BASIC PRINCIPLES OF
ELECTRON TRANSFER
Electron transfer (ET) is one of the most important chemical
process in nature and it plays a central role in many biological,
physical and chemical systems.
 ET may be one of the most basic forms of chemical reaction but without it life
cannot exist.
 ET is one of the simplest forms of a chemical reaction!?!
 ET occurs in photosynthetic reaction center where transfer of electrons is used
to create charge imbalance across a membrane, originating a proton pumping
mechanism to produce ATP
 ET respiration and detoxification
 ET at the metal surface with oxygen is responsible of the corrosion.
 ET in solid state electronics (the control of the ET in semiconductors) - the
new area of molecular electronics depends critically on the understanding and
the control of the transfer of electrons in and between molecules.
Electron Transfer
some fields related to the ET
Rudolph A. Marcus
Born: July 21, 1923 (age 91)
Montreal, Quebec
American, Canadian
Alma mater: McGill University
Nobel Prize in Chemistry: 1992
outer-sphere electron transfer was
based on a transition-state theory
approach
The Marcus theory of electron transfer was then extended to include
inner-sphere electron transfer (is ET) by Noel Hush and Marcus
 Nobel Prize in 1992 (the first formulation in 1956)
 “isotopic exchange reactions” (self-exchange reactions) and later
“cross reactions”
Marcus theory of ET
 for the reaction-rate of electron transfer: AB + C → A + BC
 Bill Libby ´s publication (Franck-Condon principle)
 in an ET from one reacting ion or molecule to another, the two new
ions or molecules formed are in the wrong environment of the solvent
molecules since the nuclei do not have time to move during the fast
electron jump
 “solvatation energy barrier” for the ET process
*radioactive isotope
Fe(CN)6
4simple
ET reactions

 233*2
*FeFeFeFe
Fe(CN)6
3-
Typical nuclear configurations for reactants,
products and surrounding solvent molecules in
reaction . The longer M-OH2 bond length in the
state +2 is indicated schematically by the larger
ionic radius.
           

2
63
3
63
3
63
2
63 NHCoNHCoNHCoNHCo

 233*2
*FeFeFeFe
 the reaction is really slow, in contrast to the picture of a ET governed by the
“solvatation energy barrier”
 there is a dramatic difference in the equilibrium Co-N bond length in the 3+ and 2+
ions so that each ions would be formed in a very “foreign”configuration of the
vibrational coordinates
 ET implies changing in the chemical structure of the reactants
 the foreign environment for the new electronic state after the electronic jump
can be seen as an energetic barrier for the ET process.
Marcus theory of ET
Classical theory of ET
RTEA
ek /

2
xy 
2
)ax(by 
2
2
2
2
4a
ab
xy
)( 

2
4


)( G
E
RTG
ek 
 42
/)(
RTEA
Aek /

RT/)G(r
ecek 
 42
[0;0]
[a;b]
A
B
baaxxxy  222
2
a
ab
x
2
2


λ……the reorganization energyr …..the distance between A and Dβ…..the interaction between A and D
 transfer from a donor (D) to an acceptor (A) during a transition state
 distance between the D and A
 the probability that an ET will decreases with increasing distance
 factors that control the rate constant (kET) involved in a unimolecular ET
 the probability of ET is identified by the term kT/h, include: the distance D-A
complex; the Gibbs energy of activation(ΔG) ; and the reorganization of
energy (λ).
Questions:
• The physical meaning of the reorganization energy λ?
• The relationship between changes in ΔG and λ, when the rate constant is the maximum.
• The increase of the rate constant when we go to more negative values ​​of ΔG?
• If we increase the distance between the electron donor and the acceptor of 5 angstroms,
we find that the rate constant (ET) decreased 10 times. What is the value of β?
• At 37° C the rate constant was twice 0° C. Assuming that the equilibrium constant of the
reaction is about 100 (in favor of product) and in the temperature range of approximately
independent of temperature, which is the reorganization energy of reaction (ΔG = ΔG0)?
Marcus theory of ET discussion
Odpovědi:
The physical meaning of the reorganization energy λ?
…konfigurace jader reaktantů – konfigurace jader produktů bez ET…
The relationship between changes in ΔG and λ, when the rate constant is the maximum
...... RT/)G(r
ecek 
 42 00 G
spontánní ET
The increase of the rate constant when we go to more negative values ​​of ΔG?
00 G
If we increase the distance between the electron donor and the acceptor of 5 angstroms, we find
that the rate constant (ET) decreased 10 times. What is the value of β?
RT/)G(r
ecek 
 42
RT/)G()r(
ecek 
 45 2
At 37° C the rate constant was twice 0° C. Assuming that the equilibrium constant of the
reaction is about 100 (in favor of product) and in the temperature range of approximately
independent of temperature, which is the reorganization energy of reaction (ΔG = ΔG0)?
V oblasti ET v proteinech a dalších biochemických materiálech…….charakterizace ET
1
2
4 RT/)G(r
ecek 
 2
2
4 RT/)G(r
ecek 

