DOC.
50
323
§2.
Diffusion and random
motion
of
molecules
The
molecular
theory
of
heat affords
yet
another
standpoint
from which
the
process
of
diffusion
can
be
viewed.
The
process
of
random
motion, which
is
what
the heat
content of
a
substance
must
be
considered
to
be,
will
cause
the individual molecules
of
a
liquid
to
change
their position in the
most
random
manner
imaginable.
This,
as
it
were,
haphazard
meandering
of the
molecules of
the dissolved substance in
a
solution will
have
as a
consequence
that the initial
nonuniform
distribution
of
concentration will
gradually
give
way
to
a
uniform
one.
We
will
now
consider this
process
in
somewhat
greater
detail,
limiting
ourselves
again
to
the
case
considered
in
§1,
where only
diffusion in
one
sin-
gle
direction,
namely
in the direction
of
the axis (x-axis) of the
cylinder
Z
has to be
taken into
account.
We
imagine
that
we
know
the x-coordinates
of
all dissolved molecules
at
a
certain time
t, and
also
at
time
t
+
r,
where
r
denotes
a
time
interval
so
short that
the
concentrations in
our
solution
change
very
little
during
it.
During
this time
r,
the x-coordinate
of
the
first dissolved
molecule
will
change by
a
certain
quantity
A1
on
account
of
the
random
thermal
motion,
that of
the second molecule will
change
by
A2,
etc. These displacements
A1,
A2,
etc.,
will
be
in
part
negative
(directed
to
the
left)
and in
part
positive (directed
to
the
right). Furthermore,
the
magnitude
of these displacements will
vary
from molecule
to
molecule.
But
since
we
assume,
as
before, that the solution is dilute, this
displacement
is
determined
only
by
the
surrounding
solvent, while the
rest of
the
dissolved
molecules
has
no
appreciable
effect; for that
reason,
these
displacements
A
will
on
the
average
be
of
equal
magnitude
in parts of
the solution
having
differing concentrations, and will
be
just
as
often
positive
as
negative.
We
now
want
to
see
how much
of
the substance diffuses
through
the unit
cross
section
of
our
solution
during
time
r
if
we
know
the
magnitude
of
the
displacements
A
in
the
direction
of the
cylinder
axis
experienced
on
the
average by
the dissolved molecules.
To
simplify
this
consideration,
we
will
assume
that all molecules
undergo
an
equally
large displacement
A,
with half
of the molecules
undergoing
the
displacement
+A,
(i.e.,
to
the right),
and
the other half the
displacement
-A
(i.e.,
to
the
left).
We
thus replace
the
individual
displacements
A1,
A2,
etc.,
by
their
mean
value
A.
[8]