fundamental & applications of electrochemistry

MSE 673 IIT Kanpur

Instructor: Dr. Shobit Omar

Lecture 21

Venue: Online Lecture

Week: 7

Materials Science & Engineering Department

Fundamental & Applications of

Electrochemistry

MSE 673 IIT Kanpur

Fundamental & Applications of Electrochemistry Week 7 Lecture 21

Linearized i-t Curve

* 2 1/2( ) exp( ) ( )f Oi t FAk C H t erfc Ht=

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For small values of H2t, the factor exp(H2t)erfc(Ht1/2) can be linearized.

2

1/2

2exp( ) ( ) 1

xx erfc x

−For small values of x

1/2*

1/2

2( ) 1f O

Hti t FAk C

= −

R is initially absent

The forward rate constant can be determined from the intercept of the

plot between i versus t1/2.

* 2 1/2( ) exp( ) ( )f Oi t FAk C H t erfc Ht=

1/2 1/2

f b

O R

k kH

D D= +Please remember

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(1 )

2 1/2( ) [exp exp ]exp( ) ( )F F

RT RToi t i H t erfc Ht

−−

= −

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For small values of H2t, the factor exp(H2t)erfc(Ht1/2) can be linearized.

2/1

2 21)()exp(

xxerfcx −For small values of x

)()exp(]exp[exp)( 2/12

)1(

HterfctHiti RT

F

RT

F

o

−−

−=

−−=

−−

2/1

2/1)1(2

1]exp[exp)(

Ht

iti RT

F

RT

F

o

Both O and R are

initially present

• A plot of i at t =0 versus η can be used to obtain io• The intercept of the plot between i versus t1/2 can provide kinetically-

controlled current free of mass transfer effects.

For small values of Ht1/2 and η

−−=

2/1

2/121)(

Ht

F

RTiti o

Complete linearized

form

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Sampled Current Voltammetry for

Quasireversible Electrode Reaction

Earlier we have derived

1/2 1/2

f b

O R

k kH

D D= +

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Sampled Current Voltammetry for

Quasireversible Electrode Reaction

( )1/2

1/2 1/2 1/21 1

f fO

O R O

k kDH

D D D

= + = +

( )

* 2 1/2

* 1/21/2 1/2 2 1/2

1/2 1/2

( ) exp( ) ( )

( ) exp( ) ( )1

f O

O O

i t FAk C H t erfc Ht

FAC Di t H t H t erfc Ht

t

=

=+

Assuming, 1/2Ht = ( )1( ) ( )

1

dii t F

=+

)()exp()( 22/1

1 erfcF =

Inputting kf

from here

Earlier we have derived

1/2 1/2

f b

O R

k kH

D D= +

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Simplified Expression for i(t) Response

( ))(

1)( 1

F

iti d

+=

A compact representation of

the way in which the current in

potential step experiment

depends on time and potential

This equation hold for all

the kinetic regimes, i.e.

reversible, quasireversible

and irreversible.

2/1Ht=

)()exp()( 22/1

1 erfcF =

Small value of λ implies a

strong kinetic influence on the

current, and a large value of λ

corresponds to the situation

where the kinetics are facile

and the response is controlled

by diffusion

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Reversible Reaction

( ))(

1)( 1

F

iti d

+=

For a reversible reaction, λ is very large & therefore F1(λ) is always close to unity

( )+=

1)( diti

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Sampled Current Voltammetry

( )1/2

1/21

f

O

k

D

= +At a fixed sampling time ( ), λ

becomes

• At a very positive E relative to Eo/, θ is very large. As a result, i is

nearly zero since the denominator becomes far greater than

numerator in the simplified expression.

Vice-versa

occurs in

oxidation

reactions

• At a very negative E relative to Eo/, kf is very large which makes

the value of F1(λ) close to unity and the value of i approximately

equal to id.

( )1( )

1

dii F

=+

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Sampled Current Voltammetry

( )1/2

1/21

f

O

k

D

= +At a fixed sampling time ( ), λ

becomes

( )

1( )1

dii F

=+

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Sampled-Current Voltammograms

for Various Kinetics

The sampled current voltammogram to have a sigmoidal shape as observed

in the reversible case.

For a very facile reaction, where

DO = DR, E1/2 is equal to Eo/.

