Characterization Technique: Voltammetry

Reading: Bard & Faulkner 1st week: ch. 1 & ch. 2 (pp. 44-54)2nd week: ch. 1, ch. 3 (pp. 87-107), & ch. 6 (pp. 226-236)3rd week: ch. 1 & ch. 13

Or the topics in any electrochemistry textbook

Current vs. reaction ratei (A) = dQ/dt (C/s)Q/nF = N (mol)

n: # of electrons in reaction (2 for reduction of Cd2+)

Rate (mol/s) = dN/dt = i/nFElectrode process: heterogeneous reaction

Rate (mols-1cm-2) = i/nFA = j/nFj: current density (A/cm2)

Electrode reaction: i-E curvesPolarization: departure of the cell potential from the equilibrium potentialExtent of potential measured by the overpotential: = E - Eeq

   

   

Factors affecting electrode reaction rate and current1. Mass transfer2. Electron transfer at the electrode surface3. Chemical reactions4. Other surface reactions: adsorption, desorption, electrodeposition

Review of homogeneous kineticsDynamic equilibrium kf

O + e = R kb

Rate of the forward process vf (M/s) = kfCA

Rate of the reverse reaction vb = kbCB

Rate const, kf, kb: s-1

Net conversion rate of A & B vnet = kfCA – kbCB

At equilibrium, vnet = 0 kf/kb = K = CB/CA

*kinetic theory predicts a const conc ratio at equilibrium, just as thermodynamicsAt equilibrium, kinetic equations → thermodynamic ones→ dynamic equilibrium (equilibrium: nonzero rates of kf & kb, but equal)

Exchange velocity v0 = kf(CA)eq = kb(CB)eq

Arrhenius equation & potential energy surfaces

k = Ae–EA/RT

EA: activation energy, A: frequency factor

Transition state or activated complex→ Standard internal E of activation: ΔE‡ Standard enthalpy of activation: ΔH‡

ΔH‡ = ΔE‡ + Δ(PV)‡ ~ ΔE‡

k = Aexp(-ΔH‡/RT)

A = A′exp(ΔS‡/RT) ΔS‡: standard entropy of activation

k = A′exp[-(ΔH‡ - TΔS‡)/RT] = A′exp(-ΔG‡/RT)

ΔG‡: standard free energy of activation

Transition state theory (absolute rate theory, activated complex theory)

General theory to predict the values of A and EA

Rate constants k = κ(kT/h)e-ΔG‡/RT

κ: transmission coefficient, k: Boltzmann const, h: Planck const

Essentials of electrode reactions *accurate kinetic picture of any dynamic process must yield an equation of the thermodynamic form in the limit of equilibrium kf

O + ne = R kb

Equilibrium is characterized by the Nernst equation

E = E0′ + (RT/nF)ln(Co*/CR

*) bulk concKinetic: dependence of current on potential Overpotential η = a + blogi Tafel equation

Forward reaction rate vf = kfCO(0,t) = ic/nFACO(0,t): surface concentration. Reduction → cathodic current (ic)Backward reaction rate vb = kbCR(0,t) = ia/nFA

Net reaction rate vnet = vf – vb = kfCO(0,t) – kbCR(0,t) = i/nFA

i = ic – ia = nFA[kfCO(0,t) – kbCR(0,t)]

Butler-Volmer model of electrode kinetics Effects of potential on energy barriers Hg Na+ + e = Na(Hg)

Equilibrium → Eeq

positive potential than equilibrium

negative potential than equilibrium

One-step, one-electron process kf

O + e = R kb

Potential change from E0′ to E → energy change –FΔE = -F(E – E0′)

ΔG‡ change: α term (transfer coefficient)

ΔGa‡ = ΔG0a

‡ – (1 – α)F(E – E0′)ΔGc

‡ = ΔG0c‡ + αF(E – E0′)

kf = Afexp(-ΔGc‡/RT)

kb = Abexp(-ΔGa‡/RT)

kf = Afexp(-ΔG0c‡/RT)exp[-αf(E – E0′)]

kb = Abexp(-ΔG0a‡/RT)exp[(1 – α)f(E – E0′)]

f = F/RT

At CO* = CR

*, E = E0′

kfCO* = kbCR

* → kf = kb; standard rate constant, k0

At other potential E

kf = k0exp[-αf(E – E0′)]kb = k0exp[(1 – α)f(E – E0′)]

