ec js module ch3304 megl physical and interfacial electrochemistry 2013 new course l6 electrode...

8/17/2019 EC JS Module CH3304 MEGL Physical and Interfacial Electrochemistry 2013 New Course L6 Electrode Kinetics

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Physical and InterfacialElectrochemistry 2013

Lecture 6.Phenomenological electrode kinetics:

The Butler-Volmer Equation.

0

1exp exp

ox red

F F i i

RT RT

i i

J.A.V. Butler Max Volmer Walter Nernst Julius Tafel


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Kinetics of interfacial ET.• Estimation of equilibrium

redox potentials provides a

quantitative measure for thetendency for a specificredox reaction to occur.Kinetic information is notderived.

• ET reactions atelectrode/solution interfacesare activated processes.– ET rate effected by:

• Applied electrode potential• Temperature

– Activation energy barrierheight can be effected byapplied potential. This is incontrast to ordinarychemical reactions.

• We seek an answer to thefollowing questions:

– How can we quantitativelymodel the rate of an ETprocess which occurs at theinterface between a metallicelectrode and an aqueoussolution containing a redox

active couple?– How can kinetic informationabout ET processes bederived?

• We shall also investigate the

influence of materialtransport, and double layerstructure on interfacial ETprocesses.


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Basic concepts of electrode kinetics.• For an interfacial ET

process:

– current flow isproportional to reactionflux (rate).

• Reaction rate is proportionalto reactant concentration at

interface.• As in chemical kinetics:– the constant of

proportionality betweenreaction rate fS (molcm-

2s-1) and reactantconcentration c (molcm-3)is termed the rateconstant k (cms-1).

• All chemical andelectrochemical reactions

are activated processes.• An activation energy barrierexists which must beovercome in order that thechemical reaction may

proceed.– Energy must be suppliedto surmount theactivation energy barrier.

– This energy may be

supplied thermally or also(for ET processes atelectrodes) via theapplication of a potentialto the metallic electrode.


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The electrode potential drivesET processes at interfaces.• Application of a potential to an electrode

generates a large electric field at theelectrode/solution interface which reducesthe height of the activation energy barrierand thereby increases the rate of the ETreaction.

• Hence the applied potential acts as a drivingforce for the ET reaction.

• We intuitively expect that the currentshould increase with increasing drivingforce. This can be understood using a simplepictorial approach.


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ne-

P

Q

Oxidation or de-electronation.

P = reductant (electron donor)

Q = Product

Electron sink electrode

(Anode).

ne-

A

B

Electron source electrode

(Cathode).

Reduction or electronation.

A = oxidant (electron acceptor)

B = Product

Interfacial electron transfer at electrode/solution interfaces:oxidation and reduction processes.

• In electrolysis we use an applied voltage

to perform chemistry at a M/S interface.• The applied voltage drives the chemicalreaction which does not occur spontaneously.• The current flowing across the M/Sinterface is a measure of the rate of

the chemical transformation at the interface.

• The greater the applied voltage,the larger the resulting currentflow, and the greater the rateof the chemical reaction.

• The rate at which charge ismoved across the M/S interface= the rate at which chemistryis accomplished at the M/S

interface.

nFAf

dt

dN nFA

dt

dqi

Current (A)

Charge(C)

# electrons transferred

FaradayConstant (Cmol-1)

Electrode area (cm2)

Amount of Material (mol)

Reaction flux (rate)mol cm-2s-1

Time (s)


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In this section we consider , in a very qualitative way, thekinetic description of a simple single step, single electrontransfer process. A typical example might be the Fe2+(aq)/ Fe3+(aq) redox reaction at a Pt electrode in an aqueouselectrolyte solution . The latter type of reaction istermed an outer sphere electron transfer process sinceno bonds are broken or made during the course of the

reaction. Consequently, we neglect complicating factorssuch as diffusional transport of reactants and products ,and adsorption effects.How are the electron transfer kinetics examinedexperimentally ? It is obvious that one must do themeasurement using an electrochemical cell containing atleast two electrodes. Indeed it is conventional to use athree electrode arrangement as outlined across. Theelectrode of interest is termed the working or indicatorelectrode. The redox chemistry occurs at the interfacebetween the working electrode and the electrolyticsolution. A second electrode , termed the counter orauxiliary electrode, is required to complete the circuit .Current is passed in a circuit containing the working andthe counter electrodes. Finally , one wishes to determine

the potential difference across the working electrode/ solution interface. The latter cannot be experimentallydetermined.

What one can do however, is to measure changesin the potential of the working electrode withrespect to a third electrode, the referenceelectrode, placed in the solution near the

working electrode. The latter will have a veryhigh impedance to current flow and so thepotential of the reference electrode can beconsidered to be constant, irrespective of thecurrent passed through the working and counterelectrodes. Thus, the measured change inpotential between the working and reference

electrodes will be equal to the change inpotential at the working electrode / solutioninterface. Hence in a typical experiment apotential waveform is applied to the workingelectrode , chemistry is done, and the resultantcurrent response is monitored using a suitableoutput device .

Workingelectrode

Referenceelectrode

Current is passed betweenworking and counterelectrodes. The potentialis measured betweenworking and referenceelectrodes/.

Counter (auxillary)Electrode.


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-

EF

LUMO

HOMO

LUMO

HOMO

Redox couplein solution

Electronenergy

n e-

Metallic electrode

Energy of electronsin metal increases upon application of a

potential more negativethan the thermodynamicequilibrium value.

A net reduction (cathodic)current flows from metal toLUMO levels of redox activespecies in solution.

Pictorial explanation ofcurrent flow due to reduction.


