Physical Chemistry Third Edition

13.5 Nonequilibrium Electrochemistry 601

For a planar electrode and for the assumed concentration profile of Figure 13.16,

Fick’s law of diffusion, Eq. (10.2-4), gives the diffusion flux of zinc ions as

J(Zn^2 +)D(Zn^2 +)

[

Zn^2 +

]

b−

[

Zn^2 +

]

s

d

(13.5-5)

whereD(Zn^2 +) is the diffusion coefficient of Zn^2 +ions, where [Zn^2 +]bis the concen-

tration in the bulk of the solution, and where [Zn^2 +]sis the concentration at the surface

of the electrode. Thecurrent densityj(current per unit area of electrode surface) is

jnFJ(Zn^2 +)nFD(Zn^2 +)

[

Zn^2 +

]

b−

[

Zn^2 +

]

s

d

(13.5-6a)

wherenis the number of electrons reacting per ion (n2 in the case of Zn^2 +ions).

If a large electrolytic current flows, the concentration of zinc ions at the surface

will be small because the ions are rapidly reduced at the surface. The current density

approaches a limit for large counter e.m.f., in which case the surface concentration

approaches zero:

jlim

nFD

[

Zn^2 +

]

b

d

(13.5-6b)

If the potential is increased enough, water can be reduced to form hydrogen gas, so the

limiting value in Eq. (13.5-7) must be estimated from potentials that are not sufficient

to reduce water or any reducible species other than zinc ions.

If there is stirring, the zinc ions are brought to the electrode by convection as well

as by diffusion. If convection predominates, the quotientD(Zn^2 +)/din Eq. (13.5-7) is

replaced by the rate of convection (volume of solution brought to unit area of electrode

per second) calledm, themass transport coefficient. With a combination of convection

and diffusionD(Zn^2 +)/dis replaced by a weighted sum ofD(Zn^2 +)/dand the rate of

convection.

If we regard the boundary layer as a concentration cell, the analogue of Eq. (8.3-7)

gives the electric potential difference across the boundary layer. This potential differ-

ence is the concentration overpotential, denoted byηconc:

|ηconc|

RT

nF

ln

([

Zn^2 +

]

[ b

Zn^2 +

]

s

)

(13.5-7)

Using Eqs. (13.5-6) and (13.5-7) and assuming that activity coefficients are equal to

unity we obtain the relation for the concentration overpotential:

|ηconc|−

RT

nF

ln

(

1 −

j

jlim

)

(13.5-8)

Exercise 13.20

Show that Eq. (13.5-8) follows from Eqs. (13.5-6) and (13.5-7).