what

follows,

I

shall outline the

train of

reasoning

in

Planck's

investiga-

tions in

a manner

that

will

bring

out

clearly

the connection

with

our

problem.

It

is

possible

to arrive at

a

theory

of the law of

cavity

radiation

at

thermal

equilibrium

(the

law

of

black-body

radiation)

by doing

a

theoretical

analysis

to

determine

the

density

and

composition

at

which

the radiation

is

in statistical

equilibrium

with

an

ideal

gas, given

the

presence

of

structures

that

make

an

energy

exchange

between the radiation and the

gas possible.

One

such structure

is

a

material

point

bound

to

a

point

in

space by

forces

proportional

to its

displacement

from this

point (oscillator); we

shall

assume

that the

material

point

of

the

oscillator is

provided

with

an

electric

charge.

Let thermal

radiation,

an

ideal

gas,

and

oscillators

of the

kind

indicated

be

enclosed

in

a

volume bounded

by

perfectly reflecting

walls.

By

virtue of their electric

charges,

the

oscillators must

emit

radiation and

continually

receive

new

momentum from

the radiation

field.

On the other

hand,

the material

point

of

the individual

oscillator

collides with

gas

molecules

and in this

way exchanges energy

with

the

gas.

The

oscillators

thus

bring

about

an

energy

exchange

between the

gas

and

the

radiation,

and the

energy

distribution of the

system

in

the

state

of

statistical

equilibrium is

completely

determined

by

the total

energy,

if

we assume

that

oscillators

of

all

frequencies are present.

In

an

investigation

based

on

Maxwell's

electrodynamics

and

on

the

mechanical

equations

for the motion of the material

point

of the

oscillator,

Planck has

now

shown

that-assuming

that

only

oscillator

and

radiation

are

present,

but

not

the gas-the

following

relation

exists

between the

mean

kinetic

energy

Ev

of

an

oscillator

of

frequency

v,

and

the radiation

density

uv2

1

M.

Planck,

Vorl.

über

d.

Theorie

der

Wärmestrahlung, pp.

104-166.

2

We

assume

here

an

oscillator with

three

degrees

of

freedom.

[4]