ChE 4520/5520: Electrode
Kinetics
Gerardine G. Botte
Objective/Introduction
•  In previous chapters we have calculated the thermodynamics feasibility of
an electrochemical reaction
•  In this chapter we will learn how to evaluate the kinetics feasibility
•  We will learn about different models that are used to simulate
electrochemical reaction rates
•  Electrochemical reactions rates offer advantages to chemical reactions
–  e.g., a 1 V change could represent a different of 10 order of magnitude in
temperature
–  They can be controlled easier than chemical reactions by adjusting the
potential difference
•  One of the disadvantages of electrochemical reactions is that their
mechanisms are much more complicated than for chemical reactions
•  We will neglect any mass transfer limitations in this chapter (which means
we will have plenty of reactants at the surface of the electrodes)
2
Outline
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  Interface Role
•  Electric Double Layer
–  Helmholtz model
•  Electrode Kinetics Models
–  Butler-Volmer Equation
–  Tafel Equation
•  Reference Electrodes
3
1
Interface Role
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  Electrode kinetics are governed by the
potential difference across a thin (order
10 A) layer adjacent to the electrode
surface
•  This layer is called the double-layer
•  Potential difference across the thin layer
is about 0.1 V
•  Large magnitude of electric field (106 V/
cm)
4
Interface Role
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  Large driving force for the
electrode reaction
•  Because of the large electric field
we will have charge separation in
the double layer
•  Electroneutrality condition does not
apply in the double layer region
5
Interface Role
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
–  Helmholtz
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  At equilibrium (thermodynamics
relationships are used) there’s no
current applied
•  When current is applied the potential will
deviate from equilibrium
•  The difference between the potential
and the equilibrium potential is called
the overpotential (or surface
overpotential)
6
2
Interface Role
•  The surface overpotential is given by:
•
•
Interface Role
Electric Double
Layer
ηs = φ − φ 0
–  Helmholtz
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
Where:
hs: surface overpotential, V
f: potential due to the current, V
f0: equilibrium potential, also called E or U
(obtained from equilibrium relationships), V
7
Issues with Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  Electrode reactions are heterogeneous.
This implies that a conductive surface
must be in contact with the electrolyte
•  This arrangement produces a number of
issues (we need a clean surface to
evaluate pure kinetics):
–  Film formation
–  Changes in electrode microstructure
–  Electrolyte contamination
–  All of them cause variations in currentpotential measurements
8
Experimental Solutions
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  Careful electrode surface
preparation (e.g., polish the
surface, in corrosion we need a
rough surface instead)
•  Electrolyte purification (e.g., deoxygenation of the electrolyte)
•  Control of mass transport to the
electrode surface (mixing)
9
3
Typical Kinetics Experimental
Set-ups
•
•
1-reference electrode
Interface Role
Electric Double
Layer
2-gas in
3-gas out
–  Helmholtz
•
4-Luggin capillary
Electrode Kinetics
Models
5-platinum counter electrode
–  Butler-Volmer
–  Tafel
•
6-rotating electrode
Reference
Electrodes
7-temperature probe
8-pH electrode
9-working electrode.
