Chapter 4 Electrochemical kinetics at electrode / solution interface and electrochemical overpotential
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Transcript Chapter 4 Electrochemical kinetics at electrode / solution interface and electrochemical overpotential
Chapter 4
Electrochemical kinetics at electrode / solution
interface and electrochemical overpotential
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.
2. Kinetic aspect
If electrode reaction is slow and electrochemical
equilibrium is broken. Different potential corresponds
to different activation energy.
4.1 Effect of potential on activation energy
4.1.1 basic concepts
For Elementary unimolecular
process
A
kf
kb
B
rf k f c A
Rate expressions
rb kb cB
rnet rf rb k f c A kb cB
rnet 0;
kb (cB ) eq k f (c A ) eq
At equilibrium
K
kf
kb
(cB ) eq
(c A )eq
Exchange rate of reaction
Some important empirical formula:
Arrhenius equation
k Ae
Ea
RT
According to Transition State Theory:
kT
k
e
h
G
RT
Corresponding to steric factor
in SCT
For electrode reactions
For reversible state
Ox ne
0
C
RT
E E o '
ln o0
nF CR
Nernst equation
For irreversible state
Tafel equation
How to explain these empirical formula?
kf
kb
Red
Standard free energy
Activated complex
Gc
In electrochemistry,
electrochemical
potential was used
instead of chemical
potential (Gibbs
free energy)
Ga
Reactant
re G
product
Reaction coordinate
Potential curve described by Morse empirical equation
Ox ne
kf
kb
Red
4.1.2 net current and exchange current
ia
Cu
Fe3+
Cu2+
ic
ic
ia
Fe2+
Ox ne
kc
ka
Red
ic
rc kc COx (0, t )
nFA
ia
ra ka CRed (0, t )
nFA
Net current:
ic nFAkc COx (0, t )
ia nFAka CRed (0, t )
Net current:
i ic ia nFA[kc cOx (0, t ) ka cRed (0, t )]
i ic ia nFA[kc cOx (0, t ) ka cRed (0, t )]
If cOx = cRed = activity = 1 at re
At equilibrium condition
Then i net = 0
ic ia i0 standard exchange current
Gc
kT RT
kc
e
h
Ga
kT RT
ka
e
h
4.1.3 effect of overpotential on activation energy
Ox ne
Red
ΔGc,0
ΔGa,0
ΔGa
c
ΔG
transfer coefficient
FΔ
F Δ
1
F Δ
polarization
ir re
ΔG nFE nFΔ
ΔG ΔG0 nFΔ
Ox ne
Red
ΔGc,0
c , re
a ,0
ΔG
ΔGa
c
ΔG
FΔ
F Δ
nFΔ
nFΔ
nFΔ
c ,ir
a ,ir
F Δ
Fraction of applied potential alters activation energy for
oxidation and for reduction
Anode side
cathode side
Ga Ga,0 nFΔ
Gc Gc,0 nFΔ
0
tan FE / x
nFΔ
x
nFΔ
deuce
tan (1 ) FE / x
tan
tan tan
is usually approximate to 1/2
4.1.4 Effect of polarization on reaction rate
Marcus theory: transition state theory
Gc ,0 nF
k BT
kc c
exp(
)
h
RT
G
k BT
nF
c ,0
c
exp(
) exp(
)
h
RT
RT
nF
kc ,0 exp(
)
RT
Ga,0 nF
k BT
ka a
exp(
h
nF
ka ,0 exp(
)
RT
RT
)
ic kc cOx (0, t ) nFcOx (0, t )kc ,0 exp(
nF
RT
)
No concentration polarization
ic ic ,0 exp(
ia ia ,0 exp(
nF
RT
nF
RT
If initial potential
is 0, then
)
)
2.3RT
2.3RT
