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7/30/2019 Lecture 8 Electro-Kinetics
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Electro-Kinetics
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Description of Electrochemical
Techniques
The technique is named according to theparameters measured
E.g.
Voltammetry measure current and voltage
Potentiometry measure voltage
Chrono-potentiometry measure voltage with
time (under an applied current) Chrono-amperometry measure current with
time (under an applied voltage)
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Electro-Kinetics
Movement of Ions
Butler Volmer Equation
Rotating Disc Electrode
Rotating Cylinder Electrode
Voltammetry
Cyclic Voltammetry
Chrono-potentiometry
Chrono-amperometry
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Movement of Ions in Solution
Diffusion Movement under a concentration gradient. If
an electrochemical reaction occurs the current due to
this reaction is called, id , the diffusion current.
Migration or Transport Movement of ions under an
electric field due to coulombic forces. If anelectrochemical reaction occurs the current due to this
reaction is called, im , the migration current.
Convection Movement due to changes in density at the
electrode solution interface. This occurs due to depletionor addition of a species due to the electrochemical
reaction.
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The Capacitance Current
The charging or capacitance current, ic , isdue to the
presence of the electrical double layer and it is alwayspresent. This current, of course, is not related to any
movement of ions.
Ic = Cdl x V
Where:
Cdl = the capacitance of the electrical double layer
V = voltage scan rate
The capacitance current makes its presence felt whenmeasuring charge transfer (Faradaic) processes at
concentrations of 10-5 M.
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Diffusion
Molecular diffusion,often called simplydiffusion, is a nettransport of
molecules from aregion of higherconcentration to oneof lower
concentration byrandom molecularmotion.
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Migration or Transport
Is the fraction of current carried by the ions.
For example in a solution of copper sulphate the transportnumber of Cu2+ is 0.4 and that of SO4
2- = 0.6.
t++ t- = 0.4 + 0.6 = 1
Since the migration current depends on the ionic strength of
the solution it is usually eliminated by addition of excess of aninert supporting electrolyte (100 1000 fold excess inconcentration)
The current is carried by the inert supporting electrolyte (e.g.NaCl , KNO
3etc) because the ions produced do not undergo
any electrochemical reaction the transport current iseffectively removed.
In excess inert supporting electrolyte, the current measureddue to the electro-active species of interest is due only to
diffusion which can be related to mass transfer.
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Voltammetry the following example shows how the migration
current is eliminated. Pb2+ + 2e Pb0
The supporting electrolyte Ensures diffusion control of limiting currents by eliminating
migration currents
Table: Limiting currents observed for 9.5 x 10-4 M PbCl2 as a
function of the concentration of KNO3 supporting electrolyte
Molarity
of KNO3
IlA
0 17.6
0.001 12.0
0.005 9.8
0.10 8.45
1.0 8.45
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Voltammetry
The example shown is for the reduction of Pb
2+
at aninert mercury electrode.
Pb2+ + 2e Pb(Hg)
At low inert electrolyte concentration a large fraction
of the total current is due to the migration current,
i.e. the currents due to the electrostatic attraction of
ions to the electrode.
For solution 1: i migration im 17.6 8.45 = 9.2 A
i diffusion id = 8.45 A
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Ficks First Law of Diffusion
Fick's first law relates the diffusive flux to theconcentration field, by postulating that the flux goesfrom regions of high concentration to regions of lowconcentration, with a magnitude that is proportional
to the concentration gradient (spatial derivative). Inone (spatial) dimension, this is
x
DJ
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Ficks First Law of Diffusion
where
J is the diffusion flux in dimensions of [(concentration of
substance) length2 time-1], example mole (M) m-2 s-1. J measures the amount of substance that will flow
through a small area during a small time interval.
D is the diffusion coefficient or diffusivity in dimensionsof [length2 time1], example m2 s-1
(for ideal mixtures) is the concentration in dimensionsof [(concentration of substance) length3], example M m-3
xis the position [length], example m
xDJ
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Ficks First Law of Diffusion
D is proportional to the squared velocity of thediffusing particles, which depends on thetemperature, viscosity of the fluid and the size of theparticles according to the Stokes-Einstein
relationship. In dilute aqueous solutions the diffusion coefficients
of most ions are similar and have values that at roomtemperature are in the range of 0.6x10-9 to 2x10-9m2/s.
For biological molecules the diffusion coefficientsnormally range from 10-11 to 10-10 m2/s.
