Physical Chemistry Third Edition

606 13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics

–60

–10

–5

0

5

10

–40 –20 0 20

/mV

j/j

0

ja

jnet

jc

40 60

Figure 13.19 The Anodic Current, the Cathodic Current, and the Net Current at an

Electrode as a Function of the Overpotential (Schematic).

Exercise 13.23

Carry out the steps of algebra to obtain Eq. (13.5-21).

Equation (13.5-21) is a fundamental equation of electrode kinetics. It indicates that

the current is proportional to the exchange current, but that it has a fairly complicated

dependence on the overpotential. Figure 13.19 shows the anodic current, the cathodic

current, and the net current for a hypothetical electrode reaction. Ohm’s law is not

obeyed except for small values of the overpotential.

If stirring is so efficient that the bulk and surface concentrations are equal, the

concentrations cancel out of Eq. (13.5-21), which is then known as theButler–Volmer

equation:

j

j 0

exp

[

( 1 −α)nF η

RT

]

−exp

[

−αnF η

RT

]

(13.5-22)

There are two important limiting cases. One is the case of small overpotential, in which

case the exponents in Eq. (13.5-21) or Eq. (13.5-22) are small, and the approximation

ex≈ 1 +x

can be used. Application of this approximation to the Butler–Volmer equation gives

j≈j 0

[

1 +

( 1 −α)nF η

RT

− 1 +

αnF η

RT

]

≈

j 0 nF η

RT

(13.5-23)

so that Ohm’s law is obeyed. The factorRT /(j 0 nF) has the dimensions of resistance

and is sometimes called thecharge transfer resistance.

The second limiting case is that of large overpotential. Consider the case that the

overpotential is large and negative. The second term on the right-hand side of the