Volume 2: The Swiss Years: Writings, 1900-1909 Page 346 (382 of 692)

33.
"Supplement"
to
"A
New Determination
of
Molecular Dimensions"
[Einstein 1906c]
Dated
Bern,
January
1906
Published
8 February
1906
In:
Annalen
der
Physik
19 (1906):
305-306.
This
supplement
is
appended
to the version
of
Document
15
which
appears
in
Annalen
der
Physik
19 (1906):
289-305.
The
journal
version,
also dated 30
April
1905,
was
received
19
August
1905.
Revised version
published
in Einstein
1922,
p.
40.

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Extracted Text (may have errors)

33.
"Supplement"
to
"A
New Determination
of
Molecular Dimensions"
[Einstein 1906c]
Dated
Bern,
January
1906
Published
8 February
1906
In:
Annalen
der
Physik
19 (1906):
305-306.
This
supplement
is
appended
to the version
of
Document
15
which
appears
in
Annalen
der
Physik
19 (1906):
289-305.
The
journal
version,
also dated 30
April
1905,
was
received
19
August
1905.
Revised version
published
in Einstein
1922,
p.
40.

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DOC.
32
THEORY
OF BROWNIAN
MOTION
345
nential distribution law
to
include the
case
of
ex-
ternal forces
acting on
the
gas
molecules
(Boltz-
mann
1868; see
also Boltzmann
1896,
§
19).
[11]
A
square
root
sign over
the second
term
of
the
equation
is
missing.
[12]
The
development
of
the
ultramicroscope
by
Siedentopf
and
Zsigmondy
had,
in
fact,
shifted the lower limit
of
observability
to
ca.
10-6
cm
(Siedentopf
and
Zsigmondy
1903).
For
a contemporary
discussion
of
microscopic
ob-
servability, see
Cotton
and
Mouton
1906,
chap.
1.
[13]
For
an
elaboration
of Einstein's
study
of
Brownian motion under the influence
of
an
elas-
tic
force,
including a proposed experimental
verification, see, e.g.,
Smoluchowski
1913;
see
also
Fürth
1922,
p.
65,
fn.
15.
[14]
The
following argument
is
presented
in
greater
detail in Einstein 1905i
(Doc. 14),
§
1
and
§
2.
[15]
Einstein 1905k
(Doc.
16).
Einstein
pre-
sumably
meant to
refer
to
Einstein 1905i
(Doc.
14),
§
1
and
§
2, as
corrected
in
Einstein
1922,
and,
in
particular,
to
p.
136.
[16]
Planck
1900a.
This
paper
does
not,
how-
ever, give
the formula for
black-body
radiation
to
which Einstein referred. This formula
is
given,
e.g., in
Planck
1901a, which
is
cited
by
Einstein elsewhere for this formula
(see, e.g.,
Einstein
1905i
[Doc. 14],
p.
136).
[17]
See
Planck
1901b.
[18]
For
a
discussion
of Einstein's
views
on
this fundamental
imperfection, see
the editorial
note,
"Einstein's
Early
Work
on
the
Quantum
Hypothesis,"
pp.
139-141.
[19]
In
1908,
Perrin
reported on experiments
showing
that the vertical distribution
of
granules
of
gamboge
in
a liquid
is
exponential
(Perrin
1908a).
Perrin mentioned
Einstein's
name,
but
gave a
derivation
of
the
exponential
distribution
that differs from
Einstein's.
[20]
The basic results
of
this section
are gener-
alizations
of
those derived in Einstein 1905k
(Doc. 16),
§
4.
[21]
The
integral
should extend from 0
to oo.
[22]
This value for N
is
close
to
the value de-
rived in
Einstein
1906c
(Doc. 33);
in Einstein
1922,
the
more
accurate value
6
x
1023 per
mole
is
given
instead. For
a
discussion
of
the
discrep-
ancy
between these
values,
and its
origin,
see
the editorial
note,
"Einstein's
Dissertation
on
the Determination
of
Molecular Dimensions,"
§
V,
pp.
179-182.
[23] Kirchhoff
1897
(see,
in
particular, p. 380);
for
Einstein's
previous use
of Stokes's
formula
for the derivation
of
the
mean square displace-
ment in Brownian
motion,
see
Einstein 1905k
(Doc.
16),
§
3
and
§
5.
[24]
See Kirchhoff
1897,
in
particular,
pp.
375-376.
[25]
Using
the values for
R
and N
given
on
p.
378,
and the value
of
k
given
in Einstein
1905j
(Doc. 15),
p.
21,
one
obtains the result
given by
Einstein. In
1909,
Perrin
performed an experi-
mental
test
of Einstein's
formula
(see
Perrin
1909a).
He worked with
granules
of
resin,
hav-
ing a
diameter
of
ca.
13
jll,
which contained
small inclusions that enabled him to follow their
rotational motion. The
experimental
values he
found
are
in
good agreement
with those
pre-
dicted
by
Einstein's
formula. See also
note 6.
[26]
See the
previous
discussion
on p.
375.
For
an
account
of
this and other
closely
related
prob-
lems,
see
De Haas-Lorentz
1913, chap.
7
(in
particular, pp.
87-88). Fluctuations
of
the
po-
tential difference between the
plates
of
a con-
denser
are
treated
in
Einstein 1907b
(Doc. 39).
Einstein's
study
of
charge
and
potential
fluctua-
tions
was
the
starting point
of his
attempt
to
de-
velop
methods for the
measurements
of
small
quantities
of
electricity
(see
Einstein 1908a
[Doc. 48]
and Vol.
5,
the editorial
note,
"Ein-
stein's 'Maschinchen'
for the Measurement of
Small
Quantities
of
Electricity").
[27]
For
a closely
related
discussion,
see
Ein-
stein 1907c
(Doc. 40).
[28]
The
equation
should be
"da/dz
=
ß0".
[29]
This restriction
was
removed later
by
in-
cluding
the threshold value
\lB
of
the time in
a
description
of
Brownian motion valid for all
time intervals
(see
the discussion in
Fürth
1922,
pp.
60-61, fn.
8).

