Chapter 4 Electrochemical kinetics at electrode / solution...

Chapter 4

Electrochemical kinetics at electrode / solution

interface and electrochemical overpotential

Valid only for reversible cell or for electrode at

electrochemical reversibility.

inet =0

ln ox

red

aRT

nF a

(2) Nernst equation:

ln i

ii

RTa

nF

Dependent of electrode potential on species activities

lnc d

C D

a b

A B

a aRTE E

nF a a

1) Two important empirical relations

0. Brief introduction

(2) Tafel equation:

The point of intersection of the extrapolation on the

line = 0 is log i0.

A is in fact the at j = 1 A cm-2.

= a + b log j

Valid only for special irreversible process when > 118 mV.

1) Two important empirical relations

0. Brief introduction

Effect of potential on electrode reaction

1. Thermodynamic aspect

If electrode reaction is fast and electrochemical equilibrium

remains, i.e., Nernst equation is applicable. Different

potential corresponds to different surface concentration—

concentration/diffusion control.

2. Kinetic aspect

If electrode reaction is slow and electrochemical equilibrium

is broken. Different potential corresponds to different

activation energy—activation/electrochemistry control.

Chapter 4 Electrochemical kinetics at electrode

f f A

b b B

net f b f A b B

r k c

r k c

r r r k c k c

Rate expressions

At equilibrium

0;

( ) ( )

( )

( )

net

b B eq f A eq

f B eq

b A eq

r

k c k c

k cK

k c

4.1.1 basic concepts

Exchange rate of reaction

For Elementary unimolecular opposing

process

4.1 Effect of potential on activation energy


Kinetic equilibrium

constant

4.1.1 basic concepts



For electrochemical reactions:

; f c

f f A c f Ox

i ir k c r k c

nFA nFA c f Oxi k nFAc

Rea f di k nFAc

0 ,0 ,0

net c a

c a

i i i

i i i

At equilibrium conditions:

exp aEk A

RT

Arrhenius equation

Some important empirical formula:

According to Transition State Theory:

expkT G

kh RT

Corresponding to steric factor in

SCT



kT/h term corresponds to the frequency

factor

kTk K

h

Ox e Redf

b

k

kn

Potential curve described by Morse empirical equation

In electrochemistry,

electrochemical potential

was used instead of

chemical potential (Gibbs

free energy)



4.1.2 net current and exchange current

Ox e Redc

a

k

kn

Net current:

Ox (0, )c ci nFAk c t

Red (0, )a ai nFAk c t

Ox Red[ (0, ) (0, )]c a c ai i i nFA k c t k c t



Ox Red[ (0, ) (0, )]c a c ai i i nFA k c t k c t

At equilibrium condition

0 ,0 ,0c ai i i

cG

RTc

kTk e

h

aG

RTa

kTk e

h

If cOx = cRed = activity = 1 at re

Then i net = 0 standard exchange current





For unequilibrium conditions

ir re Δ ΔG nFE nF

0Δ Δ ΔG G nF

polarization

4.1.3 effect of overpotential on activation energy

transfer coefficient



Fraction of applied potential

alters activation energy for

oxidation and for reduction

ΔF

ΔF

ΔF

Δ cG

,0Δ cG

Δ aG

,0Δ aG

Ox en Red

ΔnF

,c re

,c ir,a ir

ΔnF

ΔnF

,0 Δa aG G nF Anode side

,0 Δc cG G nF cathode side



tantan

tan

/)1(tan

/tan

xFE

xFE

x

ΔnF

ΔnF

0

is usually approximate to 1/2



,0

,0

,0

exp( )

exp( )exp( )

exp( )

cB

c c

cB

c

c

G nFk Tk

h RT

Gk T nF

h RT RT

nFk

RT

4.1.4 Effect of polarization on reaction rate

,0

,0

exp( )

exp( )