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Metallic

electrode

+

EF

LUMO

HOMO

LUMO

HOMO


Electronenergy

n e-

Energy of electronsin metal decreases uponapplication of a potential

more positive than thethermodynamic equilibriumvalue.

A net anodic (oxidation)current flows from theHOMO level of the redoxspecies in solution to themetallic electrode.

Pictorial explanation of current

flow due to oxidation.


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Consider the situation depicted in the previous twoslides, one for a net reduction process and the other ora net oxidation process.

In this pictures (which in effect distill some verycomplex quantum mechanical considerations) weindicate, in a very schematic way, the filled and emptyelectronic states in the metallic electrode and thehighest occupied and lowest unoccupied energy levels ofthe donor species A and acceptor species B in thesolution. The demarcation line between filled and empty

electronic states in the metal is designated the Fermienergy EF. Now if species A and B are both present insolution and if no external potential is applied to theworking electrode, then after a certain time a steadyopen circuit potential termed the equilibrium potentialEe may be measured. Indeed the value of thispotential will depend on the logarithm of ratio of the

concentrations of A and B via the Nernst equation asdiscussed previously in lecture 4. Under such conditionswe may set EF = Ee.

In contrast, when the working electrode becomespositively charged via application of an externalpotential more positive than Ee then the energy of the

electrons in the metal will be lowered and EF shiftsdownwards in energy. If the applied potential issufficiently positive then a stage will be reached suchthat EF is lower in energy than the HOMO level of thedonor species A and one can obtain a net flow ofelectrons from the donor to the metal. An anodicoxidation current flows.

Conversely, if a potential more negative thanthe equilibrium value is applied to the electrode

then the energy of the electrons in the filledlevels of the metal will be raised. A stage willbe reached when EF is now higher in energythan the LUMO level of the acceptor speciesB and electrons will be transferred from themetal to B in solution. A cathodic reductioncurrent will flow. We shall consider thispicture in greater detail in the next chapterwhen we discuss the microscopic quantummechanical approach to interfacial electrontransfer.


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•

A survey of electrochemical

reaction types.• Electrochemical reactions

are usually complex multistep

processes involving thetransfer of more than oneelectron.

• In this course we focus onsimple single step ETprocesses involving thetransfer of a singleelectron.

• The kinetics of simple ET

processes can be understoodusing the activated complextheory of chemical kinetics(see SF Kinetics notes).


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Factors effecting the current/potential

response at electrode/solution interfaces.• The current observed at an

electrode/solution interfacereflects two quantities:– Charging of electrical double

layer : non Faradaic chargingcurrent iC

– Interfacial ET acrosselectrode/solution interface :Faradaic current iF.

• We initially focus attention onthe Faradaic component.

• The Faradaic current iF in turncan have components:– at low potentials : arising from

rate determining interfacial ET ,

– at high potentials : arising frommaterial transport (MT) due todiffusion mechanisms.

– These components can bequantified in terms ofcharacteristic rate constants :k0 (units: cms-1) for ET and kD(units: cms-1) for MT.

• Our aim is to be able tounderstand the shape that thegeneral current/potential curve

adopts and to be able tointerpret the voltammogram interms of ET and MT effects.

ET

MT


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CDL

R CT

i

iC

iF

R S

Electrode

Solution

Simple equivalent circuit representation of

electrode/solution interface region.

Faradaiccurrent

DL chargingcurrent

F C iii

Resistance of solution

Double layer chargingcurrent always presenin addition toFaradaic current inelectrochemicalmeasurements.


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Temperature effects in chemical kinetics.• Chemical reactions are activated processes : they require an

energy input in order to occur. The same is true forelectrochemical reactions which occur at electrode/solutioninterfaces.

• Many chemical reactions are activated via thermal means. In

contrast and in addition electron transfer reactions may also beactivated via application of an electrode potential which alters theFermi energy of the electrons in the electrode material.

• The relationship between rate constant k and temperature T isgiven by the empirical Arrhenius equation (refer back to CH1101for details). The form of this equation is independent of the

physical model used to explain it. The exact interpretation of thethe pre-exponential factor A and the Activation energy Term EA

• The activation energy EA is determined from experiment, by

measuring the rate constant k at a number of differenttemperatures. The Arrhenius equation is then used to constructan Arrhenius plot of ln k versus 1/T. The activation energy

is determined from the slope of this plot.

RT

E Ak Aexp

Pre-exponentialfactor

dT

k d RT T d

k d R E A

ln

/1

ln 2

k ln

T

1

R

E Slope A

ActivationEnergy

The activation energy measures the sensitivityOf the rate constant to changes in temperature.


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Reaction coordinate

E n e r g y

U0

E

E’

R

P

TSVan’t Hoff expression:0

2

ln c

P

d K U

dT RT

Standard change in internalenergy:

0U E E

k

k

c

R

k

k

P

K

0

2

ln lnln

P

d k d k d k U

dT k dT dT RT

2

2

ln

ln

d k E

dT RT

d k E

dT RT

This leads to formaldefinition of ActivationEnergy.


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In some circumstances the Arrhenius Plot is curved which implies thatthe Activation energy is a function of temperature.Hence the rate constant may be expected to vary with temperatureaccording to the following expression.

expm E k aT RT

2 2

2

ln ln ln

ln A

A

E k a m T

RT

d k m E E RT RT E mRT

dT T RT

E E mRT

We can relate the latter expression to the Arrhenius parameters A and EA

as follows.

Henceexp expm m A A

m m

E E k aT e A

RT RT

A aT e

Svante August

Arrhenius

Arrhenius equation: more elaborate situations.


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A morecomprehensive

Picture ofchemicalreactivity isgiven byTransition State

Theory.

Refer to earlier lecturesPresented in thisModule by Prof. Bridge.


