Chapter 5
Mechanism of complex electrode reaction
5.1.1 B-V equation for multi-electron process
For a di-electron reaction
Ox + 2e Red
Its mechanism can be described by
0
0
Ox 1e X
X+1e Red
a
b
i
i
At stable state d[X]
0dt
0 x0x
exp( ) exp( )2
a aa c c
F Fcii
RT RTc
0 x0x
exp( ) exp( )2
b ab c c
F Fcii
RT RTc
0 0
( ) ( )exp exp
1 12exp exp
a b a bc c
b ac c
a b
F F
i RT RTF F
RT RTi i
If 0 0a bi i
0 (1 )2 exp( ) exp( )b a
b c c
F Fi i
RT RT
0
0
Ox 1e X
X+1e Red
a
b
i
i
0 (1 )2 exp( ) exp( )b a
b c c
F Fi i
RT RT
Therefore 0 02 bi i
1
2b
2b
Consider a multi-step electrochemical process proceeding via the following mechanism
5.1.2 important consideration
Ox 'e Ox '
Ox ' e Red'
Red'+ ''e Red
n
n
Net result of steps preceding rds
(r.d.s.)
Net result of steps following rds
Note: n’+n’’+1 = n
0 0 ' 0 'O ' R '(0, ) exp( ( ) (0, ) exp( ( )rds rds rds
nF nFi nFAk c t c t
RT RT
Since preceding step is in equilibrium, one can write
'
'
ln'
o Opre
O
cRT
n F c
0 ''
'exp ( )O O pre
n Fc c
RT
Similarly, the succeeding reaction is also assumed to be
fast, i.e., at equilibrium
0 ' 'ln''
Rpost
R
cRT
n F c
0 '
'
'exp ( )O
preO
c n F
c RT
0 ''
''exp ( )R R post
n Fc c
RT
0 0 ' 0 'O ' R '(0, ) exp( ( ) (0, ) exp( ( )rds rds rds
nF nFi nFAk c t c t
RT RT
0 ''
''exp ( )R R post
n Fc c
RT
Replacing above
0 ' 0 'O
0
0 ' 0 'R
'exp ( ) exp( ( )
'' exp ( ) exp( ( )
pre rds
rds
post rds
n F nFc
RT RTi nFAk
n F nFc
RT RT
0 0 ' 0 'exp ' exp ( ' )c rds pre rds
F Fi k n n
RT RT
0 0 ' 0 'exp '' exp ( '' )a rds post rds
F Fi k n n
RT RT
O R(0, ) (0, )c ai nFA k c t k c t
0 ''
'exp ( )O O pre
n Fc c
RT
After very laborious algebra, one can show that
0 00
(0, ) (0, )( ' ) ( '' )exp expO R
O R
c t c ti n F n F
i RT RTc c
This equation correctly accounts for influence of redox pre-
equilibrium on measured value of Tafel slop for the reaction
scheme.
Tafel slope is not Tafel slope of rate determining step, ,
rather it is (n’+)
without considering concentration effects
0
( ' ) ( '' )exp exp
n F n Fi i
RT RT
By making comparison with
0 exp expnF nF
i iRT RT
The effect of potential change on activation energy of the
cathodic and anodic reaction differ from that of simple
electrochemical reaction
', '' 1n n
At small overpotentials, i.e., in the linear regime:
0
nFi i
RT
Therefore, charge transfer resistance for multi-step is:
0ct
RTR
nFi
The exchange current is n times that of the current of the r.d.s.