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Ficks First Law of Diffusion
In two or more dimensions we must use, , the del orgradient operator, which generalises the first derivative,obtaining
J = -D
The driving force for the one-dimensional diffusion is thequantity -/x
which for ideal mixtures is the concentration gradient. Inchemical systems other than ideal solutions or mixtures,the driving force for diffusion of each species is the
gradient of chemical potential of this species. Then Fick'sfirst law (one-dimensional case) can be written as:
xRT
DcJ ii
1
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Ficks First Law of Diffusion
where the index i denotes the ith species, c is the concentration (mol/m3),
R is the universal gas constant (J/(K mol)),
T is the absolute temperature (K), and
is the chemical potential (J/mol).
xRT
DcJ ii
1
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Butler-Volmer Equation
The Butler-Volmer equation is one of the
most fundamental relationships in
electrochemistry. It describes how the
electrical current on an electrode depends onthe electrode potential, considering that both
a cathodic and an anodic reaction occur on
the same electrode:
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Butler-Volmer Equation
where:
I = electrode current, Amps
Io= exchange current density, Amp/m2
E = electrode potential, V Eeq= equilibrium potential, V
A = electrode active surface area, m2
T = absolute temperature, K
n = number of electrons involved in the electrode reaction
F = Faraday constant R = universal gas constant
= so-called symmetry factor or charge transfer coefficientdimensionless
The equation is named after chemists John Alfred Valentine Butler and
Max Volmer
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Butler-Volmer Equation
The equation describes two regions: At high overpotential the Butler-Volmer equation
simplifies to the Tafel equation
E Eeq = a blog(ic) for a cathodic reaction
E Eeq = a + blog(ia) for an anodic reaction
Where:
a and b are constants (for a given reaction and
temperature) and are called the Tafel equationconstants
At low overpotential the Stern Geary equation
applies
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Current Voltage Curves for Electrode Reactions
Without concentrationand therefore masstransport effects tocomplicate theelectrolysis it is possibleto establish the effects of
voltage on the currentflowing. In this situationthe quantity E - Ee reflectsthe activation energyrequired to force current ito flow. Plotted below are
three curves for differingvalues ofio with = 0.5.
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Voltammetry
Although the Butler Volmer Equation predicts, thatat high overpotential, the current will increaseexponentially with applied voltage, this is often notthe case as the current will be influenced by masstransfer control of the reactive species.
Take the following example of the reduction of ferricions at a platinum rotating disc electrode (RDE).
Fe3+ + e = Fe2+
The rotation of the electrode establishes a welldefined diffusion layer (Nernst diffusion layer)
The contribution of the capacitance current will alsobe demonstrated in this example.
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Effect of the Capacitance Current in Voltammetry. The reduction of Ferric Chloride
is carried out in the presence of 1 M NaCl to eliminate the migration current.
Slope due to ic
Applied Potential -Ve
Current
10-5 M Fe3+ Fe3+ + e Fe2+
Current
10-3 M Fe3+ Fe3+ + e Fe2+
Applied Potential -Ve
(a)
(b)
Note that the iE curve in Fig. (a)
is recorded at a much higher
sensitivity than in Fig. (b).
ild
ild
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Charging Current or Capacitance
Current
Note that due to the presence of the electrical
double layer a charging or capacitance current
is always present in voltammetric
measurements.
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Butler-Volmer Equation
where:
I = electrode current, Amps
Io= exchange current density, Amp/m2
E = electrode potential, V
Eeq= equilibrium potential, V
A = electrode active surface area, m2
T = absolute temperature, K
n = number of electrons involved in the electrode reaction
F = Faraday constant R = universal gas constant
= so-called symmetry factor or charge transfer coefficientdimensionless
The equation is named after chemists John Alfred Valentine Butler andMax Volmer
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Butler Volmer Equation
While the Butler-Volmer equation is valid over the fullpotential range, simpler approximate solutions can beobtained over more restricted ranges of potential. Asoverpotentials, either positive or negative, become largerthan about 0.05 V, the second or the first term of equation
becomes negligible, respectively. Hence, simple exponentialrelationships between current (i.e., rate) and overpotentialare obtained, or the overpotential can be considered aslogarithmically dependent on the current density. Thistheoretical result is in agreement with the experimentalfindings of the German physical chemist Julius Tafel (1905),and the usual plots of overpotential versus log current densityare known as Tafel lines.
The slope of a Tafel plot reveals the value of the transfercoefficient; for the given direction of the electrode reaction.