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We begin by noting that chemical reactions arethermally activated processes. Consequently in orderfor chemical to occur, the reactant species mustinitially come together in a molecular encounter andthen gain enough thermal energy to subsequently passover the activation energy energy barrier . Werecall that the activation energy is simply the energyrequired to bring the reactants to some criticalconfiguration from which they can rearrange to form

products. It is clear that during the course of achemical reaction bonds will be stretched and brokenin the reactants and new bonds will be formed. Hencewe see that the potential energy of the system willvary during the course of the reaction. Hence a morecomplete and rigorous description involves theexamination of potential energy changes during the

course of a chemical reaction. This approach leads tothe development of multidimensional potential energysurfaces where the potential energy of the system isplotted as a function of various bond distances andbond angles.

Consider the following simple reaction between an

atom A and a diatomic molecule BC in the gas phase :

A BC ABC * AB C

In the latter expression the reactants are transformed toproducts via formation of a high energy activated complex[ABC]* . If a potential energy surface is to be calculated forthis reaction then the potential energy V should be plottedas a function of the two bond distances RAB and RBC and thebond angle = .

Of course, this procedure would necessitate use of a fourdimensional diagram. However to make matters less

complicated, one usually fixes one of the parameters, say ,at a particular value, and then one examines thecorresponding three dimensional surface defined by the axesV(R), RAB and RBC . The latter is a particular three dimensionalcut from the four dimensional energy hypersurface. Aparticular three dimensional potential energy surfacecorresponding to a fixed value is presented in the next

slide. We note that the course of the reaction is representedby a trajectory on the potential energy surface from P to Q,where P denotes the classical ground state of the diatomicmolecule BC and Q represents the ground state of thediatomic molecule AB. Note that the variation of V(R) withbond distance R is described by the Morse potential energyfunction :

ABC

V R Do 1 exp a R Re 2

where Do denotes the bond dissociation energy , Re isthe equilibrium bond separation distance, and a is theanharmonicity constant.


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Now the system will tend to describe a trajectory ofrelatively low energy along the potential energy surface.Typically, this pathway involves two valleys meeting at a

saddle point or col (labelled †) located in the interior of thepotential energy surface. This pathway is outlined in moredetail in the figure. Hence we see that in order for thesystem to pass from the reactant state to the product state,it will tend to travel along the bottom of the first valley(reactant region), over the col, and down into the secondvalley (product region). This minimum energy trajectory,

termed the reaction coordinate , q, is illustrated as a dashedline in the figure presented across. This represents themost probable pathway along the three dimensional energysurface for the P / Q transformation. Finally, a cross sectionthrough this minimum energy path, known as a reactionprofile is illustrated in the fig. outlined below. We payparticular attention to the point at the top of the V(q)profile. This point corresponds to the saddle point on thepotential energy surface, and corresponds not only to aposition of maximum energy with respect to the reaction co-ordinate, but also defines a position of minimum energy withrespect to trajectories at right angles with respect to thereaction coordinate.

Transition state

Ea

E n e r g

y

Reaction coordinate

Activated complex

Hence the activated complex or transition state theoryfocuses attention on the chemical species at the saddle point,i.e., at the point where the reactants are just about totransform into products. At this point one has the activatedcomplex which is that special configuration of atoms of asystem in transit between reactants and products.


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Reaction co-ordinate

Henry Eyring1901-1981

Developed (in 1935) the

Transition State Theory(TST)orActivated Complex Theory(ACT)

of Chemical Kinetics.


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Our presentation to date has been couched inqualitative terms. We can of course get more

quantitative but this has already been coveredadequately earlier on in the module by Prof.Bridge. Now we have stated that chemicalactivation occurs via collisions between moleculesand that reaction involves formation of a discreteactivated complex of transient existence. The

latter species represents a configuration in whichthe reactant molecules have been brought to adegree of closeness and distortion, such that asmall perturbation due to a molecular vibration inan appropriate direction will transform thecomplex to products. The overall rate of reactionis then equal to the rate of passage of theactivated complex through the transition state.The Activated Complex Theory has beendiscussed earlier on but we will deal with a specialcase of a unimolecular reaction at an electrodesurface. Since the reaction is heterogeneous wemust include a characteristic reaction layer

thickness which we label (the length of which isof the order of a molecular diameter).Hence for an interfacial electron transferreaction we have the following.

k

k BT

h exp

G 0

RT

Where denotes the electrochemical Gibbs energy ofactivation. Note that the term has got units of s-1

and is expressed in cm.Hence k ' has got units of cm s-1 as it should.

0G


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Phenomenological electrode kinetics

In this section we consider the rate of electrochemical

reactions. This is the domain of kinetics . Thermodynamics isvery useful, it is a well developed subject, but it can only tellus whether a given reaction will occur in a spontaneousmanner . In short thermodynamics sets the ground rules : itdetermines whether a given reaction is energetically feasible. If we wish to be able to determine how quickly an electrodereaction occurs we need to apply chemical reaction kinetics .

The rate or velocity of an electrochemical reaction isexpressed in electrical terms . Reaction rate is expressed asa current i . The quantity of charge passed across aninterface can be directly related to the amount of materialthat has undergone chemical reaction as outlined via thefollowing expression

q nFAN k

In the latter expression q denotes the charge passed , n isthe number of electrons transferred , A is the geometricsurface area of the electrode, F is the Faraday constantand Nk represents the amount of species k chemicallytransformed per unit area of the electrode surface .hence the rate of reaction is given by :

k dN dqi nFA nFAf dt dt

This is a very important relationship . It expressesthe connection between the rate at which charge ismoved across the interface and the rate at which

chemistry is done at the interface . The quantity fƩdenotes the reaction flux or reaction velocity of theheterogeneous electrochemical process , and isexpressed in units of mol cm-2 s-1 . We recall that thevelocity of a chemical reaction is proportional to theconcentration of the reactants , and that the constantof proportionality is termed the rate constant . So it

is also with heterogeneous electrochemical processesat interfaces . We consider a simple electrochemicaltransformation which occurs irreversibly at anelectrode/solution interface .The fluxor reaction velocity will be given by the followingexpression :

0 E f k a

In the latter we note that a0 denotes theinterfacial concentration of the reactant species A(units : mol cm-3 ) and k’E represents theheterogeneous electrochemical rate constant (units: c m s-1) .

