# Chapter 4 Electrochemical kinetics at electrode / solution interface and electrochemical overpotential.

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Chapter 4 Electrochemical kinetics at electrode / solution interface and electrochemical overpotential

Effect of potential on electrode reaction
Thermodynamic aspect If electrode reaction is fast and electrochemical equilibrium remains, i.e., Nernst equation is applicable. Different potential corresponds to different surface concentration. 2. Kinetic aspect If electrode reaction is slow and electrochemical equilibrium is broken. Different potential corresponds to different activation energy.

4.1 Effect of potential on activation energy
4.1.1 basic concepts For Elementary unimolecular process Rate expressions At equilibrium Exchange rate of reaction

Some important empirical formula:
Arrhenius equation According to Transition State Theory: Corresponding to steric factor in SCT

For electrode reactions
For reversible state Nernst equation For irreversible state Tafel equation (1905) Overpotential How to explain these empirical formula?

Potential curve described by Morse empirical equation
Activated complex Reactant product Reaction coordinate Standard free energy In electrochemistry, electrochemical potential was used instead of chemical potential (Gibbs free energy)

4.1.2 net current and exchange current
Fe3+ Fe2+ Net current: Net current:

If cOx = cRed = activity = 1 at re
At equilibrium condition standard exchange current Then i net = 0

4.1.3 effect of overpotential on activation energy
Ox Red Na(Hg)x Na+ + e Ox Na(Hg)x Na+ + e The energy level of species in solution keeps unchanged while that of the species on electrode changes with electrode potential. Red Na+ + e Na(Hg)x

polarization transfer coefficient

Fraction of applied potential alters activation energy  for oxidation and  for reduction
Anode side cathode side

 is usually approximate to 1/2
x  is usually approximate to 1/2

4.1.4 Effect of polarization on reaction rate
Marcus theory: transition state theory

No concentration polarization
If initial potential is 0, then

At equilibirum

4.2 Electrochemical polarization
4.2.1 Master equation Master equation

Theoretical deduction of Nernst equation from Mater equation
At equilibrium Nernst equation

4.2.2 Butler-Volmer model and equation
Butler-Volmer equation

4.2.3 discussion of B-V equation
1) Limiting behavior at small overpotentials Current is a linear function of overpotential

Charge transfer resistance
Cathode Anode Net current  / V i / A False resistance

2) Limiting behavior at large overpotentials
Cathode Anode Net current  / V i / A One term dominates Error is less than 1% When cathodic polarization is larger than 118 mV

Taking logarithm of the equation gives:
Making comparison with Tafel equation One can obtain

The typical Tafel slope
At 25 oC, when n = 1,  = 0.5 The typical Tafel slope -100 -200 -300 300 200 100

Tafel plot:   log i plot
re

4.2.4 determination of kinetic parameters
For evolution of hydrogen on Hg electrode

active dissolution active dissolution: n n Ag+ /Ag 0.5 Hg2+ /Hg 0.6
i lgi active dissolution: n n Ag+ /Ag 0.5 Hg2+ /Hg 0.6 1.4 Cu2+ /Cu 0.4 1.6 Zn2+ /Zn 0.47 1.47

transfer coefficient

Anode side cathode side

Butler-Volmer equation
Master equation Nernst equation Butler-Volmer equation Tafel equation

4.2.5 Exchange current density
1) The exchange current of different electrodes differs a lot Electrode materials solutions Electrode reaction i0 / Acm-2 Hg 0.5 M sulfuric acid H++2e– = H2 510-13 Cu 1.0 M CuSO4 Cu2++2e– = Cu 210-5 Pt 0.1 M sulfuric acid 110-3 110-3 M Hg2(NO3) M HClO4 Hg22++2e– = 2Hg 510-1

2) Dependence of exchange currents on electrolyte concentration
Electrode reaction c (ZnSO4) i0 / Acm-2 Zn2++2e– = Zn 1.0 80.0 0.1 27.6 0.05 14.0 0.025 7.0 High electrolyte concentration is need for electrode to achieve high exchange current. Use of Ag/AgCl electrode.

