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;
At equilibrium
kb (cB ) eq  k f (c A ) eq
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:
G 
kT  RT
k 
e
h
Corresponding to steric factor
in SCT
For electrode reactions
For reversible state
Ox  ne

kf
kb
Red
0
C
RT
E  E o '
ln o0 Nernst equation
nF CR
For irreversible state
  a  b log i
Tafel equation (1905)
Overpotential
How to explain these empirical formula?
Potential curve described by Morse empirical equation
Ox  ne

kf
kb
Red
Standard free energy
Activated complex
Gc
Reactant
re G
Ga
product
Reaction coordinate
In electrochemistry,
electrochemical
potential was used
instead of chemical
potential (Gibbs
free energy)
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
ic  nFAkc COx (0, t )
Net current:
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
kc  k a
Gc
kT  RT
kc  
e
h
Ga
kT  RT
ka  
e
h
4.1.3 effect of overpotential on activation energy
Ox
Red
Na+ + e
Na(Hg)x
Ox
Na(Hg)x
Na+ + e
Red
Na+ + e
Na(Hg)x
The energy level of
species in solution keeps
unchanged while that of
the species on electrode
changes with electrode
potential.
polarization   ir  re
ΔG  nFE  nFΔ
Ox  ne
ΔG  ΔG0  nFΔ
Red
ΔGc,0

c
ΔG
F Δ
F  Δ

a
ΔG
ΔGa,0
F Δ
transfer
coefficient
F  Δ
F Δ
   1
Fraction of applied potential alters activation energy 
for oxidation and  for reduction
 c , re
 nFΔ
nFΔ
 nFΔ
 c ,ir
Anode side
cathode side
 a ,ir
Ga  Ga,0   nFΔ
Gc  Gc,0   nFΔ
 0
tan  FE / x
 nFΔ

x


 nFΔ
 
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
 nF 
ka   a
exp(
)  ka ,0 exp(
)
h
RT
RT
ic  kc cOx (0, t )  nFcOx (0, t )kc,0 exp(
 nF 
RT
)
No concentration polarization
ic  ic,0 exp(
 nF 
If initial potential
is 0, then
RT
)
ia  ia,0 exp(
 nF 
RT
)
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 equilibirum
ia ,0  ic ,0  i0
2.3RT ic
c 
lg
 nF
i0
2.3RT ia
a 
lg
 nF
i0

lg ic
2.3RT
2.3RT
c  
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
inet
 nF 
 nF  

 nFk0 cOx (0, t )exp(
)  cRed (0, t )exp(
)
RT
RT 

At equilibrium
inet  0
cOx (0, t )  c ;
0
Ox
c exp(
0
Ox
 nF 
RT
)c
0
cOx
nF

exp(
 )
0
RT
cRed
    
cRed (0, t )  c
0
Red
0
Red
exp(
 nF 
RT
)
0
RT cOx
 
ln 0
nF cRed
0
RT cOx
  
ln 0
nF cRed
Nernst equation
4.2.2 Butler-Volmer model and 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 
Charge transfer resistance
nF

  RT
i
i0
i/A
nF
Rct 
RTi0
Cathode
Net current
/V
False resistance
Anode
2) Limiting behavior at large overpotentials
 nF
 nF 

i  i0 exp(
 )  exp(
 )
RT
RT


i/A
Cathode
Net current
/V
One term dominates
exp(
 nF
Anode
)
 nF 
RT
 exp 
   1%
 nF
 RT 
exp(
)
RT
Error is less than 1%
  118 mV
When cathodic polarization is larger than 118 mV
inet  ic  ia  ic
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
At 25 oC, when n = 1,  = 0.5
2.3RT
b
 nF
b  118 mV
The typical Tafel slope
lg i
118 mV
118 mV
lg i0
300
200
100
0
-100
-200
-300
 / mV
Tafel plot:   log i plot
log i0
re
4.2.4 determination of kinetic parameters
2.3RT
a
lg i0
 nF
2.3RT
b
 nF
For evolution of hydrogen on Hg electrode
  1.40  0.118lg i
  0.5
i0  1.6 1012 A  cm2
i0  nFcOx (0, t )kc,0 exp(
 nF
RT
)
k  5  1013 cm  s 1
active dissolution
lgi
i

