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Electrode Kinetics

1

Background

• Consider the reaction given below:
A

B (1)

• Let kf and kb are the rate constants of the
forward and backward reactions
2

Reaction rates
• Rate of the forward reaction is given by:
Rf = kf * CA (2)
• Rate of the backward reaction is given by:
Rb = kb * CB (3)

3

Net reaction rate
• The net rate of reaction is given by:
Rnet = Rf – Rb
• Thus, Rnet can be written as:
Rnet = kf * CA - kb * CB (4)

4

Equilibrium
Equilibrium is defined as the point at which the
net reaction rate is zero
• From equation 4, we obtain an equilibrium
concentration ratio

kf / kb = K = CB /CA (5)
• K is a constant, and is called the equilibrium
constant (K)

5

Requirement
Every kinetic theory requires that the
kinetic equations collapse to
thermodynamic relations at equilibrium
• Equation similar to (5) required of any
kinetic theory

6

Thermodynamics vs
...

A - pre exponential factor
8

Activation energy
• Above relationship - called Arrhenius
relationship
• Activation energy : Energy barrier that has
to be surmounted by the reactants before
they can be converted to product
• Larger the activation energy - more energy
needed by reactants to surmount barrier

9

Significance
k = A e -Ea/RT (6)
• Exponent term (e -Ea/RT ) - a probabilistic
feature - represents the probability that the
energy barrier will be surmounted
...
i
• Electrode kinetic theory - must provide
information about the dependence of current
on electrode potential
• When a current is passed, some
electrochemical change occurs at the
electrode

15

Overvoltage
• Due to irreversibility - sustaining a given
current requires that a penalty be paid in
terms of electrode potential – penalty called
overvoltage
η = Eeq - E (9)
Eeq - expected (equilibrium) electrode potential
E - actual electrode potential
16

Tafel equation
• Experimental observation: At low currents, the
current is exponentially related to the
overpotential

η= a + b log i (10)
• a and b are constants , i is the current density
• Called the Tafel equation
• Any successful electrode kinetic theory must also
explain Tafel behaviour – 2nd requirement of an
electrode kinetic theory
17

Recap
1
...
Any electrode kinetic theory must also
explain Tafel behaviour
3
...

18

Kinetics of Electrode Reactions

19

Rates of forward and backward reaction
O + ne = R (7)
• For the above reaction, the rate of the forward reaction is given
by:
Rf = kf CO (0, t) = ic /nF (11)
Where CO (0, t) is the surface concentration of O
• Similarly the rate of the backward reaction is given by:
Rb = kb CR (0, t) = ia /nF (12)
Recall that reaction rate and current are interlinked – also
recall that reduction occurs at the cathode and oxidation at
the anode
20

Net reaction rate
• The net reaction rate (and hence the net
current) is given by
Rnet = Rf – Rb = i/ nF = [ic – ia]/ nF
= [kf CO (0, t) - kb CR (0, t)] (13)

21

Potential dependence of kf and kb
• Both kf and kb are potential dependent
functions
• The forward reaction (a reduction) is an
electron accepting process - the rate
increases as the electrode potential becomes
more negative
• This is because the electrode can give away
electrons more easily

22

• The opposite trend is seen for the backward
(oxidation) reaction - the rate increases as
the electrode potential becomes more
negative
This potential dependence needs to be
qualified to obtain a true picture of
electrochemical kinetics

23

Potential dependence of kf and kb

Value of k

kf
kb

More positive E
24

At equilibrium
• It is possible to adjust the electrode
potential and O and R concentrations make
net reaction rate zero
...
5 = 1- α
...

30

Influence on kinetics

Will return to this picture after detailed
understanding of kinetic expressions

31

Standard rate constants
• Equation 19 on integration gives:

ln (1/ kf) = αFE/RT+ c (21)
• If kf = kf° @ E = E°

kf = kf° e{- [αF/RT] (E-E°)} (22 - a)
Similarly:

kb = kb° e{[(1-α)F/RT][(E-E°)} (22 -b)

32

• kf° and kb° are termed the standard (or
conditional) rate constants
• If the concentrations of O and R are equal, ,
and the potential is maintained at E° to
prevent current flow:
from equation 15, kf° = kb°
Can be replaced by a single symbol k°
...

• Measure of the “kinetic facility” of the
redox couple –
• The larger the value of k°, the faster
equilibrium will be attained
...
g
...

• The largest measured values of these
constants are on the order of 1-10 cm/s
...

