# Lecture 24

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Chapter 25

Electron transfer in heterogeneous systems
(Processes at electrodes)
25.8 The electrode-solution
interface
• Electrical double layer

Helmholtz layer model
Gouy-Chapman Model
• This model explains why
measurements of the
dynamics of electrode
processes are almost
always done using a large
excess of supporting
electrolyte.
The Stern model of the electrode-
solution interface
• The Helmholtz model
overemphasizes the
rigidity of the local solution.

• The Gouy-Chapman
model underemphasizes
the rigidity of local solution.

• The improved version is
the Stern model.
The electric potential at the interface

1. Outer potential
2. Inner potential
3. Surface potential

The potential difference
between the points in
the bulk metal (i.e.
electrode) and the bulk
solution is the Galvani
potential difference
which is the electrode
potential discussed in
chapter 7.
The origin of the distance-
independence of the outer potential
The connection between the
Galvani potential difference and the
electrode potential

• Electrochemical potential (û)
û = u + zFø

• Discussions through half-reactions
25.9 The rate of charge transfer
• Expressed through flux of products: the amount of material produced over a
region of the electrode surface in an interval of time divided by the area of the
region and the duration of time interval.
• The rate laws
Product flux = k [species]
The rate of reduction of Ox,
vox = kc[Ox]
The rate of oxidation of Red,
vRed = ka[Red]

• Current densities:
jc = F kc[Ox] for Ox + e- → Red
ja = F ka[Red] for Red →Ox + e-
the net current density is:
j = ja – jc = F ka[Red] - F kc[Ox]
The activation Gibbs energy
•   Write the rate constant in the form suggested by activated complex theory:


k  B e   G / RT

                       
j  FBa [Red]e  Ga / RT    FBc [Ox]e  Gc / RT

•   Notably, the activation energies for the catholic and anodic processes
could be different!
The Butler-Volmer equation

Variation of the Galvani potential difference across the electrode –
solution interface
The reduction reaction, Ox + e → Red
 Gc   Gc (0)  F

 Gc   Gc (0)  F

 Ga   Ga (0)  (1   ) F

The parameter α is called the transient coefficient and lies in the range 0 to 1.

Based on the above new expressions, the net current density can be expressed
as:
                                             
j  FBa [RED ]e   Ga ( 0) / RT e (1 ) F / RT  FBc [Ox]e   Gc ( 0) / RT e F / RT
F
assuming f 
RT
  Ga ( 0 ) / RT (1 ) f
then     ja  FBa [RED ]e                        e

jc  FBc [Ox]e   Gc ( 0) / RT e f
j  j a  jc
• Example 25.1 Calculate the change in
cathodic current density at an electrode
when the potential difference changes
from ΔФ’ to ΔФ

• Self-test 25.5 calculate the change in
anodic current density when the potential
difference is increased by 1 V.
Overpotential
• When the cell is balanced against an external source, the
Galvani potential difference, , can be identified as the
electrode potential.

• When the cell is producing current, the electrode potential
changes from its zero-current value, E, to a new value, E’.

• The difference between E and E’ is the electrode’s
overpotential, η.
η = E’ – E

• The ∆Φ = η + E,
• Expressing current density in terms of η
ja = j0e(1-a)fη     and      jc = j0e-afη
where jo is called the exchange current density, when ja = jc
• The butler-Volmer
equation:
j = j0(e(1-a)fη - e-afη)

• The lower
overpotential limit
( η less than 0.01V)

• The high
overpotential limit
(η ≥ 0.12 V)
The low overpotential limit
• The overpotential η is very small, i.e. fη <<1

• When x is small, ex = 1 + x + …

• Therefore ja = j0[1 + (1-a) fη]
jc = j0[1 + (-a fη)]
• Then j = ja - jc = j0[1 + (1-a) fη] - j0[1 + (-a fη)]
= j0fη
• The above equation illustrates that at low overpotential limit, the
current density is proportional to the overpotential.

• It is important to know how the overpotential determines the property
of the current.
Calculations under low overpotential
conditions
• Example 25.2: The exchange current density of a Pt(s)|H2(g)|H+(aq)
electrode at 298K is 0.79 mAcm-2. Calculate the current density
when the over potential is +5.0mV.
Solution: j0 = 0.79 mAcm-2
η = 5.0mV
f = F/RT =
j = j0fη

• Self-test 25.6: What would be the current at pH = 2.0, the other
conditions being the same?
The high overpotential limit
• The overpotential η is large, but could be positive or
negative !
• When η is large and positive
jc = j0e-afη = j0/eafη becomes very small in comparison to ja
Therefore j ≈ ja = j0e(1-a)fη
ln(j) = ln(j0e(1-a)fη ) = ln(j0) + (1-a)fη

• When η is large but negative
ja is much smaller than jc
then j ≈ - jc = - j0e-afη
ln(-j) = ln(j0e-afη ) = ln(j0) – afη
• Tafel plot: the plot of logarithm of the current density
against the over potential.
Applications of a Tafel plot
• The following data are the anodic current through a platinum
electrode of area 2.0 cm2 in contact with an Fe3+, Fe2+ aqueous
solution at 298K. Calculate the exchange current density and the
transfer coefficient for the process.
η/mV 50           100     150   200     250
I/mA    8.8       25      58    131     298

Solution: Asked to calculate j0 and α
First, I needs to be converted to J
second: choose
ln(j) = ln(j0e(1-a)fη ) = ln(j0) + (1-a)fη
• Self-test 25.7: Repeat the analysis using
the following cathodic current data:
η/mV       -50 -100 -150 -200 -250
I/mA       -0.3 -1.5 -6.4 -27.61 -118.6

• In general exchange currents are large when the redox
process involves no bond breaking or if only weak bonds
are broken.

• Exchange currents are generally small when more than
one electron needs to be transferred, or multiple or
strong bonds are broken.
The general arrangement for
electrochemical rate measurement

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