ChE 455/555
10
Typical Kinetics Experimental
Set-ups
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  The electrode rotates to have control of
the mass transfer limitations
•  The flow profile is known in this type of
systems
•  Classical arrangements:
–  Rotating disk electrode (use for laminar
flows)
–  Rotating cylinder electrode (use for turbulent
flows)
11
More Issues with Electrode
Kinetics
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  Accurate measurement of the potential
is difficult
•  Because we can’t measure an absolute
value for the potential, we are forced to
use a reference electrode
•  Reference electrodes are chosen based
on: reversibility, stability, and
convenience (cost)
•  Suitable placement of the reference
electrode is another issue (need to
correct for ohmic differences)
12
4
Electric Double Layer
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  When we apply a potential to an
electrode the charges that
accumulate at the surface of the
electrode attract opposite charges
from the electrolyte
•  We expect to have a distribution of
charges in order to balance the
charges at the surface with the
charges from the electrolyte
13
Electric Double Layer
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
–  Helmholtz
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  There are different models to
determine the effect (or to simulate
the effect) of the double layer:
–  Helmholtz model
–  Gouy and Chapman model
–  Stern model
14
Helmholtz Model
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  It was developed in
Electrode Electrolyte
1879
+ S •  It’s the simplest model
S
for double layer
+ S •  Two parallel layers of
S
+ S charges are separated
S
by solvent molecules
S
+
•  The distance (d)
S
S
represents the outer
+
Helmholtz plane
d
•  Fixed distribution of layer
S: solvent
(charges)
15
5
Helmholtz Model
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
•  Double layer model as a simple parallel
plate capacitor
CH =
–  Helmholtz
–  Butler-Volmer
–  Tafel
Dε 0
d
Eq. 1
C: capacitance per unit area (F/m2 or mF/cm2)
D: dielectric constant (or relative permittivity)
D: Separation between charges, cm
e0: Permittivity of free space, 8.8542x10-14 F/cm
16
Helmholtz Model
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•  The potential distribution is a linear
function between the two layers of charge
–  Helmholtz
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
φ=
q
C
Eq. 2
q: charge
•  Because the distribution of the charges
does not change the potential is
constant, also C is constant
17
Helmholtz Model
•
•
Interface Role
Electric Double
Layer
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Helmholtz
–  Butler-Volmer
–  Tafel
•  For water D≈10, d≈10 A, then:
C≈
10 (8.9 x10−14 F / cm )
10−7 cm
C ≈ 10µ F / cm2
≈ 8.9 x10−6 F / cm 2
18
6
Helmholtz Model
•  Experimental
measurements use a NaF
solution in a mercury
electrode
•  Mercury offers a uniform
surface and NaF is not
adsorbed
•  Capacitance in the right
order of magnitude but it is
not constant
Fig. 5.2 Capacitance vs. potential relative to
the point of zero charge for a NaF solution on •  Only for high concentrations
a mercury electrode at 25oC (experimental
the capacitance tends to be
data)
constant
19
Gouy-Chapman Model
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  It was developed in 1910
•  It’s analogous to the DebyeHückel theory
Electrode
+
•  The thickness of the double layer
represents a compromise
between electrical forces
(tending to maintain the ordering)
and thermal forces (tending to
make the arrangement random)
•  No fixed charges
•  Significant deviation from
electroneutrality occurs on the
Debye length
Electrolyte
-
+
-
+
+
-
+
l
20
Gouy-Chapman
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
•  The capacitance is given by
cG −C =
ε
⎛ zFφ0 ⎞
cosh ⎜
⎟
λ
⎝ 2 RT ⎠
Eq 3
–  Butler-Volmer
–  Tafel
21
7
Gouy-Chapman Model
Higher electrolyte
concentration
Lower
electrolyte
concentration
Fig. 5.4 Capacitance vs.
potential relative to the
point of zero charge for a
NaF solution, calculated
using Gouy-Chapman
model (Eq. 3)
•  It’s good at
potentials near the
zero charge region
•  At potentials more
than 0.5 V in either
direction the
observed flattering of
Fig. 5.2 Capacitance vs.
the capacitance is
potential relative to the
not predicted
point of zero charge for a
NaF solution on a
mercury electrode at 25oC
(experimental data)
22
Stern Model
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Combines the Helmholtz
model and the GouyChapman model
Electrode
+
•  Some of the charge is fixed (d
region) and some of the charge
is diffuse or spread out
•  The total length of the boundary
layer is given by the fixed region
plus the diffuse region
Electrolyte
S
+
S
+
S
+
S
+
S
-
S
S
-
S
S
-
d
S: solvent
23
Stern Model
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Because the capacitances are in
series the capacitance of the
double layer is given by:
1
1
1
=
+
Eq. 4
CS CH CG −C
24
8
Stern Model
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
•  The smallest capacitance is the
one that governs the behavior of
the system:
–  If CH>>CG-C then CS≈CG-C
–  If CH<<CG-C then CS≈CH
Eq. 5
Reference
Electrodes
25
Stern Model
Stern
Model
Experimental
data
Fig. 5.6 Capacitance of 0.001 M NaF vs.
