Physical Chemistry Third Edition

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 forηgives

η

RT

αnF

ln(j 0 )−

RT

αnF

ln(|j|) (13.5-25)

Equation (13.5-25) is of the form of theTafel 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 exten-

sively 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−→2M+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+]Gdenotes the concentration of hydrogen ions at the Guoy plane (outer

Helmholtz plane) andθdenotes 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+]be−Fφ^1 /RT (13.5-30)

where [H+]bdenotes the concentration of H+in the bulk of the solution and whereφ 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+]bexp

[

−Fφ 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.