Physical Chemistry Third Edition

13.5 Nonequilibrium Electrochemistry 603

(a)

Reaction coordinate

–(1–a)nF(– 8 )

• nF(– 8 )
  
  
  • nF(– 8 )
  
  
  
  

V

(b) (c)

Reaction coordinate

V

Reaction coordinate

V

–(1–a)nF(– 8 )

Figure 13.18 The Potential Energy as a Function of a Reaction Coordinate.(a) Without applied potential. (b) With a negative applied
potential. (c) Figure to show the effect of the symmetry factor. After Bard and Faulkner,op. cit., p. 94.

The maximum in the potential energy is partly due to the increase of the potential

energy of the positive ions as they approach the positive electrode, and partly due to the

energy required to remove electrons from the ions. The decrease of potential energy

as the reaction proceeds past the maximum in Figure 13.18a is due to the binding

of the electrons into the electrode. We regard the state of high potential energy as a

transition state and interpret the potential energy to reach the maximum from the left

as the activation energy for the oxidation half-reaction, denoted byEa,ox. The potential

energy required to reach the maximum from the right is the activation energy for the

reverse (reduction) half-reaction, denoted byEa,red. The rate constants of the forward

and reverse half-reactions can be represented by the Arrhenius formula of Eq. (12.3-1):

koxAoxe−Ea,ox/RT (13.5-10a)

kredArede−Ea,red/RT (13.5-10b)

where we assume that the preexponential factorsAoxandAredare temperature-

independent.

If the system is at equilibrium and if each half-reaction is first order, we can write

for unit area of the electrode

oxidation rate per unit areakox[R]eq

kred[O]eq

reduction rate per unit area (13.5-11)

where [O] is the concentration of the oxidized species (Fe^3 +in our example) and [R]

is the concentration of the reduced species (Fe^2 +in our example) at the surface of

the electrode. At equilibrium at the standard-state voltage [R] and [O] are equal and

kredkox. We denote this value of the rate constants byk◦eq:

k◦eqAoxexp(−E◦a,ox/RT)Aredexp(−E◦a,red/RT) (13.5-12)

where the superscript◦means that the values apply in the standard state at equilibrium.

Let us now change the counter e.m.f. in the external circuit, thus changing the electric

potential in the electrode fromφ◦(the value ofφcorresponding toEE◦)toanew

valueφ. The potential energy ofnmoles of electrons in the electrode is changed by an