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Butler-Volmer Equation
ialoverpotentcathodichighat
exp
ialoverpotentanodichighat
1exp
0`
0`
RT
nFii
RTnFii
cc
aa
ia and icare
the exhange
current
densities for
the anodicand cathodic
reactions
These equations can be rearranged to give
the Tafel equation which was obtained
experimentally
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Butler Volmer Equation - Tafel Equation
nb
in
a
iba
in
in
in
in
i
nF
RTi
nF
RT
o
aaa
a
c
cc
c
c
cc
c
059.0
ln059.0
and
log
equationTafelknownwelltheisequationThe
processanodicfor theC25atlog059.0
log059.0
processcathodicfor theC25atlog059.0
log059.0
lnln
0
0
0
0
0
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Tafel Equation
The Tafel slope is an intensive parameter and does not
depend on the electrode surface area.
i0 is and extensive parameter and is influenced by the
electrode surface area and the kinetics or speed of the
reaction.
Notice that the Tafel slope is restricted to the number of
electrons, n, involved in the charge transfer controlled
reaction and the so called symmetry factor, .
n is often = 1 and although the symmetry factor can varybetween 0 and 1 it is normally close to 0.5.
This means that the Tafel slope should be close to 120
mV if n = 1 and 60 mV if n = 2.
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Tafel Equation
We can write:
ii
nF
RTb
iibiinF
RT
log303.2ln
slopeTafelthe303.2
where
lnorln 00
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Current Voltage Curves for Electrode Reactions
Without concentrationand therefore masstransport effects tocomplicate theelectrolysis it is possibleto establish the effects of
voltage on the currentflowing. In this situationthe quantity E - Ee reflectsthe activation energyrequired to force current ito flow. Plotted below are
three curves for differingvalues ofio with = 0.5.
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Tafel Equation
The Tafel equation can be also written as:
where
the plus sign under the exponent refers to an anodicreaction, and a minus sign to a cathodic reaction, n isthe number of electrons involved in the electrodereaction k is the rate constant for the electrodereaction, R is theuniversal gas constant, F is the
Faraday constant. kis Boltzmann's constant, Tis theabsolute temperature, e is the electron charge, and is the so called "charge transfer coefficient", thevalue of which must be between 0 and 1.
http://en.wikipedia.org/wiki/Faraday_constanthttp://en.wikipedia.org/wiki/Universal_gas_constanthttp://en.wikipedia.org/wiki/Boltzmann's_constanthttp://en.wikipedia.org/wiki/Boltzmann's_constanthttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Boltzmann's_constanthttp://en.wikipedia.org/wiki/Electron_chargehttp://en.wikipedia.org/wiki/Electron_chargehttp://en.wikipedia.org/wiki/Electron_chargehttp://en.wikipedia.org/wiki/Electron_chargehttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Electron_chargehttp://en.wikipedia.org/wiki/Electron_chargehttp://en.wikipedia.org/wiki/Absolute_temperaturehttp://en.wikipedia.org/wiki/Boltzmann's_constanthttp://en.wikipedia.org/wiki/Faraday_constanthttp://en.wikipedia.org/wiki/Universal_gas_constant7/30/2019 Lecture 8 Electro-Kinetics
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Tafel Equation
The following equation was obtainedexperimentally
Where:
= the over-potential
i= the current density
a and b = Tafel constants
iba log
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Tafel Equation
Applicability
Where an electrochemical reaction occurs in two half reactions onseparate electrodes, the Tafel equation is applied to each electrodeseparately.
The Tafel equation assumes that the reverse reaction rate isnegligible compared to the forward reaction rate.
The Tafel equation is applicable to the region where the values ofpolarization are high. At low values of polarization, the dependenceof current on polarization is usually linear (not logarithmic):
This linear region is called "polarization resistance" due to its formalsimilarity to Ohms law
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Stern Geary Equation Applicable in the linear region of the Butler Volmer
Equation at low over-potentials
resistanceonpolarisatimeasuredthe
3.2
constantTafelthe
Where
a
a
iE
R
B
R
Bi
p
c
c
p
corr
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Tafel Equation
Overview of the terms
The exchange current is the current at equilibrium,i.e. the rate at which oxidized and reduced speciestransfer electrons with the electrode. In other words,the exchange current density is the rate of reaction
at the reversible potential (when the overpotential iszero by definition). At the reversible potential, thereaction is in equilibrium meaning that the forwardand reverse reactions progress at the same rates.This rate is the exchange current density.
