ELECTROCHEMICAL METHODS Fundamentals and Applications - Allen.J.Bard

30 • Chapter 1. Introduction and Overview of Electrode Processes
C o
(x =
Figure 1.4.1 Concentration profiles (solid lines) and diffusion layer approximation (dashed
lines), x = 0 corresponds to the electrode surface and 8 0
is the diffusion layer thickness.
Concentration profiles are shown at two different electrode potentials: (7) where C 0
(x = 0)
is about CQ/2, (2) where C 0
(x = 0) « 0 and i = i h
Since 8Q is often unknown, it is convenient to combine it with the diffusion coefficient to
produce a single constant, YHQ — DQ/8Q, and to write equation 1.4.4 as
~ C o
(x = 0)] (1.4.5)
The proportionality constant, m o
, called the mass-transfer coefficient, has units of
cm/s (which are those of a rate constant of a first-order heterogeneous reaction; see
Chapter 3). These units follow from those of v and CQ, but can also be thought of as
volume flow/s per unit area (cm 3 s" 1 cm" 2 ). 11 Thus, from equations 1.4.1 and 1.4.5
and taking a reduction current as positive [i.e., / is positive when CQ > CQ(X = 0)], we
obtain
nFA = m o[C% - C o
(x = 0)] (1.4.6)
Under the conditions of a net cathodic reaction, R is produced at the electrode surface, so
that C R
(x = 0) > CR (where CR is the bulk concentration of R). Therefore,
= m R
[C R
(x = 0) - C*] (1A7)
11
While m 0
is treated here as a phenomenological parameter, in more exact treatments the value of m 0
can
sometimes be specified in terms of measurable quantities. For example, for the rotating disk electrode,
m 0
= 0.62D /3 o
(o l/2 v~ l/e , where со is the angular velocity of the disk (i.e., 2тг/, with/as the frequency in
revolutions per second) and v is the kinematic viscosity (i.e., viscosity/density, with units of cm 2 /s) (see
Section 9.3.2). Steady-state currents can also be obtained with a very small electrode (such as a Pt disk
with a radius, r 0
, in the \xva range), called an ultramicroelectrode (UME, Section 5.3). At a disk UME,
m 0
= 4D o
/7rr 0
.