Note that the corresponding units for a first orderchemical reaction occurring in the bulk of anaqueous solution are s-1 . The difference in unitsused in the electrochemical situation reflects theheterogeneous nature of the process . Charge istransferred between two phases of differingcomposition .

A Bne

(1)

(2)

(3)


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Now we come to the crux of the matter .The heterogeneous electrochemical rateconstant possesses a characteristic that is

not exhibited by ordinary chemical rateconstants . Its magnitude is stronglyinfluenced by the magnitude of the potential applied to the electrode . Put another way :the rate of the electrode process is stronglyinfluenced by the energy of the electronsnear the Fermi level in the electrode

material . By varying the energy of theelectrons in the metallic electrode, one cansignificantly alter the rate of chemicaltransformation at the interface .This is a very significant result . We seethat the electrode potential acts as a drivingforce for interfacial electron transfer . Itis the objective of phenomenologicalelectrode kinetics to derive a quantitativerelationship between the heterogeneousrate constant and the driving force forelectron transfer .

All that has been said so far applies to anychemical reaction . However for electron transferprocesses at interfaces , the externally appliedpotential affects the interfacial potential

difference or electric field at the interface , andthis in turn has a marked effect on the activationenergy barrier . It is the effect of the externallyapplied potential on the activation energy barrierwhich differentiates an electrochemical reactionfrom an ordinary chemical reaction . The rate ofthe latter can only be affected by temperature .

Electrochemical reactions are not only temperaturedependent, they are also potential dependent . Theheterogeneous electrochemical rate constantdepends on applied potential according in thefollowing manner.

Electrochemical reactions are thermallyactivated processes . In order for any chemicalreaction to occur the reactant species mustinitially come together in a molecular encounterand then gain enough thermal energy tosubsequently pass over the activation energybarrier . It is clear that during the course of achemical reaction bonds will be stretched andbroken in the reactants and new bonds will be

formed when the products are generated .

k k F E E

RT

k E0

0

0exp exp

In the latter expression k0 denotes the standardelectrochemical rate constant which is a measure ofthe kinetic facility of an electrode process . If k0 islarge than the ET kinetics will be fast and vice versa .The parameter is called the symmetry factor and

determines how much of the input electrical energyfed into the system will affect the height of theactivation energy barrier for the electrode process .Note that will exhibit values between 0 and 1 and inmany cases will be close to 0.5 . Note that the positivesign in the exponential refers to an oxidation processand the negative sign to a reduction process .

(4)


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It is also possible to express the information contained ineqn.4 in the following way since kinetic expressions inelectrochemistry may be expressed either in terms of rate

constants or in terms of currents . Both representations areequivalent .

i iF

RT

0

exp

In the latter expression i0

is termed the exchange currentdensity (note that current density is simply the currentdivided by the area of the electrode and so has units of Acm-2). This quantity is related to the standard heterogeneousrate constant k0 via :

i FAk a b00

0 0

1

Note also that denotes the overpotential . Thelatter quantity is defined as the extra potential ,over and above that dictated by thermodynamics,which one must apply in order to make an electrodereaction occur at a net finite rate . Hencesymbolically the overpotential is defined as :

e eE i E

(5)

(6)

(7)

In the latter expression we recall that E(i)

represents the potential measured with respectto a chosen reference electrode when a netcurrent i is flowing and Ee is the potentialrecorded at equilibrium .

We can see that the overpotential can be thoughtof as an example of the concept that one cannot

get something for nothing .

It is not possible for an electrode reaction totake place at a significant rate under conditionsat which thermodynamics says it may occur . Thedeviation of actuality from the thermodynamicexpectation is the overpotential . Overpotential

has to be inputted into the system in the form ofelectrical energy in order to make the electrodereaction proceed at a finite rate . The larger therate , the greater is the overpotentialcontribution necessary .

Hence we note that:E(i) = -

ref

and Ee

= e

- ref

,where ref denotes the Galvani potential of thechosen reference electrode .


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Let us now assume that the interfacial ET reaction is reversible . Hence we can write :

A Bk'E

k'-E

where we assume that the forward process is a de-electronation oxidation process and the reverse

reaction is an electronation reduction process .Both processes involve the transfer of a singleelectron . The net flux is given by :

0 0 E E f k a k b

and the heterogeneous electrochemical rateconstants are given by :

k k

k k

E

E

0

0 1

exp

exp

In the latter expressions we introduce thenormalised potential as :

F E E

RT

0

0

Alternatively we can state

i i

F

RT

F

RT

0

1exp exp

This is the Butler-Volmer equation which is thefundamental equation of electrochemical ET kinetics .

We recall that electrochemical equilibrium implied, on amicroscopic level, that the electronation and de-electronation fluxes were in balance . No net currentspassed across the electrode/electrolyte interface .Now if a net current is passed, this balanced situationis perturbed , a net current will flow in one specific

direction . One has a departure from equilibrium . Thegreater the departure from equilibrium , the larger willbe the observed net reaction rate and the larger willbe the overpotential . Positive overpotentialscorrespond to oxidation processes , negativeoverpotentials to reduction processes . Zerooverpotential corresponds to equilibrium . This can be

summarised as follows.

