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BASIC PRINCIPLES OF ELECTRODE PROCESSES
Heterogeneous kinetics
II
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Electron Transfer Reactions
í
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O
C ÍLU
^4 absorption F fluorescence
state
ground state
r interatomic distance
xy
Potential energy curves for the ground state and an excited state of a diatomic molecule
Franck- Condon principle
Electron-transfer processes must satisfy the Frank-Condon 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.
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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
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Electron transfer in biochemistry
A     + Bre(j ^~~^ ^red "^     ox       ^ox 1S me oxidized form of A (the oxidant in the reaction shown)
Bred is the reduced form of B (the reductant).
For such an electron transfer, one may consider two half-cell reactions!
l.A0X+ ze_H Ared......e.g., Fe+++ + e- ^ Fe++
2.B +zeoBd
For each half reaction:
E = E0f -RT/zF (In [red]/[ox])
e.g., for the first half reaction:
E = E°' -RT/zF (In [ared]/[aox])                    [ared] = [a J,.. E = E°'
OXJ
E°f is the mid-point potential, or standard redox potential. It is the potential at which [oxidant] = [reductant] for the half reaction. For an electron transfer:
AF°f = F°f             — F°f               = F°f                 — F°f
íaľj          ľj    (0Xjdant)      rj    (reductant)      ^    (e- acceptor)      ^    (e- donor)
AGof = - nFAEof
Exercise: Consider ET of 2 electrons from NADH to oxygen:
a. %02 + 2H+ + 2 e --> H20                         E°' = + 0.815 V
b. NAD+ + 2H+ + 2 e - -» NADH + H+           E°» = - 0.315 V
Subtracting reaction b from reaction a:
Vi02 + NADH + H+ -> H20 + NAD+            AE°' = + 1.13 V
AG0' = - nFAE0' = - 2(96485 Joules/Volt • mol)(1.13 V) = - 218 kJ/mol
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The solid metallic electrode
q>M (Galvani potential) % (surface potential)
\|/ (Volta potential)
<p   = \|/  + %
Eletfffifo íSfía
(pM (Galvani potential - inner potential) is associated with with EF
\|/ (Volta potential - outer potential) is associated with the potential
outside the electrode's electronic distribution X (surfacepotential)        EF = Eredox-e%

'■■ ^' ""■■.■■ "^ . ^t"^
..-,.
>.í..V- "-■: . V-:..'-^
ř-=.v ■»,                '       II... ŕ.-   .    .     -il^-V-l.^.V"^
From without potential insertion on solid phase
over / below Er        more positive / more negative ^electrode polarization, overpotential
F   = F              =F
v   __           i~r          w-i                     p          polarization        polarizační
=>77 = ED-Er
P        r           F  = F              = F
r          equilibrium        rovnovážny
=> 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
T] = overpotential = přepětí

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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
v^- none                                              jj = 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)
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*■     ■   ■ .
DYNAMIC ELECTROCHEMISTRY
O + ze - <*
^
desorption
O'
chemical reaction
-►      O
R
hydration
desolv
z.e~
adsorption
O
ads
R
Electron transfer
ads^ adsorption
chemical reaction *-----------   R
desorptio
R'
desolv
"   o
dehydration
sol v
dehydration
hydration
R
solv
Electrode      Electrode surface
Bulk solution
Electrode kinetics - can be controlled by electrode potential
Velocity of electrode process
dN
v =
Faraday: It=  NnF=Q
dt
= for transformation of / mol of substance with a charge of z, charge of nF coulomb is consumed ;   F = 96484 coulomb/mol For transformation of dn mol ö/substance at time dt, a current / is consumed                              ,AT
I = nF----= nFv
dt
Heterogeneous reactions - velocity to surface unit:
Current density        .      /         ^ d N 1             A = area
i = — = nF----------
A            dt  A
Experimental dependences:
T\   =f(j)                                        j   =f(T\)
polarization curves                     current-potential curves


h".-

Activation overpotential
Ox + we   ( .rf(l) Red
Kox
Mz++e~^-^M(z-1)+
ox
z+
Mz^+ne   < Vrt) M
3+
FeJ++e"<-?aL>Fe
2+
ox
ox
ox           ox     red
red           red     ox
J
red
Z7 dNox             Z7
=nF——=nFv dt
red
/*   -nF —— = nFv
J OX                                          r j
at
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An expression for the rate of electrode reaction
Arrhenius
r    *rr*\
k = A'exp
\
AH RT
r *~#\
Ä= A exp
J
\
R
J
Gibbs-Helmholtz
k =A exp
ŕ AH*-TAS*^
\
RT
J
í     ,^*\
=A exp
\
AG
RT
J
Ox + ne  <■
>Red
£->
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&^lfe^j$!3S*'     . .L-i&b
An expression for the rate of electrode reaction
a is a coefficient of charge transfer =
symmetry coefficient
a+ac=l
a= a;                      a = 1 - a
AG*c=AG:o+acnFE
AGt=AG:o-aanFE
A(p for reduction .... acE A(p for oxidation .... aa E = (1- ac) E
Effect of a change in
applied electrode
potential on the
reduction of Ox to Red
G
I
AG/
AG   *
CO
E = E(neg.)
E = 0V
AG/
AG   *
i   a.o
..y...
Reaction coordinate
Reaction coordinate
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.
an = 1.5       =^>      a=0.75
ľ US  ......rate - determining step
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Balance of electrode process
J
ox
m                                                                          m
J red        J
vox       red
0
j0   is charge current density
Jo = Jred=nFCoxAred   eXP
AGt, + anF(E-E°)
red
RT
Jo = h,=nFcredA0X exp
AG:x-{l-a)nF(E-E°)
RT