lg ic ,0
lg ic
nF
nF
2.3RT
2.3RT
lg ia ,0
lg ia
nF
nF
2.3RT
2.3RT
c re
lg ic ,0
lg ic
nF
nF
2.3RT
2.3RT
a re
lg ia ,0
lg ia
nF
nF
At equilibrium
ia ,0 ic ,0 i0
2.3RT ic
c
lg
nF
i0
2.3RT ia
a
lg
nF
i0
c
lg ic
2.3RT
2.3RT
lg i0
lg ic
nF
nF
0 re
lg i
lg i0
lg ia
a
2.3RT
2.3RT
lg i0
lg ia
nF
nF
4.2 Electrochemical polarization
4.2.1 Master equation
ic nFcOx (0, t )kc ,0 exp(
ia nFcRed (0, t )ka ,0 exp(
nF
RT
nF
RT
)
)
inet ic ia
nFcOx (0, t )kc ,0 exp(
nF
) nFcRed (0, t ) ka ,0 exp(
nF
RT
RT
nF
nF
nFk0 cOx (0, t ) exp(
) cRed (0, t ) exp(
)
RT
RT
Master equation
)
Theoretical deduction of Nernst equation from Mater equation
nF
nF
inet nFk0 cOx (0, t ) exp(
) cRed (0, t ) exp(
)
RT
RT
At equilibrium inet 0
0
cOx (0, t ) cOx
;
0
cOx
exp(
nF
RT
0
cRed (0, t ) cRed
0
) cRed
exp(
0
cOx
nF
exp(
)
0
RT
cRed
nF
RT
)
0
RT cOx
ln 0
nF cRed
0
RT cOx
Nernst equation
ln 0
nF cRed
Butler-Volmer equation
inet
nF
nF
nFk0 cOx (0, t ) exp(
) cRed (0, t ) exp(
)
RT
RT
nF
nF
i i0 exp(
) exp(
)
RT
RT
Butler-Volmer equation
nF
nF
i i0 exp(
) exp(
)
RT
RT
4.2.3 discussion of B-V equation
1) Limiting behavior at small overpotentials
exp(
nF
RT
) 1
nF
RT
i
nF nF
i i0 1
1
RT
RT
nF nF
nF
i i0 1
1
i0
RT
RT
RT
Current is a linear function of overpotential
i
nF
RT
i
i0
nF
Rct
RTi0
Charge transfer resistance
False resistance
i/A
Cathode
Net current
/V
Anode
2) Limiting behavior at large overpotentials
nF
nF
i i0 exp(
) exp(
)
RT
RT
i/A
Net current
/V
One term dominates
exp(
nF
)
nF
RT
exp
1%
nF
RT
exp(
)
RT
Anode
Error is less than 1%
118 mV
At cathodic polarization larger than 118 mV
inet ic ia ic
Cathode
i i0 exp(
nF
RT
)
Taking logarithm of the equation gives:
lg i lg i0
nF
2.3RT
2.3RT
2.3RT
lg i0
lg i
nF
nF
Making comparison with Tafel equation
One can obtain
2.3RT
a
lg i0
nF
a b lg i
2.3RT
b
nF
b
At 25 oC, when n = 1, = 0.5
2.3RT
nF
b 118 mV
The typical Tafel slope
lg i
118 mV
118 mV
lg i0
300
200
100
0
-100
-200
-300
/ mV
log i0
re
Tafel plot: log i plot
4.2.4 determination of kinetic parameters
2.3RT
a
lg i0
nF
2.3RT
b
nF
For evolution of hydrogen over Hg electrode
1.40 0.118lg i
0.5
i0 1.6 1012 A cm 2
i0 nFcOx (0, t )kc ,0 exp(
nF
RT
)
k 5 1013 cm s1
4.2.5 Exchange current density
1) The exchange currents of different electrodes differ a lot
Electrode
materials
solutions
Electrode
reaction
i0 / Acm-2
H++2e– = H2
510-13
Cu2++2e– = Cu
210-5
H++2e– = H2
110-3
Hg
0.5 M sulfuric acid
Cu
1.0 M CuSO4
Pt
0.1 M sulfuric acid
Hg
110-3 M Hg2(NO3)2 Hg22++2e– = 2Hg
+ 2.0M HClO4
510-1
2) Dependence of exchange currents on electrolyte
concentration
Electrode reaction
Zn2++2e– = Zn
c (ZnSO4)
i0 / Acm-2
1.0
0.1
0.05
0.025
80.0
27.6
14.0
7.0
High electrolyte concentration is need for electrode to achieve
high exchange current.