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Tafel Equation
The Tafel slope is measured experimentally; however,
it can be shown theoretically when the dominantreaction mechanism involves the transfer of a singleelectron that
Tis the absolute temperature,
R is the gas constant
is the so called "charge transfer coefficient", thevalue of which must be between 0 and 1.
FRTb
303.2
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Levich Equation
The Levich Equation models the diffusion and solution
flow conditions around a rotating disc electrode (RDE). Itis named after Veniamin Grigorievich Levich who firstdeveloped an RDE as a tool for electrochemical research.It can be used to predict the current observed at an RDE,in particular, the Levich equation gives the height of thesigmoidal wave observed in rotating disk voltammetry.The sigmoidal wave height is often called the Levichcurrent.
In work at a RDE the electrode is usually rotated quite
fast (1000 rpm) in order to establish a well defineddiffusion layer.
The scan rate is relatively slow typically 2-5 mV s-1
http://en.wikipedia.org/wiki/Veniamin_Grigorievich_Levichhttp://en.wikipedia.org/wiki/Veniamin_Grigorievich_Levich7/30/2019 Lecture 8 Electro-Kinetics
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Current Voltage Curve at a RDE
It is important to remember that in order to determine the
diffusion current and the mass transfer coefficient usingvolatmmetry, excess inert supporting electrolyte must be
present to eliminate the migration current
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Levich Equation The Levich Equation is written as:
where
iL is the Levich current
n is the number ofelectrons transferred in the half reaction
Fis the Faraday constant A is the electrode area
D is the diffusion coefficient (see Fick's law of diffusion)
wis the angular rotation rate of the electrode
vis the kinematic viscosity Cis the analyte concentration
While the Levich equation suffices for many purposes,improved forms based on derivations utilising more terms inthe velocity expression are available.[1][2]
http://en.wikipedia.org/wiki/Electronshttp://en.wikipedia.org/wiki/Half_reactionhttp://en.wikipedia.org/wiki/Faraday_constanthttp://en.wikipedia.org/wiki/Fick's_law_of_diffusionhttp://en.wikipedia.org/wiki/Kinematic_viscosityhttp://en.wikipedia.org/wiki/Analytehttp://calctool.org/CALC/chem/electrochem/levichhttp://calctool.org/CALC/chem/electrochem/levichhttp://en.wikipedia.org/wiki/Analytehttp://en.wikipedia.org/wiki/Kinematic_viscosityhttp://en.wikipedia.org/wiki/Fick's_law_of_diffusionhttp://en.wikipedia.org/wiki/Faraday_constanthttp://en.wikipedia.org/wiki/Half_reactionhttp://en.wikipedia.org/wiki/Electrons7/30/2019 Lecture 8 Electro-Kinetics
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Rotating Disc Electrode
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Levich Equation
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Levich Equation
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Levich Equation It is important to note that the layer of solution immediately
adjacent to the surface of the electrode behaves as if it werestuck to the electrode. While the bulk of the solution is being
stirred vigorously by the rotating electrode, this thin layer of
solution manages to cling to the surface of the electrode and
appears (from the perspective of the rotating electrode) to bemotionless.
This layer is called the stagnant layer in order to distinguish it
from the remaining bulk of the solution. The act of rotation
drags material to the electrode surface where it can react.Providing the rotation speed is kept within the limits that
laminar flow is maintained then the mass transport equation is
given by the Levich equation.
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Levich Equation RDE
The Levich equation takes into account both the rate of
diffusion across the stagnant layer and the complex solution
flow pattern. In particular, the Levich equation gives the
height of the sigmoidal wave observed in rotated disk
voltammetry. The sigmoid wave height is often called the
Levich current, iL, and it is directly proportional to the analyteconcentration, C. The Levich equation is written as:
iL = (0.620) n F A D2/3 w1/2 v1/6 C
where w is the angular rotation rate of the electrode
(radians/sec) and v is the kinematic viscosity of the solution(cm2/sec). The kinematic viscosity is the ratio of the solution's
viscosity to its density.
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Current Voltage Curve at a RDE
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Ilimiting vs (electrode rotational velocity)
Le ich Eq ation RDE
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Levich Equation - RDE
The linear relationship between Levich current and the squareroot of the rotation rate is obvious from the Levich plot. A
linear least squares fit of the data produces an equation forthe best straight line passing through the data. The specificexperiment shown, the electrode area, A, was 0.1963 cm2, theanalyte concentration, C, was 2.55x106 mol/cm3, and thesolution had a kinematic viscosity, v, equal to 0.00916cm2/sec. After careful substitution and unit analysis, you cansolve for the diffusion coefficient, D, and obtain a value equalto 4.75x106 cm2/s. This result is a little low, probably due tothe poor shape of the sigmoidal signal observed in thisparticular experiment.