Net oxidation : >

Net reduction : <

Equilibrium : =

e

e

e

0

0

0


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Transport and kinetics at

electrode/solution interface• We consider two

fundamental processes whenconsidering dynamic eventsat electrode/solutioninterfaces:– Reactant /product transport

to/from electrode surface– Electron transfer (ET)

kinetics at electrodesurface.• We first consider the

kinetics of interfacial ETfrom a classical, macroscopic

and phenomenological (nonquantum) viewpoint.• This approach is based on

classical Transition StateTheory, and results in the

Butler-Volmer Equation.

DMT

ETK


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Reactantstate Product

state

Transition state Activated complex

*0G

Reaction coordinate

E n

e r g y

• In electrochemistry the rateconstant k varies with

applied potential E becausethe Gibbs energy of activationG* varies with applied potential.

nFAf

dt

dN nFA

dt

dqi

0' ck f ET Reaction

Fluxmol cm-2 s-1

Interfacialreactantconcentrationmol cm-3

HeterogeneousET rate constant

cm s-1

= 0

finite* G F

Total addedElectrical energy

F

Amount ofBarrier lowering

F GG ** 0

overpotentialSymmetryfactor

Application of a finite overpotential

lowers the activation energy barrierby a fixed fraction .

N E E

ThermodynamicNernst potential

Appliedpotential


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We use the result of TST to obtain a value forthe ET rate constant.

0' ck f ET

RT G Z k ET *exp'

h

T k

Z

B

Transmissioncoefficient

Characteristic ET

distance (moleculardiameter).

F GG ** 0

Electrochemical Gibbs energy

of activation

RT

F k

RT

F

RT

G Z k

ET

ET

exp

exp*

exp'

0

0

The important result is that the rate constant forheterogeneous ET at the interface depends in amarked manner with applied electrode potential.As the potential is increased the larger will be the

rate constant for ET.

overpotential

Symmetryfactor

nFAf i

Apply Eyring eqn. from TST


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Butler-Volmer Equation.For the moment we neglect the fact that mass

transport may be rate limiting and focus attentionon the act of electron transfer at the electrode/solution interface.We examine the kinetics of a simple ET processin which bonds are not broken or made, involving the

transfer of a single electron in a single step.

)()(

)()(

aq Aaq B

aq Baq A

e

e

1expexp0ii

Oxidationcomponent

Reductioncomponent

Net rate

Exchangecurrent

Symmetryfactor

Normalisedpotential

• OxidationandReductionprocesses aremicroscopicallyreversible.• Net current i at interfacereflects a balance between

iox and ired .

BV equation

RT

E E F

RT

F N

overpotentialThermodynamicNernst potential

baFAk i 10

0

Standard rateconstant

Exchangecurrent

Appliedpotential

• Exchange currentprovides a measure

of kinetic facilityof ET process.

red ox iii

• Symmetry factor determineshow much of the input electricalenergy fed into the system willaffect the activation energybarrier for the redox process.Note 0< < 1 and typically = 0.5.


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F /RT

-6 -4 -2 0 2 4 6

= i /

i 0

-15

-10

-5

0

5

10

15

1expexp0ii

exp0iii ox

1exp0iii red

2sinh2

2/1

0

ii

Tafel Region

Linear Ohmicregion

Tafel Region

Delineating the regions of the simpleButler-Volmer Equation.

oxidation reduction

The Butler-Volmerequation describesthe shape of the

current density /overpotentialcharacteristic of aninterfacial ETreaction . Typical iversus curves arepresented across fortypical values of the

exchange currentdensity . Note thatwhen = 0 the netcurrent i density iszero and the reactionis at equilibrium .The current densityis seen to rise

rapidly when theoverpotentialdeviates from itsequilibrium value ofzero . The rate ofcurrent increase withoverpotentialdepends on the

magnitude of theexchange currentdensity and hence onthe value of theheterogeneousstandard rateconstant .


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F /RT

-6 -4 -2 0 2 4 6

=

i / i 0

-15

-10

-5

0

5

10

15

red ox iii

Net oxidation

Net Reduction

Equilibrium

ox

red ox

ii

ii

red

oxred

ii

ii

0

i

ii red ox

The situation interms ofpotential energy

curves of thetype used intransition statetheory arepresentedacross.

Th h f h / i l


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F /RT

-6 -4 -2 0 2 4 6

=

i / i 0

-100

-80

-60

-40

-20

0

20

40

60

80

100

9.0

7.0

5.0

1.0

3.0 When differs from0.5 the i vs curvebecomes asymmetrical.

The shape of the current/potential curvedepends on the numerical value of the

symmetry factor.

A i ti t th BV ti (I)


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Approximations to the BV equation (I).High overpotential Tafel Limit.

red ox ii

RT

F

RT

F ii

1

expexp0

If >> 0 then i iox : net de-electronationor oxidation.

RT

F iii ox

exp0

RT

F ii

0

lnln

If 120 mV.

At high overpotentials we assumethat the ET reaction occurs inthe forward direction and thereaction occurring in the reversedirection can be neglected.

This results in the derivation of alogarithmic relationship betweencurrent and overpotential.A plot of ln i vs is linear. This iscalled a Tafel plot.

Evaluation of the slope of the linearTafel region enables the symmetryFactor to be evaluated, whereas the exchangeCurrent i0 is obtained from the intercept at = 0.

f l Pl l


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Tafel Plot Analysis

= F/RT

-8 -6 -4 -2 0 2 4 6 8

l o g ( i / i 0 ) = l o g

0.001

0.01

0.1

1

10

100

Linear TafelRegion

Linear TafelRegion

Reduction

Oxidation

Tafel approximationnot valid at lowoverpotentialExchange

Current evaluatedAt = 0

ilog

F

RT

id

d b

F

RT

id

d b

C

A

1

303.2

log

303.2log


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Approximations to the BV equation (II).The low overpotential linear limit.

red ox ii

RT

F

RT

F ii

1

expexp0

Taylor expansion at small x.

