Lecture 8 Electro-Kinetics

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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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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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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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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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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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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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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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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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_constant


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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_Levich


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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/Electrons


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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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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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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/levich


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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

Lecture 8 Electro-Kinetics

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