DOC.
50
321
(2)
v =
k,
where
IK
is
a
constant which
we
will call the
frictional resistance
of
the
molecule. In
general,
this frictional resistance
cannot
be
determined
by
theoretical methods.
But
if
we are
allowed
to regard
the molecule
approxi-
mately
as a
sphere
that is
large
compared
with
a
molecule of
the solvent, then
we
can
determine the frictional resistance
of the dissolved molecule
by
the
methods of
ordinary
hydrodynamics,
in
which
the molecular
constitution
of
the
liquid
is
not
taken into
account. Within
the limits of validity of
ordinary
hydrodynamics, a
sphere
moving
in
a
liquid
obeys
equation
(2), where
we
put
(3)
=
6
i7]p.
Here
tj
denotes the coefficient of
viscosity
of the
liquid,
and
p
the
radius of the
sphere.
If
we can assume
that the molecules of
a
dissolved
substance
are
approximately
spherical and large
compared
with the
molecules of
the solvent,
then
equation
(3)
may
be
applied to
the individual dissolved
molecules.
Now we can
calculate the
amount
of dissolved substance
diffusing
through
a cross
section of the cylinder
per
unit time.
The
unit
volume
contains
v
gram-molecules,
which amounts to
vN
real molecules,
where
N
denotes the
number
of real molecules
in
one
gram-molecule.
If
a
force
K
is distributed
over
these
vN
molecules contained in the unit
volume,
it will
impart
a
velocity
to
them
that is
vN
times smaller than
the velocity
it
would be able
to impart
to
a
single
molecule if it acted
upon
the latter alone.
Taking
into
account equation
(2),
we
get
therefore for
the velocity
v
that
the force
K
can
impart to
vN
molecules
«-
1
K
v
-
w
m
In the
case
considered,
K
is
equal
to
the osmotic force exerted
on
the
vN
molecules contained in the unit
volume, which
we
determined
before,
so
that
we
get
from
this,
using equation
(1),