potential relative to the point of zero charge
at 25oC. The experimental data (circles)
agrees very well with the model (Stern
Model, Eq. 4)
•  Assuming that CH is
constant
•  The Stern model
predicts the
experimental data
very well
26
Consequences of the DoubleLayer
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Species outside the Helmholtz
region are too distant to react
•  The driving force for the reaction is
the potential drop across the
Helmholtz region rather than the
potential drop across the whole
double layer
27
9
Consequences of the DoubleLayer
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Concentration at the bulk is different to
the concentration at the surface of the
electrode
•  When we study kinetics we need only
the intrinsic effect of kinetics (need to
eliminate the effect of the double layer)
–  Add a non-reacting supporting electrolyte to
the solution
–  This increases the CG-C, then the overall
capacitance is approximated by the CH
28
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  In ordinary kinetics
we express the
progress of a
reaction by plotting
the reaction
coordinate vs. the
energy of the
species (assuming
transition state
theory)
Ea
29
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Let us consider one elementary
step electrochemical reaction:
k
c


O+ + e− 

R
k
a
Eq. 6
Where
O+: oxidized species
R: reduced species
kc: cathodic reaction rate constant
ka: anodic reaction rate constant
30
10
Electrode Kinetics
•  A more negative potential
(more positive energy) tends
to promote reduction
•  At progressively more
negative potential, the
energy of the oxidized
species is increased
•  f3: reduction is favored
•  f1: oxidation is favored
•  f2: equilibrium potential, no
net reaction takes place
Eac2
Eac1
31
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Consider the case where we start
an experiment at the potential f1
and we reduce it to f2
•  The activation energy for the first
process (Eac1) is higher than for the
second process (Eac2)
32
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  We can express the activation
energy for the second process as a
function of the first process by:
Gc 2 = Gc1 + β nF (φ2 − φ1 )
Eq. 7
•  Where
–  b is the symmetry factor (transfer coefficient)
represents the fraction of energy that has
been used to reduce the activation energy
of the reaction
33
11
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Similarly the activation energy for
the anodic process (which
increases) can be expressed by:
Ga 2 = Ga1 − (1 − β ) nF (φ2 − φ1 )
Eq. 8
n: is the number of electrons transferred in
the reaction. For elementary steps n is most
of the time 1, it is unusual to have more than
1 electron involved in an elementary step
34
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
•  The form of our kinetic expression is the
same as that for chemical reactions
(using an Arrhenius dependence of
temperature):
⎛ −G ⎞
k = k ' exp ⎜
⎟
⎝ RT ⎠
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
Eq. 9
Where
k’: is the a constant, cm/s
G: is the free energy of activation
35
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
•  The rate of electrochemical reaction is
directly proportional to the current
density:
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
r=
Where
i
⎛ −G ⎞
= k 'c exp ⎜
⎟
nF
⎝ RT ⎠
Eq. 10
r: reaction rate, mol/s cm2
i: current density, A/cm2 (the area is the electrode
surface area)
c: is the reactant concentration (mol/cm3)
36
12
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  For the general anodic reaction given in