0
( ' )expc c
n Fi i
RT
At higher negative polarization
At higher negative polarization0
( '' )expa a
n Fi i
RT
0
( ' )expc c
n Fi i
RT
0 c= log +- log( ' ) ( ' )c
RT RTi i
n F n F
For a multi-electron reaction
Ox + ne Red
Its mechanism can be described by 0
0
0
0
0
0
0
1
1 2
2 1
1
1
2 1
1
Ox 1e X
X +1e X
X +1e X
X +1e X (rds)
X +1e X
X +1e X
X +1e Red
a
b
b
b
b
b
b
i
i
i
j j
i
j j
i
j j
i
n n
i
n
Steps before rds, with higher i0 at equilibrium
Steps after rds, with higher i0 at equilibrium
0( 1) ( )
exp( ) exp( )j jj c c
j F n j Fi ni
RT RT
Therefore 0 0ji ni
1j j
n
j n j
n
0( 1) ( )
exp( ) exp( )j jj c c
j F n j Fi ni
RT RT
2 0j c
Fi n i
RTAt small overpotential
2 00 ji n i
At higher overpotential
0( 1)
exp jc j c
ji ni
RT
For cathodic current
0( 1)
exp ja j a
ni ni
RT
For anodic current
5.1.3 Stoichiometric number multi-electron process
5.2 surface transitions reactions:
Ob
Surface region
Bulk solution
Os
Mass transfer
O*Chem. rxn
O* Desorption/ adsorption
R*
EC rxn
R*
Desorption/ adsorption
RbRsMass transfer
Chem. rxn
5.2 Homogeneous proceding surface reactions
place
homogeneous ( region close to electrode surface)
heterogeneous ( adsorption, desorption, new phase formation )
time
Foregoing / preceding
Post, succeeding
parallel
Electrochemical -chemical (EC)
Chemical-Electrochemical (CE)
Classification of couple electrode homogeneous :1) Mechanism with single electrochemical step
(1) CE – preceding reaction
e.g. Reduction of formaldehyde on mercury
Dominant, no EC rxn.
2 2
2
CdX Cd X
Cd 2e Cd(Hg)
f
b
k
k
A difficult to be reduced CE A H HA Pne
1) Mechanism with single electrochemical step
(2) EC – following reaction
NH2HO NHO + 2H+ + 2e
NHO + H2O OO + NH3
O
O
e-
O
O
H2OO
O
EC
H+ + M +e M H2 M H 2M + H2
H+ + M H + e M + H2
Possible proceeding/succeeding reactions:
dissociation, complexities, dimerization, isomerization ,
formation of new phase (gas bubble, metal plating, conversion
layer).
EC
EE
For evolution of hydrogen
1) Mechanism with single electrochemical step
(3) ECcat – catalytic reaction
Fe3+ + 1e- Fe2+
2Fe2+ + H2O2 Fe3+ + 2H2O+ 2H+ 2
*
*
O O
O + R
f
b
k
k
ne
For CreEre reaction as f
b
kK
k
If K <1, then O is the main reactant which can be reduced at potential 2, while O* is easier to be reduced at potential 1 than O. This means at 2, both O and O* can be reduced.
* ,d O dI I
At 1, For fast chemical kinetics, O* can be replenished in time:
* ,,d O dO dI I I
Limiting kinetic current Ik
5.2 Reaction mechanism-proceeding reaction
At 1, For slow chemical kinetics:
1 2
At 2, For slow chemical kinetics:
* ,,d O dO dI I I
Curves I and II can be described by normal diffusion current when O and O* become totally depleted, respectively.
Curve III is different.
At electrode surface, the concentration gradient of O and O* can be described as:
*
2
2O O
O f O b O
c cD k c k c
t x
* *
* *
2
2O O
f O bO O
c CD k C k C
t x
At stable state: 0Oc
t
*
0Oc
t
*O Oc cIf: At 1
*
2
20O
O f O b O
cD k c k c
x
Very small
No concentration polarization of O at electrode surface.
* *
* *
2
02
O Of O bO O
c CD k C k C
t x
For O* at complete concentration polarization, its boundary conditions are:
* (0, ) 0O
c t At x = 0, At x = , *
0( , ) OOc t Kc
* *
*
0 1 exp b
O OO
kc c x
D
*
*
*
0
0
O b
OOx
c kc x
x D
Therefore, the concentration gradient at electrode surface is:
surface concentration:
*
*
*
0
0
O b
OOx
c kc x
x D
The thickness of reaction region
* *
*
0
0
O O
bO
x
c D
kc
x
* *
*
0
0
O O
bO
x
c D
kc
x
Less than the effective thickness diffusion layer, why?