           

2
62
3
62
3
62
2
62 OHFeOHFeOHFeOHFe
Fe-O in Fe(II) is 2.21
Fe-O in Fe(III) is 2.05
Marcus theory of ET
PARABOLIC
CURVES
Displaced Harmonic Oscilator (DHO)
Energy Gap Hamiltonian (EGH)
Describing ET
(e- and H+)
D – donor A – akceptor
Marcus theory of ET
potential energy
D
A
potential energy
inner shell
(vibrational modes)
outer shell
D + A = D+ + A-
Franck- Condon principle
Electron Transfer Reactions
Potential energy curves for the
ground state and an excited state of
a diatomic molecule
vibrational states
r interatomic distance
A absorption
F fluorescence ET processes must satisfy the FranckCondon
restrictions, i.e.
a) the act of electron transfer (ET) is much
shorter than atomic motion (femto-seconds)
b) the consequences are that no angular
momentum can be transferred to or from the
transition state during electron transfer, there
is also restrictions in changes in spin.
c) an electronic transition is most likely to occur
without changes in the positions of the nuclei
in the molecular entity and its environment.
The resulting state is called a
Franck-Condon state, and the transition
involved, a vertical transition
ground state
excited state
Jablonski diagram
The Franck-Condon Principle
According to the Franck-Condon principle,
the most intense vibronic transition is from
the ground vibrational state to the
vibrational state lying vertically above it.
Transitions to other vibrational levels also
occur but with lower intensity
In the QM version of the FC-principle, the
molecule undergoes a transition to the upper
vibrational state that ‘most closely resembles’
the vibrational wavefunction of the vibrational
ground state of the lower electronic state. The
two wavefunctions shown here have the greatest
overlap integral of all the vibrational states of
the upper electronic state
most closely resembles
Marcus theory of ET
 provides a thermodynamic and kinetic framework for describing one electron
outer-sphere electron transfer
 ET is a mechanistic description of the thermodynamic concept of redox wherein
the oxidation states of both reaction partners change
 Inner-sphere electron transfer
redox centers are covalently linked via any bridge during the ET
 Outer-sphere electron transfer
redox centers are not covalently linked via any bridge during the ET
Five steps of an outer sphere reaction
1. reactants diffuse together out of their solvent shells => precursor complex
(requires work = wr)
2. changing bond lengths, reorganize solvent => activated complex
3. Electron transfer
4. Relaxation of bond lengths, solvent molecules => successor complex
5. Diffusion of products (requires work = w )
Classes of electron transfer
Outer Sphere Electron-Transfer
Electron Transfer Reactions in Solutions
In principle all outer sphere mechanism involves electron transfer (ET)
from reductant to oxidant with the coordination shells or spheres of each
staying intact. That is one reactant becomes involved in the outer or
second coordination sphere of the other reactant and an electron flows
from the reductant to oxidant. Such a mechanism is established when
rapid ET occurs between two substitution complexes.
Elementary Steps in the Outer Sphere Mechanism
a) Formation of a precursor (cage) complex
b) Chemical activation of the precursor, electron
transfer (ET) and relaxation of the successor complex
c) Dissociation to the separated products
the rate observed is
determined by KA and kel
this step is always
considered to be fast
kobs = KAkel
Important factors are:
(i) Solvent reorganization
(ii) Electronic structure
(iii) M-L reorganization
small
Electron Transfer Reactions in Solutions
An inner sphere mechanism is one in which the reactant and
oxidant share a ligand in their inner or primary coordination
spheres the electron being transferred across a bridging group.
Inner Sphere Electron-Transfer
H+
d orbitals
t2g …the representations of the
d-orbitals of transition metalseg and t2g …. 2- en 3dimensional
representations
in the 48-fold cubic point
group Oh. The g in the
subscript denotes inversion
symmetry.
Franck- Condon principle
Fe(II)
Fe(III)
The solid metallic electrode
φM (Galvani potential)
 (Volta potential)
 (surface potential)
Electrode Solution
φ
x
φM (Galvani potential – inner potential) is associated with with EF
 (Volta potential – outer potential) is associated with the potential
outside the electrode´s electronic distribution
 (surface potential) EF = Eredox – e 
φ =  + 
Electrode reactions
Electron transfer followed by
chemical reaction
Copper deposition at a
Cu electrode
A single electron transfer
reaction
Applied voltage
Representation of the Fermi-level in a metal at three different
applied voltages
Schematic representation of the reduction of a species (O) in
solution
Animation of the reduction of a species (O) in solution
At voltage E the formation of the species O is thermodynamically
favored
E1
Applied voltage
P
R
The key to driving an electrode reaction is the application of a potential
potentialchargeenergy 
     VCJ 
   