Eq.6 (first order reaction), we substitute
Eq. 8 into Eq. 10
ra =
⎧ G − (1 − β ) nFφ ⎫
ia
= ka' cR exp ⎨− a
⎬
nF
RT
⎩
⎭
We have assumed a reference potential,
therefore the subscripts 1 and 2 have been
dropped
Eq. 11
37
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
•  We can redefine the reaction
constant including the activation
energy at our reference potential:
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
ra =
Reference
Electrodes
⎧ (1 − β ) nFφ ⎫
ia
= ka cR exp ⎨
⎬
nF
RT
⎩
⎭
Eq. 12
38
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Similarly for the cathodic reaction:
rc =
ic
⎧ − β nFφ ⎫
= kc cO exp ⎨
⎬
nF
⎩ RT ⎭
Eq. 13
•  The net current density (i=ia-ic) is the
difference between the anodic and
cathodic current densities (Eq. 12-Eq. 13)
r = ra − rc =
⎧ (1 − β ) nFφ ⎫
i
⎧ − β nFφ ⎫
= ka cR exp ⎨
⎬ − kc cO exp ⎨
⎬
nF
RT
⎩ RT ⎭
⎩
⎭
Eq. 14
39
13
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
•  At equilibrium the net current
density is zero, but the rates of the
anodic and cathodic reaction are
not zero. The magnitude of both (ia
and ic) are the same and this is
called exchange current density (i0)
Reference
Electrodes
40
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  If we designate the equilibrium
potential as f0
⎧⎪ (1 − β ) nFφ 0 ⎪⎫
⎧ −β nFφ 0 ⎫
i0
= ka cR exp ⎨
⎬ = kc cO exp ⎨
⎬ Eq. 15
nF
RT
⎩ RT ⎭
⎩⎪
⎭⎪
•  Taking the logarithm of Eq. 15 and
rearranging:
φ0 =
RT ⎛ kc ⎞ RT ⎛ CR ⎞
ln ⎜ ⎟ −
ln ⎜
⎟
nF ⎝ ka ⎠ nF ⎝ CO ⎠
Eq. 16
41
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Substituting Eq. 16 into Eq. 14 and
using the definition of overpotential:
⎧⎪ (1 − β ) nF ⎛
i
RT kc RT CO ⎞ ⎪⎫
= ka cR exp ⎨
ln +
ln
⎜η s +
⎟⎬
nF
nF ka nF CR ⎠ ⎭⎪
⎝
⎩⎪ RT
⎧⎪ − β nF ⎛
RT kc RT CO
− kc cO exp ⎨
ln +
ln
⎜η s +
nF ka nF CR
⎪⎩ RT ⎝
⎞ ⎫⎪
⎟⎬
⎠ ⎪⎭
Eq. 17
42
14
Electrode Kinetics
•  Rearranging Eq. 17:
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
⎡
⎧ (1 − β ) nF ⎫
⎧ −β nF ⎫⎤
i = nFkc1− β k a− β c1O− β c Rβ ⎢exp ⎨
ηs ⎬ − exp ⎨
ηs ⎬⎥ Eq. 18
⎩ RT
⎭⎦⎥
⎩ RT
⎭
⎣⎢
•  Eq. 18 is a general kinetics expression for the
first order elementary step given in Eq. 6
•  The concentration of the reactants are at the
surface of the electrode
•  The cathodic and anodic kinetic constants can
be evaluated at equilibrium from the exchange
current density:
kc =
i0
i
Where the superscript 0 represents
and ka = 0
nFCO0ChE 455/555 nFCR0 equilibrium conditions
43
Electrode Kinetics
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
•  Substituting the kinetic constants
into Eq. 18 we obtain:
1− β
⎛c ⎞
i = i0 ⎜ O0 ⎟
⎝ cO ⎠
⎛ cR ⎞
⎜ 0⎟
⎝ cR ⎠
β
⎡
⎧ (1 − β ) nF ⎫
⎧ − β nF ⎫⎤
ηs ⎬ − exp ⎨
η s ⎬⎥
⎢exp ⎨
⎩ RT
⎭⎥⎦
⎢⎣
⎩ RT
⎭
–  Butler-Volmer
–  Tafel
Eq. 19
44
Butler-Volmer Equation
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Redefining the transfer coefficients for the
anodic and cathodic components as:
α a = (1 − β ) n
αc = β n
•  And assuming the concentration at the surface
is equal to the concentration at the bulk which
will be the case of equilibrium condition, then
Eq. 19 becomes:
⎡
⎧α F ⎫
⎧ −α F ⎫⎤
i = i0 ⎢exp ⎨ a ηs ⎬ − exp ⎨ c η s ⎬⎥
⎩ RT ⎭
⎩ RT
⎭⎦
⎣
Eq. 20
Eq. 20 is known as the Butler-Volmer Equation
45
15
Butler-Volmer Equation
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Three variables aa, ac, and i0 need to be
determined to use Butler-Volmer
Equation
•  Butler-Volmer equations gives a good
representation of experimental data for
many systems
•  The exchange current density is a
strong function of temperature
•  When the exchange current density is