O2 + 4 H+ + 4e = H2O
• The oxygen reduction reaction - a 4 electron
process - has a lower value of k° than the
hydrogen oxidation reaction – increased
complexity
36

Sluggish cathode kinetics
• k° for oxygen reduction << k° for hydrogen oxidation
• Note however that forward and backward rate constants
can still be large

kf = k° e{- [αF/RT] (E-E°)} (22-a)
kb = k° e{[(1-α)F/RT][(E-E°)} (22-b)

• Thus, for small k° s, need a larger overpotential to
maintain a given k
• Larger overpotential – lower net cell voltage - less
efficient fuel cell
Urgent need to improve oxygen reduction kinetics –
increase k°
37

The Butler – Volmer Theory

38

End Result
i = nF k° [CO (0, t) e{- [αnF/RT] (E-E°)}
- CR (0, t) e{[(1-α)nF/RT](E-E°)}] (25)
(n – stands for no
...
of electrons transferred
This formulation is called the Butler – Volmer
formulation of electrode kinetics
42

B-V Theory
•

This formulation links four important
parameters:
- Faradaic current
- Electrode potential
- Concentration of reactant
- Concentration of product
...
H
...
H
...
rearranges to give the Nernst equation

44

Tafel kinetics
• For very large overpotentials (ή = E - E°),
one or the other of the exponents on the
R
...
S
...
28 reduces to:
e (nF/RT)[Eeq-E°] = CO* /CR* (29)

• Restatement of the Nernst equation (8)
• Though the net current is zero, faradaic
activity is still in progress at the electrode
surface
An equal magnitude of anodic and cathodic
current flows
48

This current density is referred to as the
exchange current (io)
• Thus the exchange current may be written
as:
io = nFk°CO* e{- [αnF/RT] (Eeq-E°)} (30)

49

e (nF/RT)[Eeq-E°] = CO* /CR* (29)
• Raise both sides of equation 29 to the power –α,
substitute for the term e{- [αnF/RT] (Eeq-E°)} to yield:
io = nFk°CO*(1-α) CR*α (31)
• In the case where CO* = CR* = C:
io = nFk°C (32)
...
α (0
...
- Derivation
• Dividing the current density (eqn
...
31), we have:
i/io = [CO (0, t) / CO *]e{- [αnF/RT] (E-E°)}[ CO * / CO *]α
- [CR (0, t)/ CR*] e{[(1-α)nF/RT](E-E°)}[ CO * / CO *] -(1-α) (33)
using equation 29 to substitute for [ CO * / CO *]α and
[ CO * / CO *] -(1-α) :
i/io = [CO (0, t) / CO *]e{- [αnF/RT] ή}
- [CR (0, t)/ CR*] e{[(1-α)nF/RT] ή }] (34)
Where ή = E - Eeq
54

Typical current – overpotential curve

55

Effect of exchange current density

Const
...
5);
(a) io= 10 –3 A/cm2; (b) io= 10 –6 A/cm2); (c) ) io= 10 –9 A/cm2

56

Effect of the transfer coefficient

io = const
...
(34) becomes:

i = io[e{- [αnF/RT] ή} - e{[(1-α)nF/RT] ή }] (35)
• Known as the Butler - Volmer equation
Used as a good approximation to eqn
...

• For very small values of η the B-V equation can be
written in a linear form as:

i = - io F ή/RT (36)
• Linear relation between current and overpotential
• Ratio of the two - units of resistance - called the charge
transfer resistance (Rct)
...
39 reduces to:
ή = [RT/(1-α)nF] ln i - [RT/(1-α)nF] ln io (40)
• Fits the Tafel formulation (eqn
...

61

Tafel plots
• A plot of log i vs
...

• Comprises an anodic branch and a cathodic
branch
• Slopes of (1- α)nF/2
...
3 RT
respectively
Sharp deviation from linearity as ή
approaches zero – due to breakdown of
assumptions
62

Tafel plot

63

Mass transport effects
• Equations (35 – 40) assumed no mass
transport effects
• Mass transport effects are frequently present
in electrochemical systems
• Need for corrections to account for mass
transport effects

64

Recall from the electrochemistry notes (eqn
...


67

• Recall: RT/Fio represents the resistance to
charge transfer (37)
• Thus, eqn
...
43
• When io >> il,c and il,a:
- Overpotential occurs mainly due to
mass transport effects even at very
close to equilibrium
- Kinetics of the system are very fast

• If the reverse is true:
- Kinetics of the reaction is very slow
- The overpotential close to equilibrium
is due to poor charge transfer
...
44 on rearrangement gives:
ή = [RT/αnF] ln (io/il
...
c - i)/i] (45)

• Equation can be used to fit a mass transport
corrected Tafel plot
• Can be used to obtain kinetic parameters in
systems having mass transport effects

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