Volume 2: The Swiss Years: Writings, 1900-1909 Page 203 (239 of 692)

DOC.
15
MOLECULAR
DIMENSIONS 203
Published
by
K. J.
Wyss,
Bern. Dated
Bern,
30
April
1905.
[1]
For
Einstein's earlier
attempts
to obtain
a
doctorate,
see
the editorial
note,
"Einstein's
Dissertation
on
the Determination
of
Molecular
Dimensions,"
§
III,
pp.
173-176.
[2]
The
dissertation
was formally
submitted
on
20
July
1905
(see
Einstein
to
Rudolf Martin
of
this
date).
[3]
Alfred Kleiner
was
Professor
of
Physics at
the
University
of Zurich. Einstein submitted the
dissertation
to
him. At
Kleiner's
request,
Hein-
rich
Burkhardt,
Professor
of
Mathematics at the
University
of
Zurich,
checked the calculations in
Einstein's
dissertation
(see
the Gutachten über
das
Promotionsgesuch
des Hrn.
Einstein,
20-24
July 1905,
SzZSa, U
110
e 9).
[4]
Einstein had stated
earlier
that he intended
to dedicate
his
dissertation to
Grossmann;
see
Einstein to
Mileva
Maric,
19
December
1901
(Vol. 1,
Doc.
130).
[5]
For
a
discussion
of
methods for determin-
ing
molecular dimensions known
at
that
time,
see
the editorial
note,
"Einstein's
Dissertation
on
the Determination
of
Molecular Dimen-
sions,"
§
II, pp.
170-173.
[6]
Einstein's
argument closely
follows Kirch-
hoff 1897,
Lecture
10, pp.
95-108. For Ein-
stein's
reading
of
Kirchhoff
1897, see
Einstein
to
Mileva
Maric,
29
July
1900 and
1
August
1900
(Vol.
1,
Docs. 68 and
69).
[7]
The
assumptions
made for the
liquid cor-
respond
to those introduced in Kirchhoff 1897
on p.
374.
[8]
For
an argument
in which similar
boundary
conditions
for
the
hydrodynamical equations are
used, see Kirchhoff 1897,
pp.
378-379.
[9]
The
following equations are
valid
if
the
ve-
locities
are
assumed
to
be
infinitely
small,
and if
the motion
is
stationary (see
Kirchhoff 1897,
p.
374).
For
a
discussion
of
the Navier-Stokes
equations, see
Brush
1976,
book
2,
§
12.3,
pp.
432-443.
[10] Kirchhoff 1897,
Lecture 26.
[11]
A
factor
k is missing
on
the
right-hand
side
of
the last
equation
on
this
line;
this
error
is
cor-
rected in Einstein 1906a.
[12]
Kirchhoff
1897,
pp.
378-379.
The Kirch-
hoff
text
begins:
"From
the
equations
9) [corre-
sponding
to
(4)
in
Einstein's
text]
follows
Ap
=
0;
if
one assumes
p
according to
this condition
. .
."
("Aus den
Gleichungen
9)
folgt Ap
=
0;
hat
man
dieser
Bedingung gemäss
p
ange-
nommen
. .
.")
and then continues
as quoted by
Einstein.
[13]
The denominator
on
the
right-hand
side
should be
8£2;
this
error
is
corrected in Einstein
1906a.
[14]
The denominator
of
the first term
on
the
right-hand
side should be
8£2;
this
error
is
cor-
rected in
Einstein
1906a.
A
reprint
of
this article
in the Einstein Archive shows
marginalia
and
in-
terlineations in
Einstein's
hand,
the first
of
which
refer to this
and the
following equation.
The term
"+g1/p"
was
added
to
the
right-hand
side
of
the
equation
for
V
and then canceled.
These
marginalia
and interlineations
are pre-
sumably part
of Einstein's
unsuccessful
attempt
to find
a
calculational
error; see
note
26
below,
and also the editorial
note,
"Einstein's
Disser-
tation
on
the Determination
of
Molecular Di-
mensions,"
§
V, pp.
179-182.
On this
as-
sumption, they
date from late 1910.
[15]
The
equation
for
u'
should be
(as
corrected
8|r
in
Einstein
1906a):
u'
=
-2c
S1/p/SE
.
In the
reprint
mentioned in
note 14,
the first derivative with
respect
to
£
was changed
to
a
second
derivative
and then
changed
back
to
a
first derivative.
At
the bottom
of
the
page,
the
following equations
are
written:
b
=
-1/12
P5a
c
=
-5/12
P3a
g
=
2/3 P3a.
[16]
The numerator
of
the last
term
in the
curly
parentheses
should be "82
(1/p)",
as
corrected
in
Einstein
1906a.
[17]
In
Einstein's
reprint
(see note 14),
a
factor
2/3
was
added
to
the first term
on
the
right-hand
side
of
this
equation
and
then canceled.
[18] "Sn/SE"
should be
"Sp/SE"
as
corrected in
Einstein 1922.
[19]
The factor
preceding
the first
parenthesis
should
be,
as
corrected in Einstein 1906a:
P
3
-
5/2
-
P5
[20]
The
equations
should be:
u
=
U, v
=
V,
w
=
W.
[21]
Xe,
Xn, XI,
should
be
Xe,
etc.,
as cor-
rected in
Einstein
1906a;
analogous
corrections
apply
to
the
subsequent
two
equations.
[22]
For the
following equations,
see
Kirchhoff
1897,
p.
369.
[23]
In Einstein's
reprint
(see
note
14),
the term