very large, the reactions is said to be
reversible
46
Butler-Volmer Equation
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  When two reactions take place
simultaneously, on the same
electrode surface, we can use the
Butler-Volmer equation for each of
them
•  We will have to determine the
individual parameters for both
reactions
47
Linear form of Butler-Volmer
Equations
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  One of the disadvantages of the
Butler Volmer equation is that the
overpotential can’t be expressed
implicitly
•  To confront this several
approximations have been made
–  Small surface overpotential
–  Large surface overpotential
48
16
Linear form of Butler-Volmer
Equations
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  When the overpotential is very
small, the exponential term in Eq.
20 can be expanded using
Maclaurin series, neglecting some
of the terms in the series:
i=
i0 (α a + α c ) F
ηs
RT
Eq. 21
49
Linear form of Butler-Volmer
Equations
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Eq. 21 is the linear form of the Butler-Volmer
Equation
•  The current density is a function of only one
parameter (i0 and the transfer coefficients can
be defined as one constant)
•  It is used to model systems operating at low
current densities
•  It’s often used when the overpotential is 10mV
or less
•  If the current density does not vary widely
(±30%), the linear expression can be used
even in the high current density
50
Tafel Equation
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  If the overpotential is large and positive,
the second term in Eq. 20 can be
neglected:
⎛α F ⎞
i = i0 exp ⎜ a η s ⎟
⎝ RT ⎠
Eq. 22
•  If the overpotential is large and negative,
the first term in Eq. 20 can be neglected:
⎛ α F ⎞
i = −i0 exp ⎜ − c η s ⎟
⎝ RT ⎠
Eq. 23
51
17
Tafel Equation
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Eqs. 22 and 23 are known as Tafel equations
•  Taking the logarithm of Eq.22 and rearranging:
Eq. 24
η s = B log i − A
B=
2.303RT
αa F
A=
2.303RT
log i0
αa F
•  The constant B is called the
Tafel Slope
•  Use of the Tafel approximation
depends on the error that can
be tolerated
•  It is general used when the
overpotential is at least 50 to
100 mV
•  The Tafel slope varies between
30 to 300 mV/decade
52
Tafel Equation
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Values of the exchange current
density and the transfer coefficient
are obtained experimentally
•  Plot overpotential vs. log(i). The
slope of the line will give the
transfer coefficient, and the
intercept will give the exchange
current density
53
Example 1
•  Solve problem 3 of chapter 5 in
your text book
54
18
Example 2
•  Solve problem 2 of chapter 5 in
your text book
55
Reference Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  So far we have learned how to
estimate kinetic expressions as a
function of the overpotential
•  We have also learned that the
overpotential is given by:
ηs = φ − φ 0
56
Reference Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  When using the overpotential equation,
we need to make sure that the potential
that we measure is only due to the
electrochemical reaction
•  One of the ways to accomplish that is by
using reference electrodes
•  Before discussing more details about
reference electrodes, we will present a
discussion in the factors that affect the
potential
57
19
Contributions to the Potential
in Galvanic Cells
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  The maximum potential that we can
measure in a galvanic cell is the
equilibrium potential.
