Page 29 - Analytical Electrochemistry 2d Ed - Jospeh Wang

14 FUNDAMENTAL CONCEPTS

nents at E is the exchange current (i ) which is directly proportional to the standard
eq
o
rate constant:

i ˆ i ˆ i ˆ nFAk C 1-25†
a
c
o
where i and i are the cathodic and anodic components, respectively.
a
c
The exchange current density for common redox couples (at room temperature)
2
can range from 10 6 mAcm 2 to A cm . Equation (1-24) can be written in terms of
the exchange current to give the Butler±Volmer equation:

i ˆ i exp anFZ=RT† exp‰ 1 a†nFZ=RTŠ 1-26†
0
where Z ˆ E E eq is called the overvoltage (i.e., the extra potential beyond the
equilibration potential leading to a net current i). The overvoltage is always de®ned
with respect to a speci®c reaction, for which the equilibrium potential is known.
Equation (1-26) can be used for extracting information on i and a, which are
0
important kinetic parameters. For suf®ciently large overvoltages (Z > 118 mV=n),
one of the exponential terms in equation (1-26) will be negligible compared with the
other. For example, at large negative overpotentials, i i and equation (1-26)
c
a
becomes
i ˆ i exp anFZ=RT† 1-27†
0
and hence we get

ln i ˆ ln i anFZ=RT 1-28†
0
This logarithmic current±potential dependence was derived by Tafel, and is known
as the Tafel equation. By plotting log i vs. Z one obtains the Tafel plots for the
cathodic and anodic branches of the current±overvoltage curve (Figure 1-8). Such
plots are linear only at high values of overpotentials; severe deviations from linearity
are observed as Z approaches zero. Extrapolation of the linear portions of these plots
to the zero overvoltage gives an intercept that corresponds to log i ; the slope can be
0
used to obtain the value of the transfer coef®cient a. Another form of the Tafel
equation is obtained by rearrangement of equation (1-28):

Z ˆ a b log i 1-29†

with b, the Tafel slope, having the value of 2.303 RT=anF.For a ˆ 0:5 and n ˆ 1,
this corresponds to 118 mV (at 25 C). Equation (1-29) indicates that the application

of small potentials (beyond the equilibrium potential) can increase the current by
many orders of magnitude. In practice, however, the current could not rise to an
in®nite value due to restrictions from the rate at which the reactant reaches the
surface. (Recall that the rate-determining step depends upon the potential region.)