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&ö = standard velocity constant (members independence on E)
.0
k    =AredeXP
AGtd-anFE RT
o
= Ax exP
AG*ox+{l-a)nFE RT
o
Jo =
0
nFcoxk exp
an F E RT
eq
0
fnFCredk    eXP
Jred~ J i
o
f      Aé~*\
k= A exp
v
AG RT
J
(l-a)nF E RT
eq
J ox    Ja
determination of E
eq



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Current - overpotential 77 crossing
7 = |yOT I + LAJ = " F (Vox ~Vred) = * F Kx ^red ^Fk
red     ox
Jo exP	(l-a)nFt]j
	RT      J
	
J o eXP	anFí] 1 RT J
	
ox
Ja
J red    Je
r
j = j0<exp
K
(l-a)nFf] RT
-exp
anFrj RT
Butler- Volmer equation for electrode process, where rds is charge transfer
Ry-.*

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Butler-Volmer equation
an F rj
In j = In L -
RT
In ja = In j 0 +
(l -a)nF ?]
RT
V  j, ka and kc depends exponentially on potential
V   linear free energy relationship the parameters: I and E
V  Eeq gives the exchange current I0= j0A ^standard rate constant
V   electrode as a powerful catalyst
V   for transport the Tafel law must be corrected



s".
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¥ the observed current is proportional to the difference between the rate of the oxidation and reduction reactions at the electrode surface
[Red] [Ox\
I=nFA(ka[Redl-kc[Oxl)
concentrations of Red and Ox next to the electrode
v = k [Redl - K [Ox]
*
ka[Red\  ; kc\Ox\     do not grow indefinitely - limited by the
transport of species to electrode
Id   diffusion-limited current
Polarization curves without overpotential
Ja
-E
Red = Ox + e-
E   equilibrium potential
J =Ja + ("A)
Polarization		curves	with activati		ion ( 2	)verpotential	
			-	1			
• Ja	Red =	Ox + e				/"	
• j				f^^'              i	'		
				Vi			E
							
				^	%	-------w	
Je						Ox + e" = Red	
Ratio dependence of current density and change current density on overpotential for different or values
	Ja ' h	0,75 /     / °'5 ^— °'25
-V      ^	'                 Je 'U	+TJ
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Polarization curve: r/ =f(j)   or I-E curves: j =f(r/)
nF
a) Small values of overpotential     j — jQ-------ľj     Development of e x function
RT

= hnF RT
\$ j
RT
hnF
= R
p
R = polarization resistance
b) Large values of overpotential
negative rj —» process of reduction    In jc = In j0 —
an F n RT
positive 7] —> process of oxidation      In j = In j +
\l-a)n F rj
RT
Tafel relations
r/ = a + b ln\j
rj = a' +b' log \j
Tafel diagram for cathode and anode current density
a = 0,25       j0 = 0,1 mA
logy«
E
eq
-n