* * * *
1/2 1/2 0 1/2 1/2 0( )k b fO O O OI nFD k c nFD Kk c
* * *
1/2 1/2 0( )sk bO O O
I nFD k c c
At incomplete polarization:
The limiting current resulted for CE mechanism is usually much larger than that of merely diffusion control kinetics, Why?
When = 0 V, c0 = 1 mM , A= 1cm2, DA = DB = DC = 10-5 cm2
/s, K =103, kf = 10-2 s-1, kb =10 s-1, T =25 , at scan rates ,℃ v of
(1) 10 V/s; (2) 1 V/s; (3) 0.1 V/s; (4) 0.01 V/s.
Cyclic voltammograms for the CE case.
A B;
B + e - C
When K=10-3, kf =10-2 s-1 kb = 10 s-1, v=0.01~10 V s-1, = 26 ~ 0.026.
v / Vs-1 lg control effect of preceding
10 -1.6 DP Less effect (1)
1 -0.6 KI
0.01 1.6 KP Depends on cre not on diffusion
0 2 4 6 8 10
0.5
1.0
1.5
2.0
v
,
,
p a
p c
i
i
1/ 2 1( )K
Some diagnostic criteria for a CE situation .
1) ip /v1/2 will decrease as v increases
2) ipa /ipc will become large for small K or for large v
The first wave The first wave corresponds the reduction corresponds the reduction of Cdof Cd2+2+ which is governed which is governed electrochemically, while electrochemically, while the second wave the second wave corresponds to reduction of corresponds to reduction of CdXCdX--. Wave III is oxidation . Wave III is oxidation of Cd(Hg) which is of Cd(Hg) which is governed by diffusion. governed by diffusion.
2 2
2
CdX Cd X
Cd 2e Cd(Hg)
f
b
k
k
Both O and O* can be reduced
O e Rn R S O Pk
Assuming [S] >> [O]
Electrocatalysis
5.3 Reaction mechanism-succeeding/parallel reaction
3 2Fe e Fe
2 32 2
1Fe + H O == Fe OH
2
5.3.1 For EreCcat
Catalytic decomposition of hydrogen peroxide
S is the substrate whose concentration is usually much higher than that of O and R. Therefore, I mainly depends on Id, O.
3 2Fe e Fe 2 32 2
1Fe + H O == Fe OH
2
O e Rn R S O Pk
Assuming [S] >> [O]
0O RO R O O
c cJ J D D
x x
2
20O
O f b O f total
cD k k c k c
x
2
20O
O f b O f total
cD k k c k c
x
Solution is
0 1 expO total
O
f b
xc c
D
k k
0
0
total O O
O f b f
x
c D Dc k k kx
When Concentration of R is very low
1/ 21/ 2 0,c d O f b OI nFD k k c
Catalytic current at complete concentration polarization
Catalytic current at other polarization
1/ 21/ 2 0,
sc d O f b O OI nFD k k c c
1/ 2/i
Increasing
i
1/ 2
diffusionECcat
Here both behaviors going on: we are consuming Red with rate constant k, this will shift the ratio [Red]/[Ox]. So we expect the half wave potential to shift. But, we also are generating Ox with rate k. So we expect the wave to get bigger.
5.3.2 For EreCir reaction
EC
- *
k*
E O + e R
C R R
*
2O O
O R2 Rt f b
c cD k c k c
x
* *
* *
2
R RR2R Rt f b
c cD k c k c
x
For ECir mechanism:
* * *
1/ 2 1/ 2 0
R R R
sk fI nFD k c c
If is negligible
The kinetic current is
*
0
Rc
* *
1/ 2 1/ 2
R R
sk fI nFD k c
The thickness of the reactive layer *R
f
D
k
for the EC reaction when the electron transfer reaction is reversible and the chemical rate constant kEC is extremely large
EreCir
The reduction in size of the reverse peak occurs since much of the R produced electrochemically is destroyed by the chemical step.
