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)
where D(Zn 2+ ) is the diffusion coefficient of Zn 2+ ions, where [Zn 2+ ] b is the concentration in the bulk of the solution, and where [Zn 2+ ] s is the concentration at the surface
of the electrode. The current density j (current per unit area of electrode surface) is
j nFJ(Zn
2+ ) nFD(Zn
2+ )
Zn 2+
b
−
Zn 2+
s
d
(13.5-6a)
where n is the number of electrons reacting per ion (n 2 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:
j lim
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 quotient D(Zn 2+ )/d in Eq. (13.5-7) is
replaced by the rate of convection (volume of solution brought to unit area of electrode
per second) called m, the mass transport coefficient. With a combination of convection
and diffusion D(Zn 2+ )/d is replaced by a weighted sum of D(Zn 2+ )/d and 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 difference is the concentration overpotential, denoted by η conc :
|η conc |
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:
|η conc | −
RT
nF
ln
1 −
j
j lim
(13.5-8)
Exercise 13.20
Show that Eq. (13.5-8) follows from Eqs. (13.5-6) and (13.5-7).
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)
where D(Zn 2+ ) is the diffusion coefficient of Zn 2+ ions, where [Zn 2+ ] b is the concentration in the bulk of the solution, and where [Zn 2+ ] s is the concentration at the surface
of the electrode. The current density j (current per unit area of electrode surface) is
j nFJ(Zn
2+ ) nFD(Zn
2+ )
Zn 2+
b
−
Zn 2+
s
d
(13.5-6a)
where n is the number of electrons reacting per ion (n 2 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:
j lim
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 quotient D(Zn 2+ )/d in Eq. (13.5-7) is
replaced by the rate of convection (volume of solution brought to unit area of electrode
per second) called m, the mass transport coefficient. With a combination of convection
and diffusion D(Zn 2+ )/d is replaced by a weighted sum of D(Zn 2+ )/d and 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 difference is the concentration overpotential, denoted by η conc :
|η conc |
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:
|η conc | −
RT
nF
ln
1 −
j
j lim
(13.5-8)
Exercise 13.20
Show that Eq. (13.5-8) follows from Eqs. (13.5-6) and (13.5-7).