i


active dissolution:
n
n
Ag+ /Ag
0.5
0.5
Hg2+ /Hg
0.6
1.4
Cu2+ /Cu
0.4
1.6
Zn2+ /Zn
0.47
1.47
ia  nFAka CRed (0, t )
ic  nFAkc COx (0, t )
Gc
Ga
kT  RT
kc  
e
h
Ox  ne
kT  RT
ka  
e
h
Red
ΔGc,0
ΔGc
F Δ
F  Δ

a
ΔG
ΔGa,0
F Δ
transfer
coefficient
F  Δ
F Δ
   1
 c , re
 nFΔ
nFΔ
 nFΔ
 a ,ir
 c ,ir
Anode side
cathode side
Ga  Ga,0   nFΔ
Gc  Gc,0   nFΔ
ic  nFcOx (0, t )kc ,0 exp(
 nF 
RT
)
ia  nFcRed (0, t )ka,0 exp(
 nF 
RT
)
Master equation
inet
 nF 
 nF  

 nFk0 cOx (0, t )exp(
)  cRed (0, t )exp(
)
RT
RT 

Nernst equation
0
RT cOx
  
ln 0
nF cRed
Butler-Volmer equation
 nF
 nF 

i  i0 exp(
 )  exp(
 )
RT
RT


Rct 
nF
2.3RT
2.3RT
lg i0 
lg i
Tafel equation  
RTi0
 nF
 nF
4.2.5 Exchange current density
1) The exchange current of different electrodes differs a lot
Electrode
materials
solutions
Hg
0.5 M sulfuric acid
Cu
1.0 M CuSO4
Pt
0.1 M sulfuric acid
Hg
110-3 M Hg2(NO3)2 +
2.0M HClO4
Electrode reaction
i0 / Acm-2
H++2e– = H2
510-13
Cu2++2e– = Cu
210-5
H++2e– = H2
110-3
Hg22++2e– = 2Hg
510-1
2) Dependence of exchange currents on electrolyte
concentration
Electrode reaction
c (ZnSO4)
i0 / Acm-2
1.0
80.0
0.1
27.6
0.05
14.0
0.025
7.0
Zn2++2e– = Zn
High electrolyte concentration is need for electrode to
achieve high exchange current.
Use of Ag/AgCl electrode.
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, basic characteristic of
reference 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 Acm-2.
If exchange current of the electrode i0 > 10~100 Acm-2, it is
difficult for the electrode to be polarized.
When i0 < 10-8 Acm-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.
Influence of impurity
If an impurity undergoes reduction at electrode
*
I Red  I Red
 IOx
If
If
I0  I Red
*
*
I0  I Red
The influence of impurity
on equilibrium is negligible.
*
I Red
 IOx  IRed
Oxidation of electrode and reduction of impurity take place.
There is net electrochemical reaction.
Single/couple electrode and Mixed potential
 / mV

lg i
Icorro
Electrode with exchange current less than 10-4 A cm-2 is hard
to attain equilibrium potential.
4.3 Diffusion on electrode kinetic
When we discuss situations in 4.2, diffusion polarization is
not take into consideration.
When diffusion take effect :
G0.c
kBT
 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
 nF
i
c )
C  C (1  ) i  (1  )i0 exp(
id
RT
id
s
i
0
i
id  i
i
 nF
(
) exp(
c )
i0
id
RT
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
The total polarization comprises of tow terms: electrochemical
term and diffusion term.
Discussion :
id
RT
i RT
c 
ln 
ln
 nF i0  nF id  i
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
RT
i
c 
ln 0
 nF i
diffusion No ec
is invalid
id
RT
d 
ln(
)
nF
id  i
i  id
i


log i
lg i
118 mV
118 mV
lg i0
300
200
100
0
-100
-200
-300
 / mV
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 (ideal unpolarizable electrode)
When id >>i0, diffusion control
diff
id
1
id
2
ec
1/2
re  1/ 2

