+ - E

Determination of E 0 rev &
e.g., +Ag/AgCl/HCl/H 2
/Pt- cell-configuration
+ -
left electrode: AgCl + e - Ag + Cl -
E
rev(Ag/
AgCl)
E
0
rev(Ag/ AgCl)
RT
F
ln
a
Cl
Ag
AgCl
Cl - H +
Pt
right electrode: H + + e - 0.5 H 2
0 RT
E
E
ln
a
rev(H2
/ H ) rev(H2
/ H )
F
H
aqueous HCl
H 2
overall cell voltage:
E
rev(cell)
E
0
rev(Ag/ AgCl)
E
0
rev(H
2
/ H
)
E
0
rev(Ag/ AgCl)
E
0
rev(H
2
0V
/ H
)
RT
F
ln a
Cl
RT
F
ln
a
H
with:
a
a a
( HCl)
H
Cl
E
rev(cell)
E
0
rev(Ag/ AgCl)
2RT
F
ln a
(HCl)
E
0
rev(Ag/ AgCl)
2RT
F
ln c
HCl
2RT
F
ln
(HCl)
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 50

Determination of E 0 rev &
e.g., +Ag/AgCl/HCl/H 2
/Pt- cell-configuration
+ -
a) E 0 rev(Ag/AgCl) :
Ag
AgCl
Cl - H +
Pt
E
2RT
ln c
F
E
2RT
ln
F
0
rev(cell)
HCl rev(Ag/ AgCl)
(HCl)
aqueous HCl
H 2
considering that:
(see pg. 48)
ln
I c
( HCl)
HCl
determination of E 0 rev(Ag/Ag/Cl)
b) (HCl)
:
ln
F
2RT
0
E
rev(Ag/ AgCl)
Erev(cell)
ln
HCl
( HCl)
c
from: C.H. Hamann & W. Vielstich; Electrochemie; Wiley-VCH (2005)
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 51

Diffusion Potentials
+ -
E
rev(left)
E
0
rev(left)
0.5 H 2
H + + e - right
RT
F
ln
a
H
left
e - H +
0.5 H 2
H + + e -
metal
HCl (0.1M)
HCl (0.01M)
H +
metal
E
rev(right)
E
0
rev(right)
RT
F
ln
a
H
H 2 H 2
E
cell
E
rev(left)
E
rev(right)
E
0
rev(left)
E
0
rev(right)
RT
F
ln
a
a
H
H
left
right
RT
F
ln
a
a
H
H
left
right
0
left: H + + e - 0.5 H 2
cathodic reaction
decrease of c HCl
Cl - move to the right
Cl - flux t Cl-
H + flux t H+
+ -
right: 0.5 H 2
H + + e -
anodic reaction
increase of c HCl
H + move to the left
E
cell
E
rev(left)
E
rev(right)
E
Nernst
E junction
RT a
H
E
junction
ln
F a
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 52
H
left
right
left
right

Diffusion Potentials
derivation of E junction
lengthy, but in the simplest case for a 1:1 electrolyte:
E
junction
t
t
left
right
ln
a
left
a
right
if t +
t -
E junction
0V (e.g.: KCl: t + = 0.49; KNO 3 : t + = 0.51)
more precise evaluation: Henderson equation (s. chapter 3.2.4 in Hamann & Vielstich)
RT
F
use of a salt-bridge with concentrated KCl or KNO 3
minimizes E junction
from: C.H. Hamann, A. Hamnett, W. Vielstich; Electrochemistry; Wiley-VCH (2007)
from: A.J. Bard & L.R. Faulkner;
Electrochemical Methods; Wiley (1980)
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 53

electrochemical cells, thermodynamics, reference electrodes
ionic conduction in electrolytes / activity coefficients
electrochemical kinetics
mass transport / diffusion effects
cyclic voltammetry / rotating disk electrodes / -electrodes
AC impedance
thermoelectrica and solid electrolytes
spectroscopic methods
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 54

Electrocatalysis
heterogeneously catalyzed electrochemical reactions are a complex sequence of possibly
many steps: adsorption of reactants, desorption of products, solvation, etc.
in this sequence of processes, the rate-determining steps (rds) may be different on
different electrocatalyts
o
O + ne - R
R ; E rev(O/R)
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 55

Overpotentials
overpotentials: deviation from E rev
as current is passed through the electrode
activation overpotential, act
, is the overpotential in absence
Cl H 2
2H + + 2e -
2
+ 2e - 2Cl -
of ohmic losses and conc. gradients
metal
metal
E cell
aqueous HCl
Cl 2 H 2
Cl -
act
where:
E E
electrode
rev
act
0 for anodic process
act
0 for cathodic process
from: C.H. Hamann, A. Hamnett, W. Vielstich; Electrochemistry; Wiley-VCH (2007)
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 56

Driving Force for Electron Transfer
consider activated transition state: A + B (AB)
r
f
k
0
f
c
A
c
B
exp
G
f
RT
cathodic (reduction) reaction:
ox + ne - (Me) red
anodic (oxidation) direction:
red ox + ne - (Me)
potential getting more negative
= 2
– 1

