Kinetics of electrode reactions (Ch. 3)

Transcription

1 Kinetics of electrode reactions (Ch. 3) Review of homogeneous kinetics Dynamic equilibrium. Arrhenius equation. Transition state theory Essentials of electrode reactions Butler-Volmer model of electrode kinetics 1-step, 1-e process. Standard rate const. Transfer coefficient Implications of Butler-Volmer model for 1-step, 1-e process Exchange current. Current-overpotential equation. Exchange current plots. Very facile kinetics & reversible behavior. Effects of mass transfer Multistep mechanisms Microscopic theories of charge transfer ( 생략 ) Marcus theory

2 Review of homogeneous kinetics Dynamic equilibrium k f O + e = R k b Rate of the forward process v f (M/s) = k f C A Rate of the reverse reaction v b = k b C B Rate const, k f, k b : s -1 Net conversion rate of A & B v net = k f C A k b C B At equilibrium, v net = 0 k f /k b = K = C B /C A *kinetic theory predicts a const conc ratio at equilibrium, just as thermodynamics At equilibrium, kinetic equations thermodynamic ones dynamic equilibrium (equilibrium: nonzero rates of k f & k b, but equal) Exchange velocity v 0 = k f (C A ) eq = k b (C B ) eq

3 Arrhenius equation & potential energy surfaces k = Ae EA/RT E A : 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

4 Transition state theory (absolute rate theory, activated complex theory) General theory to predict the values of A and E A Rate constants k = κ(kt/h)e -ΔG /RT κ: transmission coefficient, k: Boltzmann const, h: Planck const

5 Essentials of electrode reactions *accurate kinetic picture of any dynamic process must yield an equation of the thermodynamic form in the limit of equilibrium k f O + ne = R k b Equilibrium is characterized by the Nernst equation E = E 0 + (RT/nF)ln(C o* /C R* ) bulk conc Kinetic: dependence of current on potential Overpotential η = a + blogi Tafel equation Forward reaction rate v f = k f C O (0,t) = i c /nfa C O (0,t): surface concentration. Reduction cathodic current (i c ) Backward reaction rate v b = k b C R (0,t) = i a /nfa Net reaction rate v net = v f v b = k f C O (0,t) k b C R (0,t) = i/nfa i = i c i a = nfa[k f C O (0,t) k b C R (0,t)]

6 Butler-Volmer model of electrode kinetics Effects of potential on energy barriers Hg Na + + e = Na(Hg) Equilibrium E eq positive potential than equlibrium negative potential than equilibrium

7 One-step, one-electron process k f O + e = R k b Potential change from E 0 to E energy change FΔE = -F(E E 0 ) ΔG change: α term (transfer coefficient) ΔG a = ΔG 0a (1 α)f(e E 0 ) ΔG c = ΔG 0c + αf(e E 0 ) k f = A f exp(-δg c /RT) k b = A b exp(-δg a /RT) k f = A f exp(-δg 0c /RT)exp[-αf(E E 0 )] k b = A b exp(-δg 0a /RT)exp[(1 α)f(e E 0 )] f = F/RT

8 At C O* = C R*, E = E 0 k f C O* = k b C R* k f = k b ; standard rate constant, k 0 At other potential E k f = k 0 exp[-αf(e E 0 )] k b = k 0 exp[(1 α)f(e E 0 )] Put to i = i c i a = nfa[k f C O (0,t) k b C R (0,t)] Butler-Volmer formulation of electrode kinetics i = FAk 0 [C O (0,t)e -αf(e E0 ) -C R (0,t)e(1 α)f(e E0 ) k 0 : large k 0 equilibrium on a short time, small k 0 sluggish (e.g., 1 ~ 10 cm/s) (e.g., 10-9 cm/s) k f or k b can be large, even if small k 0, by a sufficient high potential

9 The transfer coefficient (α) 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

10 Implications of Butler-Volmer model for 1-step, 1-electron process Equilibrium conditions. The exchange current At equilibrium, net current is zero i = 0 = FAk 0 [C O (0,t)e -αf(eeq E0 ) -C R (0,t)e(1 α)f(eeq E0 ) e f(eeq E0 ) = C O* /C R * (bulk concentration are found at the surface) This is same as Nernst equation!! (E eq = E 0 + (RT/nF)ln(C O* /C R* )) 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 i a = i c ) exchange current (i 0 ) i 0 = FAk 0 C O* e -αf(eeq E0 ) = FAk 0 C O* (C O* /C R* ) -α i 0 = FAk 0 C O *(1 α) C R *α i 0 is proportional to k 0, exchange current density j 0 = i 0 /A

