390 DOC. 38 THEORY OF

SPECIFIC

HEAT

Published

in Annalen

der

Physik

22

(1907):

180-190.

Dated

Bern,

November

1906, re-

ceived

9 November

1906,

published

28 Decem-

ber

1906

(for January 1907).

[1]

Einstein

1905i

(Doc.

14)

and

Einstein

1906d

(Doc. 34).

The second work

was pub-

lished

in

1906.

[2]

Einstein

had looked

for such

a

link in

1901.

See Einstein

to

Mileva

Maric,

23 March

1901

(Vol.

1,

Doc.

93).

[3]

Evidently,

this

note

is

a

reference to

p.

181,

fn.

1.

[4]

This condition

assures

the

validity

of

Liou-

ville's

theorem for this

system.

See the

editorial

note,

"Einstein

on

the Foundations

of

Statistical

Physics,"

p.

52.

[5]

Einstein

1903

(Doc. 4).

[6]

This

equation

is

equivalent

to Einstein

1903

(Doc. 4),

p.

176,

eq.

(3).

[7]

Planck

1900a.

[8]

This formula does not

appear on p.

99

of

Planck

1900a.

It

can

be derived from

equations

on pp.

99 and 111. Planck first

published

the

equation

in

Planck

1900e,

p.

241.

[9]

When

expressed as

px,

the

energy density

per

unit

wavelength,

this

is

now

called the

Ray-

leigh-Jeans

law

(see Rayleigh

1900, 1905a,

1905b;

Jeans

1905a,

1905b).

Planck

1906c,

p.

159,

calls it the

"Rayleigh

radiation

law"

("Rayleighsches Strahlungsgesetz").

[10]

Einstein

noted this in Einstein 1905i

(Doc.

14), p.

136.

[11]

These

sections

of Planck

1906c include

the derivation

of

the distribution law. A

common

feature

is

Planck's

division

of

energies

distrib-

uted

over

individual oscillators

into

integral

multiples

of

hv. For

Einstein's

review

of

Planck

1906c,

see

Einstein

1906f

(Doc. 37).

[12]

The

procedure

outlined in the

next

few

sentences

is

similar

to

Einstein's

procedure

in

Einstein

1906d

(Doc. 34).

[13]

A factor

of

E

is

missing

from the

exponen-

tial in the denominator

of

the first

equality.

[14]

As in Einstein 1905i

(Doc.

14)

and Ein-

stein

1906d

(Doc. 34),

ß

corresponds

to Nh/R

=

h/k,

where

h

and

k

are

Planck's and Boltzmann's

constants,

respectively.

[15]

Planck had obtained

an equivalent

result in

Planck

1906c,

p.

157,

eq.

(231),

from his

expression

for

the

radiation's

entropy

S,

eq.

(227),

and the relation dS/dU

=

1/T,

where U

is

the

average

oscillator

energy.

[16]

For further discussion

of

this

assertion,

see

Einstein 1907h

(Doc.

45),

pp.

372-373.

[17]

A model

of

this

type

is

used in Boltzmann

1871b and in Boltzmann

1876,

pp.

556-559, for

calculating

the

specific

heat of

a

monatomic

solid. The results

are equivalent

to the

expres-

sions

given

here

near

the

top

of

p.

185.

[18] According

to the

Dulong-Petit rule,

for

a

number

of

solid monatomic

elements the

prod-

uct

of

the atomic

weight

and

the atomic

specific

heat is

a

constant.

Neumann

extended the Du-

long-Petit

rule to

compounds (Neumann,

F. E.

1831):

for

chemically

similar

compounds,

the

product

of

the

molecular

specific

heat and the

molecular

weight

is

a

constant.

According to

the

Kopp

rule

(Kopp

1864),

each

element contrib-

utes the

same

amount

to

the

specific

heat

of

a

compound as

its

specific

heat

as

a

free element.

Using

this

rule, one

can

calculate the molecular

specific

heat

of

a

substance from the atomic

spe-

cific heats

of

its constituents. For

a

brief

state-

ment

of

the

Kopp

rule,

see p.

188,

where it

is

called the F.

Neumann-Kopp

rule. For

a

contem-

porary

discussion

of

these

rules,

see

Winkel-

mann

1906a. For

an

earlier

reference to the

rules,

see

Einstein

to

Mileva

Maric,

23 March

1901 (Vol.

1,

Doc.

93).

[19]

H. F. Weber observed this

phenomenon

for the three elements in

question

in

Weber,

H.

F. 1875. Einstein

may

have learned

of Weber's

results while at the ETH. See Vol.

1,

the edito-

rial note,

"Einstein

as a

Student

of

Physics,

and

His Notes

on

H. F.

Weber's

Course,"

pp.

60-

62.

[20]

These assertions

are supported by

data

given

in

Landolt and

Börnstein

1905,

pp.

387-

392.

[21]

Drude

1904a.

[22]

These results

are

stated

on

p.

682

of

Drude

1904a.

[23]

According

to

Landolt and

Börnstein

1905,

p.

611,

the

wavelengths

of

ultraviolet

light range

from

0.1u

to

0.36u.

[24] According to

Landolt and

Börnstein

1905,

p.

611,

the

wavelengths

of infrared

light range

from

0.81u

to

61.1u.

[25]

Drude

1904a and 1904b. In Einstein

1907d

(Doc. 42)

Einstein corrected this and the

following

sentence.

[26]

For later

attempts

to

obtain

the infrared

proper

frequencies

from the

physical properties

of

a

solid,

see

Lindemann

1910 and Einstein

1911b.

[27]

This

important prediction

was

tested and

confirmed

by

Nernst

(see

Nernst

1911a, 1911b,

1911c).

He showed not

only

that the

specific