!21exp

!21exp

2

2

x x x

x x x

RT

F

RT

F

RT

F

RT

F

RT

F

RT

F

11

11

exp

1exp

RT

F

ii

RT

F

RT

F

RT

F ii

0

0 11

We note that the logarithmicTafel behaviour breaks down as 0.

Tafel behaviour is characteristicof totally irreversible (hard driven)ET kinetics and will only be validif the driving force for the

electrode process is very largewhich will be the case at highoverpotentials.In the limit of low overpotentials( < 10 mV) the exponential termsin the BV equation may be simplifiedvia use of a Taylor expansion to producea linear relationship between currentand overpotential (Ohm’s Law).

Linearapproximation

Li i


38/51

= F/RT

-2 -1 0 1 2

=

i / i 0

-2

-1

0

1

2

Linear i versus relation

• We introduce the concept of acharge transfer resistance RCT asbeing a measure of the resistance

to ET across the metal/solutioninterface.• Since RCT and i0 are inverselyproportional to each other, then alarge value of RCT implies a small

value of i0 and vice versa.• Large RCT implies sluggish ET kinetics,and small RCT implies facile ET kinetics.• Useful idea : physical act of ET modelledin terms of a resistor of magnitude RCT .

• Hence ET process described using anelectrical equivalent circuit element.

RT

F ii 0

F i

RT

di

d R

i

CT

00

CT iR

R CT

Interfacial ET

Ohms Law

Charge TransferResistance


39/51

Transport effects in electrode

kinetics• Influence of reactant transport (logistics) becomes important

when the applied overpotential becomes very large.

• The current/potential curve “bends over” and a current plateauregion is observed.• This observation is explained in terms of rate control via

diffusion of the reactant species to the electrode surface.• We consider a two step sequence : diffusion to the site of ET

at the electrode/solution interface followed by the act of ETitself.• When the overpotential is very large, the driving force for

interfacial ET is very large, and so ET becomes facile and henceno longer controls the rate.

• Matter transport via diffusion (i.e. getting the reactant speciesto the region of reaction) becomes rate limiting.


40/51

Transport and kinetics at electrode/solution

interfaces

Bulk Phase

ReactantReactant at

electrode

Product at

electrode

DMT

ETK

Bulk Phase

ReactantReactant at

electrode

Product at

electrode

DMT

ETK

Bulk Phase

ReactantReactant at

electrode

Product at

electrode

DMT

ETK

Bulk Phase

ReactantReactant at

electrode

Product at

electrode

DMT

ETK

Slow rate determiningDiffusive mass transport,

Fast interfacial ET kinetics

Fast diffusive masstransport, slow ratedetermining interfacial

ET kinetics

Large overpotentialsituation :Mass TransportControl

Low overpotentialsituation:Charge TransferControl

Transport and kinetics D ff l d


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0ck f ET

D

k

cc

ET

1

0

= F(E-E0)/RT

-6 -4 -2 0 2 4 6

=

i / i D

0.0

0.5

1.0

ET D

D ET

k k

ck k

f

ck ck f D ET

111

ET

MT

ET & MT

exp0k k ET Normalised

potential

Dk D

Transport and kineticsat electrodes.

Net flux

nF

J

nFA

i f

Current density

Diffusion layer approximation used.

0000

cck cc D

ck

dx

dc D f D ET

cki

f D D

DMass transport corrected Tafel Equation


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1 1

ET D D D

D D ET D

ET ET

k k c k c f i f

k k nFA k k k k

1 D D

ET

f k

f k

00

1 1 exp D D D DF E E i f f f k

i f f k RT

00exp ET

F E E k k

RT

ck nFA

f D D

00 0

ln ln ln ln D D D DF E E f f i i k k

f i k RT k

This is one form of the mass transport corrected Tafel equation. We see that a plot of ln (iD-i/i)vs – is linear . The slope of this plot yields - F/RT whereas the intercept directly

yields ln (kD/k0). Since kD may be readily evaluated, then the standard rate constant k0 maybe determined. This form of Tafel plot has been much used in the literature.

0

E E

)ln(0k

k D

i

ii Dln

RT

F S

MT correctedTafel plot

p q

Other useful data analysis strategies.


43/51

Other useful data analysis strategies.The Tafel analysis may be applied with good accuracy to obtaini0

and when the exchange current density for the electrodeprocess is low (typically when i0 < 10-3 A cm-2). In contrast , thelow overpotential linear approximation is useful when the exchangecurrent density is large (i0 > 10-3 A cm-2).

i0 FAk 0a

1 b

ln i0 ln FAk 0 1 ln a ln b

Hence we see that a plot of ln i0 versus either ln a∞ or ln b∞ shouldbe linear with slope producing a value for the symmetry factor .

ln i0

ln a

b

1 ln i0 lnb

a

Bne A

We note that the Butler-Volmer equation may be used to determinei0 and regardless of the magnitude of the overpotential using thefollowing procedure.

i i0 exp F

RT

exp F

RT

exp F

RT

i0 exp F

RT

1 exp F

RT

ln i

1 exp F

RT

ln i0 F

RT

Hence we see that a plot of ln i 1 exp F RT versus

should be linear, the slope yielding and the intercept at = 0 yielding i0 .

Allen-Hickling Equation

The molecular interpretation of electron


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The molecular interpretation of electron

transfer.

