Potential Sweep Methods (Ch. 6)

Transcription

1 Potential Sweep Methods (Ch. 6) Nernstian (reversible) systems Totally irreversible systems Quasireversible systems Cyclic voltammetry Multicomponent systems & multistep charge transfers

2 Introduction Linear sweep voltammetry (LSV) Cyclic voltammetry (CV)

3 Nernstian (reversible) systems Solution of the boundary value problem O + ne = R (semi-infinite linear diffusion, initially O present) E(t) = E i vt Sweep rate (or scan rate): v (V/s) Rapid e-transfer rate at the electrode surface C O (0, t)/c R (0, t) = f(t) = exp[nf (E i -vt-e 0 )/RT] σ = (nf/rt)v i = nfac O *(πd O σ) 1/2 χ(σt)

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5 Peak current and potential Peak current: π 1/2 χ(σt) = i p = (F 3 /RT) 1/2 n 3/2 AD O 1/2 C O *v 1/2 At 25 C, for A in cm 2, D O in cm 2 /s, C O* in mol/cm 3, v in V/s i p in amperes i p = (2.69 x 10 5 )n 3/2 AD O 1/2 C O *v 1/2 Peak potential, E p E p = E 1/ (RT/nF) = E 1/2 28.5/n mv at 25 C Half-peak potential, E p/2 E p/2 = E 1/ (RT/nF) = E 1/ /n mv at 25 C E 1/2 is located between E p and E p/2 E p E p/2 = 2.20(RT/nF) = 56.5/n mv at C For reversible wave, E p is independent of scan rate, i p is proportional to v 1/2

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7 Spherical electrodes and UMEs Spherical electrode (e.g., a hanging mercury drop) φ(σt): tabulated function (Table 6.2.1) i = i(plane) + nfad O C O *φ(σt)/r 0 For large v in conventional-sized electrode i(plane) >> 2 nd term Same for hemispherical & UME at fast scan rate For UME at very small v: r 0 is small i(plane) << 2 nd term voltammogram is a steady-state response independent of v v << RTD/nFr 0 2 r 0 = 5 μm, D = 10-5 cm 2 /s, T = 298 K steady-state voltammogram at v < 1 V/s r 0 = 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

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9 cf. For potential sweep (Ch.1) Linear potential sweep with a sweep rate v (in V/s) E = vt E = E R + E C = ir s + q/c d vt = R s (dq/dt) + q/c d If q = 0 at t = 0, i = vc d [1 exp(-t/r s C d )] - Current rises from 0 and attains a steady-state value (vc d ): measure C d

10 Effect of double-layer capacitance & uncompensated resistance Charging current at potential sweep i c = AC d v Faradaic current measured with baseline of i c i p varies with v 1/2, i c varies with v i c more important at faster v i c /i p = [C d v 1/2 (10-5 )]/[2.69n 3/2 D O 1/2 C O* ] At high v & low C O* severe distortion of the LSV wave R u cause E p to be a function of v

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12 Totally irreversible systems Solution of the boundary value problem k f Totally irreversible one-step, one-electron reaction: O + e R Where k f = k 0 e αf(e(t) E0 ), E(t) = E i vt i/fa = D O ( C O (x, t)/ x) x=0 = k f (t)c O (0, t) k f (t)c O (0, t) = k fi C O (0, t)e bt Where b = αfv & k fi = k 0 exp[-αf(e i E 0 )] i = FAC O *D O 1/2 v 1/2 (αf/rt) 1/2 χ(bt) χ(bt) (Table 6.3.1). i varies with v 1/2 and C O * For spherical electrodes i = i(plane) + FAD O C O *φ(bt)/r 0

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14 Peak current and potential Maximum χ(bt) at π 1/2 χ(bt) = Peak current i p = (2.99 x 10 5 )α 1/2 AC O* D O 1/2 v 1/2 n-electron process with RDS: n in right side Peak potential α(e p E 0 ) + (RT/F)ln[(πD O b) 1/2 /k 0 ] = -0.21(RT/F) = mv at 25 C Or E p = E 0 -(RT/αF)[ ln(d O 1/2 /k 0 ) + ln(αfv/rt) 1/2 ] E p E p/2 = 1.857RT/αF = 47.7/α mv at 25 C E p : ftn of v for reduction, 1.15RT/αF (or 30/α mv at 25 C) negative shift for tenfold increase in v i p = 0.227FAC O* k 0 exp[-αf(e P E 0 )] i p vs. E p E 0 plot at different v: slope of αf and intercept proportional to k 0 n-electron process with RDS: n in right side

15 Quasireversible systems For one-step, one-electron system k f O + e = R k b For the quasireversible one-step, one-electron case (5.5.3, p. 191) i/fa = D O ( C O (x, t)/ x) x=0 = k f C O (0, t) k b C R (0, t) Where k f = k 0 e αf(e E0 ) & k b = k 0 e (1 α)f(e E0 ), f = F/RT The shape of peak & peak parameters ftns of α & Λ Λ = k 0 /(D O 1-α D Rα fv) 1/2 Or for D O = D R = D Λ = k 0 /(Dfv) 1/2 Current i = FAD O 1/2 C O* f 1/2 v 1/2 Ψ(E) Ψ(E) (Fig ): Λ > 10 approach to the reversible

16 Ψ(E) I. Λ = 10 II. Λ = 1 III. Λ = 0.1 IV. Λ = 0.01 Dashed line: reversible

17 Cyclic voltammetry (0 < t λ) E = E i vt (t > λ) E = E i 2vλ + vt Nernstian systems i-t curve at different E λ

18 i-e curve (CV) at different E λ (1) E λ (1) E 1/2 90/n, (2) E 1/2 130/n, (3) E 1/2 200/n mv, (4) after i pc 0 i pa /i pc = 1 for nernstian regardless of scan rate, E λ (> 35/n mv past E pc ), D

19 i pa /i pc kinetic information If actual baseline cannot be determined, i pa /i pc = (i pa ) 0 /i pc (i sp ) 0 /i pc Reversal charging current is same as forward scan, but opposite sign ΔE p = E pa E pc ~ 2.3RT/nF (or 59/n mv at 25 C)

20 Quasireversible systems Wave shape & ΔE p ftns of v, k 0, α & E λ If E λ > 90/n mv beyond cathodic peak small E λ effect Ψ = Λπ -1/2 = [k 0 (D O /D R ) α ]/(πd O fv) 1/2 (1) Ψ = 0.5, α = 0.7, (2) Ψ = 0.5, α = 0.3, (3) Ψ = 7, α = 0.5, (4) Ψ = 0.25, α = 0.5

21 For 0.3 < α < 0.7 ΔE p independent of α; depend only on Ψ estimating k 0 in quasireversible systems ΔE p vs. v ΔE p vs Ψ

22 Multicomponent systems & Multistep charge transfers O & O system

23 Method for obtaining baselines Constant E after 1 Sweep stop beyond E p1

24 In vivo applications of LSV & CV e.g., rat brain