326
THEORY
OF
BROWNIAN MOTION
(8)
A
-
We see
from
this formula
that the path
travelled
on
the
average
by a
molecule
is
not
proportional
to
the
time1,
but
to
the
square
root
of
the time.
This is
due to
the fact that the
paths
travelled
in
two
consecutive time units
are
not
always
to
be
added,
but just
as
frequently
are
to
be
subtracted.
The
displacement experienced
on
the
average
by a
molecule
on
account
of
random
molecular
motion
can
be
calculated
according
to
equation
(7a)
from
the
coefficient of
diffusion,
or
according
to equation
(8)
from the force
of
resistance
IK
offered
to
a
forced
motion proceeding
with velocity
v
=
1.
If
the dissolved molecule
is
spherical
and large
compared
with the
molecule of the
solvent,
we can
substitute
for
IK
in equation
(8)
the
value
given
in
equation
(3),
so
that
we
get
(8a)
A
-
RT
1
(T
A
~
7T
3^
This
equation permits
us
to
calculate the
displacement
average2
A
from
the
temperature
T,
the viscosity of the
solvent
?/,
and
the molecular
radius
p.
But
according to
the molecular-kinetic
concept,
there exists
no
fundamental difference
between
a
dissolved
molecule and
a
suspended
corpuscle.
We
must
therefore consider
equation (8a) to be
valid
for
any
kind
of
suspended
spherical
particles
as
well.
We
now
calculate the
path
A
travelled
on
the
average
by a
particle
with
a
diameter of
1
micron
in
1
second in
a
particular direction
in
water at
room
temperature.
We
put
[12]
R
=
8.31.107
,
T
=
290,
N
=
6.1023,
n
=
0.0135,
p
=
0.5.10-4,
[11]
T=1.
1Cf. A.
Einstein, Zeitsch. f. Elektrochemie
6
(1907).
2To
be
more
precise,
the
square root
of the
mean
value
of
A2.