We have presented a brief analysis of thefundamentals of electrode kinetics in terms ofcurrent/overpotential relationships . Theanalysis was based on macroscopic orphenomenological considerations . However aproper understanding of interfacial electrontransfer requires the adoption of a

microscopic perspective . This requiresquantum mechanics .

The sharp rise in current density with increasingoverpotential as presented in the current vsoverpotential curve can be understood as follows .

Here we present a molecular interpretation ofinterfacial electron transfer . It is a quantummechanical process and a quantitative developmentrequires some sophisticated mathematics .Consequently we shall adopt the lazy mansapproach and present a qualitative pictorialpresentation of the essentials . Consider the

situation depicted in the next two slides which wehave shown previously.

In these pictures we indicate , in a very schematic manner,the filled and empty electronic states in the metallicelectrode and the highest occupied and lowest unoccupiedenergy levels of the donor species A and acceptor species Bin the solution . The demarcation line between filled andempty electronic states in the metal is designated the Fermienergy EF . Now if the redox species A and B are bothpresent in the solution and if no external potential is appliedto the metal, then as previously noted, an equilibrium

potential Dfe or Ee will be set up reflecting the balancedFaradaic activity across the interface . The value of thelatter potential will depend on the logarithm of the ratio ofthe activities of A and B via the Nernst equation aspreviously discussed . Under such conditions we may set EF =Ee . In contrast, when the electrode becomes positivelycharged via application of an external potential more positive

than Ee , then the energy of the electrons in the metal islowered and EF shifts downward in energy . If the appliedpotential is sufficiently positive then a stage will be reachedsuch that EF is lower in energy than the HOMO level of thedonor species A and one obtains a net flow of electrons fromdonor species to metal . An anodic oxidation current flows .Conversely, if a potential more negative than the equilibrium

value is applied to the electrode, then the energy of theelectrons in the filled levels of the metal will be raised . Astage will be reached when EF is now higher in energy thanthe LUMO level of the acceptor species B and electrons willbe transferred from the metal to species B in solution . Acathodic reduction current will flow .

Energy of electronsin metal decreases upon

A net anodic (oxidation)l


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Metallic

electrode

+

EF

LUMO

HOMO

LUMO

HOMO


Electronenergy

n e-

in metal decreases uponapplication of a potential

more positive than thethermodynamic equilibriumvalue.

current flows from theHOMO level of the redox

species in solution to themetallic electrode.

Pictorial explanation of current

flow due to oxidation.

Energy of electronsin metal increases

A net reduction (cathodic)fl f l


46/51

-

EF

LUMO

HOMO

LUMO

HOMO


Electronenergy

n e-

Metallic electrode

in metal increases upon application of a

potential more negativethan the thermodynamicequilibrium value.

current flows from metal toLUMO levels of redox active

species in solution.

Pictorial explanation of

current flow due to reduction.

Hence we see that the overpotential for the production of a It is possible (Bockris and Khan 1983) to


47/51

Hence we see that the overpotential for the production of adefinite reaction rate at a metal/solution interface is in factthe shift of the Fermi Level in the metal from the valuewhich it had when the electrode reaction was taking place atequilibrium .

The discussion has focused on the reactant species , theelectron donor or electron acceptor . Bonds becomestretched or activated , the potential energy of the systemchanges and, at a critical configuration (the transition state)the electron is somehow magically transferred . The

following question arises : how does the electron reach theacceptor species in solution , or conversely, how does theelectron pass from the donor to the metal ?

The answer proposed from classical physics is that theelectron would be emitted from the metal, passing directlyover the potential energy barrier at the metal/solution

interface . However it can be shown that an electron emissionof this type would only generate a very tiny current , far lessthan that observed experimentally . The problem is not unlikethe situation of a particle decay : radioactive substances dodecay , a particles leave the nucleus although classically theyare not meant to do so . What does happen is that theelectron (and indeed the a particle) does not pass over the

potential energy barrier, but actually tunnels through thebarrier from their levels in the metal to the vacant levels ofthe acceptor ion in solution .

Hence we state :the fundamental act of interfacial electrontransfer is a quantum event, governed by the

rules of quantum mechanics .

quantitatively evaluate the tunnelling probabilityPT by solving from first principles orapproximately (via the WKB approximation ) the

pertinent Schrodinger equation provided that asuitable barrier geometry is proposed . Typicalbarrier shapes used in calculations of this typeare rectangular or parabolic or those of theEckart type . For instance if we assume thatthe electron tunnels through a rectangularpotential energy barrier and described by a

potential energy function of the form :U x x

U x U x a

U x x a

( )

( )

( )

0 0

0

0

0

In the latter a represents the width of therectangular barrier and U0 denotes the barrierheight ,

T o t a l e n e r g y o f p a r t i c l e

I IIIII

BarrierHeight U0

Transmission of particleThrough barrier

Incidence of particle

on barrier

Barrier width a

Distance0 a

The latter expressions lead to the remarkable prediction

Now if the energy E of the transferring electron isless than U0 , detailed solution of the pertinent


48/51

In particular when the barrier isvery high (U0 / E >> 1) and wide(so that ka >> 1) we can writethat

and so the tunnelling probabilityPT reduces to :

sinh expka ka 2

p m pthat a quantum particle exhibits a certain finite (althoughsmall) probability of leaking through a potential energybarrier which is completely opaque from the viewpoint ofclassical physics (since E < U0) . Indeed the parameter

provides a measure of the ‘opacity’ of the barrier , and inthe classical limit PT will be very small due to the fact thatthe opacity parameter will be very large .