–  The equilibrium potential is only obtained
when no current (or very small current) flows
through the circuit
•  When a current flows through the circuit
the potential measure will always be
smaller than the equilibrium potential
58
Contributions to the Potential
in Galvanic Cells
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
•  The decrease in the potential of the cell
is due to several limitations, and this is
often called “Potential loss, Eloss”
•  Therefore, the potential of a cell is
defined as:
φ = E − Eloss
–  Butler-Volmer
–  Tafel
Eq. 24
Where
E: equilibrium potential
F: potential of the cell
59
Contributions to the Potential
in Galvanic Cells
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  The potential limitations include:
–  Surface overpotential limitations, due to
kinetics limitations
–  Concentration overpotential, due to diffusion
and convection limitations in the electrolyte
(also known as liquid-junction potential)
–  Ohmic drop, due to the mobility limitations
(ion interactions)
–  Solid diffusion limitations, due to diffusion
limitations in porous electrodes, e.g.,
electrodes that the ones used in lithium ion
batteries
60
20
Contributions to the Potential
in Galvanic Cells
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Accounting for all the limitations the
potential loss is given by:
Eloss = Δφohm + ηs ,a + ηs ,c + ηcn + ηsd
Eq. 25
Where:
Δφohm : Ohmic drop
ηs,a : Anodic surface overpotential
ηs,c : Cathodic surface overpotential
ηcn : Concentration overpotential
ChE 455/555
ηsd : Solid diffusion
overpotential
61
Contributions to the Potential
in Electrolytic Cells
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  In an electrolytic cell the minimum
potential that we need to apply for the
reaction to take place is the equilibrium
potential
•  Therefore the potential in an electrolytic
cell is given by:
φ = E + Eloss
Eq. 26
•  The potential loss is calculated using
Eq. 25.
62
Reasons for Using Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Measurement of the potential at
equilibrium conditions is relatively
easy. All we need to make sure is
that the current that flows through
the circuit is very small
•  Such determination depends on
the use of a counter-electrode
having a known reversible potential
63
21
Reasons for Using Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Under load the measurement of the potential
respect to a reference electrode becomes more
complicated (due to the Eloss). In addition we
will have significant reactions at both
electrodes
•  Then a simple two electrode approach is not
longer satisfactory for making accurate
measurements
•  A technique to overcome this problem is to use
a third electrode into the electrolyte
64
Reasons for Using Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
V
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
Counter Reference Working
electrode electrode electrode
•  The reference electrode
should be place very close
to the working electrode
•  In theory it should be
placed just outside the
electrical double layer
•  However, the electrical
field of the electrode can
affect the measurement of
the overpotential
•  A rule of thumbs suggest to
place the reference
electrode at least 4
diameters (of the reference
electrode) away from the
working electrode
65
Reasons for Using Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  An approach to
avoid the effect of
the reference
electrode field on the
working electrode is
to use a luggin
capillary
•  Ohmic drop in the
capillary tube is
small because the
current that flows
through it is very
small
66
22
Reasons for Using Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Sometimes a second reference
electrode is added to measure the
ohmic drop between two points in the
solution. This approach is known as the
four electrode arrangement
•  Another use of reference electrodes is to
measure current distributions in a cell
having a non uniform current distribution
67
Reasons for Using Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Summarizing the reasons for using
reference electrodes are:
–  Accurate measurement of surface
overpotentials
–  Measurement of ohmic drops in
solution
–  Measurement of current distribution
68
Types of Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  When choosing a reference electrode we have
the following criteria
–  Reproducibility
–  Stability
–  Small temperature sensitivity
•  Sometimes a reference electrode of the same
type as the working electrode is chosen, this is
known as a pseudo reference electrode:
–  Avoids:
•  Contamination problems
•  Liquid junction potential problems
–  The problem with doing this is that sometimes the
results are not reproducible and they are more
difficult to generalize
69
23
Types of Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  Generally the reference electrode
should be chosen that is reversible
to one of the ions in solution.
