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The electrode potential and the surface concentrations of О and R are described by an equation of the Nernst form, regardless of the current flow.
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A plot of log i vs. η
known as a
Tafel plot
can obtain the values of α and i
0
^
At large negative overpotentials
The plots
deviate
sharply from
linear behavior as η approacheszero, because the back reactionscan no longer be regarded asnegligible
Real Tafel plots for the Mn(IV)/Mn(III) system in concentrated acid
The deviations from linearity at very largeoverpotentials: by
mass transfer
The deviations at very low overpotentials: by nonnegligible amounts of a
reverse
reaction
Can be rewritten as
Take the log of both sides
Make a plot of log [i/(1 - e
fη
)] vs. η
: obtain an intercept of log i
0
and a slope of -αF/2.3RT
Let us reconsider the
Butler-Volmer equation for quasi-reversible cases
as follows
Let us reconsider the current-overpotential equation as follows
: the
Butler-Volmer equation
Let us consider its behavior when
i
0
becomes very large
compared to any current of
interest
The ratio i/i
0
then approaches zero
, and we can rearrange the limiting form of
equation
Net current flows
because the surface concentrations are not at equilibrium with
the bulk
mass transfer continuously moves material
to the surface, where it must be
reconciled to the potential by electrochemical change ▪^
Previously, a system that is always at equilibrium is termed a reversible System
an electrochemical system in which the
charge transfer interface is always at
equilibrium
is also called a
reversible
(or, alternatively, a nernstian) system
At extreme η (blue box),
the current approaches the limiting current
the current is
limited by mass transfer
We assume here that
stirring is ineffective at the electrode surface
so the solution velocity term need not be considered at x = 0. ▪^
This simplified treatment is based on the idea that
a stagnant layer of thickness δ
O
exists at the electrode surface (Nernst diffusion layer), with stirring
maintaining the
concentration of О at
O
O
Since we also assume that there is an excess of
supporting electrolyte
migration is not important
the
rate of mass transfer
is proportional to the
concentration gradient at the
electrode surface
as given by the first (diffusive) term in the equation:
The proportionality constant,
m
, called theO
mass-transfer coefficient
, has units of
cm/s ▪^
Can also be thought of as
volume flow/s per unit area (cm
3
-1s
cm
Thus, from the following equations and taking a
reduction current as positive
[i.e., i is
positive when
O
(x = 0)], we obtainO
The
largest rate of mass transfer
of О occurs
when
(x = 0) = 0O
or more precisely, when Co (x = 0) <<
O
*, so that
(x = 0)O
The
value of the current
under these conditions (maximum current)
is called
the limiting current, i
,^ l
where
When the limiting current flows,
the electrode process is occurring
at the maximum rate
possible for a given set of
mass-transfer conditions,
О is being reduced
as fast as it can be brought to the electrode surface
Or for the particular case when
The
values of
O
(x = 0) and
(x = 0) are functions of electrode potential, ER
. (Nernst
equation: ch. 2) ▪^
Under the conditions of a net cathodic reaction,
R is produced at the electrode surface,
so that
(x = 0) >R
Therefore,
If the kinetics of
electron transfer are rapid
the concentrations of О and R at the electrode surface can be assumed to be
at
equilibrium
with the electrode potential, as governed by the
Nernst equation
for the
half-reaction ▪^
Let us derive the steady-state
i-E curves for nernstian reactions
under several
different conditions.
a plot of log[(i
a plot of i - E
a slope of nF/2.3RT
an E-intercept of Е
1/
When
both members of the redox couple exist
in the bulk, we must distinguish between
a cathodic limiting current, i
l,c
, when C
(x = 0)O
and an anodic limiting current, i
l,a
, when C
(x = 0)R
The limiting anodic current naturally reflects the
maximum rate
at which R can be
brought to the electrode surface for conversion to O
Sign convention
: cathodic currents are taken as positive and anodic ones as negative
