Physical Chemistry Third Edition

604 13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics

amount equal to−nF(φ−φ◦), wherenis the number of electrons transferred in the

reaction (n1 in our example). If the maximum in the potential energy curve remains

at the same position, the activation energy for the reduction increases bynF(φ−φ◦)

while the activation energy of the oxidation remains unchanged. However, the ordinary

case is that, as shown schematically in Figure 13.18b, the position of the maximum is

lowered, but by a magnitude smaller than|nF(φ−φ◦)|.

We define a parameterαsuch that the position of the minimum is lowered by

(1−α)nF(φ−φ◦), lowering the activation energy of the oxidation by the same amount.

The activation energy of the reduction is increased by the amountαnF(φ−φ◦). The

parameterαis called thetransfer coefficientor thesymmetry factor. The name “sym-

metry factor” is used because its value, which generally lies between 0 and 1, is related

to the shape (symmetry) of the curve in Figure 13.18a.

Figure 13.18c shows in an oversimplified way how the shape of the curve affects

the value ofα. For this illustration we assume that the position of the right side of the

curve is determined solely by the electric potential at the electrode, whereas the left

side of the curve is determined solely by chemical factors that are unaffected by the

potential of the electrode, and assume also that the two sides of the curve meet at a

cusp. When the electric potential is increased by an amountφ−φ◦, the entire right

side is lowered by an amountnF(φ−φ◦). The peak drops by an amountnF(φ−φ◦)/ 2

(corresponding toα 1 /2) if the slopes of the two sides are equal in magnitude, and

by a different amount if the slopes have different magnitudes. The parameterαis thus

a measure of the asymmetry of the curve in the diagram.

Exercise 13.21

Show by sketching graphs similar to Figure 13.18c thatα< 1 /2 if the left side of the curve is

steeper than the right, and thatα> 1 /2 if the right side is steeper.

Assuming that the preexponential factors in Eq. (13.5-10) do not depend on the electric

potential in the electrode, we write

koxAoxexp

[

−

Ea,ox−( 1 −α)nF(φ−φ◦)

RT

]

k◦exp

[

( 1 −α)nF(φ−φ◦)

RT

]

(13.5-13a)

Similarly,

kredk◦exp

[

−αnF(φ−φ◦)/RT

]

(13.5-13b)

Let us now investigate the case of equilibrium at a potential not necessarily equal to

φ◦. The magnitudes ofja, the anodic current per unit area, andjc, the cathodic current

per unit area of electrode, are given by

|ja|nFkox[R]s (13.5-14a)

|jc|nFkred[O]s (13.5-14b)

where the concentrations at the surface of the electrode enter in these equations. At

equilibrium, the net current is zero and the surface concentrations are equal to the bulk

concentrations, so that