2 02 2

m U a P

ka

E

U

E

U

T

14

1

2

0 0

1

sinh k

m U E

2 02

pSchrodinger equation results in the followingexpression for the tunnelling probability which welabel PT :

Barrier Type Barrier Heigth (wrtFermi Level /eV

Barrier Width/Å TunnelingProbability

Parabolic Barrier 5 5 1.2 x10-4

10 10 9.1 x10-12

15 15 5.2 x10-21

20 20 6.0 x10-32

RectangularBarrier

5 5 1.0x10-5

10 10 8.8x10-15

15 15 1.5x10-26

20 20 1.7x10-40

PE

U

E

U ka

E

U kaT

16

1 2

16

20 0 0

exp exp

We note that the tunneling probability is very sensitive to the barrierparameter.

For a parabolic barrier one can show thatThe tunneling probability takes the following formWhere L = barrier width and U0 is barrier maximum.

21/2

0exp 2T L

P m U E h

A further consideration must be noted . In order forl t t lli t th l t st t


49/51

electron tunnelling to occur, the electron must move toan acceptor energy level exactly equal in energy tothat of the electron in the metal . In other words the

electron transfer process is said to be radiationless .These acceptor states are of a special kind : they aresaid to be vibrationally excited . We can quote aconcrete example at this point . In the electronationof a hydrated proton, which is one of the constituentsteps in the multistep hydrogen evolution reaction,The latter reaction is often used as a prototype in the

fundamental investigation of electrode processes .

In the latter expression M denotes a vacant metal siteand MH an adsorbed hydrogen atom , one supposesthat the O-H bonds which receive electrons are inexcited vibrational states . The degree to which thisexcitation energy exceeds the ground state energy of

the reaction is a contribution to the activation energy .Hence molecular activation via bond stretching is avery important contribution to the net activationenergy . One can also show that the solvation of theion also plays an important role . The ion in solution hasassociated with it a solvation shell .The solvation state of the reduced and oxidised formof a redox couple is different . Hence the solvationshell of the reactant species must be configured in acertain way to allow the electron transfer reaction tooccur . This re-organization of the solvation shell alsorequires energy and so will also be a contributor to theenergy of activation .

These ideas form the basis for theMarcus theory of interfacial electron transfer.

H3O+ + e

- + M MH + H2O

The Marcus model assumes that the electron transfer rateconstant is given by :

Detailed development of this Marcus model which isbased on the transition state theory of chemical


50/51

g y

where G* is the Gibbs energy of activation . The pre-exponential factor is given by

(Sutin 1982 ; Hupp and Weaver 1984 ) where e is theelectronic transmission coefficient and represents theprobability with which electron transfer will occur once thetransition state has been formed ( and so is equivalent to thetunnelling probability) . Typically . Also n (units :s-1 ) is the nuclear frequency factor and is defined as thefrequency at which the configuration of nuclear co-ordinatesappropriate for electron transfer is attained . In other

words it represents the rate at which the reactant species inthe vicinity of the transition state is transformed intoproducts .This factor depends on the mechanism of activation and willinclude contributions from solvent re-organization and bondstretching . Its value is typically in the range 0.5 - 1.0 x 1013

s-1 . Finally n is termed the nuclear tunnelling factor and is

a quantum mechanical correction which becomes important atlow temperature . All of these factors have been discussed indetail in a recent review (Weaver 1987) .

k AG

k T

E

B

exp

A e n n

0 1 e

based on the transition state theory of chemicalreactions results in the following expressions for theheterogeneous electron transfer rate constants :

k Ae

k T

k Ae

k T

E

B

E

B

exp

exp

2

2

4

4

where A is a pre-exponential factor the nature ofwhich has been discussed by Hupp and Weaver(1983) and denotes the re-organization energywhich consists of both an outer sphere and aninner sphere component reflecting the two

contributions (solvent shell re-organization andbond stretching) to the activation energy . Thelatter quantity can be evaluated approximately andis typically 0.5 to 1.5 eV

We note that if

the quadratic term in the energy of activationpresented in the Marcus-Hush expressionsoutlined above may be expanded to first order ine and the heterogeneous electron transferrate constant for the forward de-electronation

process admits the form given on the left handside across.

e

k A

e

k T A

k T

e

k TE B B B

exp exp exp

2

4 4 2

This expression is just the simple Butler-Volmer equationh h f h l l f ½ As utlined in fi ure 2 23 we can c nsider a number


51/51

with the symmetry factor having a numerical value of ½ .More generally according to the Marcus formulation thesymmetry factor or transfer coefficient is given by :

G

G

G0

01

2 1

2

where we note that G* is the activation energy for electron

transfer and G0 represents the standard Gibbs energychange on proceeding from the reactant to the product statewhich for an electrochemical reaction is given by - e . TheGibbs energy of activation is given by (Albery 1975) :

G

G G*

4 1

4

0 2 0 2

The expression for the symmetry factor presented in theexpression above contains some important information . Ingeneral the latter quantity is a measure of the location ofthe transition state along the reaction co-ordinate .

As outlined in figure 2.23 we can consider a numberof simple situations . If, for example the transitionstate is symmetric then G0 = 0 and = ½ .

Alternatively when G0 is negative the electrontransfer proceeds in an energetically downhill mannerand < ½ and the transition state is “reactant like” .Finally when G0 is positive , the electron transferreaction is uphill and > ½ and the transition state is“product like” .

The latter statements are a quantitative way ofexpressing the so called Hammond Postulate.

Note also that the facility of electron transfer willdepend on the magnitude of the re-organizationenergy . When

the electron transfer kinetics are slow and as notedin eqn.2.52. On the other hand when l is small and onehas fast electron transfer then the following threecases must be considered . When then .Alternatively, if

then 1 . Finally, if

then 0 .

G e0

G0

G 0

ec js module ch3304 megl physical and interfacial electrochemistry 2013 new course l6 electrode...

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