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TI  =   f(k)	j:	= 10-6A.cm-	2; n = 1; c0x= ImM; az		= 0.5; T=298 K	
k' (cm.s"1)	ío-3	io-4	io-6	jO-io	io-14	
ri(V)	0.0002	0.003	0.12	0.59	1.06	
reversible x irreversible process/ current density /j
1   metal	'hydrogen V v /	Moxygen (V)
Ag	0.48	0.58
Au	0.24	0.67
Cu	0.48	0.42
Hg	0.88	-
Ni	0.56	0.35
1 Pt( smoothed)	0.02	0.72
| Pt(platinized)	0.01	0.40        |
j = 10"3A.cm-2  T = 298 K
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DIFFUSION: the natural movement of species in solution without the effects of the electrical field
I. Fick law:  the natural movement of species isolution without the
effects of the electrical field
dc.
dx D
J = -D
dc dx
dNi            dct
'- = -DA    l-
dt             dx
concentration gradient
6i,
diffusion coefficient [cmV] ; 10~5- 10"6 in aqeous solutions
II. Fick law: What is the variation of concentration with time ???
dc        d2Ct
= D-----               D = const
dt
d x2
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Laplace operator in various coordinate systems
Coordinates
Cartesian
Laplace operator
d        d        d
+ — +
dx      dy     dz
Cylindrical
did       d
+-----+ —
dr     r dip     dx
Spherical
a    i a       1
+        +
dr     r 3(p     r sin (p d@
Cartesian
^U
x
Cylindrical
x
Spherical
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For any coordinate system
J = -DWc
V  Laplace operator
3c
= DV2c      Laplace transformation =LT
LT for Fick's second law under conditions of pure diffusion control
id
■* 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
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Diffusion-limited current: planar and spherical electrodes
Mass transport
No reaction
Reaction of all species reaching the electrode
I
t
Potential step to obtain a
diffusion-limited current of
the electroactive species
t = 0
planar electrode.....semi-infinite linear diffusion
/
I = nFAD
Boundary conditions
t = 0              c0 = c^
t >0            lim c = c „
t > 0 and
de dx
\
x = 0
>-
c0 = 0
no electrode reaction bulk solution
diffusion-limited current I.
dc _     d2c d t        dx
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r
c=c \\-erfc
OO                                      t/
^
—I>V
x
x
{D,r
>
J
Cottrell equation
1/2
I(t)=Id(t) =
nFAD>  c
OO
(Kt)
1/2
to planar electrode
linear diffusion
I(t)=Id(t)= nFADc^
to spherical electrode
spherical diffusion
* Small t        (spherical diffusion--------►   linear diffusion)
4» large t     (the spherical diffusion dominates, which represents a steady state current)
Microelectrodes
|Li-electrodes   and  ultra-(Li-electrodes
*  small size at least one dimension   0.1-0.5 |im
*  steady state                                                   nFAD C
* high current density                                -lj                            : 2Kľlrľ0JJCc
*  low total current (% electrolysis is small)
* interference from natural convection is negligible (supporting electrolyte)
Diffusion overpotential			
Ci,0 1	ô		
			
		solution	CÍ,oo
electrode	/         1		
Co			
distance from electrode         x			
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Diffusion overpotential
Solitary electrode process is in balance
E. =E°+ —
n.F
C;
lne
(y=l ^d:   = c,)
E_n=E?+ —
c „0
lne
n:F
i, 0
(E°s = standard potential)
overpotential required for getting over of concentration
difference
nd=ECi>0-E   =—- In^-
n:F      C:





I. Fick law:
dt              dx
Nernst diffusion layer ô
in stationary state
dN,     7               dN;       ^ . c,. - c,. 0
1 =konst.  =>      l =-DA-     lA
dx
dt
I
1 dN
Faraday:       7 _^__„ F
h    A        '     A dt
i   ___
n\FD -
c, -c
i, 0
^Red
nFD
Red
KOx -
nFD
Ox
Red
Ox
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ni
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in limit:
Ho= °
Jhim=niFD(c/$
J
9
J
d,l
h O               l              7 ___          Z
1,0
1,0
$- = i-J-
C:
1,0
m
J
d,l
Diffusion overpotential
nd=Ecl,o-Ecl=—c, fo—
n:F       C:
RT ,
Vd=—vln n: F
r
.   N
1-
J
V
Jd,l J
Polarization curves for diffusion controlled processes
E=E°+— lnKRed RT ln^
J
KOx   nF
J»i'2
dl
Diffusion overpotential - polarography
F    — F  +
Polarization curves for diffusion controlled processes
;.........;
0>j 73 i
-n

cat
m
Jd,l,anod
+ j A/nr2
mV
Jujim
j-*--"--"-?-»
ed
+ TJ
J D, lim, o
x
"J
A/nr2
+ T|
mV
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Jaroslav Heyrovsky
* Dec. 20,1890, Prague, Bohemia, Austro-Hungarian Empire [now Czech Rep.] f March 27,1967, Prague, Czechoslovakia
						
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Jaroslav Heyrovsky was an inventor of the Polarographie method, father of electroanalytical chemistry, recipient of the Nobel Prize (1959). His contribution to electroanalytical chemistry can not be overestimated. All voltammetry methods used now in electroanalytical chemistry originate from polarography developed by him.
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Half-wave potential, limited diffusion current
x.-
DME
^ tune
Mn+ + ne- +Hg   = M(Hg) amalgam
a) V2 wave potential (Ey2) characteristic of Mn+ %
b) height of either average current maxima (i avg)
or top current max (i max) is ~ analyte concentration
c) size of i max is governed by
rate of growth of DME > drop time (t, sec) rate of mercury flow (m, mg/s) diffusion coefficient of analyte (D, cm2/s) number of electrons in process (n) analyte concentration (c, mol/ml)
Limiting diffusion current
K.cs Jdu.il .ľuiiĽii:
J_____j_____I             J
-ü.3                   -n.ň
-09
-L.2
ŕppliŕti panuji. V *«, SCE
llkovič equation
(id)max = 0.706nD1/2m2/3t1/6c (id)avg = 0.607nD1/2m2/3t1/6c
■m«      . tr
Instrumentation, common techniques
Autolab
Ecochemie
Utrecht
The Netherlands
VA-Stand 663
Metrohm
Zurich
Switzerland
Electrochemical
analyzer
AUTOLAB
Autolab 30
|®§8Ü9III6<^
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Thank you for your attention