2.3RT ic
c
lg
nF
i0
nF
i RTi0
2.3RT ic
c
lg
nF
i0
When i0 is large and i << i0, c is small.
When i0 = , c=0, ideal nonpolarizable
electrode
When i0 is small, c is large.
When i0 = 0, c = , ideal polarizable
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 potential on electrode kinetics
Ox ne
kc
ka
Red
Gc
k BT
ic k
COx (0, t ) exp(
)
h
RT
Ga
k BT
ia k
CRed (0, t ) exp(
)
h
RT
Shift of potential
Gc Go,c nF
1 keeps
Ga Ga nF constant
The nature of potential -dependence of rate
ic nFAk f CO x (0, t )
G0.c
k BT
nF
nFACOx (0, t )k
exp(
) exp(
)
h
RT
RT
nF
i0,c exp(
)
RT
At equilibrium: cox(0,t)= cox0 i0,c=i0,a=i0
i0 exp(
nF
RT
)
Master equation:
inet
nF
nF
nFk0 cOx (0, t ) exp(
) cRed (0, t ) exp(
)
RT
RT
Master equation:
inet
nF
nF
nFk0 cOx (0, t ) exp(
) cRed (0, t ) exp(
)
RT
RT
At equilibrium inet 0
0
cOx (0, t ) cOx
;
0
RT cOx
ln 0
nF cRed
0
cRed (0, t ) cRed
Nernst equation
Master equation:
inet
nF
nF
nFk0 cOx (0, t ) exp(
) cRed (0, t ) exp(
)
RT
RT
Butler-Volmer equation
nF
nF
i i0 exp(
) exp(
)
RT
RT
Butler-Volmer equation
nF
nF
i i0 exp(
) exp(
)
RT
RT
at small overpotentials
nF
Rct
RTi0
nF
i i0
RT
Charge transfer resistance
at large overpotentials
118 mV
inet ic ia ic
2.3RT
2.3RT
lg i0
lg i
nF
nF
2.3RT
a
lg i0
nF
Tafel equation
2.3RT
b
nF
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 :
G0.c
k BT
nF
ic nFACOx (0, t )k
exp(
) exp(
)
h
RT
RT
COx (0, t ) 0 k BT
G0.c
nF
ic nFA
COx k
exp(
) exp(
)
0
h
RT
RT
COx
inet
COx (0, t )
nF
ic
i0 exp(
c )
0
RT
COx
inet
COx (0, t )
nF
ic
i0 exp(
c )
0
RT
COx
At high cathodic polarization
i
C C (1 )
id
s
i
0
i
i
nF
i (1 )i0 exp(
c )
id
RT
id i
i
nF
(
) exp(
c )
i0
id
RT
id i
i
nF
(
) exp(
c )
i0
id
RT
Taking logarithm yields
Therefore:
id i
i
nF
ln ln(
)(
c )
i0
id
RT
id
RT
i RT
c
ln
ln
nF io nF id i
Electrochemical term
Diffusion term
At this time the total polarization comprises of tow terms:
electrochemical term and diffusion term.
id
RT
i RT
c
ln
ln
nF i0 nF id i
Discussion :
1. id >> i >> i0
No diffusion
ec polarization
At small polarization :
i
At large polarization:
i
c
nF
i i0
RT
0
c
RT
i
c
ln
nF i0
id
RT
i RT
c
ln
ln
nF i0 nF id i
2. id i << i0
diffusion
No ec
RT
i
c
ln 0
nF i
is invalid
Id
RT
d
ln(
)
nF
id i
i id
i
log i
id
RT
i RT
c
ln
ln
nF i0 nF id i
3. id i >> i0 both terms take effect
4. i << i0, id no polarization
id
diff
1
id
2
ec
1/2
0 1/ 2
When id >>i0
id
RT
i RT
c
ln
ln
nF i0 nF id i
1
id
RT
RT
2
ln
ln 2
nF
i0
nF
id
RT
RT
RT
ln
ln id
ln i0
nF i0 nF
nF
d1/ 2
RT
d ln i0 id nF
id
diff
id
RT
i RT
c
ln
ln
nF i0 nF id i
ec
1/ 2
Id
RT
ln
nF i0
lg id
lg i
118 mV
lg i0
400
300
200
100
0
-100
-200 -300 -400
Tafel plot under diffusion polarization
/ mV
Tafel plot with diffusion control:
i0 << i < 0.1 id
Electrochemical polarization
i between 0.1id 0.9id mixed control
i >0.9 id
diffusion control
How to overcome mixed / diffusion control?
