128
MOVEMENT
OF SMALL PARTICLES
[11] §3.
Theory
of
diffusion of
small
suspended
spheres
Suppose
that
suspended
particles
are
randomly
distributed
in
a
liquid.
We
wish
to
investigate
their
state
of
dynamic
equilibrium
under the
assumption
that
a
force
K,
which
depends
on
the
position
but
not
on
the
time, acts
on
the individual
particles.
For
the
sake of simplicity,
we
will
assume
that the
force is
everywhere
in
the direction of the X-axis.
If the
number
of
suspended
particles
per
unit
volume
is
v,
then in the
case
of
thermodynamic
equilibrium
v
is
such
a
function
of
x
that the
variation
of
the free
energy
vanishes for
an
arbitrary virtual
displacement
§x
of the
suspended
substance.
Thus
[12]
&F
=
6E
-
28S
=
0.
Let
us assume
that the liquid
has
a cross
section
1
perpendicular
to
the
X-axis, and
that it is
bounded
by
the
planes
x
=
0
and
x
=
l.
We
then
have
SE
d
Kvbxdx
0
and
[13]
6
S
=
d
"
v
dbx
,
_
R
d
0
R nsr
dx
~'
w
J0
mbxdx
Hence,
the
equilibrium
condition
sought
is
(1)
-^
+
7rl
=
°
or
d
Kv
"
-Jx -
°-
The
last
equation states
that the force
K
is balanced
by
the forces of
osmotic
pressure.
We
use
equation
(1) to
determine the coefficient
of
diffusion
of
the
suspended
substance.
The
state
of
dynamic
equilibrium that
we
have
just
considered
can
be
conceived
as a
superposition
of
two processes proceeding
in
[14] opposite
directions,
namely,