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