Put to i = ic – ia = nFA[kfCO(0,t) – kbCR(0,t)]

Butler-Volmer formulation of electrode kinetics

i = FAk0[CO(0,t)e-αf(E – E0′) - CR(0,t)e(1 – α)f(E – E0′)

k0: large k0 → equilibrium on a short time, small k0 → sluggish (e.g., 1 ~ 10 cm/s) (e.g., 10-9 cm/s)

kf or kb can be large, even if small k0, by a sufficient high potential

The transfer coefficient (α)α: a measure of the symmetry of the energy barrier

tanθ = αFE/xtanφ = (1 – α)FE/x

→α = tanθ/(tanφ + tanθ)

Φ = θ & α = ½ → symmetrical

In most systems α: 0.3 ~ 0.7

Implications of Butler-Volmer model for 1-step, 1-electron processEquilibrium conditions. The exchange currentAt equilibrium, net current is zero

i = 0 = FAk0[CO(0,t)e-αf(Eeq – E0′) - CR(0,t)e(1 – α)f(Eeq – E0′)

→ ef(Eeq – E0′) = CO*/CR

* (bulk concentration are found at the surface)

This is same as Nernst equation!! (Eeq = E0′ + (RT/nF)ln(CO*/CR

*))“Accurate kinetic picture of any dynamic process must yield an equation of thethermodynamic form in the limit of equilibrium”

At equilibrium, net current is zero, but faradaic activity! (only ia = ic)→ exchange current (i0)

i0 = FAk0CO*e-αf(Eeq – E0′) = FAk0CO

*(CO*/CR

*)-α

i0 = FAk0CO*(1 – α) CR

*α

i0 is proportional to k0, exchange current density j0 = i0/A

Current-overpotential equationDividing

i = FAk0[CO(0,t)e-αf(E – E0′) - CR(0,t)e(1 – α)f(E – E0′)]

By i0 = FAk0CO*(1 – α) CR

*α

→ current-overpotential equation i = i0[(CO(0,t)/CO

*)e-αfη – (CR(0,t)/CR*)e(1 – α)fη]

cathodic term anodic termwhere η = E - Eeq

Approximate forms of the i-η equation(a) No mass-transfer effectsIf the solution is well stirred, or low current for similar surface conc as bulk

i = i0[e-αfη – e(1 – α)fη] Butler-Volmer equation

*good approximation when i is <10% of il,c or il,a (CO(0,t)/CO* = 1 – i/il,c = 0.9)

For different j0 (α = 0.5): (a) 10-3 A/cm2, (b) 10-6 A/cm2, (c) 10-9 A/cm2 → the lower i0, the more sluggish kinetics → the larger “activation

overpotential”((a): very large i0 → engligible activation overpotential)

(a): very large i0 → engligible activation overpotential → any overpotential:“concentration overpotential”(changing surface conc. of O and R)

i0 → 10 A/cm2 ~ < pA/cm2

The effect of α

(b) Linear characteristic at small ηFor small value of x → ex ~ 1+ x i = i0[e-αfη – e(1 – α)fη] = -i0fη

Net current is linearly related to overpotential in a narrow potential range near Eeq

-η/i has resistance unit: “charge-transfer resistance (Rct)”

Rct = RT/Fi0

(c) Tafel behavior at large η i = i0[e-αfη – e(1 – α)fη] For large η (positive or negative), one of term becomes negligiblee.g., at large negative η, exp(-αfη) >> exp[(1 - α)fη]

i = i0e–αfη

η = (RT/αF)lni0 – (RT/αF)lni = a + blogi Tafel equation

a = (2.3RT/αF)logi0, b = -(2.3RT/αF)

(d) Tafel plots (i vs. η) → evaluating kinetic parameters (e.g., i0, α)

anodic cathodic

Effects of mass transferPut CO(0,t)/CO

* = 1 – i/il,c and CR(0,t)/CR* = 1 – i/il,a

to i = i0[(CO(0,t)/CO*)e-αfη – (CR(0,t)/CR

*)e(1 – α)fη]

i/i0 = (1 – i/il,c)e-αfη – (1 – i/il,a)e(1 – α)fη

i-η curves for several ratios of i0/il

Multistep mechanismsRate-determining electron transfer- In electrode process, rate-determining step (RDS) can be a heterogeneous to electron-transfer reaction→ n-electrons process: n distinct electron-transfer steps → RDS is always a one-electron process!! one-step, one-electron process 적용 가능 !!

