Page 86 - 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

164 Electrochemistry

These relationships can be combined to give an expres- Marcus/Hush/Gerischer type) are discussed in Bard and
sion for the net current density ( j), which by definition Faulkner (2001).
is equal to j a − j c in terms of the activation overpoten-
tial (substitute η + E e = E), which is referred to as the
Butler-Volmer equation II. ELECTRODE POTENTIALS
AND POTENTIOMETRY
j = j a − j c = j o {exp[(1 − α)n a Fη/RT ]
Use of the potential of a galvanic cell to measure the con-
− exp[−αn a Fη/RT ]}. (18)
centration of an electroactive species developed later than
Both k c and k a vary exponentially with potential a number of other electrochemical methods. In part, this
difference: wasbecausearationalrelationbetweenelectrodepotential
k c = (k c ) E=0 exp[−αn a EF/RT ], (19) and the concentration of an electroactive species required
the development of thermodynamics, and, in particular, its
k a = (k a ) E=0 exp[(1 − α)n a EF/RT ]. (20) application to electrochemical phenomena. The work of
At equilibrium ( j = 0) and for the special case when J. Willard Gibbs in the 1870s provided the foundation for
(C ox ) o = (C red ) o , the Nernst equation. The latter provides a quantitative re-
lationship between potential and the ratio of effective ther-
(k c ) E=0 = (k a ) E=0 = k s . (21) modynamic concentrations [activities] for a redox couple
([ox]/[red]) and is the basis for potentiometry and poten-
This simple rate constant (k s ) is the defined value of k c
◦
and k a at the formal potential, E . Then, tiometric titrations. The utility of potentiometric measure-
ments for the characterization of ionic solutions was es-
(k c ) E=0 = k s exp[−αn a F( E − E )/RT ] (22)
◦
tablished with the invention of the glass electrode in 1909
(k a ) E=0 = k s exp[(1 − α)n a F( E − E )/RT ] (23) for a selective potentiometric response to hydronium-ion
◦
concentrations. Another milestone in the development of
j = nFk s {(C red ) o exp[(1 − α)n a F
potentiometric measurements was the introduction of the
× ( E − E )/RT ] hydrogen electrode for the measurement of hydronium-
◦
ion concentrations; one of many important contributions
◦
− (C ox ) o exp[−αn a F( E − E )/RT ]}.
by Professor Joel Hildebrand. Subsequent development
(24) of special glass formulations has made possible electrodes
that are selective to different monovalent cations. The idea
Redox couples in aqueous solutions at room temperature is so attractive that intense effort has led to the develop-
have k s values that range from about 1.0 cm s −1 down to ment of electrodes that are selective for many cations and
−1
essentially zero (<10 −10 cm s ); e.g., k s = 0.1cm s −1 anions, as well as several gas- and bioselective electrodes.
III
II
3−
for Fe (CN) /Fe (CN) 4− and 10 −5 cm s −1 for
6 6 The use of these electrodes and the potentiometric mea-
II
III
2+
3+
Cr (OH 2 ) /Cr (OH 2 ) .
6 6 surement of pH continue to be among the most important
o
At the equilibrium potential, E = E e , j = 0, C =
ox applications of electrochemistry.
b
o
b
C , and C o = C , where C denotes the concentration
ox red red
b
of a species at the electrode surface, whereas C its bulk
concentration. Then A. Principles and Fundamental Relations
C b
bx = exp[−nF( E − E )/RT ] (25) Potentiometric measurements are based on thermody-
◦
C b namic relationships and, more particularly, the Nernst
red
equation which relates potential to the concentration of
and
electroactive species. For our purposes, it is most conve-
b
C ox
◦
E e = E + (RT/nF)ln nient to consider the redox process that occurs at a single
C b
red electrode, although two electrodes are always essential
for an electrochemical cell. However, by considering each
◦
= E + (0.05915/n) log ([ox]/[red ]) (26)
electrode individually, the two electrode processes are eas-
(Nernst equation for a half reaction). ily combined to obtain the entire cell process. Further-
Also, at the equilibrium potential ( E e ), more, confusion can be minimized if the half reactions
b (1−α) b α for electrode processes are written in a consistent man-
j o = nFk s C ox C red
(27) ner. Here, these are always reduction processes with the
= nFk s C (when C ox = C red ). oxidized species reduced by n electrons to give a reduced
species,
More modern models and treatments for the dy-
namics of electrochemical electrontransfer (of the ox + ne − red. (28)