 C
J
V 
From without potential insertion on solid phase
over / below Er more positive / more negative
electrode polarization, overpotential
  = Ep - Er
 begins lead an electrode process
As well as each electrode process, it consists of more
follow steps – levels
rds - rate determining step
a most slow step
Ep = E polarization = Epolarizační
Er = E equilibrium = Erovnovážný
 = overpotential = přepětí
Overvoltage - Overpotential
Electrode reactions steps
Substance crossing from within of electrolyte to a level of
maximal approximation  transport (diffusion)
overpotential three transport mechanisms
 migration – movement of ions through solution by electrostatic attraction to
charged electrode
 diffusion – motion of a species caused by a concentration gradient
 convection – mechanical motion of the solution as a result of stirring or flow
2. Adsorption (localization) of ions or molecules in space
of electric double layer
3. Dehydration (desolvation)
 absolute
 partial
 none  = overpotential = přepětí
4. Chemical reactions on a metal surface, coupled with
making of intermediates capable of obtaining or losing of
electrons
 reaction overpotential
5. Electrode reaction - solitary electron crossing through
interface
 activation overpotential
6. Adsorption of primary product of electrochemical
process on a metal surface
7. Desorption of a primary product
8. Transport of product from a metal surface
a) Soluble product – by diffusion
(the most used style)
b) Gas products – by bubbling
c) Products can be integrated to an electrode crystal
lattice
 crystalization ( nucleation) overpotential
g) By diffusion to inside of electrode (for ex. amalgam)
without chemical transformation or breaking chemical bonds
RedneO x   
a mass tranfer coefficient
(10-16 s)
a mass tranfer coefficient
heterogeneous rate constant (m.s-1)
Franck-Condon principle
(adiabatic process)
Electrode kinetics
Phenomenological electrode kinetics: the Butler-Volmer equation
– ET rate effected by:
• Applied electrode potential • Temperature
– Activation energy barrier height can be effected by applied potential. This is
in contrast to ordinary chemical reactions.
John Alfred Valentine Butler Max Volmer
ET reactions at electrode/solution
interfaces are activated processes.
      redox0 jjαfη-exp-fηα1expjj 
RT
nF
f 
Electrode kinetics
We seek an answer to the following questions:
 How can we quantitatively model the rate of an ET process
which occurs at the interface between a metallic electrode and an
aqueous solution containing a redox active couple?
How can kinetic information about ET processes be derived?
We shall also investigate the influence of material transport, and
double layer structure on interfacial ET processes.
http://www.ceskatelevize.cz/porady/10121359557-port/501-
pribeh-kapky/video/
Basic concepts of electrode kinetics
For an interfacial ET process:
– current flow is proportional to reaction flux (rate)
– reaction rate is proportional to reactant concentration at interface.
As in chemical kinetics:
– the constant of proportionality between reaction rate v (mol.cm-2s-1)
and reactant concentration c (mol.cm-3) is termed the rate
constant k (cm.s-1)
The electrode potential drives ET processes at interfaces
 generates a large electric field at the el/sol interface
 reduces the height of the activation energy barrier
 increases the rate of the ET reaction
 increases the current
Electrode kinetics
macroscopic phenomenological approach: the formal description of electron
transfer kinetics in terms of rate equations and current - potential relationships.
It is based largely on the activated complex theory of chemical reactions.
microscopic molecular based approach: the subject of quantum electrode
kinetics. It is based on the molecular interfacial electron transfer with the
effects of the molecular structure of the reactant molecules and the
electronic band structure of the electrode (GNOME ?)
The Fe3+/Fe2+ type of reaction is termed an outer sphere electron transfer process
since no bonds are broken or made during the course of the reaction. Consequently,
we neglect complicating factors such as diffusional transport of reactants and
products, and adsorption effects.
Current is passed between working
and counter electrodes. The potential
is measured between working and
reference electrodes.
Potentiostat
Galvanostat
IE
Reference Electrode – RE
(bridge/solution)
(aqueous/nonaqueos)
Counter Electrode – CE
Auxillary Electrode – AE
Working Electrode – WE
The potential applied to the
electrode is controlled using an
electronic device called a
potentiostat.
Electrode kinetics
the electrode acts as an electron source and is termed a cathode
the electrode acts as an electron sink and is termed an anode
Electrode kinetics
Electrode kinetics
The current observed at an el/sol interface reflects two quantities:
– Charging of electrical double layer : non Faradaic charging current iC
– Interfacial ET across interface : Faradaic current iF
ET
MT
at low potentials : arising from
rate determining interfacial ET,
– at high potentials : arising from
material transport MT due to
diffusion mechanisms.
– These components can be
quantified in terms of
characteristic rate constants :
k0 (units: cms-1) for ET and kD (units: cms-1) for MT.
non Faradaic charging current iC Faradaic current iF
Electrode kinetics
Heterogeneous reactions – velocity to surface unit:
Velocity of electrode process
nFSv
dt
dN
nFSI 
S
1
dt
Nd
Fn
S
I
j 
Experimental dependences:
 = f ( j ) j = f (  )
polarization curves current-potential curves
dt
dN
v 
- for transformation of 1 mol of substance with a charge of n charge
of nF coulomb is consumed ; F = 96484 coulomb/mol
- for transformation of dN mol of substance at time dt, a current
I is consumed
Current density
Faraday : I t = N nF = Q
)area(surfaceS 
 = Ep - Er
amount of analyt (mols)
Activation overpotential

