Biomedical Engineering Reference
In-Depth Information
cell. At intermediate salt concentrations this electric field can as-
sist mass transport, but at lower salt concentrations the potential
drop near the electrode becomes so small that electron transfer can
no longer take place.
Small volumes can also lead to new behavior by virtue of con-
taining a relatively small number of redox molecules. Consider a
small, solution-filled cavity connected to a reservoir. The cavity
contains an average number of molecules, ¢
N
², which is given by
¢
N
² =
CV N
A
. Here,
C
is the concentration of redox molecules,
V
is
the volume of the cavity, and
N
A
is Avogadro's number, as per our
previous notation. At any given instant in time, however, the num-
ber of redox molecules inside the cavity is not necessarily precise-
ly equal to ¢
N
²: as individual molecules diffuse in and out of the
cavity and into the reservoir, the instantaneous number of mole-
cules,
N
(
t
), can deviate from ¢
N
². In typical experiments, these
fluctuations are so small relative to ¢
N
² that they are utterly negli-
gible. In the simplest case of non-interacting particles, for example,
the fluctuations in
N
(
t
) are given by the Poisson distribution, and
the relative size of the fluctuations scales as ¢
N
²
-1/2
. It is thus seen
explicitly that the fluctuations essentially vanish as ¢
N
²
Æ
f. In
the opposite limit ¢
N
² o 1, however, the size of the fluctuations
becomes comparable to the value of ¢
N
² itself. In this case the fluc-
tuations can no longer be ignored. Note that these fluctuations are
not some extrinsic noise imposed on the system. Rather, they are
an intrinsic feature of systems in diffusive equilibrium with a res-
ervoir, as embodied in the Gibbs sum of statistical mechanics.
Thin-layer cells present an excellent opportunity for observing
these statistical fluctuations. Equation (11) can be rewritten in the
form
i
lim
= ¢
N
²(
eD/z
2
). The quantity
eD/z
2
thus represents the con-
tribution of each individual molecule to the current, and its magni-
tude increases quadratically with decreasing electrode spacing.
Shrinking
z
thus helps make the fluctuations visible in two ways: it
decreases the volume and, correspondingly, the number of mole-
cules ¢
N
², and it makes the contribution of each molecule to the
current larger, rendering fluctuations easier to detect. In the case
where the number of molecules varies very slowly compared to the
time for individual molecules to cross the cavity (given by
z
2
/2
D
),
the variations in
N
(
t
) then lead to corresponding slow fluctuations
in the instantaneous current,
i
(
t
):
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