The ways to elevate limiting diffusion current
4.4
EC methods under EC-diff mixed control
4.4.1 potential step
Using B-V equation with consideration of diffusion polarization
COx (0, t )
CRed (0, t )
nF
nF
it i0
exp(
)
exp(
)
0
0
RT
RT
CRed
COx
at high polarization . c
COx (0, t )
nF
it i0
exp(
c )
0
RT
COx
c constant
it CO(0,t)
COx (0, t )
nF
it i0
exp(
c )
0
RT
COx
nF
at low polarization :
RT
exp(
c
nF
RT
is very small
c ) (1
nF
RT
c )
CO (0, t ) CR (0, t ) nF
CO (0, t )
CR (0, t )
it i0
(
)
0
0
0
0
RT
CR
CO
CR
CO
Constant
Constant
1
1
2
it i exp( 2 t )erfc(t )
i
1
2
3 2 t
is the current density at no concentration polarization at
0
cOx (0, t ) cOx
;
t=0
0.5
0
cRed (0, t ) cRed
1
2
exp( 2t )erfc(t ) 1
i(0)= i
no concentration polarization
When
t
1
2
it t
1
it ic
1/ 2
at time right after the
potential step : it t1/2
is linear
ic
1
2
t
1/ 2
ic
it
Double-layer charge
i
EC control
Extrapolating the linear
part to y axes can obtain
2
diff control
C
Making potential jump to different can obtain i at
different . Then plot i against c can obtain i~c without
concentration polarization.
The way to eliminate concentration polarization effect
c time constant s
it > i due to charge of double layer capacitor
4.4.2 current step
i
cathodic current : 0 ic
ic
nF
ic nFk ' COx (0.t ) exp
( 0 )
RT
0
t
c at different i0
3.8.2 Current step / jump
1
2
t
ci (0, t ) ci0 1
t
1
2
ic nFk ' C [1 ( ) ]exp(
0
Ox
nF
RT
)
t
1
2
ic nFk ' C [1 ( ) ]exp(
0
Ox
nF
RT
)
1
0
nFk
'
C
RT
RT
t 2
Ox
c (t )
ln
ln[1 ( ) ]
nF
ic
nF
c (t )
t
1
2
[1 ( ) ]
transition time
c
c
c(0)
when potential step
to next rxn.
0
i= i charge
t
The slope of the linear relationship between c (t) can be used to
determine n and .
When t0 the second
term = 0
(t 0)
0
nFK ' COx
RT
ln
nF
ic
4.4.3 cyclic voltammetry (CV)
(t ) 0 vt
COx (0, t )
CRed (0, t )
nF
nF
it i0
exp(
c (t ))
exp(
c (t ))
0
0
RT
RT
CRed
COx
c (t ) (t ) 0
I
Typical CV diagram for
reversible single electrode
I Pc
Potential separation
Δ Pc Pa
I Pa
59
mV
n
For typical CV diagram of
irreversible single electrode
I
I Pc
59
Δ
mV
n
I Pa
for fast EC reaction : i << i0
controlled by diffusion
i
v
0.2
0.1
0.0
0.1
0.2
for the reversible systems , use the forward kinetics only :
i nFACO (0, t )k f (t )
can be only by numerical method:
1
2
Ox
1
2
0
i nFACOx
D (
nF
RT
1
2
1
2
) x(bt )
Nicholson-Shain equation
tramper coefficient
n – number of electrons involved in charge transfer step
1
2
x(bt )
is tabulated
x (bt) max =0.4958
For totally irreversible systems, peak potential shift with scan
rate
RT
k
nFv
Ep E (
)[0.78 ln
ln
]
F
RT
D
v
i
0.2
0.1
0.0
0.1
0.2
for slow EC reaction : ii0
( quasi reversible, irreversible)
in comparison to the same rate,
equilibrium can not establish
rapidly. Because current takes
more time to respond to the
applied voltage, Ep shift with
scan rate .