However, this is difficult to
accomplish all the time
•  The practical approach is to
choose a reference electrode that
is standard built for some
electrolyte conditions
70
Types of Reference
Electrodes
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  Typical reference electrodes
–  Hydrogen electrode
–  Calomel electrode (SCE)
–  Mercury-Mercuric oxide electrode
–  Mercury-mercurous sulfate electrode
–  Silver-Silver Chloride electrode
71
Calomel Electrode
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  It is constructed by covering a pool
of mercury with mercurous chloride
(calomel)
•  Potassium chloride is the
electrolyte for the following
reaction:
Hg2Cl2 + 2e− ⇔ 2Hg + 2Cl −
72
24
Calomel Electrode
•  The equilibrium potential is given by:
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
2.303RT
log aCl2 −
2F
E 0 = 0.268
E = E0 −
•  Commercial electrodes are commonly
prepared with three concentrations of KCl: 0.1
N, 1 N and saturated
•  The calomel electrode is best used in acid
solutions
•  Reproducibility is about 2 mV
73
Ag/AgCl Electrode
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
Reference
Electrodes
•  It is used when mercury contamination is not
allowed
•  KCl is used as the electrolyte. The most
common concentration is 4M
•  It is also used in acidic solutions
•  The reaction involved is:
AgCl + e− ⇔ Ag + Cl −
•  The equilibrium potential is given by:
E = E 0 − 0.059log aCl −
0
EChE
=455/555
0.222
74
Mercury/Mercurous electrode
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  It is used in acid solution
•  When chloride contamination is not
allowed
•  The reactions if given by:
Hg2 SO4 + 2e− ⇔ 2Hg + SO4=
2.303RT
log aSO=
4
2F
0
E = 0.615
ChE 455/555
E = E0 −
75
25
Mercury/Mercuric oxide
electrode
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
–  Butler-Volmer
–  Tafel
•
•  It is used in basic solutions
•  It uses KOH as the electrolyte
•  The reaction is given by:
HgO + H 2O + 2e− ⇔ Hg + 2OH −
Reference
Electrodes
E = E 0 − 0.059log aOH −
E 0 = 0.0986
76
Calculation of overpotentials
•
•
Interface Role
Electric Double
Layer
–  Helmholtz
–  Gouy-Chapman
–  Stern
•
Electrode Kinetics
Models
•
Reference
Electrodes
–  Butler-Volmer
–  Tafel
•  For the calculation of overpotential we need to
use Eq.
ηs = φ − φ 0
•  Because the applied potential is measured
respect to a reference electrode, the
Equilibrium potential for the reaction needs to
be expressed respect to the reference
electrode:
φ 0 = Ew − Eref Eq. 27
Where: Ew: equilibrium potential of the working electrode
Eref: equilibrium potential of the reference electrode
77
Example 3
•  The deposition of nickel takes place in a typical sulfate bath
that contains: NiSO4, H2O, NiCl2 and H3BO4
•  Typical operating conditions are T=25oC, pH=5 and nickel
concentration of 1 M
•  A SCE reports a measurement of -0.67 V. At this conditions
the ESCE=0.22V vs. SHE
•  We know that hydrogen will be evolved as a parasitic
reaction and we would like to estimate the current efficiency
•  We know that hydrogen generation and nickel deposition can
be characterized by Tafel equation according to:
⎛ i ⎞
−5 ⎟
⎝ 10 ⎠
i
⎛
⎞
= −ChE
0.1log
455/555
⎜ −4 ⎟
⎝ 10 ⎠
η Ni = −0.06 log ⎜
ηH
2
with i in
A/cm2
78
26
Summary
•  At the end of this chapter you must be able
to:
–  Use and understand the different expression to
express electrode kinetic rates
–  Calculate overpotentials
–  Correct surface overpotentials from other loss
effects
–  Know the uses and applications of reference
electrodes
–  Understand the effect of the double layer on the
electrode kinetics (e.g., what do you do
experimentally do reduce this effect?)
79
27