O + ne = R→ mechanism: O + n′e = O′ (net result of steps preceding RDS) kf

O′ + e = R′ (RDS) kb

R′ + n˝e = R (net result of steps following RDS)n′ + 1 + n˝ = n

Current-potential characteristics

i = nFAkrds0[CO′(0,t)e-αf(E – Erds 0′) – CR′(0,t)e(1 – α)f(E –Erds 0′)]

krds0, α, Erds

0′ apply to the RDS

Multistep processes at equilibriumAt equilibrium, overall reaction → Nernst equation

Eeq = E0′ + (RT/nF)ln(CO*/CR

*)

Nernst multistep processesKinetically facile & nernstian (reversible) for all steps

E = E0′ + (RT/nF)ln[CO(0,t)/CR(0,t)]

→ E is related to surface conc of initial reactant and final product regardless ofthe details of the mechanism

Mass transport-controlled reactionsModes of mass transferElectrochemical reaction at electrode/solution interface: molecules in bulk solution must be transported to the electrode surface “mass transfer”Mass transfer-controlled reaction

vrxn = vmt = i/nFA Modes for mass transport: (a) Migration: movement of a charged body under the influence of an electric field (a gradient of electric potential)(b) Diffusion: movement of species under the influence of gradient of chemical potential (i.e., a concentration gradient)(c) Convection: stirring or hydrodynamic transport

Nernst-Planck equation (diffusion + migration + convection) 

Ji(x) = -Di(Ci(x)/x) –(ziF/RT)DiCi((x)/x) + Civ(x)

 Where Ji(x); the flux of species i (molsec-1cm-2) at distance x from the surface,

Di; the diffusion coefficient (cm2/sec), Ci(x)/x; the concentration gradient at

distance x, (x)/x; the potential gradient, zi and Ci; the charge and

concentration of species i, v(x); the velocity (cm/sec) Steady state mass transfersteady state, (C/t) = 0; the rate of transport of electroactive species is equal to the rate of their reaction on the electrode surface In the absence of migration (excess supporting electrolyte),

O + ne- = RThe rate of mass transfer,

vmt (CO(x)/x)x=0 = DO(COb – CO

s)/where x is distance from the electrode surface & : diffusion layer

vmt = mO[COb – CO

s]

where COb is the concentration of O in the bulk solution, CO

s is the

concentration at the electrod surfacemO is “mass transfer coefficient (cm/s)” (mO = DO/δ)

i = nFAmO[COb – CO

s]

i = -nFAmR[CRb – CR

s]

 

largest rate of mass transfer of O when COs = 0 “limiting current”

il,c = nFAmOCOb

Maximum rate when limiting current flows

COs/CO

b = 1 – (i/il,c)

COs = [1 – (i/il,c)] [ il,c/nFAmO] = (il,c – i)/(nFAmO)

COs varies from CO

b at i = 0 to negligible value at i = il

If kinetics of electron transfer are rapid, the concentrations of O and R at the electrode surface are at equilibrium with the electrode potential, as governed by the Nernst equation for the half-reaction

E = E0´+ (RT/nF)ln(COs/CR

s)E0´: formal potential (activity coeff.), cf. E0 (standard potential)

(a) R initially absentWhen CR

b = 0, CRs = i/nFAmR 

COs = (il,c – i)/(nFAmO)

E = E0´- (RT/nF)ln(mO/mR) + (RT/nF)ln(il,c – i/i)i-E plotWhen i = il,c/2, E = E1/2 = E0´- (RT/nF)ln(mO/mR)E1/2 is independent of concentration & characteristic of O/R system

E = E1/2 + (RT/nF)ln(il,c – i/i)

(b) Both O and R initially presentSame method,

CRs/CR

b = 1 – (i/il,a)

il,a = -nFAmRCRb

CRs = -[1 – (i/il,a)] [ il,a/nFAmR] = -(il,a – i)/(nFAmR)

 Put these equations to E = E0´+ (RT/nF)ln(COs/CR

s)

 E = E0´ – (RT/nF)ln(mO/mR) + (RT/nF)ln[(il,c – i)/(i - il,a)]

When i = 0, E = Eeq and the system is at equilibriumDeviation from Eeq: concentration overpotential

