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Electroanalytical Chemistry Lecture #4 Why Electrons Transfer?

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The Metal Electrode EFEF E z E f = Fermi level; highest occupied electronic energy level in a metal

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Why Electrons Transfer EFEF E redox EFEF Net flow of electrons from M to solute E f more negative than E redox more cathodic more reducing Reduction Oxidation Net flow of electrons from solute to M E f more positive than E redox more anodic more oxidizing E E

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The Kinetics of Electron Transfer zConsider: O + ne - = R zAssume: yO and R are stable, soluble yElectrode of 3rd kind (i.e., inert) yno competing chemical reactions occur kRkR koko

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Equilibrium for this Reaction is Characterised by... zThe Nernst equation: E cell = E 0 - (RT/nF) ln (c R * /c o * ) zwhere: c R * = [R] in bulk solution c o * = [O] in bulk solution zSo, E cell is related directly to [O] and [R]

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Equilibrium (cont’d) zAt equilibrium, no net current flows, i.e., E = 0 i = 0 zHowever, there will be a dynamic equilibrium at electrode surface: O + ne - = R R - ne - = O both processes will occur at equal rates so no net change in solution composition

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Current Density, I zSince i is dependent on area of electrode, we “normalize currents and examine I = i/A we call this current density zSo at equilibrium, I = 0 = i A + i C i a /A = -i c /A = I A = -I c = I o which we call the exchange current density yNote: by convention i A produces positive current

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Exchange Current Density zSignificance? zQuantitative measure of amount of electron transfer activity at equilibrium zI o large much simultaneous ox/red electron transfer (ET) inherently fast ET (kinetics) zI o small little simultaneous ox/red electron transfer (ET) sluggish ET reaction (kinetics)

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Summary: Equilibrium zPosition of equilibrium characterized electrochemically by 2 parameters: yE eqbm - equilibrium potential, E o yI o - exchange current density

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How Does I vary with E? zLet’s consider: ycase 1: at equilibrium ycase 2: at E more negative than E eqbm ycase 3: at E more positive than E eqbm

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Case 1: At Equilibrium E = E o - (RT/nF)ln(C R * /C O * ) E - E 0 = - (RT/nF)ln(C R * /C O * ) E = E o so, C R * = C o * I = I A + I C = 0 no net current flows IAIA ICIC O R Reaction Coordinate GG

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Case 2: At E < E eqbm zE - E eqbm = negative number = - (RT/nF)ln(C R * /C O * ) ln(C R * /C O * ) is positive C R * > C O * some O converted to R net reduction passage of net reduction current IAIA ICIC O R Reaction Coordinate GG I = I A + I C < 0

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Case 2: At E > E eqbm zE - E eqbm = positive number = - (RT/nF)ln(C R * /C O * ) ln(C R * /C O * ) is negative C R * < C O * some R converted to O net oxidation passage of net oxidation current IAIA ICIC O R Reaction Coordinate GG I = I A + I C > 0

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Overpotential, zFast ET = current rises almost vertically zSlow ET = need to go to very positive/negative potentials to produce significant current zCost is measured in overpotential, = E - E eqbm fast slow Cathodic Potential, V E eqbm Cathodic Current, A E decomp

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Can We Eliminate ? What are the Sources of z = A + R + C y A, activation an inherently slow ET = rate determining step y R, resistance due to finite conductivity in electrolyte solution or formation of insulating layer on electrode surface; use Luggin capillary y C, concentration polarization of electrode (short times, stirring)

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Luggin Capillary zReference electrode placed in glass capillary containing test solution zNarrow end placed close to working electrode zExact position determined experimentally Reference Luggin Capillary Working Electrode

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The Kinetics of ET zLet’s make 2 assumptions: yboth ox/red reactions are first order ywell-stirred solution (mass transport plays no role) zThen rate of reduction of O is: - k R c o * where k R is electron transfer rate constant

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The Kinetics of ET (cont’d) zThen the cathodic current density is: zI C = -nF (k R C O * ) zExperimentally, k R is found to have an exponential (Arrhenius) potential dependence: k R = k OC exp (- C nF E/RT) ywhere C = cathodic transfer coefficient (symmetry) yk OC = rate constant for ET at E=0 (eqbm)

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, Transfer Coefficient O R Reaction Coordinate GG - measure of symmetry of activation energy barrier = 0.5 activated complex halfway between reagents/ products on reaction coordinate; typical case for ET at type III M electrode