  23
FeeFeMneM z
  
 
  1-zz
MeM
redoxox ckv 
oxredred ckv 
red
ox
red vFn
dt
dN
Fnj  ox
red
ox vFn
dt
dN
Fnj 
RedneOx    kred
kox
kred
kox
kred
kox
kred
kox
dt
dN
v  surfaceunitS 
An expression for the rate of electrode reaction








RT
ΔH
´expAk
Arrhenius








R
ΔS
expA´A

















 


RT
ΔG
expA
RT
STΔΔH
expAk
Gibbs-Helmholtz
RedneOx   
QM tunnelling of electrons: time scale is from 10-15 to 10-16 s
Nuclear motion: about 10-12 - 10-13 s
Δφ for reduction .... αc E
Δφ for oxidation .... αa E = (1- αc ) E
α is a coefficient of charge transfer =
symmetry coefficient
a+ c = 1
c=  ; a= 1 - 
An expression for the rate of electrode reaction
Reaction coordinate
E = E(neg.)
G
E = 0 V
nFE
 nFE
Ox
Red
Ga,o

Gc
 Ga

Gc,o
Effect of a change in
applied electrode
potential on the
reduction of Ox to Red
F EnαΔGΔG cc,oc  
F EnαΔGΔG aa,oa  
G
αc for the cathodic process
αa for the anodic process
  0   0.5   1G G G
Reaction coordinate
n = 1.5   = 0.75
rds ……rate - determining step
In many cases electrode processes involving the transfer
of more than one electron take place in consecutive
steps. The symmetry of the activation barrier referred to
the rate-determining step.
n
Red Red
Red
Ox
Ox
Ox
Balance of electrode process
0redox jjj 
vox = vred
j0 is charge current
density





