Page 27 - Analytical Electrochemistry 2d Ed - Jospeh Wang

12 FUNDAMENTAL CONCEPTS

Consider again the electron-transfer reaction: O ‡ ne „ R; the actual electron
transfer step involves transfer of the electron between the conduction band of the
electrode and a molecular orbital of O or R (e.g., for a reduction, from the
conduction band into an unoccupied orbital in O). The rate of the forward
(reduction) reaction, V , is ®rst order in O:
f
V ˆ k C 0; t† 1-16†
O
f
f
while that of the reversed (oxidation) reaction V , is ®rst order in R:
b
V ˆ k C 0; t† 1-17†
b b R
where k and k are the forward and backward heterogeneous rate constants,
f
b
respectively. These constants depend upon the operating potential according to the
following exponential relationships:

k ˆ k exp‰ anF E E †=RTŠ 1-18†
f

k ˆ k exp‰ 1 a†nF E E †=RTŠ 1-19†
b

where k is the standard heterogeneous rate constant, and a is the transfer coef®cient.
1

The value of k (in cm s ) re¯ects the reaction between the particular reactant and
the electrode material used. The value of a (between zero and unity) re¯ects the
symmetry of the free energy curve (with respect to the reactants and products). For
symmetric curves, a will be close to 0.5; a is a measure of the fraction of energy that
is put into the system that is actually used to lower the activation energy (see
discussion in Section 1-2.2.1). Overall, equations (1-18) and (1-19) indicate that by
changing the applied potential we in¯uence k and k in an exponential fashion.
f
b
Positive and negative potentials thus speed up the oxidation and reduction reactions,
respectively. For an oxidation, the energy of the electrons in the donor orbital of R
must be equal or higher than the energy of electrons in the electrode. For reduction,
the energy of the electrons in the electrode must be higher than their energy in the
receptor orbital of R.
Since the net reaction rate is

V net ˆ V V ˆ k C 0; t† k C 0; t† 1-20†
f
b
R
b
f
O
and as the forward and backward currents are proportional to V and V , respectively,
f
b
i ˆ nFAV 1-21†
f f
i ˆ nFAV 1-22†
b b