324
THEORY OF BROWNIAN MOTION
According
to
our
simplifying assumption,
the
plane
E
of
our
cylinder
(Fig.
95) [on
p.
320] can
be
crossed
during
time
r
from
left
to
right
only
by
those dissolved
molecules
which prior
to
interval
r
were
located left
of
E, at
a
distance
from
E
smaller than
A.
These
molecules
are
all located
between
the
planes Q1
and
E (Fig.
95).
But
since
only
half
of
these
molecules
experience
the
displacement
+A,
only
half
of
them
will
cross
the
plane E. But
one-half
of
the
dissolved
substance contained
between
Q1
and
E
amounts,
in
gram-molecules, to
^
A,
where
v1
denotes
the
mean
concentration in the
volume
Q1E,
i.e., the
concentration in the
midplane
M1.
Since the
cross
section
equals 1,
A
represents
the
volume
enclosed
between
Q1
and
E,
which,
when
multiplied
by
the
mean
concentration,
gives
the dissolved substance contained in this
volume
in
gram-molecules.
By
an
analogous
consideration
we
find that the
amount
of
dissolved
substance crossing
E
from right
to
left
during
time
r
equals
2"
^2
where
v2
denotes the concentration
in
the
midplane
M2.
The amount
of
substance
diffusing
through
E
from
left
to
right
during
time
r
is
obviously equal
to
the difference
of these
two
values, and hence
equals
(6)
iA(z/t
-
v2).
v1
and
v2
are
the concentrations in
two
cross
sections
separated
by
the
very
small
distance
A.
If
we
again
denote
a cross
section's
distance
from the
left
end
of the cylinder
by x, we
will
have
according to
the
definition of the differential quotient