DISSERTATION ON MOLECULAR DIMENSIONS 177
Einstein's other fundamental
equation
(the
third
equation on p.
21
of Einstein
1905j
[Doc. 15])
follows from
an expression
for the coefficient
of
diffusion D
of
the solute. This
expression
is
obtained from
Stokes's
law for
a sphere
of
radius
P
moving
in
a
liquid,
and
Van
't Hoff's
law for the osmotic
pressure:
R
T
1
D
=
*77^
(2)
6
it
k
N
P
where R is the
gas
constant,
T the absolute
temperature,
and N
Avogadro's
number.
The derivation
of
eq.
(1),
technically
the
most
complicated part
of
Einstein's
thesis,
presupposes
that the motion
of
the fluid
can
be
described
by
the
hydrodynamical
equations
for
stationary
flow
of
an incompressible
homogeneous
liquid,
even
in the
presence
of
solute
molecules;
that the inertia
of
these molecules
can
be
neglected;
that
they
do
not
affect each
other's
motions;
and that
they can
be
treated
as rigid spheres moving
in
the
fluid without
slipping,
under the sole influence
of
hydrodynamical
stresses.[58]
The
hydro-
dynamic techniques
needed
are
derived from
Kirchhoff 1897,
a
book that Einstein first
read
during
his
student
years.[59]
Eq.
(2)
follows from the conditions
for the
dynamical
and
thermodynamical equilibrium
of
the fluid. Its derivation
requires
the identification
of
the force
on a single molecule,
which
appears
in
Stokes's
law, with
the
apparent
force due to
the
osmotic
pressure
(see
Einstein
1905j [Doc. 15],
p.
20).
The
key
to
handling
this
problem
is
the introduction
of
fictitious
countervailing
forces. Einstein had earlier introduced such fictitious forces:
they
are
used in
Einstein
1902a
(Doc.
2)
to counteract
thermodynamical
effects in
proving
the
applicability
to diffusion
phenomena
of
a generalized
form
of
the second law
of
thermo-
dynamics;[60]
they
are
also
used in his
papers on
statistical
physics.[61]
Einstein's
derivation
of
eq. (2)
does
not
involve the theoretical tools
he
developed in
his
work
on
the statistical foundations
of
thermodynamics;
he reserved
a more
elaborate
derivation,
using
these
methods,
for his first
paper
on
Brownian
motion.[62]
Eq.
(2)
was
derived
independently,
in
somewhat
more general
form,
by
Sutherland
in 1905.[63]
To deal
with
the
available
empirical
data,
Sutherland had
to
allow for
a varying
coefficient
of
sliding
friction between the
diffusing
molecule and the solution.
The basic elements
of Einstein's method-the
use
of
diffusion
theory
and the
applica-
tion
of
hydrodynamical techniques
to
phenomena involving
the atomistic
constitution of
[58]
Einstein's
derivation
is
only
valid for
Couette
flow;
a generalization
to Poiseuille flow
is
given
in Simha 1936. For
a
discussion
of
Ein-
stein's
assumptions, see
Pais
1982,
p.
90.
[59]
See Einstein to Mileva
Maric,
29
July
1900 and
1
August
1900
(Vol.
1,
Docs. 68 and
69).
[60]
For
a
discussion
of
this
generalization, see
the editorial note,
"Einstein
on
the Nature of
Molecular Forces,"
p.
8.
[61]
See, in
particular,
Einstein 1902b
(Doc.
3),
§
10.
[62]
This
derivation, given
in
Einstein 1905k
(Doc. 16),
§
3,
is cited
in
a
footnote to Einstein
1905j (Doc. 15), p. 20,
that
was presumably
added after the former
paper
had
appeared.
In
the
same paper,
he also used
eq. (2)
to
study
the
relation between diffusion and fluctuations.
[63]
See
Sutherland
1905,
pp.
781-782.