Lecture 3
Electrochemical Kinetics
Basics
Current density
The rate of a homogeneous reaction, vr:
vr 
1 dn
 dt
1.
where n and ν are the amount of material and stoichiometric number of a
reactant or product. The unit for vr is mol s-1. Taking stoichiometric number to
be one,
vr 
dn
dt
The reaction rate of a heterogeneous process (e.g., an electrode process )
depends on the surface of the electrode at constant polarization potential. The
rate v refers to unit surface area
v 
1 dn
A dt
1
The unit for v is mol dm-2 s-1.The rate of an electrode process is proportional to
the current density
j
1 dq
A dt
The current density is proportional to rate of an electrochemical reaction and it
makes rate independent of the area of electrode.
An infinitesimal amount of charge dq is proportional to the amount of material
passed through the interface dq  zF  dn .
j
zF dn
A dt
2
From equations 1. and 2. we get
j  zF  v
3
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Lecture 3
The reaction rate of a unit surface electrode is proportional to current density.
Polarization, overvoltage
Polarization: the shift in the voltage across a cell caused by the passage of
current, departure of the cell potential from the reversible (or equilibrium or
nernstian) potential.
For a simple reversible redox reaction
Mz  ze-  M
the cathodic process is a reduction at a rate jc, cathodic current density, while the
anodic process is an oxidation at a rate ja, anodic current density.
At equilibrium a rest or equilibrium potential εe is measured against a proper
reference electrode. When electrochemical equilibrium is established
j c  j a  j0
the anodic and cathodic current densities are the same and equals to j0, the
exchange current density.
Increasing the negative potential:
jc  j a
Fig. 1. Polarization curves
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Lecture 3
The portion of potential differs from equilibrium potential is called overvoltage
(η ).
   pol   e
4.
Processes at electrodes
The rate constant of forward reaction, kf is characteristic to the cathodic
reduction process. The higher the negative polarization potential, the greater the
rate of cathodic reaction.
Fig. 2. Heterogeneous process at the electrode surface.
Mass transport:
reactants are transported from the bulk of solution to the
electrode surface by diffusion
products are transported from electrode to the bulk by
diffusion
Charge transfer: the electron jump between the metal and liquid phase.
The slowest process determines the net rate of the electrode reaction.
Mass transport and charge transfer are the most important steps in controlling
the rate of electrochemical processes.
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Lecture 3
Reaction mechanisms
1.Fast mass transport but rate limiting electron transfer kinetics.
2. Fast electron transfer but mass transport rate limiting kinetics
Normal homogeneous kinetics at a temperature is characterized a rate constant
independent of the composition of reaction mixture.
Why can an applied voltage affect the kinetics of a reaction?
Question can be answered at the level of equilibrium electrochemistry
G o   RT ln K a
and
G o  nFE o
Question can be answered at the level of electrochemical kinetics by using the
transition state theory (TST)
Consider a reduction reaction:
O + ze-  R
By TST, the species, O with gain an electron and goes through a transition state.
The energy barrier to forming this is called ΔG‡c. The c subscript denotes this to
be a cathodic reaction – that is a reduction reaction.
The free energy barrier of an electrochemical reaction is linked to the applied
potential.
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Lecture 3
G‡a(2)
G‡c(2)
G
nFE
G‡c
G‡a
O + neR
Reaction coordinate
When a potential is applied, the free energy of reactants (O + ne-) is raised by an
amount zFE where E is the applied potential.
When η = 0
Energy to reach transition state = ΔG‡c.
When η > 0
New energy to transition state = ΔG‡c(2)
The reaction proceeds faster when a negative polarization potential ε = εpol is
applied. As a result:
ΔG‡c(2) < ΔG‡c.
The converse is true for the back reaction:
R  O + ne-.
For the reduction reaction:
ΔG‡c(2) = ΔG‡c + αnFε
5.a
Remember: cathodic potential is negative.
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Lecture 3
For the back reaction (oxidation of R) we can write:
ΔG‡a (2) = ΔG‡a – (1-α)nFε
5.b
The quantity α relates to the symmetry of the energy barrier. 0 < α <1
We can write the Arrhenius equation for the forward and back reactions:
For forward (cathodic) reaction:
Gc
k f  A f exp 
RT
For backward (anodic) reaction:
Ga
k b  Ab exp 
RT
From 5.a and 5.b we get
ΔG‡c = ΔG‡c(2) - αzFε
and
ΔG‡a = ΔG‡a(2) + (1-α)zFε
So we can write:
k f  A f exp 
Gc
zF
 exp 
RT
RT
Ga
1   zF
kb  Ab exp 
 exp 
RT
RT
But since the first part of both equations are potential independent we do not
need to consider them further and can write:
k f  k 0f exp 
zF
6.a
RT
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Lecture 3
k 0f
That is:
Gc
 A f exp 
RT
and
kb  kb0 exp
1   zF
6.b
RT
Now we have two expressions that relate how the forward and back rate
constants for an electron transfer reaction at an electrode are affected by the
applied potential.
The polarization curve for charge transfer limiting process, j = f(η)
Recall Eq. 3:
j
I
 zF  v
A
The rate of a homogeneous process for reaction O + ne-  R, can be given
v ,  k ,f cox
and for a reaction at the electrode surface
v 
1
0
k f cox
A
7.
0
where the rate refers to unit surface area, and cox
is the concentration of
oxidized form at the surface.
From equations 3. and. 7.
jc  zF 
1
0
k f cox
A
for a cathodic process
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Lecture 3
ja  zF 
1
0
kb cred
A
for an anodic process
The constant area of the electrode can be included in rate constants.
The net current j can be given as
j  jc  j a
and

0
0
j  zF  k f cox
 kb cred

The mass transport is fast, therefore the surface concentrations are identical to
the concentrations in the bulk phase.
Substituting rate constants from 6.a and 6b.
1   zF 
zF

0
0
j  zF   k 0f cox
exp 
 kb0 cred
exp

RT
RT


8.
This is the Butler-Volmer equation and very important in understanding
electrode kinetics. Apart from thermodynamic activation, the exponential term
shows us the potential dependence of rate constant.
At equilibrium potential the cathodic and anodic reaction proceeds at an equal
rate:
0
j0  zF  k 0f cox
exp 
zF 0
RT
0
 zF  kb0 cred
exp
1   zF 0
RT
The j = f(η) is called polarization curve, therefore equation 8. should be
transformed
From    pol     e we have,
1   zF    e  
zF    e  0 0

0
j  zF   k 0f cox
exp 
 kb cred exp

RT
RT


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Lecture 3
which can be separated, (showing only for the cathodic part)

zF
 zF e  
0
j  zF   k 0f cox
exp 
 exp 

RT
RT



9.
Recognized in equation 9. the part of exchange current density:
zF 

j  j0   exp 

RT 

and
1   zF 
zF

j  j0   exp 
 exp

RT
RT


This is the equation of a polarization curve giving a current potential
characteristics of an electrochemical process for a metal electrode of clean
surface, crystal parameters, electrolyte composition concentration are known
and the process is charge transfer controlled.
Current vs overpotential curve, transfer coefficient α = 0.5
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