Page 106 - Analytical Electrochemistry 2d Ed - Jospeh Wang

3-6 FLOW ANALYSIS 91

mass transport controlled reactions i ˆ nFADC=d†, one obtains the limiting steady-
l
state response of ¯ow-through electrodes:

i ˆ nFAK CU a 3-32†
m
l
where K is the mass-transport coef®cient D=B†.
m
A more rigorous treatment takes into account the hydrodynamic characteristics of
the ¯owing solution. Expressions for the limiting currents (under steady-state
conditions) have been derived for various electrodes geometries by solving the
three-dimensional convective diffusion equation:

2
2
2
@C @ C @ C @ C @C @C @C
ˆ D ‡ ‡ U x ‡ U y ‡ U z 3-33†
@t @x 2 @y 2 @z 2 @x @y @ z
The resulting equations, arrived at by setting appropriate initial and boundary
conditions (depending on the particular electrode), are given in Table 3-4.
A generalized equation for the limiting-current response of different detectors,
based on the dimensionless Reynolds (Re) and Schmidt (Sc) numbers has been
derived by Hanekamp and co-workers (62):

b a
i ˆ nkFCD Sc† b Re† 3-34†
l
where k is a dimensionless constant and b is the characteristic electrode width.
In the case of coulometric detectors (with complete electrolysis), the limiting
current is given by Faraday's law:

i ˆ nFCU 3-35†
l
TABLE 3-4 The Limiting-Current Response of Various
Flow-Through Electrodes
Electrode Geometry Limiting Current Equation
Tubular i ˆ 1:61 nFC DA=r† 2=3 U 1=3
n
Planar (parallel ¯ow) i ˆ 0:68 nFCD 2=3 1=6 A=b† 1=2 U 1=2
Thin-layer cell i ˆ 1:47 nFC DA=b† 2=3 U 1=3
n
Planar (perpendicular) i ˆ 0:903nFCD 2=3 1=6 3=4 1=2
A
u
n
a
A
Wall-jet detector i ˆ 0:898nFCD 2=3 5=12 1=2 3=8 U 3=4
a ˆ diameter of inlet; A ˆ electrode area; b ˆ channel height;
C ˆ concentration (mM); F ˆ Faraday constant; D ˆ diffusion coef®cient;
n ˆ kinematic viscosity; r ˆ radius of tubular electrode; U ˆ average
1
volume ¯ow rate; u ˆ velocity (cm s ); n ˆ number of electrons.
Adapted from reference 62.