# Characterization Technique: Voltammetry

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01-Feb-2016Category

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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

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 reactionRate (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 kbRate of the forward process vf (M/s) = kfCARate of the reverse reaction vb = kbCBRate const, kf, kb: s-1Net conversion rate of A & B vnet = kfCA kbCBAt 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 = AeEA/RT

EA: activation energy, A: frequency factor

Transition state or activated complex Standard internal E of activation: E Standard enthalpy of activation: H H = E + (PV) ~ E

k = Aexp(-H/RT)

A = Aexp(S/RT) S: standard entropy of activation

k = Aexp[-(H - TS)/RT] = Aexp(-G/RT)

G: 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 = (kT/h)e-G/RT

: 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 kbEquilibrium is characterized by the Nernst equation

E = E0 + (RT/nF)ln(Co*/CR*) bulk concKinetic: dependence of current on potential Overpotential = 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 kbPotential change from E0 to E energy change FE = -F(E E0)

G change: term (transfer coefficient)

Ga = G0a (1 )F(E E0)Gc = G0c + F(E E0)

kf = Afexp(-Gc/RT)kb = Abexp(-Ga/RT)

kf = Afexp(-G0c/RT)exp[-f(E E0)]kb = Abexp(-G0a/RT)exp[(1 )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[-f(E E0)]kb = k0exp[(1 )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-f(E E0) - CR(0,t)e(1 )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 (): a measure of the symmetry of the energy barrier

tan = FE/xtan = (1 )FE/x

= tan/(tan + tan)

= & = symmetrical

In most systems : 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-f(Eeq E0) - CR(0,t)e(1 )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-f(Eeq E0) = FAk0CO*(CO*/CR*)-

i0 = FAk0CO*(1 ) CR*

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

Current-overpotential equationDividing i = FAk0[CO(0,t)e-f(E E0) - CR(0,t)e(1 )f(E E0)]

By i0 = FAk0CO*(1 ) CR*

current-overpotential equation i = i0[(CO(0,t)/CO*)e-f (CR(0,t)/CR*)e(1 )f] cathodic term anodic termwhere = E - Eeq

Approximate forms of the i- equationNo mass-transfer effectsIf the solution is well stirred, or low current for similar surface conc as bulk i = i0[e-f e(1 )f] Butler-Volmer equation

*good approximation when i is

(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

(b) Linear characteristic at small For small value of x ex ~ 1+ x i = i0[e-f e(1 )f] = -i0f

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

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

Rct = RT/Fi0

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

i = i0ef = (RT/F)lni0 (RT/F)lni = a + blogi Tafel equation

a = (2.3RT/F)logi0, b = -(2.3RT/F)

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

anodiccathodic

Effects of mass transferPut CO(0,t)/CO* = 1 i/il,c and CR(0,t)/CR* = 1 i/il,ato i = i0[(CO(0,t)/CO*)e-f (CR(0,t)/CR*)e(1 )f]

i/i0 = (1 i/il,c)e-f (1 i/il,a)e(1 )f

i- 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 !!

O + ne = R mechanism: O + ne = O (net result of steps preceding RDS) kf O + e = R (RDS) kb R + ne = R (net result of steps following RDS)n + 1 + n = n

Current-potential characteristics

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

krds0, , Erds0 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 t

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