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DOC. 33 SUPPLEMENT TO DOC.
15
347
Neue
Bestimmung
der
Moleküldimensionen.
305
abhängig
von
der Natur des
Lösungsmittels,
der
gelösten
Sub-
stanz
und der
Temperatur
herausstellen
muß, wenn unsere
Theorie den Tatsachen
entspricht.
Wir
wollen
die
Rechnung
für
wässerige Zuckerlösung
durchführen. Nach den oben
mitgeteilten Angaben
über die
innere
Reibung
der
Zuckerlösung
folgt
zunächst für 20°
C.:
NP3
=
200.
Nach Versuchen
von
Graham
(berechnet
von
Stefan)
ist
der Diffusionskoeffizient
von
Zucker
in
Wasser
bei 9,5°
C.
0,384,
wenn
der
Tag
als Zeiteinheit
gewählt
wird. Die
Zähig-
keit
des Wassers
bei
9,5°
ist
0,0135.
Wir
wollen
diese
Daten
in
unsere
Formel für
den
Diffusionskoeffizienten
einsetzen,
trotzdem sie
an
10
proz. Lösungen gewonnen
sind
und eine
genaue
Gültigkeit unserer
Formel bei
so
hohen
Konzentrationen
nicht
zu
erwarten
ist.
Wir erhalten
N
P
=
2,08.1016.
Aus
den
für
N P3 und
N
P
gefundenen
Werten
folgt,
wenn
wir
die
Verschiedenheit
von
P bei 9,5°
und
20° vernach-
lässigen,
P
=
9,9
.
10-8
cm,
N
=
2,1
.
1023.
Der für N
gefundene
Wert stimmt der
Größenordnung
nach mit den
durch
andere Methoden
gefundenen
Werten für
diese Größe
befriedigend
überein.
Bern, den 30.
April
1905.
(Eingegangen 19.
August
1905.)
Nachtrag.
In
der
neuen
Auflage
der
physikalisch-chemischen
Tabellen
von
Landolt
und
Börnstein
finden sich weit
brauchbarere
[1]
Angaben
zur
Berechnung
der Größe des Zuckermoleküls und
der
Anzahl
N
der wirklichen Moleküle
in
einem Gramm-
molekül.
Thovert
fand
(Tab.
p. 372)
für den Diffusionskoeffizienten
[2]
von
Zucker
in
Wasser
bei
18,5° C.
und der Konzentration

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