EC
O+ R
R Pk
ne
A/B * = 0 V, c0=1 mM, A =1 cm2, D = 10-5 cm2 /s, and kf = 10 s-1. The vertical
scale changes from panel to panel.
Scan rates on voltammograms
Conversion rate constant on Conversion rate constant on voltammogramsvoltammograms
http://www.nuigalway.ie/chem/Donal/Surfaces11.ppt#274,13,Catalytic
180 120 60 600
0.2
0.0
0.2
0.4
(e)
Nor
mal
ized
cur
rent
( 1/2) n / mV
= 10 0.1
0.01
0.1
0.01
Normalized current for several values of .
For small , reversible by nature. For large , no reverse current can be observed, i.e., irreversible.
k RT
v nF
0.2
0.4
0.6
0.8
1.0
lgv
I p,c/ I p,c
1) ipa / ipc will approach 1 as v
increases
2) ipc proportional to v1/2
3) pc will be displaced in the anodic direction as v decreases
(30/n mV per 10 in v)180 120 60 600
0.2
0.0
0.2
0.4
(e)
Nor
mal
ized
cur
rent
( 1/2) n / mV
= 10 0.10.01
0.1
0.01
Diagnostic Criteria for EreCir mechanism:
Electrochemical dimerizationElectrochemical dimerization
5.4.1 Conversion involving adsorption
Osol Oads
RadsRsol
solads
sol ads y y
rad 10* 0 coverage
rde 0* maximum coverage 0
* at equilibrium
0 00 0
1
1o o
pe pe
o o
i i i
1 1o at large negative polarization : rxn, fast
0o
So 01
ope
d
o
ii
When 0 1o make adsorption .id = io
0lnre o
RTconst
nF lnir o
RTconst
nF
0
ln ore ir
o
RT
nF
0, 1oo
00
(1 )ope
o
i i
0 0
1 o d
dpe o
i ii
ii
ln( )d
d
IRT
nF I I
For proceeding reaction, its polarization curves is similar to that of diffusion-control kinetics.
post kinetic :0
Repostine
Ox d R
0 0Re0, 0Re
1
1d R
post postd R
i i i
Using similar treatment : ln 1post
RT i
nF i
so ln lnopost
RT RTi i
nF nF
For succeeding reaction, its polarization curves is similar to that of electrochemistry-control kinetics.
Since R and O are confined, no diffusion
If we use the Langmuir isotherm to describe the coverages of O and R
make use of the Nernstian criterion
5.4.2 Conversion of surface species
When bO bR,
Reversible, Nernstian, Langmuir, Monolayer
Electrochemistry of LB film
Dynamics of Br electrosorption on single-crystal Ag(100)
Journal of Electroanalytical ChemistryVolume 493, Issues 1-2, 10 November 2000, Pages 68-74
* *( / ) ln Op re ad ad
R
bRTO R
nF b
y
If bR >>bOIf bO >> bR
Pre-wave post-wave
Dash line: without adsorption
Solid line: with adsorption
1
2
O+ B
B+ P
e
e
1 2
12
0
1 2
0
1 2
0
5.5 Other mechanisms
5.5.1 EreEre mechanism
Changing shapes of cyclic voltammograms for the Er Er reaction scheme at different values of E0
When > 125 mV, two peaks becomes distinguishable
Shoulder
CVs for the reduction of di-anthrylalkanes (An-(CH2)n-An) in 1:1 benzene/acetonitrile containing 0.1 M tetrabutylammonium perchlorate at a Pt electrode.
A Be 01
B Cb
f
k
k /f bK k k
C De 02
0 02 1
It is much easier for C to be reduced than A
180mV
5.5.2 EreCreEre mechanism
The figure shows the voltammogram for an ECE mechanism where the product (S) is more difficult to reduce than the starting material (O).
