Page 23 - Analytical Electrochemistry 2d Ed - Jospeh Wang

8 FUNDAMENTAL CONCEPTS

linear diffusion, i.e., a planar electrode) for these conditions results in a time-
dependent concentration pro®le,

1=2
C x; t†ˆ C b†f1 erf‰X= 4D t† Šg 1-9†
O
O
O
whose derivative with respect to x gives the concentration gradient at the surface,
@C C b†
O
ˆ 1=2 1-10†
@x pD t†
O
when substituted into equation (1-6) leads to the well-known Cottrell equation:

nFAD C b†
O
O
i t†ˆ 1-11†
1=2
pD t†
O
1=2
That is, the current decreases in proportion to the square root of time, with pD t†
O
corresponding to the diffusion layer thickness.
Solving equation (1-8) (using Laplace transform techniques) yields the time
evolution of the current of a spherical electrode:
nFAD C b† nFAD C O
O
O
O
i t†ˆ 1=2 ‡ 1-12†
pD t† r
O
The current response of a spherical electrode following a potential step thus contains
time-dependent and time-independent terms, re¯ecting the planar and spherical
diffusional ®elds, respectively (Figure 1-3), becoming time independent at long time
scales. As expected from equation (1-12), the change from one regime to another is
strongly dependent upon the radius of the electrode. The unique mass transport
properties of ultramicroelectrodes (discussed in Section 4-5.4) are attributed to the
shrinkage of the electrode radius.
1-2.1.2 Potential-Sweep Experiments Let us move to a voltammetric
experiment involving a linear potential scan, the reduction of O to R and a quiescent
solution. The slope of the concentration gradient is given by (C b; t† C 0; t††=d
O
O
where C b; t) and C 0; t† are the bulk and surface concentrations of O. The
O
O
change in the slope, and hence the resulting current, is due to changes of both
C 0; t† and d. First, as the potential is scanned negatively, and approaches the
O
standard potential (E ) of the couple, the surface concentration rapidly decreases

in accordance to the Nernst equation (equation 1-2). For example, at a potential
equal to E the concentration ratio is unity (C 0; t†=C 0; t†ˆ 1). For a potential

O
R

59 mV more negative than E , C 0; t† is present at 10-fold excess
R
(C 0; t†=C 0; t†ˆ 1=10; n ˆ 1). The decrease in C 0; t† is coupled with an
O
O
R
increase in the diffusion layer thickness, which dominates the change in the slope
after C 0; t† approaches zero. The net result is a peak-shaped voltammogram. Such
O