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Extracted Text (may have errors)

DOC.
15
MOLECULAR
DIMENSIONS 203
Published
by
K. J.
Wyss,
Bern. Dated
Bern,
30
April
1905.
[1]
For
Einstein's earlier
attempts
to obtain
a
doctorate,
see
the editorial
note,
"Einstein's
Dissertation
on
the Determination
of
Molecular
Dimensions,"
§
III,
pp.
173-176.
[2]
The
dissertation
was formally
submitted
on
20
July
1905
(see
Einstein
to
Rudolf Martin
of
this
date).
[3]
Alfred Kleiner
was
Professor
of
Physics at
the
University
of Zurich. Einstein submitted the
dissertation
to
him. At
Kleiner's
request,
Hein-
rich
Burkhardt,
Professor
of
Mathematics at the
University
of
Zurich,
checked the calculations in
Einstein's
dissertation
(see
the Gutachten über
das
Promotionsgesuch
des Hrn.
Einstein,
20-24
July 1905,
SzZSa, U
110
e 9).
[4]
Einstein had stated
earlier
that he intended
to dedicate
his
dissertation to
Grossmann;
see
Einstein to
Mileva
Maric,
19
December
1901
(Vol. 1,
Doc.
130).
[5]
For
a
discussion
of
methods for determin-
ing
molecular dimensions known
at
that
time,
see
the editorial
note,
"Einstein's
Dissertation
on
the Determination
of
Molecular Dimen-
sions,"
§
II, pp.
170-173.
[6]
Einstein's
argument closely
follows Kirch-
hoff 1897,
Lecture
10, pp.
95-108. For Ein-
stein's
reading
of
Kirchhoff
1897, see
Einstein
to
Mileva
Maric,
29
July
1900 and
1
August
1900
(Vol.
1,
Docs. 68 and
69).
[7]
The
assumptions
made for the
liquid cor-
respond
to those introduced in Kirchhoff 1897
on p.
374.
[8]
For
an argument
in which similar
boundary
conditions
for
the
hydrodynamical equations are
used, see Kirchhoff 1897,
pp.
378-379.
[9]
The
following equations are
valid
if
the
ve-
locities
are
assumed
to
be
infinitely
small,
and if
the motion
is
stationary (see
Kirchhoff 1897,
p.
374).
For
a
discussion
of
the Navier-Stokes
equations, see
Brush
1976,
book
2,
§
12.3,
pp.
432-443.
[10] Kirchhoff 1897,
Lecture 26.
[11]
A
factor
k is missing
on
the
right-hand
side
of
the last
equation
on
this
line;
this
error
is
cor-
rected in Einstein 1906a.
[12]
Kirchhoff
1897,
pp.
378-379.
The Kirch-
hoff
text
begins:
"From
the
equations
9) [corre-
sponding
to
(4)
in
Einstein's
text]
follows
Ap
=
0;
if
one assumes
p
according to
this condition
. .
."
("Aus den
Gleichungen
9)
folgt Ap
=
0;
hat
man
dieser
Bedingung gemäss
p
ange-
nommen
. .
.")
and then continues
as quoted by
Einstein.
[13]
The denominator
on
the
right-hand
side
should be
8£2;
this
error
is
corrected in Einstein
1906a.
[14]
The denominator
of
the first term
on
the
right-hand
side should be
8£2;
this
error
is
cor-
rected in
Einstein
1906a.
A
reprint
of
this article
in the Einstein Archive shows
marginalia
and
in-
terlineations in
Einstein's
hand,
the first
of
which
refer to this
and the
following equation.
The term
"+g1/p"
was
added
to
the
right-hand
side
of
the
equation
for
V
and then canceled.
These
marginalia
and interlineations
are pre-
sumably part
of Einstein's
unsuccessful
attempt
to find
a
calculational
error; see
note
26
below,
and also the editorial
note,
"Einstein's
Disser-
tation
on
the Determination
of
Molecular Di-
mensions,"
§
V, pp.
179-182.
On this
as-
sumption, they
date from late 1910.
[15]
The
equation
for
u'
should be
(as
corrected
8|r
in
Einstein
1906a):
u'
=
-2c
S1/p/SE
.
In the
reprint
mentioned in
note 14,
the first derivative with
respect
to
£
was changed
to
a
second
derivative
and then
changed
back
to
a
first derivative.
At
the bottom
of
the
page,
the
following equations
are
written:
b
=
-1/12
P5a
c
=
-5/12
P3a
g
=
2/3 P3a.
[16]
The numerator
of
the last
term
in the
curly
parentheses
should be "82
(1/p)",
as
corrected
in
Einstein
1906a.
[17]
In
Einstein's
reprint
(see note 14),
a
factor
2/3
was
added
to
the first term
on
the
right-hand
side
of
this
equation
and
then canceled.
[18] "Sn/SE"
should be
"Sp/SE"
as
corrected in
Einstein 1922.
[19]
The factor
preceding
the first
parenthesis
should
be,
as
corrected in Einstein 1906a:
P
3
-
5/2
-
P5
[20]
The
equations
should be:
u
=
U, v
=
V,
w
=
W.
[21]
Xe,
Xn, XI,
should
be
Xe,
etc.,
as cor-
rected in
Einstein
1906a;
analogous
corrections
apply
to
the
subsequent
two
equations.
[22]
For the
following equations,
see
Kirchhoff
1897,
p.
369.
[23]
In Einstein's
reprint
(see
note
14),
the term