O R S T
If the product (S) is more easy to reduce, slightly different behaviour is seen
0.00 0.60
0.00
4.00
3.00
2.00
1.00
3.00
2.00
1.00
Cur
rent
=
0
(a)E
0.10
II
I
Cur
rent
=
0.4
0
0.00 0.60
0.00
4.00
3.00
2.00
1.00
2.00
1.00
(c)E
0.10 III
I
IV
Cur
rent
=
0.0
5
0.00 0.60
0.00
3.00
2.00
1.00
2.00
1.00
(b)E
0.10 III
I
IV
II
0.00 0.60
1.00
5.00
4.00
3.00
2.00
2.00
1.00
0.00C
urre
nt
= 0
(d)E
0.10III
I
IV
CVs for the EreCirEre case obtained by digital simulation for E10
= 0.44 V, E20 =
0.20 V for different values of =(kb/v)(RT/F); n1=n2=1.(a) =0 (unperturbed
Nernstian reaction ); (b) 0.05 ;(c) 0.40 ;(d) 2.
The ECE mechanismThe ECE mechanism
Figure 5.5 – CV of sample B67 in 3 10-3 mol dm-3 4-aminophenol & 0.5 mol dm-3 H2SO4
Various scan rates, Ag dag contact, geometric area of working electrode = 20 mm2
CV of 4-Aminophenol
cyclic voltammograms recorded
using a highly doped diamond
electrode in an aqueous solution
containing 3 10-3 mol dm‑3 4-
aminophenol ( C6H4(OH)(NH2) ),
and 0.5 mol dm‑3 sulphuric acid
(H2SO4).4-aminophenol is an
aromatic organic molecule, which
may undergo a two step oxidation.
The cyclic voltammograms show
two oxidation peaks and two
reduction peaks per scan
http://www.chm.bris.ac.uk/pt/diamond/mattthesis/chapter5.htm
CCrere E Erere
(as above)(as above)
CCrere E Eirir
Diffusion equation (all Diffusion equation (all xx and and tt))ReactionReactionCaseCase
Y Of
b
k
k
O Rne
2Y Y
Y f Y b O
c cD k c k c
t x
2
O OO f Y b O
c CD k C k C
t x
2
2R R
R
c cD
t x
Y Of
b
k
k
O Rne
Summarization
(as above, with (as above, with kkbb = 0) = 0)
(equation for (equation for ccYY not required ) not required )
EEre re CCirir
EEre re CCrere O Rne
R Yf
b
k
k
2
2O O
O
c cD
t x
2
R RR f R b Y
c cD k c k c
t x
2
Y YY f R b Y
c cD k c k c
t x
O Rne
R Yfk
EEre re CC2ir2ir O Rne
2R Xfk
2
2O O
O
c cD
t x
22
2R R
R f R
c cD k c
t x
The zones are DP, pure diffusion: DM, diffusion modified by equilibrium constant of preceding reaction: KP pure kinetics: and KI, intermediate kinetics.
Here,
f bk k RT
v nF
CreEre reaction diagram with zones for different types of
electrochemical behavior as a function of K and (defined in
the following table).
Treatment depend on scan rate and on particular technique :
Dimensionless Parameters for Voltammetric Methods
Technique TimeParameter (s)
Dimensionless Kinetic parameter, , for
CE EC EC
Chronoamperometryand polarography
t (kf +kb ) t k t k Cz* t
Linear sweep and cyclic voltammetry
1/ [(kf +kb )/v] (RT/nF)
(k/v )(RT/nF)
[(kcz* )/v]
(RT/nF)
Chronopotentiometry (kf +kb ) k k Cz*
Rotating disk electrode 1/ (kf +kb )/ k/ k cz* /
for large kf and kb, p will be displaced as a function of v .
