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
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 equlibrium
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/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
(a) 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)
(d) Tafel plots (i vs. η) → evaluating kinetic parameters (e.g., i0, α)
anodic cathodic
e.g., real Tafel plots for Mn(IV)/Mn(III) system in concentrated acid
- At very large overpotential: mass transfer limitation
Exchange current plots
i0 = FAk0CO*e-αf(Eeq – E0′)
→ logi0 = logFAk0 + logCO* + (αF/2.3RT)E0′ - (αF/2.3RT)Eeq
A plot of logi0 vs. Eeq at const CO* → linear with a slope of –αF/2.3RT
→ obtaining α and i0
Another way to determining α
i0 = FAk0CO*(1 – α) CR* α
→ logi0 = logFAk0 + (1 – α)logCO* + αlogCR*
(∂logi0/∂logCO*)CR* = 1 – α and (∂logi0/∂logCR*)CO* = α
Or from i0 = FAk0CO*(1 – α) CR* α
→ [dlog(i0/CO*)]/[dlog(CR*/CO*)] = α
Not require holding CO* or CR* constant
Very facile kinetics and reversible behavior
i/i0 = (CO(0,t)/CO*)e-αfη – (CR(0,t)/CR*)e(1 – α)fη
At very large i0 (big standard rate constant k0) → i/i0 → 0
CO(0,t)/CR(0,t) = (CO*/CR*)ef(E - Eeq)
Put Nernst eqn: ef(Eeq – E0′) = CO*/CR* (Eeq = E0′ + (RT/nF)ln(CO*/CR*))
CO(0,t)/CR(0,t) = ef(Eeq – E0′) ef(E - Eeq) = ef(E – E0′)
Rearrangement
E = E0′ + (RT/F)ln[CO(0,t)/CR(0,t)]
Potential vs. surface concentration regardless of the current flow
No kinetic parameters due to very facile kinetics
Effects of mass transfer
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
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 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
Quasireversible and irreversible multistep processes
pp. 111-115