 


RT
)E-α n F (EΔG
expAF cnjj red
redoxred
0
0
 
RT
)E-n F (EαΔG
expAn F cjj ox
oxredox 




 

 0
0
1
k0 = standard velocity constant
(members independence on E)
 


















 





 


RT
EFnα)-(1
expkcFn
RT
EFnα
expkcFnj
RT
EFnα1ΔG
expA
RT
EFnαΔG
expAk
eq0
red
eq0
ox0
0
ox
ox
0
red
red
0
jred = jc
jox
= ja
+








RT
ΔG
expAk
determination of Eeq
Current - overpotential  crossing
 







 


RT
αnFη
expj-
RT
nFηα1
expj
ckFn-ckFn)v(vFnjjj
o0
oxredredoxredoxredox
jred = jcjox = ja
 













 

RT
αnFη
exp-
RT
nFηα1
expjj 0
Butler- Volmer equation for electrode process,
where rds is charge transfer RT
nF
f 
     αfη-exp-fηα1expjj 0 
     αfη-exp-fηα1expjj 0 
RT
nF
f  Butler- Volmer equation
 
TR
ηFnα-1
jlnjln 0a 
TR
ηFnα
jlnjln 0 c
Butler-Volmer equation
 j, ka and kc depends exponentially on potential
 linear free energy relationship the parameters: I and E
 Eeq gives the exchange current jo standard rate constant
 for transport the Tafel law must be corrected
 electrode as a powerful catalyst
 the observed current is proportional to the difference
between the rate of the oxidation and reduction reactions at
the electrode surface
    )Oxk-RednFA(kI ca 
 Red
 Ox concentrations of Red and Ox next to the electrode
    Oxk;Redk ca
do not grow indefinitely – limited by the
transport of species to electrode
    Oxk-Redkv ca
Id - diffusion-limited current
jc
Eeq
Red = Ox + eOx
+ e- = Red
ja
-E E
ja
jc
j = ja
j0
j0
Ep
Polarization curves without overpotential
Eeq equilibrium
potential
j = ja + (- jc)
Polarization curves with activation overpotential
1
Red = Ox + eOx
+ e- = Red
ja
jc
E
Eeq
1
2
3
2 3
j
Polarization anodiccathodic
hydrogen overvoltage oxygen overvoltage
1. Hydrogen ions are absorbed from the solution onto the anode surface.
2. ET occurs from the electrode, the hydrogen ions to form hydrogen.
3. The hydrogen atoms form hydrogen gas molecules.
4. Hydrogen gas bubbles are formed.
0,5
ja /
j0
+-
jc / j0
0,25
0,75
Ratio dependence of current density and change current
density on overpotential for different  values
 













 

RT
αnFη
exp-
RT
nFηα1
expjj 0
Butler- Volmer equation for
electrode process,
where rds is charge transfer
     fηαex p-fηα1ex pjj 0 
RT
nF
f 
Polarization curve:  = f ( j ) or I-E curves: j = f ( )
a) Small values of overpotential
b) Large values of overpotential
η
RT
nF
jj 0
TR
Fnj
η
j 0
0η









p
0
R
Fnj
TR
j
η








positive   process of oxidation
negative   process of reduction
Development of e-x function
 
TR
ηFnα-1
jlnjln 0a 
TR
ηFnα
jlnjln 0 c
Rp = polarization resistance
Tafel relations
jlnbaη  jlogyxη 
Activation Polarization ηact
Tafel diagram for cathode and anode current density
 = 0.25 j0 = 0.1 mA
log jc
log ja
log j0
log j
+
- 
Eeq
ko
TR2.303
Fnα
 
TR2.303
Fnα-1
α
TR
ηFnα
jlnjln 0 c
 
TR
ηFnα-1
jlnjln 0a 
53
I
E
totally polarized
totally depolarized
Eo or Er
η
typical case
η = overvoltage = E − Eo or E - Er
η = f (I )
(eta)
polarization =
ohmic polarization
activation polarization
concentration polarization
η = ηohm + ηact + ηconc
abc
Polarization
A
V
voltage or
current source
W
E
R
EA
0 L
x
ohm
0 0 0
d d
(0) ( ) d
L L L
i x I x IL
L IR
A A
          
    
potentials in solution
2a
WE
r
ohm 2
d d d
( ) ( ) d
( ) 2 2a a a a
i r I r I r I
a IR
A r r A
   