Electrochemical techniques

Potential step experimentTypes of techniquesPotentiostat: control of potential

Basic potential step experiment: O + e → R (unstirred solution, E2: mass-transfer (diffusion)-limited value (rapid kinetics → no O on surface))

chronoamperometry (i vs. t)

-Series of step experiments (between each step: stirring for same initial condition)

4, 5: mass-transfer (diffusion)-limited (no O on electrode surface)) sampled-current voltammetry (i(τ) vs. E)

Potential step: E1 → E2 → E1 (reversal technique) double potential step chronoamperometry

IntroductionMost widely used technique Applying a continuously time-varying potential working electrode -oxidation/reduction reactions of electroactive species-adsorption of species-capacitive current due to double layer charging-mechanisms of reactions-identification of species-quantitative analysis of reaction rates-determination of rate constants Two forms: linear sweep voltammetry(LSV) & cyclic voltammetry(CV)

Voltammetry

Potential Sweep Methods

IntroductionLinear sweep voltammetry (LSV)

Cyclic voltammetry (CV)

Nernstian (reversible) systemsSolution of the boundary value problemO + ne = R (semi-infinite linear diffusion, initially O present) E(t) = Ei – vt Sweep rate (or scan rate): v (V/s)Rapid e-transfer rate at the electrode surface

CO(0, t)/CR(0, t) = f(t) = exp[nF (Ei - vt - E0′)/RT]

i = nFACO*(πDOσ)1/2χ(σt)σ = (nF/RT)v

Peak current and potentialPeak current: π1/2χ(σt) = 0.4463

ip = 0.4463(F3/RT)1/2n3/2ADO1/2CO*v1/2

At 25°C, for A in cm2, DO in cm2/s, CO* in mol/cm3, v in V/s → ip in amperes

ip = (2.69 x 105)n3/2ADO1/2CO*v1/2

Peak potential, Ep

Ep = E1/2 – 1.109(RT/nF) = E1/2 – 28.5/n mV at 25°C

Half-peak potential, Ep/2

Ep/2 = E1/2 + 1.09(RT/nF) = E1/2 + 28.0/n mV at 25°C

E1/2 is located between Ep and Ep/2

|Ep – Ep/2| = 2.20(RT/nF) = 56.5/n mV at °C

For reversible wave, Ep is independent of scan rate, ip is proportional to v1/2

Spherical electrodes and UMEsSpherical electrode (e.g., a hanging mercury drop)

i = i(plane) + nFADOCO*φ(σt)/r0

φ(σt): tabulated function (Table 6.2.1)

For large v in conventional-sized electrode → i(plane) >> 2nd termSame for hemispherical & UME at fast scan rate

For UME at very small v: r0 is small → i(plane) << 2nd term→ voltammogram is a steady-state response independent of v→ v << RTD/nFr0

2

r0 = 5 μm, D = 10-5 cm2/s, T = 298 K → steady-state voltammogram at v < 1 V/s r0 = 0.5 μm → steady-state behavior up to 10 V/s

Transition from typical peak-shaped voltammograms at fast v to steady-statevoltammograms at small v

cf. For potential sweep (Ch.1)Linear potential sweep with a sweep rate v (in V/s) E = vtE = ER + EC = iRs + q/Cd

vt = Rs(dq/dt) + q/Cd

If q = 0 at t = 0, i = vCd[1 – exp(-t/RsCd)]

- Current rises from 0 and attains a steady-state value (vCd): measure Cd

Effect of double-layer capacitance & uncompensated resistanceCharging current at potential sweep

|ic| = ACdv

Faradaic current measured with baseline of ic

ip varies with v1/2, ic varies with v → ic more important at faster v

|ic|/ip = [Cdv1/2(10-5)]/[2.69n3/2DO1/2CO

*]

At high v & low CO* → severe distortion of the LSV wave

Ru cause Ep to be a function of v

Totally irreversible systemsSolution of the boundary value problem kf

Totally irreversible one-step, one-electron reaction: O + e → R

i = FACO*DO1/2v1/2(αF/RT)1/2χ(bt)

χ(bt) (Table 6.3.1). i varies with v1/2 and CO*

i/FA = DO(∂CO(x, t)/∂x)x=0 = kf(t)CO(0, t)