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The Kinetics of ET (cont/d) Substituting: zI C = - nF (k R c o * ) = = - nF c 0 * k OC exp(- C nF E/RT) Since oxidation also occurring simultaneously: zrate of oxidation = k A c R * zI A = (nF)k A C R *

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zk A = k OA exp(+ A nF E/RT) zSo, substituting I A = nF C R * k OA exp(+ A nF E/RT) zAnd, since I = I C + I A then: zI = -nF c O * k OC exp(- C nF E/RT) + nF c R * k OA exp(+ A nF E/RT) I = nF (-c O * k OC exp(- C nF E/RT) + c R * k OA exp(+ A nF E/RT)) The Kinetics of ET (cont’d)

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zAt equilibrium (E=E eqbm ), recall I o = I A = - I C zSo, the exchange current density is given by: nF c O * k OC exp(- C nF E eqbm /RT) = nF c R * k OA exp(+ A nF E eqbm /RT) = I 0

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The Kinetics of ET (cont’d) zWe can further simplify this expression by introducing (= E + E eqbm ): zI = nF [-c O * k OC exp(- C nF ( + E eqbm )/RT) + c R * k OA exp(+ A nF ( + E eqbm )/ RT)] zRecall that e a+b = e a e b zSo, I = nF [-c O * k OC exp(- C nF /RT) exp(- C nF E eqbm /RT) + c R * k OA exp(+ A nF / RT) exp(+ A nF E eqbm / RT)]

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zAnd recall that I A = -I C = I 0 So, I = I o [-exp(- C nF /RT) + exp(+ A nF / RT)] This is the Butler-Volmer equation The Kinetics of ET (cont’d)

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The Butler-Volmer Equation zI = I o [- exp(- C nF /RT) + exp(+ A nF / RT)] zThis equation says that I is a function of: y yI 0 y C and A

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The Butler-Volmer Equation (cont’d) zFor simple ET, C + A = 1 ie., C =1 - A zSubstituting: I = I o [-exp(( A - 1)nF /RT) + exp( A nF / RT)]

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Let’s Consider 2 Limiting Cases of B-V Equation z1. low overpotentials, < 10 mV z2. high overpotentials, > 52 mV

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Case 1: Low Overpotential zHere we can use a Taylor expansion to represent e x : e x = 1 + x +... zIgnoring higher order terms: I = I o [1+ ( A nF /RT) - 1 - ( A - 1)nF / RT)] = I o nF /RT zI = I o nF /RT so total current density varies linearly with near E eqbm

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Case 1: Low Overpotential (cont’d) zI = (I o nF/RT) intercept = 0 slope = I o nF/RT zNote: F/RT = 38.92 V -1 at 25 o C

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Case 2: High Overpotential zLet’s look at what happens as becomes more negative then if I C >> I A zWe can neglect I A term as rate of oxidation becomes negligible then I = -I C = I o exp (- C nF /RT) zSo, current density varies exponentially with

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Case 2: High Overpotential (cont’d) zI = I o exp (- C nF /RT) zTaking ln of both sides: ln I = ln (-I C ) = lnI o + (- C nF/RT) which has the form of equation of a line zWe call this the cathodic Tafel equation zNote: same if more positive then ln I = ln I o + A nF/RT we call this the anodic Tafel equation

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Tafel Equations zTaken together the equations form the basis for experimental determination of yI o y c y A zWe call plots of ln i vs. are called Tafel plots ycan calculate from slope and I o from y- intercept

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Tafel Equations (cont’d) zCathodic: ln I = lnI o + (- C nF/RT) y = b + m x zIf C = A = 0.5 (normal), for n= 1 at RT slope = (120 mV) -1

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Tafel Plots In real systems often see large negative deviations from linearity at high due to mass transfer limitations , V E eqbm ln |i| _ + Cathodic Anodic ln I o High overpotential: ln I = lnI o + ( A nF/RT) Mass transport limited current Low overpotential: I = (I o nF/RT)

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EXAMPLE: zCan distinguish simultaneous vs. sequential ET using Tafel Plots yEX: Cu(II)/Cu in Na 2 SO 4 xIf Cu 2+ + 2e - = Cu 0 then slope = 1/60 mV xIf Cu 2+ + e - = Cu + slow ? Cu + + e - = Cu 0 then slope = 1/120 mV xReality: slope = 1/40 mV viewed as n = 1 + 0.5 = 1.5 xInterpreted as pre-equilibrium for 1st ET followed by 2nd ET

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Effect of on Current Density z A = 0.75 oxidation is favored z C = 0.75 reduction is favored

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Homework: zConsider what how a Tafel plot changes as the value of the transfer coefficient changes.

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