Help

204
DOC.
15
MOLECULAR DIMENSIONS
" +
5/6 p3
Ae/p3"
is added
to
the
right-hand
side
of
this
equation.
After the last
terms
of
the second
and
third
equations,
series of dots
are
added.
These interlineations
are presumably
related to
the
marginal
calculations
indicated in note
14.
[24]
In
Einstein's
reprint
(see
note
14),
the
term
" +
5/6
P3
AE/p3"
is
added to the
right-hand
side of
this
equation.
This interlineation
is
presumably
related to the
marginal
calculations indicated in
note
14.
[25]
In
Einstein's
reprint
(see
note
14),
the
term
" -
5/3
kP3A
(1/p3
-
9
E/p5)"
is
added
to
the
right-
hand side
of
this
equation.
This addition
is
pre-
sumably
related
to the
marginal
calculations
re-
ferred to
in
note 14.
[26]
This
equation
and the
subsequent one are
incorrect.
Apart
from minor
errors, they
contain
a
calculational
error bearing on
the numerical
factors. In Einstein
1906a,
" +
25" in front
of
the last term in the
equation
for
X{
is
changed
to
"-25".
In
Einstein's
reprint
(see
note
14),
the
factor
"£2"
in the last
term
on
the
right-hand
side
of this
equation
is
corrected to "EE", and
the factor
"n2" in front of the
parenthesis
in the
last term
on
the
right-hand
side
of
the
equation
for
Xn
is corrected
to
£n.
The calculational
error
that is
also contained in these
equations,
and
some
of its
consequences, are
corrected in Ein-
stein 1911d. The corrections
are integrated
into
the
text
of
the
reprint
of
this
paper
in Einstein
1922. For
an
account
of the
discovery
of
this
er-
ror
and
of
its
correction,
see
the editorial note,
"Einstein's
Dissertation
on
the Determination
of
Molecular
Dimensions,"
§
V, pp.
179-182.
The correct
equations are:
X7)
=
+
5
kP3
(A
+ B)^j]
-
25kP3
+
BV2
+
C£2)
P7
Xi =
+
5
kP3
(A
+ C)tt
-
25kP3
tt{Ai;2
+
BV2
+
CI2)
P7
[27] "-10"
should be
replaced by
"-5,"
and
"25"
by
"20"
(see previous note).
[28]
The third
"+"
sign
should be
replaced
by
"="
as
corrected in Einstein 1922.
"-10"
should
be
replaced by
"-5,"
and
"20"
by
"15"
(see
note
26).
[29]
In
Einstein's
reprint (see
note
14),
the
fac-
tor
"4/15"
was
changed
to
"8/15"
and
then
changed
back to
"4/15".
[30]
"4/15"
should be
replaced by
"8/15"
as
corrected in
Einstein's
reprint
(see
note
14).
[31]
This
equation
should be
(see
note
26):
W
=
8/3
tt
R3
k
82
+
4/3
it
P3
k
82
=
2b2k(V
+
l/2).
[32]
"8"
should be
"82".
This correction
is
made in
Einstein's
reprint (see
note
14).
[33]
It follows from the correction to
eq. (7)
that the
dissipated energy
is
actually
increased
by
half
this amount. The statement in the text
is
only partially
corrected in Einstein
1922;
the
amount
is
correctly given
but still
described
as a
diminution. The final sentence
of
this
paragraph,
which
no longer applies
to the corrected calcu-
lation,
is
omitted
from
Einstein 1922.
[34]
The
point
should be denoted
by xv, yv,
zv,
as
corrected in Einstein 1906a.
[35]
The second denominator should be
Pv3
as
corrected in Einstein 1906a.
[36]
The heat
production
per
unit time
is
actu-
ally
increased
by 82
k
f.
The correct
equations
are
thus
(see
note
26):
W
=
2
h2
k +
nh2
k&,
and W
=
2
S2
k
(1
+
cp/2).
[37]
The
following
two sentences
are
revised in
Einstein 1922:
"In
order to calculate from
equa-
tion
(7b)
the coefficient
of
friction
of
an
inhomo-
geneous
mixture
of
fluid and
suspended spheres
(in
the
following
called
"mixture"
for
short)
which
we
are
examining,
we
must further take
into consideration that
A, B,
C
are
not
values
of
the
primary
dilations
of
the motion
of
fluid
rep-
resented in
equation
(8);
we
want to
designate
the
primary
dilations
of
the mixture
as
A*, B*,
C*"
("Um
aus
Gleichung (7b)
den
Reibungs-
koeffizienten
der
von uns
betrachteten in-
homogenen
Mischung
von
Flüssigkeit
und
sus-
pendierten Kugeln
(im
Folgenden
kurz
"Mischung"
genannt) zu
berechnen,
müssen
wir noch
berücksichtigen,
daß
A, B,
C nicht die
Werte der
Hauptdilatationen
der
in
Gleichungen
(8)
dargestellten Flüssigkeitsbewegung
sind;
wir
wollen die
Hauptdilatationen
der
Mischung A*,
B*,
C*
nennen").
[38]
In this and the
following
two
equations,
the
sign
after "="
should be
"+."
The third
equation
should have "zv" instead
of
"x"
in the
numerator;
the latter correction
is made in Ein-
stein 1906a.