1
At half wave potential i  id
2
id
RT
i RT
c 
ln 
ln
 nF i0  nF id  i
1
id
id
RT
RT
RT
2

ln

ln 2 
ln
 nF
i0
 nF
 nF i0

RT
RT
ln id 
ln i0
 nF
 nF
 d1/ 2 
RT



d
ln
i
 nF
0 id

The half wave potential depends on both id and i0
diff
id
1
id
2
id
RT
i RT
c 
ln 
ln
 nF i0  nF id  i
ec
1/ 2 1/ 2

1/ 2

ec
lgi0
diff
lgid
id
RT

ln
 nF i0
lg i
118 mV
118 mV
lg i0
300
200
100
0
-100
-200
-300
 / mV
Tafel plot without diffusion polarization
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
Question:
How to overcome mixed / diffusion control?
please summarize 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
At constant c, it  cOx(0,t)
at low polarization :
exp(
 nF
RT
 )  (1 
 nF
RT
 nF
RT
c
is very small
)
 cOx (0, t )
cRed (0, t )
 nF
 nF 
it  i0 
exp(
) 
exp(
 )
0
0
RT
RT
cRed
 cOx


cR (0, t )  
 cO (0, t ) cR (0, t ) nF  cO (0, t )

it  i0 




 
0
0
0
0
RT 
cR
cO
cR  

 cO

Constant for
potential step
Numerical solution:
1
2
it  i exp( 2t )erfc(t )
K c*
K a*
  1/ 2  1/ 2
DOx
DRed
i is the current density at no concentration polarization at 
0
c
(0,
t
)

c
That is Ox
Ox ;
0
cRed (0, t )  cRed
1
At t = 0
1
2
exp( 2t )erfc(t )  1
0.5
1
i(0)= i
no concentration polarization
2
3 2 t
When
1
2
t  1
it  t

it  ic 
it
i 
t
ic
Double-layer charge
EC control
Extrapolating the linear
part to y axes can obtain
i c

1
2
1/ 2 
1/ 2
at time right after the
potential step : it t1/2 is
linear

2
diff control
C
t1 / 2
Making potential jump to different  can obtain i at
different . Then plot i against c can obtain i~c
without concentration polarization.
The way can be used to eliminate concentration
polarization.
it
c  time constant s
i 
Double-layer charge
EC control
it > i due to charge of
double layer capacitor
diff control
C
t1 / 2
i
4.4.2 current step
cathodic current : 0  ic
 nF 
ic  nFk ' COx (0.t )exp 

 RT 
1


2
t


ci (0, t )  ci0 1    
   


ic
0
t
Record c at different ic
t
1
2
0
ic  nFk ' COx
[1  ( ) ]exp(

 nF
RT
)
0
nFk ' COx
RT
RT
t 12
c (t )  
ln

ln[1  ( ) ]
 nF
ic
 nF

constant
  transition time when potential steps to next reaction.
c
t
1
2
c (t )  [1  ( ) ]

0
nFk ' COx
RT
RT
t 12
c (t )  
ln

ln[1  ( ) ]
 nF
ic
 nF

c
c(0)

t
0
i= icharge
c  (t )
c
t
The slope of the linear par of c
(t) can be used to determine n
and .
1
2
ln[1  ( ) ]

When t0 the second term = 0
c(0)
( t  0)
0
t
1
2
ln[1  ( ) ]
0
nFk ' COx
RT

ln
 nF
ic
4.4.3 cyclic voltammetry (CV)
 (t )  0  vt c (t )   (t )  0
 COx (0, t )