The kinematic viscosity is the ratio of the absolute viscosity of
a solution to its density. Absolute viscosity is measured inpoises (1 poise = gram cm1 sec1). Kinematic viscosity ismeasured in stokes (1 stoke = cm2 sec1). Extensive tables ofsolution viscosity and more information about viscosity unitscan be found in the CRC Handbook of Chemistry and Physics.
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Cyclic Voltammetry
Cyclic Voltammetry is carried out at a stationary
electrode.
This normally involves the use of an inert disc
electrode made from platinum, gold or glassy carbon.
Nickel has also been used.
The potential is continuously changed as a linear
function of time. The rate of change of potential with
time is referred to as the scan rate (v). Compared to a
RDE the scan rates in cyclic voltammetry are usuallymuch higher, typically 50 mV s-1
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Cyclic Voltammetry
Cyclic voltammetry, in which the direction of the
potential is reversed at the end of the first scan. Thus,
the waveform is usually of the form of an isosceles
triangle.
The advantage using a stationary electrode is that theproduct of the electron transfer reaction that occurred in
the forward scan can be probed again in the reverse
scan.
CV is a powerful tool for the determination of formalredox potentials, detection of chemical reactions that
precede or follow the electrochemical reaction and
evaluation of electron transfer kinetics.
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Cyclic Voltammetry
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Cyclic Voltammetry
For a reversible
process
Epc Epa = 0.059V/n
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The Randles-Sevcik equation Reversible systems
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The Randles-Sevcik equation Reversible systems
n = the number of electrons in the redox reaction
v= the scan rate in V s-1 F = the Faradays constant 96,485 coulombs mole-1
A = the electrode area cm2
R = the gas constant 8.314 J mole
-1
K
-1
T= the temperature K
D = the analyte diffusion coefficient cm2 s-1
21
4463.0 RTnFvDnFACip ACDvnip 212123510687.2
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The Randles-Sevcik equation Reversible systems
As expected a plot of peak height vs the square root of the scan rate
produces a linear plot, in which the diffusion coefficient can be obtained
from the slope of the plot.
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Cyclic Voltammetry
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Cyclic Voltammetry
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Cyclic Voltammetry
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Cyclic Voltammetry Stationary Electrode
Peak positions are related to formal potential of redox
process
E0 = (Epa + Epc ) /2
Separation of peaks for a reversible couple is 0.059/n volts
A one electron fast electron transfer reaction thus gives
59mV separation
Peak potentials are then independent of scan rate
Half-peak potential Ep/2 = E1/2 0.028/n
Sign is + for a reduction
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Cyclic Voltammetry Stationary Electrode
The shape of the voltammogram depends on the transfer
coefficient
When deviates from 0.5 the voltammograms become
asymmetric -cathodic peak sharper as expected from Butler
Volmer eqn.
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Web Sites
http://calctool.org/CALC/chem/electrochem/l
evich
http://www.calctool.org/CALC/chem/electroc
hem/cv1
T f l E i
http://calctool.org/CALC/chem/electrochem/levichhttp://calctool.org/CALC/chem/electrochem/levichhttp://www.calctool.org/CALC/chem/electrochem/cv1http://www.calctool.org/CALC/chem/electrochem/cv1http://www.calctool.org/CALC/chem/electrochem/cv1http://www.calctool.org/CALC/chem/electrochem/cv1http://calctool.org/CALC/chem/electrochem/levichhttp://calctool.org/CALC/chem/electrochem/levich7/30/2019 Lecture 8 Electro-Kinetics
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Tafel Equation
The Tafel slope is an intensive parameter and does not
depend on the electrode surface area.
i0 is and extensive parameter and is influenced by the
electrode surface area and the kinetics or speed of the
reaction.
Notice that the Tafel slope is restricted to the number of
electrons, n, involved in the charge transfer controlled
reaction and the so called symmetry factor, .
n is often = 1 and although the symmetry factor can varybetween 0 and 1 it is normally close to 0.5.
This means that the Tafel slope should be close to 120
mV if n = 1 and 60 mV if n = 2.
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Tafel Equation
We can write:
ii
nF
RTb
iibiinF
RT
log303.2ln
slopeTafelthe303.2
where
lnorln 00
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Evans Diagrams
i
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Evans Diagrams
E Di
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Evans Diagrams