When i0 is large and i << i0, c is small.
When i0 = , c=0, ideal nonpolarizable electrode, basic characteristic of reference electrode. When i0 is small, c is large. When i0 = 0, c = , ideal polarizable electrode

The common current density used for electrochemical study ranges between 10-6 ~ 1 Acm-2.
If exchange current of the electrode i0 > 10~100 Acm-2, it is difficult for the electrode to be polarized. When i0 < 10-8 Acm-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.

Influence of impurity If an impurity undergoes reduction at electrode
The influence of impurity on equilibrium is negligible. If If Oxidation of electrode and reduction of impurity take place. There is net electrochemical reaction.

Single/couple electrode and Mixed potential
Icorro Electrode with exchange current less than 10-4 A cm-2 is hard to attain equilibrium potential.

4.3 Diffusion on electrode kinetic
When we discuss situations in 4.2, diffusion polarization is not take into consideration. When diffusion take effect :

At high cathodic polarization
Therefore: Electrochemical term Diffusion term The total polarization comprises of tow terms: electrochemical term and diffusion term.

Discussion : 1. id >> i >> i0 No diffusion ec polarization At large polarization: At small polarization : c i i 0 c

2. id  i << i0 diffusion No ec is invalid i  id i log i

3. id  i >> i0 both terms take effect
300 200 100 -100 -200 -300 3. id  i >> i0 both terms take effect 4. i << i0, id no polarization (ideal unpolarizable electrode)

When id >>i0, diffusion control
ec 1/2 At half wave potential The half wave potential depends on both id and i0

id diff ec diff lgid ec lgi0

Tafel plot without diffusion polarization
-100 -200 -300 300 200 100 Tafel plot without diffusion polarization -100 -200 -300 300 200 100 400 -400 Tafel plot under diffusion polarization

Tafel plot with diffusion control:
i0 << i < 0.1 id Electrochemical polarization i between 0.1id  0.9id mixed control i >0.9 id diffusion control Question: How to overcome mixed / diffusion control? please summarize the ways to elevate limiting diffusion current

4.4 EC methods under EC-diff mixed control
potential step Using B-V equation with consideration of diffusion polarization at high polarization c At constant c, it  cOx(0,t)

at low polarization : is very small Constant for potential step

Numerical solution: i(0)= i
is the current density at no concentration polarization at  That is 1 2 3 0.5 2 t At t = 0 i(0)= i no concentration polarization

at time right after the potential step : it t1/2 is linear
When diff control EC control Double-layer charge it C i at time right after the potential step : it t1/2 is linear Extrapolating the linear part to y axes can obtain

The way can be used to eliminate concentration polarization.
Making potential jump to different  can obtain i at different . Then plot i against c can obtain i~c without concentration polarization. The way can be used to eliminate concentration polarization. diff control EC control Double-layer charge it C i c  time constant s it > i due to charge of double layer capacitor

4.4.2 current step i ic cathodic current : 0  ic t
Record c at different ic current step cathodic current : 0  ic constant   transition time when potential steps to next reaction.

i= icharge t c c c(0) The slope of the linear par of c (t) can be used to determine n and . c c(0) When t0 the second term = 0

4.4.3 cyclic voltammetry (CV)
for reversible single electrode I Potential separation

for the reversible systems , use the forward kinetics only :
can be solved only by numerical method: Nicholson-Shain equation for fast EC reaction : i << i0 controlled by diffusion  tramper coefficient n – number of electrons involved in charge transfer step 0.1 i v 0.0 0.1 0.2 0.2 is tabulated x (bt) max =0.4958

For irreversible single electrode
i

For totally irreversible systems
peak potential shift with scan rate for slow EC reaction : ii0 ( quasi reversible, irreversible) in comparison to the same rate, equilibrium can not establish rapidly. Because current takes more time to respond to the applied voltage, Ep shift with scan rate . 0.1 i v 0.0 0.1 0.2 0.2

Dependence of p on 

4.5 effect of 1 potential on EC rate :
1=0, validate at high concentration or larger polarization G = nF x 1 effect of 1: 1. on concentration 2.  =   1

This means 1 has same effect on the forward (reduction) and reverse (oxidation) reaction.