CRed (0, t )
 nF
 nF
it  i0 
exp(
c (t )) 
exp(
c (t )) 
0
0
RT
RT
CRed
 COx

for reversible single electrode
I
iPc
Potential separation
59
Δ      mV
n
c
P
iPa

a
P
for the reversible systems , use the forward kinetics only :
i  nFACO (0, t )k f (t )
can be solved only by numerical method:
1
2
Ox
1
2
i  nFAC D  (
0
Ox
 nF
RT
1
2
1
2
)  x(bt )
for fast EC reaction : i << i0
controlled by diffusion
Nicholson-Shain equation
 tramper coefficient
i
n – number of electrons involved
in charge transfer step
1
2
 x (bt )
v
is tabulated
x (bt) max =0.4958
0.2
0.1
0.0
0.1
0.2
For irreversible single electrode
i
I Pc
I Pa

59
Δ  mV
n
For totally irreversible systems
RT
k
 nFv
Ep  E  (
)[0.78  ln
 ln
]
F
RT
D
peak potential shift with scan rate
v
i
0.2
0.1
0.0
0.1
0.2
for slow EC reaction : ii0
( 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 .
Dependence of p on 
1/ 2
 D 
   Ox 
 DRe d 
1/ 2
nF 

 k /   DOx
v
RT 

 29  kv 1/ 2
4.5 effect of 1 potential on EC rate :
1=0, validate at high concentration

or larger polarization
1
G = nF
x
effect of 1:
1. on concentration
2.  =   1
zi F
C  C exp(  1 )
RT

i
0
i
( n  zO ) F
 nF
ic  nFAk C exp(
 )exp(
1)
RT
RT
0
Ox
ia  nFAk
0
Re d
0
Ox
0
Re d
C
( n  zO ) F
 nF
exp(
 )exp(
1)
RT
RT
This means 1 has same effect on the forward (reduction) and
reverse (oxidation) reaction.
zO   n
RT
i  ic  ia    const 
ln i 
1
 nF
n
zO   n
RT
c  const 
ln i 
1
 nF
n
zO   n
RT
c  const 
ln i 
1
 nF
n
zO   n
1  0
n
When zO <0 ( minus ), n zO is large, therefore, for
anion reduced on cathode , 1 effect is more significant.
When zO  n
1 made c shift positively
so: if 1 increases, i decreases
z0   n
RT
c  const 
ln i 
1
 nF
n
if: n = zO
 = 0.5
Cu2+ +2e- = Cu
MnO4 +e = MnO42
H+ +e- = 1/2 H2
RT
c  const 
ln i  1
 nF
if :zO = 0
RT
c  const 
ln i  1
 nF
adsorption of anion slow reaction
without specific adsorption
 reduction of +1 cation
…… reduction of 1 anion
RT
c  const 
ln i  1
 nF
1 accelerates reduction of cation, slows reduction of anion
Rotation rate of RDE on reduction of 110-3 mol/L K2S2O8
without supporting electrolyte
Effect of potential of zero charge on polarization curve of
RDE for reduction of K2S2O8 without supporting electrolyte
Only when the electrode potential is near to the potential
of zero charge, 1 has large effect on the reaction rate,
while at higher polarization, 1 take less effect.
Effect of concentration of supporting electrolyte (sodium
sulfate) on the polarization curve of RDE for reduction of
K2S2O8 .
1: 0; 2: 2.8 10-3; 3: 0.1; 4: 1.0 mol/L Na2SO4
Problem: how to eliminate the effect of 1?
4.6 EC kinetics for multi-electron process
For a di-electron reaction
Ox + 2e  Red
Its mechanism can be described by
ia0
Ox  1e
X+1e
At stable state

ib0
X
Red
d[X]
0
dt

a F
a F
cx
i
0 
 ia exp(
c )  0 exp(
c ) 
2
RT
RT
cx



b F
a F
i
0  cx
 ib  0 exp(
c )  exp(
c ) 
2
RT
RT
 cx

Ox  1e
X+1e

 ( a   b ) F 
 (  a  b ) F 
exp 
c   exp  
c 
i
RT
RT





1
2
 b F  1
 a F 
exp 
c   0 exp  
c 
0
ia
 RT
 ib
 RT

0
0
i

i
If a
b
(1   b ) F
a F


i  2i exp(
c )  exp(
c ) 
RT
RT


0
b
ia0

ib0
X
Red
(1   b ) F
a F


i  2i exp(
c )  exp(
c ) 
RT
RT


0
b
Therefore
i  2i
0
0
b
  
1  b

2

b
2
For a multi-electron reaction
Ox + ne  Red
Its mechanism can be described by
Ox  1e
X1 +1e
ia0