aB

a a

a

G nFk Tk

h RT

nFk

RT

Marcus theory: transition state theory



Ox Ox ,0(0, ) (0, ) exp( )c c c

nFi k c t nFc t k

RT

,0 exp( )a a

nFi i

RT

No concentration polarization

,0 exp( )c c

nFi i

RT

,0

2.3 2.3lg lgc c

RT RTi i

nF nF

,0

2.3 2.3lg lga a

RT RTi i

nF nF

If initial potential

is 0, then



,0

2.3 2.3lg lgc re c c

RT RTi i

nF nF

,0

2.3 2.3lg lga re a a

RT RTi i

nF nF

At equilibirum

,0 ,0 0a ci i i 0

2.3lg c

c

iRT

nF i

0

2.3lg a

a

iRT

nF i



lg i

lg ai

lg ci

0lg i

0 re

0

2.3 2.3lg lgc c

RT RTi i

nF nF

0

2.3 2.3lg lga a

RT RTi i

nF nF



Ox ,0 Red ,0

0 Ox Red

(0, ) exp( ) (0, ) exp( )

(0, )exp( ) (0, )exp( )

net c a

c a

i i i

nF nFnFc t k nFc t k

RT RT

nF nFnFk c t c t

RT RT

Ox ,0(0, ) exp( )c c

nFi nFc t k

RT

Red ,0(0, ) exp( )a a

nFi nFc t k

RT

Master equation

4.2.1 Master equation

4.2 Electrochemical polarization


0 Ox Red(0, )exp( ) (0, )exp( )net

nF nFi nFk c t c t

RT RT

At equilibrium 0neti

0 0

Ox Ox Red Red(0, ) ; (0, )c t c c t c

0 0

Ox Redexp( ) exp( )nF nF

c cRT RT

0

Ox

0

Red

exp( )c nF

RTc

0

Ox

0

Red

lncRT

nF c

0

Ox

0

Red

lncRT

nF c Nernst equation

Theoretical deduction of Nernst equation from Mater equation



Butler-Volmer equation

0 Ox Red(0, )exp( ) (0, )exp( )net

nF nFi nFk c t c t

RT RT

0 exp( ) exp( )nF nF

i iRT RT

Butler-Volmer equation



1) Limiting behavior at small overpotentials

exp( ) 1nF nF

RT RT

0 1 1nF nF

i iRT RT

0 01 1nF nF nF

i i iRT RT RT

i

Current is a linear function of overpotential

i

4.2.3 discussion of B-V equation



0

nF

RT

i i

ct

0

nFR

RTi Charge transfer resistance

False resistance



2) Limiting behavior at large overpotential

0 exp( ) exp( )nF nF

i iRT RT

One term dominates

exp( )

exp 1%

exp( )

nF

nFRTnF RT

RT

Error is less than 1%

118 mV

net c a ci i i i

At cathodic polarization larger than 118 mV



0 exp( )nF

i iRT

0lg lg2.3

nFi i

RT

Taking logarithm of the equation gives:

