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

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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. 2. Kinetic aspect If electrode reaction is slow and electrochemical equilibrium is broken. Different potential corresponds to different activation energy.

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

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

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

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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)

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4.1.2 net current and exchange current Cu Cu 2+ Fe 3+ Fe 2+ Net current:

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At equilibrium condition If c Ox = c Red = activity = 1 at re Then i net = 0 standard exchange current

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

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polarization transfer coefficient

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

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x is usually approximate to 1/2

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

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No concentration polarization If initial potential is 0, then

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At equilibirum

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Master equation Master equation 4.2 Electrochemical polarization

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

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4.2.2 Butler-Volmer model and equation Butler-Volmer equation

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

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

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

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

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At 25 o C, when n = 1, = 0.5 The typical Tafel slope

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log i 0 re Tafel plot: log i plot

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

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active dissolution i lgi i active dissolution: nnnn nnnn Ag + /Ag Hg 2+ /Hg Cu 2+ /Cu Cu 2+ /Cu Zn 2+ /Zn

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transfer coefficient

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Anode side cathode side

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Master equation Nernst equation Butler-Volmer equation Tafel equation

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Electrode materials solutions Electrode reaction i 0 / A cm -2 Hg 0.5 M sulfuric acid H + +2e – = H 2 5 Cu 1.0 M CuSO 4 Cu 2+ +2e – = Cu 2 Pt 0.1 M sulfuric acid H + +2e – = H 2 1 Hg 1 M Hg 2 (NO 3 ) M HClO 4 Hg e – = 2Hg 5 ) The exchange current of different electrodes differs a lot Exchange current density

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Electrode reaction c (ZnSO 4 ) i 0 / A cm -2 i 0 / A cm -2 Zn 2+ +2e – = Zn ) Dependence of exchange currents on electrolyte concentration High electrolyte concentration is need for electrode to achieve high exchange current. Use of Ag/AgCl electrode.

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

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The common current density used for electrochemical study ranges between ~ 1 A cm -2. If exchange current of the electrode i 0 > 10~100 A cm -2, it is difficult for the electrode to be polarized. When i 0 < 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.

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

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

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

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

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Discussion : 1. i d >> i >> i 0 No diffusionec polarization At small polarization : cc i At large polarization: cc i 00

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2. i d i << i 0 diffusionNo ecis invalid i i d i log i

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3. i d i >> i 0 both terms take effect 4. i << i 0, i d no polarization (ideal unpolarizable electrode)

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When i d >>i 0, diffusion control idid diff ec 1/2 At half wave potential The half wave potential depends on both i d and i 0

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idid diff ec diff lgi d ec lgi 0

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Tafel plot without diffusion polarization Tafel plot under diffusion polarization

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Tafel plot with diffusion control: How to overcome mixed / diffusion control? please summarize the ways to elevate limiting diffusion current i 0 << i < 0.1 i d Electrochemical polarization i between 0.1i d 0.9i d mixed control i >0.9 i d diffusion control Question:

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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, i t c Ox (0,t)

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at low polarization : is very small Constant for potential step

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At t = 0 i(0)= i t is the current density at no concentration polarization at no concentration polarization Numerical solution: That is

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When at time right after the potential step : i t t 1/2 is linear Extrapolating the linear part to y axes can obtain diff control EC control Double-layer charge itit CC ii

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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. c time constant s i t > i due to charge of double layer capacitor diff control EC control Double-layer charge itit CC ii

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4.4.2 current step cathodic current : 0 i c i icic t Record c at different i c 0 transition time when potential steps to next reaction. constant

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i= i charge t cc 0 cc c (0) The slope of the linear par of c (t) can be used to determine n and . When t 0 the second term = 0 0 cc c (0)

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

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for the reversible systems, use the forward kinetics only : can be solved only by numerical method: tramper coefficient n – number of electrons involved in charge transfer step x (bt) max = is tabulated Nicholson-Shain equation for fast EC reaction : i << i 0 controlled by diffusion 0.1 i v 0.0 0.2

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For irreversible single electrode i

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For totally irreversible systems for slow EC reaction : i i 0 ( 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, E p shift with scan rate. 0.1 i v 0.0 0.2 peak potential shift with scan rate

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Dependence of p on

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4.5 effect of 1 potential on EC rate : 1 =0, validate at high concentration or larger polarization G = nF effect of 1 : 1. on concentration 2. = 1 x 11

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

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When z O n 1 made c shift positively so: if 1 increases, i decreases When z O <0 ( minus ), n z O is large, therefore, for anion reduced on cathode, 1 effect is more significant.

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if: n = z O Cu 2+ +2e - = Cu = 0.5 H + +e - = 1/2 H 2 MnO 4 +e = MnO 4 2 if :z O = 0 adsorption of anion slow reaction

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without specific adsorption reduction of +1 cation …… reduction of 1 anion 1 accelerates reduction of cation, slows reduction of anion

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Rotation rate of RDE on reduction of 1 mol/L K 2 S 2 O 8 without supporting electrolyte

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Effect of potential of zero charge on polarization curve of RDE for reduction of K 2 S 2 O 8 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.

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Effect of concentration of supporting electrolyte (sodium sulfate) on the polarization curve of RDE for reduction of K 2 S 2 O 8. 1: 0; 2: 2.8 ; 3: 0.1; 4: 1.0 mol/L Na 2 SO 4 Problem: how to eliminate the effect of 1 ?

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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

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If

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Therefore

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For a multi-electron reaction Ox + ne Red Its mechanism can be described by Steps before rds, with higher i 0 at equilibrium Steps after rds, with higher i 0 at equilibrium

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Therefore At small overpotential

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At higher overpotential For cathodic current For anodic current

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4.7 Marcus theory for electron transfer M Outer-sphere reaction M inner-sphere reaction Effect of reactant, solvents, electrode materials and adsorbed species on electrochemical reaction. Electron-transfer between two coordination compounds. No strong interaction between electrode surface and reactant. Reduction of Ru(NH 3 ) 6 3+ reactant, intermediate and product interact with electrode surface strongly. Reduction of O and oxidation of H

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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

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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

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

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For homogeneous electron transfer

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Work of assemblying reactants, i.e., ion pair + electrostatic work to bring charged species next to charged electrode, w O and w R 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 .

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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.

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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.

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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)

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

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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

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tunneling R P

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