Electroanalytical Chemistry Lecture #4 Why Electrons Transfer?

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

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

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

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]

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

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

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)

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

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

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 GG

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 GG I = I A + I C < 0

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 GG I = I A + I C > 0

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

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)

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

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

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)

, Transfer Coefficient O R Reaction Coordinate GG  - 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

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 *

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)

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

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)]

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)

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

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)]

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

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) (  A - 1)nF  / RT)] = I o nF  /RT zI = I o nF  /RT so total current density varies linearly with  near E eqbm

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

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 

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

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

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

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) 

EXAMPLE: zCan distinguish simultaneous vs. sequential ET using Tafel Plots yEX: Cu(II)/Cu in Na 2 SO 4 xIf Cu e - = 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 = = 1.5 xInterpreted as pre-equilibrium for 1st ET followed by 2nd ET

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

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