Electrokinetics
the Butler-Volmer equation is generally used to describe the overpotential, temperature,
and reactant concentration dependence of the current of an electrode reaction *) :
based on Hamann & Vielstich,
Electrochemie, Wiley (2005):
with:
o
O + ne - R
R ; E rev(O/R)
nF
RT
i i0(T,c
,c )
rf e e
O R
(1
) nF
RT
i anodic (>0) i cathodic (

Variants of the Butler-Volmer Equation
often the Butler-Volmer equation is written as log 10 & with anodic/cathodic transfer coeff.:
i
i
0( T , c
O
, c
R
)
rf
a
F
c
F
2.303
RT
2.303
RT
10 10 i0(
T , c
O , cR
)
rf
10
b
a
10
b
c
where:
b
a,c
2.303
RT
F
a,
c
is referred to as the anodic/cathodic Tafel slope, representing the
overpotential increase required for a 10x increase in current
at 25°C, b commonly ranges from 120 mV/decade ( a,c =0.5)
to 30 mV/decade ( a,c = 2)
nF
(1
) nF
special case:
RT
RT
i i
0(T,c
,c )
rf
e e
O R
here, is also referred to as transfer coefficient, even though its meaning is
different from the definitions in [2] and [3] (actually, in [1], the symmetry factor is
meant! ...detailed explanation of differences: E. Gileadi, Electrode Kinetics...VCH)
care must be taken to not mix up these two different definitions
the more general definition with a,c as transfer coefficients will be used here
[1] A.J. Bard & L.R. Faulkner, Electrochemical Methods, John Wiley & Sons (1980); [2] J. O’M. Bockris
& A.K.N. Reddy, Modern Electrochemistry, Plenum Press: (1970); [3] J.S. Newman, Electrochemical Systems, Prentice Hall (1991).
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 59

Dependence of i 0 on the Electrocatalyst
exchange current densities vary by many order of magnitudes for different electrocatalysts
e.g., for the H 2 evolution reaction (2H + +2e - H 2 ) in acid electrolytes:
Pt
Rh
Re
Au
Ir
Cu
Ni
Co
Fe
W
Pb
Tl
Sn
Zn
Cd
Bi Ag
Ga
Mo
Nb
Ti Ta
from: S. Trasatti, J. Electroanal.
Chem. 39 (1972) 163
according to Sabatier’s Principle, high reaction rates (i 0 ’s) require bonding of
the reaction intermediate (M-H) which is not too weak and not too strong
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 60

Temperature and Concentration Dependence of i 0
as any rate constant of a chemical reaction, the exchange current density depends
on temperature and reactant/products concentrations
it is frequently based on the definition used in [3]:
i
i
0 ( T , cO
, cR
)
* * *
0( T , cO
, cR
)
c
c
R
*
R
c
c
O
*
O
e
E
RT
act
with:
i
0( T
*
*
*
, c O , c R )
[A/cm 2 real]: exchange current density at defined reference temperature and
reference reactant/product (c * R , c * O ) concentrations
, [dimensionless]: reaction orders, describing the concentration dependence of i 0
E act [J/mol]: activation energy of the exchange current density
[1] A.J. Bard & L.R. Faulkner, Electrochemical Methods, John Wiley & Sons (1980); [2] J. O’M. Bockris
& A.K.N. Reddy, Modern Electrochemistry, Plenum Press: (1970); [3] J.S. Newman, Electrochemical Systems, Prentice Hall (1991).
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 61

Characteristics of the Butler-Volmer Reaction
the Butler-Volmer equation is the summation of anodic and cathodic currents; this is shown
for the example of the HOR/HER kinetics on low-index Pt single crystals:
HOR/HER on Pt(hkl) (0.05M H 2 SO 4 at 60°C)
Pt face (110) (100) (111)
i 0 [mA/cm 2 ] 1.35 0.76 0.83
b [mV/decade] 33 44 66
(Markovic et al., J. Phys. Chem. B 101 (1997) 5405))
where rf = 1 for flame-annealed Pt(hkl) single crystals
i
net
i
0( T , c
O
, c
R
)
rf
10
b
a
i anodic
10
b
c
i cathodic
at =0: i anodic
= i cathodic
=i 0(T,H2, H+)
dynamic equilibrium
2
Pt ]
current density [ mA/cm
8
6
4
2
0
-2
-4
-6
i anodic : H 2 2H + + 2e - i net
(i 0 rf )
i cathodic : 2H + + 2e - H 2
at +b a
/2: i anodic
10 i cathodic
reverse reaction negligible
-8
-40 -30 -20 -10 0 10 20 30 40
overpotential [mV]
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 62

