13.5 Nonequilibrium Electrochemistry
607
Butler–Volmer equation, Eq. (13.5-22), is much larger than the first term, and
| j| ≈ j 0 e
−αnF η/RT
(13.5-24)
Solving this equation for η gives
η
RT
αnF
ln( j 0 ) −
RT
αnF
ln(| j|)
(13.5-25)
Equation (13.5-25) is of the form of the Tafel equation, an empirical equation of
the form
|η| a + blog 10 (| j|)
(13.5-26)
The reduction of hydrogen ions at various metal electrode surfaces has been extensively studied. The mechanism of the electrode reaction can depend on the material of
the electrode. One possible mechanism is 33
(1) H
+
+ e
−
+ M M−H
(13.5-27a)
(2) 2 M−H − → 2 M + H 2
(13.5-27b)
where M denotes the metal of the electrode, and M−H denotes a chemically adsorbed
hydrogen atom. Another possible mechanism is 34
(1) H
+
+ e
−
+ M M−H
(13.5-28a)
(2) M−H + H
+
+ e
−
− → M + H 2
(13.5-28b)
In either of these mechanisms, it is possible for different metals that either the first step
or the second step is rate-limiting. If the first step of the mechanism of Eq. (13.5-27) is
rate-limiting in both directions,
(forward rate) k 1 [H
+ ] G (1 − θ)
(13.5-29)
where [H
+ ] G denotes the concentration of hydrogen ions at the Guoy plane (outer
Helmholtz plane) and θ denotes the fraction of surface sites occupied by hydrogen
atoms. If the electric potential in the bulk of the solution is called zero, then by the
Boltzmann probability distribution
[H
+ ] G [H
+ ] b e
−Fϕ 1 /RT
(13.5-30)
where [H + ] b denotes the concentration of H + in the bulk of the solution and where ϕ 1
is the electric potential at the Guoy plane.
The rate at equilibrium gives the exchange current. Using Eq. (13.5-13a) for the rate
constant,
j 0 F (1 − α)[H
+ ] b exp
−F ϕ 1
RT
k
◦ exp
(1 − α) nF (ϕ − ϕ ◦ )
RT
(13.5-31)
With the expression in Eq. (13.5-31) for the exchange current, the Butler–Volmer
equation, Eq. (13.5-22), and Eqs. (13.5-23) through (13.5-25) can be used, since it is
of the same form for a reduction as for an oxidation half-reaction.
33 K. J. Laidler, J. Chem. Educ., 47, 600 (1970) (note 1).
34 Ibid.
607
Butler–Volmer equation, Eq. (13.5-22), is much larger than the first term, and
| j| ≈ j 0 e
−αnF η/RT
(13.5-24)
Solving this equation for η gives
η
RT
αnF
ln( j 0 ) −
RT
αnF
ln(| j|)
(13.5-25)
Equation (13.5-25) is of the form of the Tafel equation, an empirical equation of
the form
|η| a + blog 10 (| j|)
(13.5-26)
The reduction of hydrogen ions at various metal electrode surfaces has been extensively studied. The mechanism of the electrode reaction can depend on the material of
the electrode. One possible mechanism is 33
(1) H
+
+ e
−
+ M M−H
(13.5-27a)
(2) 2 M−H − → 2 M + H 2
(13.5-27b)
where M denotes the metal of the electrode, and M−H denotes a chemically adsorbed
hydrogen atom. Another possible mechanism is 34
(1) H
+
+ e
−
+ M M−H
(13.5-28a)
(2) M−H + H
+
+ e
−
− → M + H 2
(13.5-28b)
In either of these mechanisms, it is possible for different metals that either the first step
or the second step is rate-limiting. If the first step of the mechanism of Eq. (13.5-27) is
rate-limiting in both directions,
(forward rate) k 1 [H
+ ] G (1 − θ)
(13.5-29)
where [H
+ ] G denotes the concentration of hydrogen ions at the Guoy plane (outer
Helmholtz plane) and θ denotes the fraction of surface sites occupied by hydrogen
atoms. If the electric potential in the bulk of the solution is called zero, then by the
Boltzmann probability distribution
[H
+ ] G [H
+ ] b e
−Fϕ 1 /RT
(13.5-30)
where [H + ] b denotes the concentration of H + in the bulk of the solution and where ϕ 1
is the electric potential at the Guoy plane.
The rate at equilibrium gives the exchange current. Using Eq. (13.5-13a) for the rate
constant,
j 0 F (1 − α)[H
+ ] b exp
−F ϕ 1
RT
k
◦ exp
(1 − α) nF (ϕ − ϕ ◦ )
RT
(13.5-31)
With the expression in Eq. (13.5-31) for the exchange current, the Butler–Volmer
equation, Eq. (13.5-22), and Eqs. (13.5-23) through (13.5-25) can be used, since it is
of the same form for a reduction as for an oxidation half-reaction.
33 K. J. Laidler, J. Chem. Educ., 47, 600 (1970) (note 1).
34 Ibid.