When zO <0 ( minus ), n zO is large, therefore, for anion reduced on cathode , 1 effect is more significant. When zO  n 1 made c shift positively so: if 1 increases, i decreases

if: n = zO Cu2+ +2e- = Cu MnO4 +e = MnO42  = H+ +e- = 1/2 H2 if :zO = 0 adsorption of anion slow reaction

 reduction of +1 cation …… reduction of 1 anion 1 accelerates reduction of cation, slows reduction of anion

Rotation rate of RDE on reduction of 110-3 mol/L K2S2O8 without supporting electrolyte

Effect of potential of zero charge on polarization curve of RDE for reduction of K2S2O8 without supporting electrolyte Only when the electrode potential is near to the potential of zero charge, 1 has large effect on the reaction rate, while at higher polarization, 1 take less effect.

Effect of concentration of supporting electrolyte (sodium sulfate) on the polarization curve of RDE for reduction of K2S2O8 . 1: 0; 2: 2.8 10-3; 3: 0.1; 4: 1.0 mol/L Na2SO4 Problem: how to eliminate the effect of 1?

4.6 EC kinetics for multi-electron process
For a di-electron reaction Ox + 2e  Red Its mechanism can be described by At stable state

If

Therefore

For a multi-electron reaction
Ox + ne  Red Its mechanism can be described by Steps before rds, with higher i0 at equilibrium Steps after rds, with higher i0 at equilibrium

Therefore At small overpotential

At higher overpotential
For cathodic current For anodic current

4.7 Marcus theory for electron transfer
Effect of reactant, solvents, electrode materials and adsorbed species on electrochemical reaction. Electron-transfer between two coordination compounds. M Outer-sphere reaction M inner-sphere reaction No strong interaction between electrode surface and reactant. Reduction of Ru(NH3)63+ reactant, intermediate and product interact with electrode surface strongly. Reduction of O and oxidation of H

Microscopic theories of electron transfer
Electron transfer reaction, a radiationless electronic rearrangement, sharing commonalities with radiationless deactivation of excited molecules. For a homogeneous redox reaction : O + R’  R + O’ Electron transfer between tow isoenergetic points ---- isoenergetic electron transfer

Franck-Condon principle:
Nuclear coordinates do not change on time scale of electronic transitions. Reactants and products share common nuclear configuration at moment of transfer. Deduce expression for standard Gibbs energy of activation as a function of structural parameters of reactant, so as to calculate rate constant of the reaction. activation

Transition state isoenergetic electron transfer g: global reaction coordinate for 1 dimensional process, related to solvation.

For homogeneous electron transfer

Work of assemblying reactants, i. e
Work of assemblying reactants, i.e., ion pair + electrostatic work to bring charged species next to charged electrode, wO and wR not considered. Improved model Predictions from Marcus theory ½ factor seems like first order term in expansion of , rest are corrections Classical Butler-Volmer theory regards  as constant, cannot predict potential dependence of .

Electron transfer occurs between empty levels of electrode (or species in solution) and filled levels of species in solution (or electrode) of the same energy. For reduction - energy of occupied level of electrode must match energy level of empty state of species in solution. For oxidation - energy of empty level of electrode must match energy level of occupied state of species in solution. Energy levels of metal and species in solution form a continuum Overall rate must be evaluated by summing or integrating over all energy matched pairs.

Since filled electrode states overlap with (empty) O states, reduction can proceed. Since the (filled) R states overlap only with filled electrode states, oxidation is blocked.

Number of electronic states of electrode in energy range E and E + dE is
area of the electrode density of states Total number of states of electrode in given energy range At absolute zero, energy of highest filled state is called Fermi level, At higher temperatures, thermal energy promotes electrons to higher levels Electron distribution given by Fermi function f(E)

concentration density function
Number concentration of R species in the range between E + dE is

Rate Constant for Reduction
Rate Constant for Oxidation FURTHER CONSIDERATIONS • Electron transfer occurs almost entirely at the Fermi level • Rate constant proportional to local rate at Fermi level. • Integrals reduce to single value

R P tunneling

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