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Chapter 4
Electrochemical kinetics at electrode / solution
interface and electrochemical overpotential
Valid only for reversible cell or for electrode at
electrochemical reversibility.
inet =0
ln ox
red
aRT
nF a
(2) Nernst equation:
ln i
ii
RTa
nF
Dependent of electrode potential on species activities
lnc d
C D
a b
A B
a aRTE E
nF a a
1) Two important empirical relations
0. Brief introduction
(2) Tafel equation:
The point of intersection of the extrapolation on the
line = 0 is log i0.
A is in fact the at j = 1 A cm-2.
= a + b log j
Valid only for special irreversible process when > 118 mV.
1) Two important empirical relations
0. Brief introduction
Effect of potential on electrode reaction
1. Thermodynamic aspect
If electrode reaction is fast and electrochemical equilibrium
remains, i.e., Nernst equation is applicable. Different
potential corresponds to different surface concentration—
concentration/diffusion control.
2. Kinetic aspect
If electrode reaction is slow and electrochemical equilibrium
is broken. Different potential corresponds to different
activation energy—activation/electrochemistry control.
Chapter 4 Electrochemical kinetics at electrode
f f A
b b B
net f b f A b B
r k c
r k c
r r r k c k c
Rate expressions
At equilibrium
0;
( ) ( )
( )
( )
net
b B eq f A eq
f B eq
b A eq
r
k c k c
k cK
k c
4.1.1 basic concepts
Exchange rate of reaction
For Elementary unimolecular opposing
process
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
Kinetic equilibrium
constant
4.1.1 basic concepts
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
For electrochemical reactions:
; f c
f f A c f Ox
i ir k c r k c
nFA nFA c f Oxi k nFAc
Rea f di k nFAc
0 ,0 ,0
net c a
c a
i i i
i i i
At equilibrium conditions:
exp aEk A
RT
Arrhenius equation
Some important empirical formula:
According to Transition State Theory:
expkT G
kh RT
Corresponding to steric factor in
SCT
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
kT/h term corresponds to the frequency
factor
kTk K
h
Ox e Redf
b
k
kn
Potential curve described by Morse empirical equation
In electrochemistry,
electrochemical potential
was used instead of
chemical potential (Gibbs
free energy)
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
4.1.2 net current and exchange current
Ox e Redc
a
k
kn
Net current:
Ox (0, )c ci nFAk c t
Red (0, )a ai nFAk c t
Ox Red[ (0, ) (0, )]c a c ai i i nFA k c t k c t
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
Ox Red[ (0, ) (0, )]c a c ai i i nFA k c t k c t
At equilibrium condition
0 ,0 ,0c ai i i
cG
RTc
kTk e
h
aG
RTa
kTk e
h
If cOx = cRed = activity = 1 at re
Then i net = 0 standard exchange current
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
For unequilibrium conditions
ir re Δ ΔG nFE nF
0Δ Δ ΔG G nF
polarization
4.1.3 effect of overpotential on activation energy
transfer coefficient
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
Fraction of applied potential
alters activation energy for
oxidation and for reduction
ΔF
ΔF
ΔF
Δ cG
,0Δ cG
Δ aG
,0Δ aG
Ox en Red
ΔnF
,c re
,c ir,a ir
ΔnF
ΔnF
,0 Δa aG G nF Anode side
,0 Δc cG G nF cathode side
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
tantan
tan
/)1(tan
/tan
xFE
xFE
x
ΔnF
ΔnF
0
is usually approximate to 1/2
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
,0
,0
,0
exp( )
exp( )exp( )
exp( )
cB
c c
cB
c
c
G nFk Tk
h RT
Gk T nF
h RT RT
nFk
RT
4.1.4 Effect of polarization on reaction rate
,0
,0
exp( )
exp( )
aB
a a
a
G nFk Tk
h RT
nFk
RT
Marcus theory: transition state theory