            
       
ηohm = “IR drop”
I
EEn
Resistance (Ohmic) polarization
Oh
d
d
m's Law
i
x

 
I
i
A

a
L
R
A


ηohm
Activation polarization
Butler
Volmer
O(soln) R(soln)± e–
   s o s oo
R O
(1 )
' '' exp exp
F F
c E E c E EI AFk
RT RT
     
       
    
Nernst
Activation
only    
1
o b b
O R act act
(1 )
' exp exp
F F
I AFk c c
RT RT
       
       
    
In = exchange current
n act n act ox rd
(1 )
exp exp
F F
I I I I I
RT RT
    
        
   
Split
       
ss
1
o b b OR
O R n nb b
R O
(1 )
' exp exp
cc F F
I AFk c c E E E E
c RT c RT
        
       
    
cabde
E
I
Iox
Ird
In
- In En
ηact
ox n act
(1 )
exp
F
I I I
RT
 
   
 At very positive E |Iox| >> |Ird|
Similarly at very negative E |Ird| >> |Iox|
     n n
(1 )
ln ln
F
I I E E
RT

  
rd n actexp
F
I I I
RT
 
     
 
     n nln ln
F
I I E E
RT

   
Tafel
equations
E
ln|I|
En
ln{In}
slope =
−αF/RT
slope =
(1−α)F/RT
Activation polarization ηact
k´ (cm.s-1)
 (V) 0.0002
10-3
10-4 10-6 10-10 10-14
0.003 0.12 0.59 1.06
 = f(k´)
reversible x irreversible process current density j
j = 10-6 A.cm-2; n = 1; cOx= 1mM;  = 0.5; T=298 K
oxygen (V)hydrogen (V)metal
Ag 0.48 0.58
Au 0.24 0.67
Cu 0.48 0.42
Hg 0.88 Ni
0.56 0.35
Pt(smoothed) 0.02 0.72
Pt(platinized) 0.01 0.40
j = 10-3 A.cm-2 T = 298 K
Activation polarization ηact
η = ηohm + ηact + ηconc
t
I
0 0
t
E
En
ηconc(t)
ηact(t)
ηohm(t)
0
t
E
En
ηconc(t)
ηact(t)
ηohm(t)
0
t
I
0 0
Three polarizations
A
Concentration polarization
R(soln) – ne O(soln)
s
o R
s
O
' ln
RT c
E E
nF c
 
   
 
Nernst’s law (holds for I ≠ 0 if ηact = 0)
W E R R
O O
ne−
transport
transport
s b
R Rc c
s b
O Oc c
I
I = 0 b
o R
n b
O
' ln
RT c
E E
nF c
 
   
 
sb
OR
conc n s b
R O
ln
cRT c
E E
nF c c
 
     
 
E
I
En
transport-limited current
limiting current
surface – s
bulk – b
ηconc
I. Fick law: the natural movement of species isolution without the
effects of the electrical field
dx
dc
DA
dt
dN ii

dx
dci
….. concentration gradient
dx
dc
DJ
D diffusion coefficient [cm2s-1] ; 10-5- 10-6 in aqeous solutions
II. Fick law: What is the variation of concentration with time ???
DIFFUSION: the natural movement of species in solution
without the effects of the electrical field
2
i
2
x
c
D
t
c





const.D 
Macroplane Electrode – Universal Solution
b s s b
R R R O O O
M
D c c D c c
nFA
         for R and O
2
2
s
b
s
s
at 0
as
c c
D
t x
c c x
c c x
I c
j D
nFA x
 

 
 
  
 
    
 
link to I through
semidifferentiation
link to E through
Nernst or ButlerVolmer
equation
semiintegration
semidifferentiation
I M Q
semiintegration
semidifferentiation
Fick's first law
c
j D
x

