Page 21 - Analytical Electrochemistry 2d Ed - Jospeh Wang

6 FUNDAMENTAL CONCEPTS

As indicated by equation (1-3), the situation is quite complex when the three
modes of mass transport occur simultaneously. This complication makes it dif®cult
to relate the current to the analyte concentration. The situation can be greatly
simpli®ed by suppressing the electromigration or convection, through the addition of
excess inert salt or use of a quiescent solution, respectively. Under these conditions,
the movement of the electroactive species is limited by diffusion. The reaction
occurring at the surface of the electrode generates a concentration gradient adjacent
to the surface, which in turn gives rise to a diffusional ¯ux. Equations governing
diffusion processes are thus relevant to many electroanalytical procedures.
According to Fick's ®rst law, the rate of diffusion (i.e., the ¯ux) is directly
proportional to the slope of the concentration gradient:

@C x; t†
J x; t†ˆ D 1-5†
@x
Combination of equations (1-4) and (1-5) yields a general expression for the current
response:

@C x; t†
i ˆ nFAD 1-6†
@x
Hence, the current (at any time) is proportional to the concentration gradient of the
electroactive species. As indicated by the above equations, the diffusional ¯ux is
time dependent. Such dependence is described by Fick's second law (for linear
diffusion):

2
@C x; t† @ C x; t†
ˆ D 1-7†
@t @x 2
This equation re¯ects the rate of change with time of the concentration between
parallel planes at points x and (x ‡ dx) (which is equal to the difference in ¯ux at the
two planes). Fick's second law is valid for the conditions assumed, namely planes
parallel to one another and perpendicular to the direction of diffusion, i.e., conditions
of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode
(where the lines of ¯ux are not parallel but are perpendicular to segments of the
sphere), Fick's second law has the form

2
@C @ C 2 @C
ˆ D ‡ 1-8†
@t @r 2 r @r
where r is the distance from the center of the electrode. Overall, Fick's laws describe
the ¯ux and the concentration of the electroactive species as functions of position
and time. The solution of these partial differential equations usually requires the
application of a (Laplace transformation) mathematical method. The Laplace
transformation is of great value for such applications, as it enables the conversion