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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

i (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/nF

Electrode process: heterogeneous reaction

Rate (mols-1cm-2) = i/nFA = j/nF

j: current density (A/cm2)

Electrode reaction: i-E curves

Polarization: departure of the cell potential from the equilibrium potential

Extent of potential measured by the overpotential: = E - Eeq

Factors affecting electrode reaction rate and current

1. Mass transfer

2. Electron transfer at the electrode surface

3. Chemical reactions

4. Other surface reactions: adsorption, desorption, electrodeposition

Review of homogeneous kinetics

Dynamic 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 thermodynamics

At 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 conc

Kinetic: dependence of current on potential

Overpotential η = a + blogi Tafel equation

Forward reaction rate vf = kfCO(0,t) = ic/nFA

CO(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

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

α: a measure of the symmetry of the energy barrier

tanθ = αFE/x

tanφ = (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 process

Equilibrium conditions. The exchange current

At 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 the

thermodynamic 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 equation

Dividing

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 term

where η = E - Eeq

Approximate forms of the i-η equation

- No mass-transfer effects
- If 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 η

For 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 negligible

e.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)

Put 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

Rate-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, Erds0′ apply to the RDS

Multistep processes at equilibrium

At equilibrium, overall reaction → Nernst equation

Eeq = E0′+ (RT/nF)ln(CO*/CR*)

Nernst multistep processes

Kinetically 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 of

the details of the mechanism

Mass transport-controlled reactions

Modes of mass transfer

Electrochemical 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 transfer

steady 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- = R

The rate of mass transfer,

vmt (CO(x)/x)x=0 = DO(COb – COs)/

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

where COb is the concentration of O in the bulk solution, COs is the concentration at the electrod surface

mO is “mass transfer coefficient (cm/s)” (mO = DO/δ)

i = nFAmO[COb – COs]

i = -nFAmR[CRb – CRs]

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

il,c = nFAmOCOb

Maximum rate when limiting current flows

COs/COb = 1 – (i/il,c)

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

COs varies from COb 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/CRs)

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

(a) R initially absent

When CRb = 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 plot

When 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 present

Same method,

CRs/CRb = 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/CRs)

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 equilibrium

Deviation from Eeq: concentration overpotential

Types of techniques

Potentiostat: 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

Voltammetry

Potential Sweep Methods

Introduction

Most 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)

Nernstian (reversible) systems

Solution of the boundary value problem

O + 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: 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 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 term

Same 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/nFr02

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-state

voltammograms at small v

cf. For potential sweep (Ch.1)

Linear potential sweep with a sweep rate v (in V/s)

E = vt

E = 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 resistance

Charging 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

Solution of the boundary value problem kf

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

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′)]

i = FACO*DO1/2v1/2(αF/RT)1/2χ(bt)

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

For spherical electrodes

i = i(plane) + FADOCO*φ(bt)/r0

Maximum χ(bt) at π1/2χ(bt) = 0.4958

Peak current

ip = (2.99 x 105)α1/2ACO*DO1/2v1/2

n-electron process with RDS: n in right side

Peak potential

α(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(DO1/2/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

If 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)

Kinetics +oxidation/reduction

v irreversibility , peak current , separation anodic and cathodic peaks

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

Various voltammetric waves appear

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

Reversible: 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 t

Electric current (unit: ampere (A)):

I = dQ/dt

Q = Idt

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

A: surface of area

Charge calculation

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=00(1 – θ) + kc0θ

Where kθ=00 is the standard rate const at the bare surface & kc0 that at the filmed portions

For completer blockage by the film, kc0 = 0

For catalysis by the filmed area, kc0 > kθ=00

Effect of adsorbed substances

Hydrogen & oxygen

CO & organics

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

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 SS

0.3 V : Cr oxidation to Cr6+

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