 

conservation law
c j
x r
 

 
zy 







x
Coordinates Laplace operator
Cartesian
Cylindrical
Spherical
xr
1








r









sinr
1
r
1
r
Laplace operator in various coordinate systems
Cartesian
x
y
z
x
r


r
Cylindrical Spherical
For any coordinate system
cDJ
cD 2
t
c



operatorLaplace
LTtiontransform aLaplace 
LT for Fick´s second law under conditions of pure diffusion control
 the potential is controlled, the current response and its variation
in time is registrated
chronoamperometry
 the current is controlled and the variation of potential with
time is registrated
chronopotentiometry
Id
Diffusion-limited current: planar and spherical electrodes
Mass transport
No reaction
Reaction of all species reaching the electrode
t
t = 0
E Potential step to obtain a
diffusion-limited current of
the electroactive species
Id
planar electrode ….. semi-infinite linear diffusion
0







dx
dc
DnFAI
Boundary conditions
t = 0 c0 = c no electrode reaction
t 0 lim c = c  bulk solution
t  0 and
x = 0
c0 = 0 diffusion –limited current Id
2
2
x
c
t
c





D
  











  21
1
Dtx
x
erfccc
21
21
)t(
cnFAD
)t(I)t(I d

 
Cottrell equation








 
0
21
11
r)Dt(
cnFAD)t(I)t(I d
to planar electrode linear diffusion
to spherical electrode spherical diffusion








 
0
21
11
r)Dt(
cnFAD)t(I)t(I d
 small t (spherical diffusion linear diffusion)
 large t (the spherical diffusion dominates, which represents a steady state current)
Microelectrodes
-electrodes and ultra--electrodes
 small size at least one dimension 0.1-0.5 m
 steady state
 high current density
 low total current (% electrolysis is small)


 DcnFr
r
cnFAD
Id 0
0
2
 interference from natural convection is negligible (supporting electrolyte)
ci,0
c0
electrode
solution
ci,
xdistance from electrode
 Nernst diffusion layer
Diffusion overpotential
  













  21
1
Dtx
x
erfccc
Diffusion overpotential
Solitary electrode process is in balance
i
i
0
ic cln
Fn
RT
EE i

i,0
i
0
i,0c cln
Fn
RT
EE i

( = 1  ai = ci)
(E0
i = standard potential)
overpotential required for getting over of concentration
difference
i
i,0
i
c,0cd
c
c
ln
Fn
RT
EEη ii

I. Fick law:
dx
dc
DA
dt
dN ii

in stationary state
δ
cc
DA
dt
dN
konst.
dx
dN i,0iii


Faraday:
δ
cc
DFn
dt
dN
S
1
Fn
S
I
j i,0i
i
i
ic



Nernst diffusion layer
Red
Red
Red
δ
DFn

Ox
Ox
Ox
δ
DFn

oi,
i
oi,
ioi,
ld, c
c
1
c
cc
j
j



ld,oi,
i
j
j
1
c
c










ld,i
d
j
j
1ln
Fn
TR
η
in limit: ci,0 = 0  jk,lim= ni F D (ci/)
Diffusion overpotential
i
i,0
i
c,0cd
c
c
ln
Fn
RT
EEη ii

j
- +
-j
-jd,l
Polarization curves for diffusion controlled processes
j
jj
ln dl
Ox




Fn
RT
ln
Fn
RT
EE Redo
c
- +
j
-j
jd,l,cat
jd,l,anod
Polarization curves for diffusion controlled processes
jd,l
E1/2
ja
jc
- E + E
Diffusion overpotential - polarography
Ox

 Redo
c21 ln
Fn
RT
EE2dljj 
Ox
Ox
Ox
δ
DFn

Red
Red
Red
δ
DFn

n act conc uE E IR       
A
W E
V
WE
act
ohm
WE
conc
R E
RE
act
RE
conc
V
R E
potentiostat
C E
Three electrodes set in voltammetry
 Steady state methods (hydrodynamic electrodes, increasing
convection, microelectrodes)
 Linear sweep voltammetry (increasing sweep rate)
 Step and pulse techniques (increasing amplitude and/or
frequency
 Impedance methods ( increasing perturbation frequency)
Methods for studying electrode reactions
timescale of electrode reactions