Where kf = k0e–αf(E(t) – E0′), E(t) = Ei – vt

→ kf(t)CO(0, t) = kfiCO(0, t)ebt

Where b = αfv & kfi = k0exp[-αf(Ei – E0′)]

For spherical electrodesi = i(plane) + FADOCO*φ(bt)/r0

Peak current and potentialMaximum χ(bt) at π1/2χ(bt) = 0.4958Peak current

ip = (2.99 x 105)α1/2ACO*DO

1/2v1/2

n-electron process with RDS: n in right side

Peak potentialα(Ep – E0′) + (RT/F)ln[(πDOb)1/2/k0] = -0.21(RT/F) = -5.34 mV at 25°C

Or Ep = E0′ - (RT/αF)[0.780 + ln(DO

1/2/k0) + ln(αFv/RT)1/2]|Ep – Ep/2| = 1.857RT/αF = 47.7/α mV at 25°C

Ep: ftn of v → for reduction, 1.15RT/αF (or 30/α mV at 25°C) negative shift for tenfold increase in v

ip = 0.227FACO*k0exp[-αf(EP – E0′)]

→ ip vs. Ep – E0′ plot at different v: slope of –αf and intercept proportional to k0

n-electron process with RDS: n in right side

i-E curve (CV) at different Eλ

(1) Eλ (1) E1/2 – 90/n, (2) E1/2 – 130/n, (3) E1/2 – 200/n mV, (4) after ipc → 0

ipa/ipc = 1 for nernstian regardless of scan rate, Eλ (> 35/n mV past Epc), D

ipa/ipc → kinetic informationIf actual baseline cannot be determined,

ipa/ipc = (ipa)0/ipc + 0.485(isp)0/ipc + 0.086

Reversal charging current is same as forward scan, but opposite sign

ΔEp = Epa – Epc ~ 2.3RT/nF (or 59/n mV at 25°C)

Quasi-reversible system

Kinetics +oxidation/reduction

v irreversibility , peak current , separation anodic and cathodic peaks

Adsorbed species

Adsorption of reagent or product on electrode voltammetric wave modified

 

Reversible reaction of adsorbed species O and R

Ip,c = (-n2F2vAO,i/4RT)

 

where O,i is surface concentration of adsorbed O. the same Ip magnitude for

oxidation

 

Ep = E0’ – (RT/nF)ln(bO/bR)

bO and bR: the adsorption energy of O and R

The value of Ep is the same for oxidation and reduction

Cyclic voltammogram for a reversible system of species adsorbed on the electrode

Multi component systemsVarious voltammetric waves appear

Cyclic votammogram in the investigation of systems of more than one component

Thin layer cellReversible: peak current v, no separation between anodic and cathodic peaks, total symmetric Ep

Irreversible:

Shape of the cyclic voltammogram obtained in a thin-layer cell for a reversible system

Electric charge (=amount of electricity) Q (unit: Coulomb, C), time tElectric current (unit: ampere (A)):

  I = dQ/dt

  Q = Idt

 Current density (unit: A/m2): i = I/A,

A: surface of area

Charge calculation

Hads → H+ + e-

Effect of adsorption of electroinactive species→ such adsorption inhibit (or poison) an electrode reaction or accelerate the

electrode reaction (e.g., hydrogen or oxygen)k0 = kθ=0

0(1 – θ) + kc0θ

Where kθ=00 is the standard rate const at the bare surface & kc

0 that at the filmed portions

For completer blockage by the film, kc0 = 0

For catalysis by the filmed area, kc0 > kθ=0

0

Effect of adsorbed substancesHydrogen & oxygenCO & organics

CV of Pt Single Crystals

Current – potential curves for a Ta/Ta2O5 electrode in 0.1M TBAP/MeCN

Electronic Structure using CV

CV of Au (111) CV of Pt (111) in H2SO4

0.1 mM H2SO4 (10 mV/S)

1.0 mM CuSO4/0.1 mM H2SO4 ( 2 mV/s)

Cu electrodeposition

Cyclic Voltammograms of 316 SS 

-1.25 V : Fe reduction (Fe3+ to Fe2+)-0.6 V : Cr reduction (Cr6+ to Cr3+) -0.8 V : Fe oxide formation (active to passive transition)0 V : Trnspassive behavior of SS0.3 V : Cr oxidation to Cr6+

In vivo applications of CVe.g., rat brain

ascorbic acid oxidation in rat brain