0

2.3 2.3lg lg

RT RTi i

nF nF

lga b i Making comparison with Tafel equation

One can obtain 0

2.3lg

RTa i

nF

2.3RTb

nF



2.3RTb

nF

At 25 oC, when n = 1, = 0.5

118 mVb The typical Tafel slope

0 -100 -200 -300300 200 100

118 mV118 mV

/ mV

lg i

0lg i



Tafel plot: log i plot



4.2.4 determination of kinetic parameters

0

2.3lg

RTa i

nF

2.3RTb

nF

1.40 0.118lg i

0.5

12 2

0 1.6 10 A cmi

For evolution of hydrogen over Hg electrode

0 Ox ,0(0, ) exp( )c

nFi nFc t k

RT

13 15 10 cm sk



Electrode

materials

solutions Electrode

reaction

i0 / Acm-2

Hg 0.5 M sulfuric acid H++2e– = H2 510-13

Cu 1.0 M CuSO4 Cu2++2e– = Cu 210-5

Pt 0.1 M sulfuric acid H++2e– = H2 110-3

Hg 110-3 M Hg2(NO3)2

+ 2.0M HClO4

Hg22++2e– = 2Hg 510-1

1) The exchange currents of different electrodes differ a lot

4.2.5 Exchange current density



Electrode reaction c (ZnSO4) i0 / Acm-2

Zn2++2e– = Zn

1.0 80.0

0.1 27.6

0.05 14.0

0.025 7.0

2) Dependence of exchange currents on electrolyte concentration

High electrolyte concentration is need for electrode to achieve

high exchange current.



0

nF

i RTi

When i0 is large and i << i0, c is small.

0

2.3lg c

c

iRT

nF i

When i0 = , c=0, ideal nonpolarizable

electrode

When i0 is small, c is large.

When i0 = 0, c = , ideal polarizable

electrode0

2.3lg c

c

iRT

nF i



The common current density used for electrochemical study

ranges between 10-6 ~ 1 Acm-2.

If exchange current of the electrode i0 > 10~100 Acm-2, it is

difficult for the electrode to be polarized.

When i0 > 10-8 Acm-2, the electrode will always undergoes

sever polarization.

For electrode with high exchange current, passing current

will affect the equilibrium a little, therefore, the electrode

potential is stable, which is suitable for reference electrode.



4.3 Diffusion on electrode kinetic

When we discuss situations in 4.2, we didn’t take diffusion

polarization into consideration

When diffusion take effect :

0.

Ox (0, ) exp( )exp( )cB

c

Gk T nFi nFAC t k

h RT RT

0Ox 0.

Ox0

Ox

(0, )exp( )exp( )cB

c

C t Gk T nFi nFA C k

h RT RTC

Ox

net 00

Ox

(0, )exp( )c c

C t nFi i i

RTC


0 (1 )s

i i

d

ic c

i

Ox

net 00

Ox

(0, )exp( )c c

c t nFi i i

RTc

At high cathodic polarization

0(1 ) exp( )c

d

i nFi i

i RT

0

( )exp( )d

c

d

i ii nF

i i RT



0

ln ln( ) ( )d

c

d

i ii nF

i i RT

ln ln d

c

o d

iRT i RT

nF i nF i i

Therefore:

At this time the total polarization comprises of tow terms:

electrochemical term and diffusion term.

Electrochemical term Diffusion term

0

( )exp( )d

c

d

i ii nF

i i RT

Taking logarithm yields



Discussion :

0

lnc

RT i

nF i

1. id >> i >> i0

No diffusion ec polarization

0

ln ln d

c

d

iRT i RT

nF i nF i i

At small polarization :

c

i

At large polarization:

c

i

0

0

nFi i

RT



ln( )d

d

d

IRT

nF i i

0lnc

RT i

nF i

2. id i << i0

diffusion No ec

0

ln ln d

c

d

iRT i RT

nF i nF i i

is invalid

i id

i

log i



3. id i >> i0 both terms take effect

4. i << i0, id no polarization

0

ln ln d

c

d

iRT i RT

nF i nF i i



When id >>i0

0

ln ln d

c

d

iRT i RT

nF i nF i i

iddiff

ec

1/2

1

2di

0 1/ 2

0

0

0

1

2ln ln 2

ln ln ln

d

d

d

iRT RT

nF i nF

iRT RT RTi i

nF i nF nF

1/ 2

0lndi

d RT

d i nF



iddiff

ec

0

ln ln d

c

d

iRT i RT

nF i nF i i

1/ 2

0

ln dIRT

nF i



Tafel plot without diffusion polarization Tafel plot under diffusion polarization



How to overcome mixed / diffusion control? The ways to elevate

limiting diffusion current

i0 << i < 0.1 id

i between 0.1id 0.9id mixed control

i >0.9 id diffusion control

electrochemical polarization

Chapter 4 Electrochemical kinetics at electrode / solution...

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