For a smaller values of ko, the

kinetics must be driven. The

observed shift in the E1/2 relative

to Eo is the extra potential which

is proportional to the required

kinetic activation.

A large displacement is observed

in the case of the totally

irreversible reaction.

Please remember the negative shift in the potential will

accelerate the reduction kinetics or forward reaction or

cathodic current.

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Totally Irreversible Reactions

The very displacement in potential that activates kf simultaneously

suppresses the kb. As a result the backward reaction becomes significantly

less important than the forward reaction.

For a smaller value of ko, a large activation of kf is required to achieve

appreciable current flow without much contribution from kb

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Totally Irreversible Reactions

The very displacement in potential that activates kf simultaneously

suppresses the kb. As a result the backward reaction becomes significantly

less important than the forward reaction.

For a smaller value of ko, a large activation of kf is required to achieve

appreciable current flow without much contribution from kb

As a result, 0b

f

k

k= = over the whole range of the voltammetric wave

( )1 1( ) ( ) ( ) ( )

1

dd

ii t F i t i F

= =

+

1/2 1/2 /

1/2 1/2

exp[ / ( )]o of

O O

k t k t F kT E E

D D

− −= =

1/2 2

1( ) exp( ) ( )F erfc =Where,

1

( )0.5 ( )

d

i tF

i= =

At this condition

o == 433.0 =t

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Calculation of ko in Totally

Irreversible ReactionsNow,

1/2 /

1/2

1/2

exp[ / ( )]0.433

o oo

O

k F kT E E

D

− −= =

1/2/

1/2 1/2

2.31ln

oo

O

kT kE E

F D

= +

One can determine the ko value if α is known

1/2/

1/2 1/2

2.31exp[ / ( )]

oo

O

kF kT E E

D

− =

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Kinetic Regimes

Reversible Regime Quasireversible Regime Irreversible Regime

By focusing on the particular value of λ at Eo/, it is possible to distinguish

the three kinetic regimes

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Kinetic Regimes

Reversible Regime Quasireversible Regime Irreversible Regime

By focusing on the particular value of λ at Eo/, it is possible to distinguish

the three kinetic regimes

Let us call the λ at Eo/ as λo

1/2 / 1/2

1/2 1/2

(1 ) exp[ / ( )](1 )

o o oo

O O

k t F kT E E k

D D

+ − −= = +

Let us assume DO = DR

2/1

2/1

2/1

2/1 2)1(

O

o

O

oo

D

k

D

k

=+=

For the reversible reaction, F1(λ) is close to unity

29.0)(1 oF 12/1

2/1

O

o

D

k Value of λo should be greater

than 2 for the reversible reaction.

θ = 1 at E = Eo/

Now,

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Irreversible Regime

For the irreversible reaction, θ is close to zero when E = E1/2

/exp[ / ( )]oF kT E E = −

Let us take the value of θ = 0.01 and estimate the E1/2 value

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Irreversible Regime

For the irreversible reaction, θ is close to zero when E = E1/2

This is the condition for

the total irreversibility

/exp[ / ( )]oF kT E E = −

Let us take the value of θ = 0.01 and estimate the E1/2 value

/

1/2

4.61( )o kTE E

F− −

But we know for the irreversible reactions,

Thus,

1/2

1/2

1 2.31ln 4.61

o

O

k

D

−

1/2

1/2

2 oo

O

k

D

=

But,

1/2/

1/2 1/2

2.31ln

oo

O

kT kE E

F D

= +

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22

31.2log −

o

Neglecting this term

log 2o −

−−

2

31.2log2log o

Value of λo should be less than

10-2α for the irreversible reaction.

Irreversible Regime

1/2

1/2

1 2.31ln 4.61

o

O

k

D

−

1/2

1/2

2 oo

O

k

D

=

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Value of λo Various Kinetic Regimes

210−o210 2 −

o 2o

Reversible RegimeQuasireversible RegimeIrreversible Regime

1/2

1/2

2 oo

O

k

D

=

•Please remember that beside the intrinsic electrode kinetics, λo value is

also dependent on sampling time.

• If the sampling time is large, than the reaction tends to become reversible.

MSE 673 IIT Kanpur

Instructor: Dr. Shobit Omar

Lecture 21

Venue: Online Lecture

Week: 7

Materials Science & Engineering Department

Fundamental & Applications of

Electrochemistry

fundamental & applications of electrochemistry

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