Page 93 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN005M-206 June 15, 2001 20:25

Electrochemistry 171

sufficient to electrolyze the electroactive species com- This relationship holds for any electrochemical process
pletely is applied to the electrode at (t = 0), the concen- that involves semi-infinite linear diffusion and is the basis
tration at the electrode surface is reduced to zero and an for a variety of electrochemical methods (e.g., polarogra-
electrode process occurs, for example, phy, voltammetry, and controlled-potential electrolysis).
Equation (50) is the basic relationship used for solid-
ox + ne − red, (45)
electrode voltammetry with a preset initial potential on
where ox and red represent an oxidized and reduced form a plateau region of the current-voltage curve. Its applica-
of an electroactive species. Passage of current requires tion requires that the electrode configuration be such that
material to be transported to the electrode surface as well semi-infinite linear diffusion is the controlling condition
as away from it. Thus, relationships must be developed for the mass-transfer process.
which involve the flux and diffusion of materials; this is
appropriately accomplished by starting with Fick’s second
law of diffusion,
B. Voltammetry
∂C (x,t) D∂C (x,t)
= , (46) 1. Polarography
∂t ∂x 2
where D represents the diffusion coefficient, C represents The most extensively studied form of voltammetry has
the concentration of the electroactive species at a dis- been polarography (first described by Heyrovsky in 1922,
tance x from the electrode surface, and t represents the with the quantitative relationships of current, potential,
amount of time that the concentration gradient has ex- and time completed by the early 1930s with the assistance
isted. Through the use of Laplace transforms with initial of associates such as Ilkovic). The potential-time depen-
and boundary conditions; dence that is used for polarographic measurements is pre-
sented in Fig. 2 (solid line). The potential is scanned from
b
for t = 0 and x ≥ 0 C = C ,
E 1 to E 2 to obtain a current response that qualitatively
b
for t ≥ 0 and x → 0 C → C , and quantitatively characterizes the electroactive species
present. The vast body of data from polarographic mea-
for t > 0 and x = 0 C = 0.
surements can be adopted by other electroanalytical meth-
Equation (46) can be solved to give a relationship for con- ods. Moreover, pulse polarographic methods and anodic
centration in terms of parameters x and t, stripping analysis, which are still used for determination
x of trace amounts of metal ions, are closely related to po-
b
C (x,t) = C erf , (47) larography. The unique characteristic of polarography is
2D 1/2 1/2
t
its use of a dropping mercury electrode, such that the elec-
b
where C is the bulk concentration of the electroactive
trode surface is continuously renewed in a well-defined
species.
and regulated manner to give reproducible effective elec-
By taking the derivative of Eq. (47) for the proper
trode areas as a function of time. The diffusion cur-
boundary condition, namely at the electrode surface
rent equation [Eq. (50)] can be extended to include a
(x = 0), the diffusion gradient at the electrode surface is
dropping mercury electrode by appropriate substitution
expressed by the relation
for the area of the electrode. Thus, the volume of the
b
∂C C drop for a dropping mercury electrode is given by the
= . (48)
∂x (0,t) π 1/2 D 1/2 1/2 relationship
t
This flux of material crossing the electrode boundary can 4 3 mt
be converted to current by the expression V = πr = , (51)
3 d
∂C (0,t)
i = nFAD , (49) where r is the radius of the drop of mercury, m is the mass
∂x
flow rate of mercury from the orifice of the capillary, t is
where n is the number of electrons involved in the elec-
the life of the drop, and d is the density of mercury under
trode reaction, F is the faraday, and A is the area of the
the experimental conditions. When this equation is solved
electrode. When Eq. (48) is substituted into this relation a
for r and the latter is substituted into the equation for the
complete expression for the current that results from semi-
area of a sphere, an expression for the area of the dropping
infinite linear diffusion is obtained (the Cottrell equation
mercury electrode drop as a function of the experimental
for a planar electrode),
parameters is obtained,
b
nFAC D 1/2
i = . (50) 1/3 2/3 2/3 2/3 −2/3
t
π 1/2 1/2 A = (4π) 3 m t d . (52)