ib0

X j 1 +1e
X j +1e
ib0

ib0
ib0

X j 1
X j (rds)
Steps before rds, with higher i0 at
equilibrium
( j  j  1) F
(  n  j) F


i  ni 0j exp(
c )  exp( j
c ) 
RT
RT


X j 1

ib0

ib0
X n  2 +1e
X n 1 +1e
X2

X j  2 +1e
X1
X n 1
Red
Steps after rds, with higher i0 at
equilibrium
( j  j  1) F
( j  n  j) F


i  ni exp(
c )  exp(
c ) 
RT
RT


0
j
Therefore
i  ni
0
0
j
At small overpotential

 j  j 1
n

F
in i
c
RT
2 0
j
i0  n i
2 0
j
j n j
n
At higher overpotential
For cathodic current
 ( j  j  1) 
ic  ni exp 
c 
RT


0
j
For anodic current
 (  j  n  1) 
ia  ni exp 
a 
RT


0
j
4.7 Marcus theory for electron transfer
Effect of reactant, solvents, electrode materials and adsorbed
species on electrochemical reaction.
Electron-transfer between two coordination compounds.
M
Outer-sphere reaction
M
inner-sphere reaction
No strong interaction
between electrode surface
and reactant.
reactant,
intermediate
and
product interact with electrode
surface strongly.
Reduction of Ru(NH3)63+
Reduction of O and oxidation of H
Microscopic theories of electron transfer
Electron transfer reaction, a radiationless electronic
rearrangement, sharing commonalities with radiationless
deactivation of excited molecules.
For a homogeneous redox reaction :
O + R’  R + O’
Electron transfer between tow isoenergetic points ---isoenergetic electron transfer
activation
Franck-Condon principle:
Nuclear coordinates do not change on
time scale of electronic transitions.
Reactants and products share common
nuclear configuration at moment of
transfer.
Deduce expression for standard Gibbs
energy of activation as a function of
structural parameters of reactant, so as
to calculate rate constant of the reaction.
Transition
state
isoenergetic electron transfer
g: global reaction coordinate for 1 dimensional process, related
to solvation.
For homogeneous
electron transfer
Work of assemblying reactants, i.e., ion pair + electrostatic
work to bring charged species next to charged electrode, wO
and wR not considered.
Improved
model
Predictions from
Marcus theory
½ factor seems like first order term in expansion of , rest are
corrections
Classical Butler-Volmer theory regards  as constant, cannot
predict potential dependence of .
Electron transfer occurs between empty levels of electrode (or
species in solution) and filled levels of species in solution (or
electrode) of the same energy.
For reduction - energy of occupied level of electrode must
match energy level of empty state of species in solution.
For oxidation - energy of empty level of electrode must match
energy level of occupied state of species in solution.
Energy levels of metal and species in solution form a continuum
Overall rate must be evaluated by summing or integrating over
all energy matched pairs.
Since filled electrode states overlap with (empty) O states,
reduction can proceed. Since the (filled) R states overlap only
with filled electrode states, oxidation is blocked.
Number of electronic states of electrode in energy range E and
E + dE is
A ( E )dE
area of the electrode
density of states
Total number of states of electrode in given energy range
E2
A  ( E )dE
E1
At absolute zero, energy of highest filled state is called Fermi
level, At higher temperatures, thermal energy promotes
electrons to higher levels Electron distribution given by Fermi
function f(E)

 E  EF  
f ( E )  1  exp 
 
kT



1
concentration density function
DR (, E)
Number concentration of R species in the range between E +
dE is
DR (, E)dE
Rate Constant for
Reduction
Rate Constant for
Oxidation
FURTHER CONSIDERATIONS
• Electron transfer occurs almost entirely at the Fermi level
• Rate constant proportional to local rate at Fermi level.
• Integrals reduce to single value
R
tunneling
P