(30/n mV per 10 times v)
'1
2
0.277 ln2
RT RT
nF nF y
1
2
ln
dRT
d v nF
10.471
1.02d
i
iK
5.5 Methods for mechanism study
Tafel Equation - “Simple” Electron Transfer
0
2.3 2.3lg lg
RT RTi i
nF nF
For a simple 1 electron process
slope = 1 / 120 mV
For a simple 2 electron process
slope = 1 / 60 mV
Assuming that ==0.5
Using the Butler-Volmer and Tafel Equation to Determine Multistep Reaction Mechanisms
Mechanism (A): Cu2+ + 2e = Cu orMechanism (B): Cu2+ + e = Cu+
Cu+ + e = Cu
For mechanism (A): n = 2 , = 0.5
Plotting logi against gives a straight line with a gradient of - [60 mV]-1.Similar arguments for reverse reaction: Cu Cu2+ + 2e, gives a straight line with a gradient of [60 mV]-1.
Mechanism B (Forward Reaction)
assume (1) is rate determining step (r.d.s.):
For mechanism (B): n = 1 , = 0.5
Hence, for Cu deposition with Cu2+ + e Cu+ (r.d.s.)
Cathodic section of Tafel plot (logi vs. )
gives a slope of - 1 / 120 [mV]
Mechanism (B): Cu2+ + e = Cu+
Cu+ + e = Cu
Mechanism B : Reverse Reaction
Reverse: Cu+ Cu2+ + e also r.d.s.
rapid step (2) in equilibrium. Can use Nernst eq. to find [Cu+]:
Tafel slope = 1 / 40 [mV]
Mechanism (B): Cu2+ + e = Cu+
Cu+ + e = Cu
5.7 determination of intermediate
Rotating Ring-Disk Electrodes
Reversal techniques are obviously not available with the
RDE, since the product of the electrode reaction, R, is
continuously swept away from the surface of the disk.
addition of an independent ring electrode surrounding the disk.
By measuring the current at the ring electrode with the potential maintained at a given value, some knowledge about what is occurring at the disk electrode surface can be obtained. For example, if the potential of the ring is held at a value at the foot of the O+ ne → R wave, product R formed at the disk will be swept over to the ring by the radial flow streams where it will be oxidized back to O (or “collected”).
The theoretical treatment of ring electrodes is more complicated than that of the RDE, since the radial mass transfer term must be included in the convective-diffusion equation.
The current at the ring electrode is given by
3
20
2r
OO r
y
ci nFD rdr
y
The solution to these equations yields the limiting ring current:
3 3 2 / 3 2 / 3 1/ 6 1/ 2 03 20.620 ( ) O Oi nF r r D v c
Levich equation for disk electrode:
2 / 3 1/ 6 1/ 2 00.620 O Oi nFAD v c
3 3 2 / 33 2
21
( )R D
r ri i
r
Notice that for given reaction conditions (co0 and ) a ring electrode
will produce a larger current than a disk electrode of the same area. Thus the analytical sensitivity of a ring electrode (i.e., the current caused by a mass-transfer-controlled reaction of an electroactive species divided by the residual current) is better than that of a disk electrode, and this is especially true of a thin ring electrode. However, constructing a rotating ring electrode is usually more difficult than an RDE.
RRDE experiments are usually carried out with a bipotentiostat,which allows separate adjustment of ED and ER.
Several different types of experiments are possible at the RRDE:
Collection experiments, where the disk-generated species is observed at the ring
Shielding experiments, where the flow of bulk electroactive species to the ring is perturbed because of the disk reaction, are the most frequent.
Example for a collection experiment: the ring (b) measures the reduction of peroxide produced at the disk (a) during the electroreduction of oxygen.
the ring current is related to the disk current by a quantity N,
the collection efficiency ; this can be calculated from the
electrode geometry, since it depends only on r1, r2, and r3 and is
independent of c, , DR,DO
R
D
iN
i
5.6.3 detection of intermediates using the same electrode (CV)
5.6.4 detection of intermediates using thin-layer cell and spectroscopy