604
13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics
amount equal to −nF (ϕ − ϕ ◦ ), where n is the number of electrons transferred in the
reaction (n 1 in our example). If the maximum in the potential energy curve remains
at the same position, the activation energy for the reduction increases by nF (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 parameter α such 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 amount αnF (ϕ − ϕ ◦ ). The
parameter α is called the transfer coefficient or the symmetry factor. The name “symmetry 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 of α. 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 amount ϕ − ϕ ◦ , the entire right
side is lowered by an amount nF (ϕ − ϕ ◦ ). The peak drops by an amount nF (ϕ − ϕ ◦ )/2
(corresponding to α 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 parameter α is thus
a measure of the asymmetry of the curve in the diagram.
Exercise 13.21
Show by sketching graphs similar to Figure 13.18c that α < 1/2 if the left side of the curve is
steeper than the right, and that α > 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
k ox A ox exp
−
E a,ox − (1 − α)nF (ϕ − ϕ ◦ )
RT
k
◦ exp
(1 − α)nF (ϕ − ϕ ◦ )
RT
(13.5-13a)
Similarly,
k red k
◦ 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 of j a , the anodic current per unit area, and j c , the cathodic current
per unit area of electrode, are given by
|j a | nFk ox [R] s
(13.5-14a)
|j c | nFk red [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
13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics
amount equal to −nF (ϕ − ϕ ◦ ), where n is the number of electrons transferred in the
reaction (n 1 in our example). If the maximum in the potential energy curve remains
at the same position, the activation energy for the reduction increases by nF (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 parameter α such 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 amount αnF (ϕ − ϕ ◦ ). The
parameter α is called the transfer coefficient or the symmetry factor. The name “symmetry 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 of α. 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 amount ϕ − ϕ ◦ , the entire right
side is lowered by an amount nF (ϕ − ϕ ◦ ). The peak drops by an amount nF (ϕ − ϕ ◦ )/2
(corresponding to α 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 parameter α is thus
a measure of the asymmetry of the curve in the diagram.
Exercise 13.21
Show by sketching graphs similar to Figure 13.18c that α < 1/2 if the left side of the curve is
steeper than the right, and that α > 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
k ox A ox exp
−
E a,ox − (1 − α)nF (ϕ − ϕ ◦ )
RT
k
◦ exp
(1 − α)nF (ϕ − ϕ ◦ )
RT
(13.5-13a)
Similarly,
k red k
◦ 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 of j a , the anodic current per unit area, and j c , the cathodic current
per unit area of electrode, are given by
|j a | nFk ox [R] s
(13.5-14a)
|j c | nFk red [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