1.15 RT 30
Ep E
mV
n F
n
E p E1/ 2 p
1.857 RT 47.7
nF
n
per decade of change
in scan rate
56.5
P P
mV
n
2
drawn - out
ip (2.99 10 )n( n)
5
1/ 2
0
Ox
1/ 2 1/ 2
AC D v
I P 2.69 10 n
5
ip COx0 lower due to
if =0.5 n= 1
i p ,ir
i p , re
0.78
3/ 2
D c
1/2
Ox
1/ 2 0
Ox
i p 0.227 FAC k exp[
0
Ox
o
RT
nF
( E p E 0 )]
lnip Ep E0 is linear with S= RT/nF, intercept is linearly
proportional to k0
4.4.4 effect of 1 potential on EC rate :
1=0, validate at high
concentration or larger
polarization
1
nF
x
effect of 1:
1.on concentration
2. = 1
zi F
C C exp( 1 )
RT
i
0
i
( n z0 ) F
nF
ic nFAk C exp(
) exp(
1)
RT
RT
0
Ox
ia nFAk
0
Re d
0
Ox
0
Re d
C
( n z0 ) F
nF
exp(
) exp(
1)
RT
RT
When z0 <0 ( minus ) n 1 large . For anion
reduced on cathode , 1 effect is more significant.
z0 n
RT
i ic ia const
ln i
1
nF
n
z0 n
RT
c const
ln i
1
nF
n
z0 n
1 0 1 made c shift positively
n
minus <0
When z0 n
plus >0
-0
so: if 1 increases, i decreases
0.001M
0.01M
0.1M 1M
0.5
1M
0.1M
0.01M
without specific adsorption
reduction of +1 cation
…… reduction of 1anion
+0.5
1 accelerates
4
2
0
+2
reduction of
cations slow reduction of anion
+4
lgi
z0 n
RT
c const
ln i
1
nF
n
if: n =z0
=0.5
Cu2+ +2e- = Cu
MnO4 +e = MnO42
H+ +e- = 1/2 H2
RT
c const
ln i 1
nF
if :z0 =0
RT
c const
ln i 1
nF
adsorption of anion slow reaction
Electrochemistry of LB film
exam:
1.Draw the potential change versus distance away from
electrode surface according to Stern electric double-layer and
indicate 1 potential
2. When the electrode was positively charged, the surface
concentration of action is still more than that of bulk solution.
Explain this phenomenon using specific adsorption model
3.The differential double layer capacitance of Cu/H2O surface
is 10-5 Fcm-2 while that for Cu/HS(CH2)11CH3 is 10-9Fcm-2
(can be taken as zero). If
the differential Cdl for a
Cu/HS(CH2)11CH3 system is measured to be 10-7 Fcm-2.
Please calculate the coverage of HS(CH2)11CH3 on copper.
4. Electro-capillary curves of Hg in KI and K2S solution are
shown in the Figure.
Please indicate the PZC of Hg on the
curves and explain the difference in
PZC.
KI
K2S
The curves coincide with each other
when potential is quite negative but
differ a lot when potential is positive,
please give explanation.
5.Tell how to determine whether or not a electrode process is
governed by diffusion. Given id for RDE can be expressed as
id= 0.62 nF Di2/31/2-1/6 ci0
Pt
6.This is a water drop with contact
angle of on Pt surface. When
potential shift negatively, plot the
change of with potential, i.e., ~ .
7.Deposition of Cu nanowire in microspore of anodic alumina
membrane (AAO) can be taken as ideal stable diffusion process. If
0
C
the thickness of (AAO) is 1m Cu =0.1 mol cm-1, DCu =105cm2
s-1. Please calculate the limiting diffusion current.
2
2
8. Convection affects diffusion. If the slope of concentration
gradient is Ci ,The effective thickness of diffusion layer E=
x x 0
and the dimity diffusion current id =
9.a typical CV peak is shown in
the figure. Please Indicate EP, EP/2,
Ere, and iP on it.
How can you determine whether or
not this electrochemical process is
electrochemical reversible?