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
Ox Ox ,0(0, ) (0, ) exp( )c c c
nFi k c t nFc t k
RT
,0 exp( )a a
nFi i
RT
No concentration polarization
,0 exp( )c c
nFi i
RT
,0
2.3 2.3lg lgc c
RT RTi i
nF nF
,0
2.3 2.3lg lga a
RT RTi i
nF nF
If initial potential
is 0, then
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
,0
2.3 2.3lg lgc re c c
RT RTi i
nF nF
,0
2.3 2.3lg lga re a a
RT RTi i
nF nF
At equilibirum
,0 ,0 0a ci i i 0
2.3lg c
c
iRT
nF i
0
2.3lg a
a
iRT
nF i
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
lg i
lg ai
lg ci
0lg i
0 re
0
2.3 2.3lg lgc c
RT RTi i
nF nF
0
2.3 2.3lg lga a
RT RTi i
nF nF
4.1 Effect of potential on activation energy
Chapter 4 Electrochemical kinetics at electrode
Ox ,0 Red ,0
0 Ox Red
(0, ) exp( ) (0, ) exp( )
(0, )exp( ) (0, )exp( )
net c a
c a
i i i
nF nFnFc t k nFc t k
RT RT
nF nFnFk c t c t
RT RT
Ox ,0(0, ) exp( )c c
nFi nFc t k
RT
Red ,0(0, ) exp( )a a
nFi nFc t k
RT
Master equation
4.2.1 Master equation
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
0 Ox Red(0, )exp( ) (0, )exp( )net
nF nFi nFk c t c t
RT RT
At equilibrium 0neti
0 0
Ox Ox Red Red(0, ) ; (0, )c t c c t c
0 0
Ox Redexp( ) exp( )nF nF
c cRT RT
0
Ox
0
Red
exp( )c nF
RTc
0
Ox
0
Red
lncRT
nF c
0
Ox
0
Red
lncRT
nF c Nernst equation
Theoretical deduction of Nernst equation from Mater equation
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
Butler-Volmer equation
0 Ox Red(0, )exp( ) (0, )exp( )net
nF nFi nFk c t c t
RT RT
0 exp( ) exp( )nF nF
i iRT RT
Butler-Volmer equation
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
1) Limiting behavior at small overpotentials
exp( ) 1nF nF
RT RT
0 1 1nF nF
i iRT RT
0 01 1nF nF nF
i i iRT RT RT
i
Current is a linear function of overpotential
i
4.2.3 discussion of B-V equation
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
0
nF
RT
i i
ct
0
nFR
RTi Charge transfer resistance
False resistance
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
2) Limiting behavior at large overpotential
0 exp( ) exp( )nF nF
i iRT RT
One term dominates
exp( )
exp 1%
exp( )
nF
nFRTnF RT
RT
Error is less than 1%
118 mV
net c a ci i i i
At cathodic polarization larger than 118 mV
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
0 exp( )nF
i iRT
0lg lg2.3
nFi i
RT
Taking logarithm of the equation gives:
0
2.3 2.3lg lg
RT RTi i
nF nF
lga b i Making comparison with Tafel equation
One can obtain 0
2.3lg
RTa i
nF
2.3RTb
nF
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
2.3RTb
nF
At 25 oC, when n = 1, = 0.5
118 mVb The typical Tafel slope
0 -100 -200 -300300 200 100
118 mV118 mV
/ mV
lg i
0lg i
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
Tafel plot: log i plot
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
4.2.4 determination of kinetic parameters
0
2.3lg
RTa i
nF
2.3RTb
nF
1.40 0.118lg i
0.5
12 2
0 1.6 10 A cmi
For evolution of hydrogen over Hg electrode
0 Ox ,0(0, ) exp( )c
nFi nFc t k
RT
13 15 10 cm sk
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
Electrode
materials
solutions Electrode
reaction
i0 / Acm-2
Hg 0.5 M sulfuric acid H++2e– = H2 510-13
Cu 1.0 M CuSO4 Cu2++2e– = Cu 210-5
Pt 0.1 M sulfuric acid H++2e– = H2 110-3
Hg 110-3 M Hg2(NO3)2
+ 2.0M HClO4
Hg22++2e– = 2Hg 510-1
1) The exchange currents of different electrodes differ a lot
4.2.5 Exchange current density
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
Electrode reaction c (ZnSO4) i0 / Acm-2
Zn2++2e– = Zn
1.0 80.0
0.1 27.6
0.05 14.0
0.025 7.0
2) Dependence of exchange currents on electrolyte concentration
High electrolyte concentration is need for electrode to achieve
high exchange current.