2
Tafel Slope Impact – HOR/HER Example
for the HOR/HER on low-index Pt single crystal faces, nearly identical i 0
’s were
reported, while the Tafel slopes varied from 33 66 mV/dec:
Pt ]
current density [ mA/cm
25
20
15
10
5
0
-5
-10
-15
-20
-25
HOR/HER on Pt(hkl) (0.05M H 2 SO 4 at 60°C)
Pt face (110) (100) (111)
i 0 [mA/cm 2 ] 1.35 0.76 0.83
b [mV/decade] 33 44 66
(Markovic et al., J. Phys. Chem. B 101 (1997) 5405))
2H + + 2e - H 2
Pt(111)
Pt(100)
Pt(110)
Pt(110)
-100 -80 -60 -40 -20 0 20 40 60 80 100
overpotential [mV]
Pt(100)
Pt(111)
H 2 2H + + 2e -
from the definition of b:
at 60ºC (333 K) and b HOR/HER
from table
2.303RT
0.033V
F
HOR , HER
2.303RT
0.066V
F
HOR , HER
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 63
the overpotential effect on current density
is the stronger, the lower the Tafel slope
near =0, i
2
1

Butler-Volmer: Linear Approximation
in the region of small overpotentials, the Butler-Volmer equation can be linearized:
for:
a
F
RT
c
F
1 and
1
RT
i.e.,
RT
F
a
and
RT
F
b
e
a
F
RT
a
F
1
RT
and
e
c
F
RT
1
c
F
RT
therefore:
i i0(
T , c , c )
O
R
rf
e
a
F
RT
e
c
F
RT
i i
0( T , c
O
, c
R
)
rf
( a
) F
c
RT
i
0( T , c
O
, c
R
)
rf
2.303
1
b
a
1
b
c
special case as defined by Bard and Faulkner:
(1
) n
a
i
i
and
0( T , c O , c )
R
rf
n
c
F n
RT
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 64

Butler-Volmer: Linear Approximation
for the previous example of HOR/HER on Pt(110):
2
Pt ]
current density [ mA/cm
6
4
2
0
-2
-4
-6
HOR/HER on Pt(hkl) (0.05M H 2 SO 4 at 60°C)
Pt face (110) (100) (111)
i 0 [mA/cm 2 ] 1.35 0.76 0.83
b [mV/decade] 33 44 66
(Markovic et al., J. Phys. Chem. B 101 (1997) 5405))
< 10% error at - < b /3
< 10% error at < b /3
-40 -30 -20 -10 0 10 20 30 40
overpotential [mV]
since:
R
ct
i
0(T,cO
,cR
)
i
rf
(
R T
F i
0(T,c
O
,c
R
)
1
rf
(
a
obtain i 0 from the slope in the
linear region, if a,c & rf are known
rf from cyclic voltammetry,
XRD, or TEM
a,c from Tafel plots
if simpler Eqn. on pg. 58 applies:
a + c = + (1-) = 1
R
ct
i
a
R T
F i
0(T,c
c
O
R T
)
F
1
,c
R
)
rf
i
c
)
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 65

Butler-Volmer: Tafel Approximation
at large overpotentials, one of the Butler-Volmer equation terms becomes negligible:
for:
1 and 1
b a
b c
i.e.,
b
and
a
b c
b
a,c
2.303RT
F
(where )
a,
c
for anodic processes (b a,c ):
i
anodic
i
0(
T , c , c )
rf
rf
O
R
ba
bc
10 10 i0(
T , cO
, cR
)
10 0.1
10
b
a
or, more commonly:
log( i ) log
0( , , )
anodic
1
i
rf
T
c
O
c
R
b
a
for cathodic processes (b a,c ): i ) logi
rf
log(
)
cathodic
0( T , c
O
, c
R
1
b
c
the log(i) vs. relationship at high is commonly referred to as Tafel equation
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 66

Butler-Volmer: Tafel Approximation
for the previous example of HOR/HER on Pt(110) and Pt(111):
HOR/HER on Pt(hkl) (0.05M H 2 SO 4 at 60°C)
Pt face (110) (100) (111)
i 0 [mA/cm 2 ] 1.35 0.76 0.83
b [mV/decade] 33 44 66
(Markovic et al., J. Phys. Chem. B 101 (1997) 5405))
log( | |i| i | ) [ mA/cm 2 Pt ]
100
10
1
0.1
(b (b a ) -1
c ) -
Pt(110)
Pt(111)
2H + + 2e - H H 2 2H + + 2e -
2
(i 0 rf )
-100 -80 -60 -40 -20 0 20 40 60 80 100
overpotential [mV]
the accuracy of the Tafel equation
at || ½b a,c is better 10%
from Tafel plots, both (i 0 rf) and
b a,c can be determined
for slow kinetics, determination of
(i 0
rf) requires extrapolation over
many orders of magnitude
large errors for ORR in PEMFCs
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 67

HOR/HER Kinetic Models
possible HOR reaction mech.:
(K. Krischer & E.R. Savinova; in: Handbook
of Het. Catalysis; Wiley (2007): ch. 8.1.1.)
Tafel: H 2
+ 2Pt 2 Pt-H
Volmer: Pt-H Pt + H + + e -
Heyrovski:
H 2
+ Pt Pt-H + H + + e -
04k - EC SS2013 (Hubert G., Tom N., Moniek T.) p. 68