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202 DOC.
15
MOLECULAR DIMENSIONS
-
21
-
n
[J
N m,
wobei
[j
die
in
der Volumeneinheit
befindliche Masse
des
ge-
lösten
Stoffes und
m
dessen
Molekulargewicht
bedeutet, so
erhält
man:
[58]
N
1»
=•
3
-
l))
4
ic
[J
V
k
Andererseits wurde
in
§
4
gefunden:
67ik
D
p.305[/ab-
hängig]
Diese beiden
Gleichungen
setzen
uns
in den Stand, die
Grössen
P
und
N
einzeln
zu berechnen, von
welchen
sich
N
als unab-
hängig von
der
Natur
des
Lösungsmittels,
der
gelösten
Substanz
und der
Temperatur
herausstellen
muss, wenn unsere
Theorie
den Tatsachen
entspricht.
Wir
wollen
die
Rechnung
für
wässerige Zuckerlösung
durch-
führen. Nach den oben
mitgeteilten Angaben
über
die
innere
Reibung
der
Zuckerlösung folgt
zunächst für 20° C:
[59]
N
P3
=
200.
Nach Versuchen
von
Graham
(berechnet von
Stefan)
ist
[60]
der
Diffusionskoeffizient
von
Zucker
in
Wasser
bei
9,50
C. 0,384,
wenn
der
Tag
als Zeiteinheit
gewählt
wird. Die Zähigkeit des
[61]
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°
vernachlässigen,
[62]
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
[63]
Grösse
befriedigend
überein.
Bern,
den
30.
April
1905.

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Page 1TEXTS click to expand contents

Page 222. "On the Thermodynamic Theory of the Difference in Potentials between Metals and Fully Dissociated Solutions of Their Salts and on an Electrical Method for Investigating Molecular Forces" click to expand contents

Page 985. "On the General Molecular Theory of Heat" click to expand contents

Page 13113. Review of Arturo Giammarco, "A Case of Corresponding States in Thermodynamics" click to expand contents

Page 14914. "On a Heuristic Point of View Concerning the Production and Transformation of Light" click to expand contents

Page 18315. A New Determination of Molecular Dimensions click to expand contents

Page 25022. Review of Paul Langevin, "On a Fundamental Formula of the Kinetic Theory" click to expand contents

Page 49650. "Elementary Theory of Brownian Motion" click to expand contents

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