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
0
nF
i RTi
When i0 is large and i << i0, c is small.
0
2.3lg c
c
iRT
nF i
When i0 = , c=0, ideal nonpolarizable
electrode
When i0 is small, c is large.
When i0 = 0, c = , ideal polarizable
electrode0
2.3lg c
c
iRT
nF i
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
The common current density used for electrochemical study
ranges between 10-6 ~ 1 Acm-2.
If exchange current of the electrode i0 > 10~100 Acm-2, it is
difficult for the electrode to be polarized.
When i0 > 10-8 Acm-2, the electrode will always undergoes
sever polarization.
For electrode with high exchange current, passing current
will affect the equilibrium a little, therefore, the electrode
potential is stable, which is suitable for reference electrode.
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
4.2 Electrochemical polarization
Chapter 4 Electrochemical kinetics at electrode
4.3 Diffusion on electrode kinetic
When we discuss situations in 4.2, we didn’t take diffusion
polarization into consideration
When diffusion take effect :
0.
Ox (0, ) exp( )exp( )cB
c
Gk T nFi nFAC t k
h RT RT
0Ox 0.
Ox0
Ox
(0, )exp( )exp( )cB
c
C t Gk T nFi nFA C k
h RT RTC
Ox
net 00
Ox
(0, )exp( )c c
C t nFi i i
RTC
Chapter 4 Electrochemical kinetics at electrode
0 (1 )s
i i
d
ic c
i
Ox
net 00
Ox
(0, )exp( )c c
c t nFi i i
RTc
At high cathodic polarization
0(1 ) exp( )c
d
i nFi i
i RT
0
( )exp( )d
c
d
i ii nF
i i RT
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
0
ln ln( ) ( )d
c
d
i ii nF
i i RT
ln ln d
c
o d
iRT i RT
nF i nF i i
Therefore:
At this time the total polarization comprises of tow terms:
electrochemical term and diffusion term.
Electrochemical term Diffusion term
0
( )exp( )d
c
d
i ii nF
i i RT
Taking logarithm yields
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
Discussion :
0
lnc
RT i
nF i
1. id >> i >> i0
No diffusion ec polarization
0
ln ln d
c
d
iRT i RT
nF i nF i i
At small polarization :
c
i
At large polarization:
c
i
0
0
nFi i
RT
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
ln( )d
d
d
IRT
nF i i
0lnc
RT i
nF i
2. id i << i0
diffusion No ec
0
ln ln d
c
d
iRT i RT
nF i nF i i
is invalid
i id
i
log i
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
3. id i >> i0 both terms take effect
4. i << i0, id no polarization
0
ln ln d
c
d
iRT i RT
nF i nF i i
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
When id >>i0
0
ln ln d
c
d
iRT i RT
nF i nF i i
iddiff
ec
1/2
1
2di
0 1/ 2
0
0
0
1
2ln ln 2
ln ln ln
d
d
d
iRT RT
nF i nF
iRT RT RTi i
nF i nF nF
1/ 2
0lndi
d RT
d i nF
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
iddiff
ec
0
ln ln d
c
d
iRT i RT
nF i nF i i
1/ 2
0
ln dIRT
nF i
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
Tafel plot without diffusion polarization Tafel plot under diffusion polarization
4.3 Diffusion on electrode kinetic
Chapter 4 Electrochemical kinetics at electrode
How to overcome mixed / diffusion control? The ways to elevate
limiting diffusion current
i0 << i < 0.1 id
i between 0.1id 0.9id mixed control
i >0.9 id diffusion control
electrochemical polarization