11 Current-overpotential equation Dividing i = FAk 0 [C O (0,t)e -αf(e E0 ) -C R (0,t)e (1 α)f(e E0 ) ] By i 0 = FAk 0 C O *(1 α) C R *α current-overpotential equation where η = E - E eq i = i 0 [(C O (0,t)/C O* )e -αfη (C R (0,t)/C R* )e (1 α)fη ] cathodic term anodic term

12 Approximate forms of the i-η equation (a) No mass-transfer effects If the solution is well stirred, or low current for similar surface conc as bulk i = i 0 [e -αfη e (1 α)fη ] Butler-Volmer equation *good approximation when i is <10% of i l,c or i l,a (C O (0,t)/C O* = 1 i/i l,c = 0.9) For different j 0 (α = 0.5): (a) 10-3 A/cm 2, (b) 10-6 A/cm 2, (c) 10-9 A/cm 2 the lower i 0, the more sluggish kinetics the larger activation overpotential ((a): very large i 0 engligible activation overpotential)

13 (a): very large i 0 engligible activation overpotential any overpotential: concentration overpotential (changing surface conc. of O and R) i 0 10 A/cm 2 ~ < pa/cm 2 The effect of α

14 (b) Linear characteristic at small η For small value of x e x ~ 1+ x i = i 0 [e -αfη e (1 α)fη ] = -i 0 fη Net current is linearly related to overpotential in a narrow potential range near E eq -η/i has resistance unit: charge-transfer resistance (R ct ) R ct = RT/Fi 0 (c) Tafel behavior at large η i= i 0 [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 = i 0 e αfη η = (RT/αF)lni 0 (RT/αF)lni = a + blogi Tafel equation a = (2.3RT/αF)logi 0, b = -(2.3RT/αF)

15 (d) Tafel plots (i vs. η) evaluating kinetic parameters (e.g., i 0, α) anodic cathodic

16 e.g., real Tafel plots for Mn(IV)/Mn(III) system in concentrated acid - At very large overpotential: mass transfer limitation

17 Exchange current plots i 0 = FAk 0 C O* e-αf(eeq E0 ) logi 0 = logfak 0 + logc O* + (αf/2.3rt)e 0 -(αf/2.3rt)e eq A plot of logi 0 vs. E eq at const C O* linear with a slope of αf/2.3rt obtaining α and i 0 Another way to determining α i 0 = FAk 0 C O *(1 α) C R * α logi 0 = logfak 0 + (1 α)logc O* + αlogc R * ( logi 0 / logc O* ) CR* = 1 α and ( logi 0 / logc R* ) CO* = α Or from i 0 = FAk 0 C O *(1 α) C R * α [dlog(i 0 /C O* )]/[dlog(c R* /C O* )] = α Not require holding C O* or C R* constant

18 Very facile kinetics and reversible behavior i/i 0 = (C O (0,t)/C O* )e -αfη (C R (0,t)/C R* )e(1 α)fη At very large i 0 (big standard rate constant k 0 ) i/i 0 0 C O (0,t)/C R (0,t) = (C O* /C R* )ef(e - Eeq) Put Nernst eqn: e f(eeq E0 ) = C O* /C R * (E eq = E 0 + (RT/nF)ln(C O* /C R* )) C O (0,t)/C R (0,t) = e f(eeq E0 ) e f(e - Eeq) = ef(e E0 ) Rearrangement E = E 0 + (RT/F)ln[C O (0,t)/C R (0,t)] Potential vs. surface concentration regardless of the current flow No kinetic parameters due to very facile kinetics

19 Effects of mass transfer Put C O (0,t)/C O* = 1 i/i l,c and C R (0,t)/C R* = 1 i/i l,a to i = i 0 [(C O (0,t)/C O* )e -αfη (C R (0,t)/C R* )e (1 α)fη ] i/i 0 = (1 i/i l,c )e -αfη (1 i/i l,a )e(1 α)fη i-η curves for several ratios of i 0 /i l

20 Multistep mechanisms 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 oneelectron process!! one-step, one-electron process 적용가능!! O + ne = R mechanism: O + n e= O (net result of steps preceding RDS) k f O + e = R (RDS) k b R + n e = R (net result of steps following RDS) n n = n Current-potential characteristics i = nfak rds0 [C O (0,t)e -αf(e Erds 0 ) C R (0,t)e (1 α)f(e Erds 0 ) ] k rds0, α, E rds 0 apply to the RDS

21 Multistep processes at equilibrium At equilibrium, overall reaction Nernst equation E eq = E 0 + (RT/nF)ln(C O* /C R* ) Nernst multistep processes Kinetically facile & nernstian (reversible) for all steps E = E 0 + (RT/nF)ln[C O (0,t)/C R (0,t)] E is related to surface conc of initial reactant and final product regardless of the details of the mechanism Quasireversible and irreversible multistep processes pp