III. Reaction Kinetics Lecture 12: Faradaic Reactions in Dilute Solutions As we begin to study electrochemical energy systems out of equilibrium, we are first concerned with the kinetics of Faradaic (charge-transfer) reactions. 1 Stochastic Theory of Reaction Rates Here we model a particle diffusing through a potential energy landscape on the molecular scale. We need UTS − U1,2 " kBT to ensure that transitions are “rare events” and thAT particles are most of the time in well-defined “states” corresponding to local energy minima. See 2009 notes for analysis of the first passage time τ for a particle to “escape” from state 1 to state 2. Here we just quote the result: < τ1 2 > = mean first passage time →UTS−U1 = τ0e kBT Or the mean escape rate per particle: 1 MIT StudentLecture 12: Faradaic reactions in dilute solutions 10.626 (2011) Bazant UTS kB Tr1 2 = r0e− →Where R0 ∝ D0 with:LminLmax D0 = a microscopic diffusivity (measuring thermal noise). Lmin = width of the energy minimum. Lmax = width of the saddle along transition path. The net reaction rate for reversible reaction 1 2 is: ↔ r = r1↔2 − r2↔1 Where: U1−U2r1 2 kBT→= e r2 1→This is a form of the de Donder Relation. The second equation does not depend on the transition state, only the local equilibrium states 1 and 2. 2 Reactions in Dilute Solutions Consider the reaction ! siRi → ! sj Pj , where reactants are in state 1 and products are in state 2. For every reaction complex going from state 1 to state 2 there is a transition state with energy UTS . The state energies are: U1 = " si,1Ui,1 U2 = " sj,2Uj,2 For the forward reaction, the number of such transitions (per volume) is si,1proportional to # ci,1 assuming a dilute solution. So we can calculate the net reaction rate (number/time per reaction site): R = R1→2 − R2→1 $% UTS −U1 UTS −U2 & R = R0 ci,si,1 1 e− kT − % cj,sj,2 2 e− kT i i 2#Lecture 12: Faradaic reactions in dilute solutions 10.626 (2011) Bazant Note we can shift energy of state 2 to pull out a constant “exchange” rate R0. Also we have: si,1 R1 2 i ci,1 U1−U2 R2→= #e kT another de Donder Relation sj,2 →1 j cj,2 In equilibrium: sj,2 ∆Ueq. =(U1 − U2)eq. '#j cj,2 ( = kB T ln si,1#i ci,1 This looks like the Nernst Equation! 3 Faradaic Reactions in Dilute Solutions 3.1 Standard form of initial and final states Consider the general half-cell Faradaic reaction, which we write in the stan-dard form " siMzi → ne−i where the reaction produces n electrons (oxidation) in the forward direction. We break this into reactants (si > 0) comprising the “reduced state” and products (si < 0) comprising the “oxidized state” of the anodic reaction. " sRj RjzR,j " sOi OiZ0,i + ne−→ j i By charge conservation, we have " sOi ZOi − n = " sRj zRj where qo = ! sOi ZOi and qR = ! sRj zRj . This allows us to express the initial and final states of the reaction (“states 1 and 2” above) as )U0U1 = UR = " sR,j R,j + zR,j eΦ* total energy of reduced state = UR 0 + q0Φ )U0U2 + neΦe = UO = " sOi R,i + zO,ieΦ* total energy of oxidized state = UO 0 + qOΦ separate electrostatic energy 3Lecture 12: Faradaic reactions in dilute solutions 10.626 (2011) Bazant 3.2 Butler-Volmer Model for the transition state The Butler-Volmer hypothesis asserts that the electrostatic energy of the transition state is a weighted average of electrostatic energies of the oxidized and reduced states: = U0UTS TS + αqRΦ + (1 − α)[q0Φ − neΦe] whereΦ 0 is electrode potential where α = transfer coefficient = weight of reduced state electrostatic energy at transition state It is typically to assume (or infer) symmetric electron transfer, α = (1 − α)= 1 .2 although the extreme values, α = 1 or α = 0, for asymmetric electron transfer are also possible. Then if we focus on the electrostatic p otential, we obtain % si,R % si,OR = ka ci,R e−[αqRΦ+(1−α)(qO Φ−neΦe)−qRΦ]/(kT ) − kc ci,O e−[αqRΦ+(1−α)(qOΦ−neΦe)−qOΦ]/(kT ) i i where we absorb UiO into anodic (oxidation) and cathodic (reduction) reac-tion rate constants, ka and kc respectively. Using charge conservation qO − qR = n, we finally express the Faradaic reaction rate in a dilute solution in the following general form 4Lecture 12: Faradaic reactions in dilute solutions 10.626 (2011) Bazant % si,R (1−α)ne∆Φ/(kB T ) % si,OR = ka ci,R e − kc ci,O e−αne∆Φ/(kB T ) i i ∆Φ ==Φ e − Φ = electro de potential − solution potential number of reactions R = per reaction site time For further reading, see O’Hare et al., Fuel Cell Fundamentals (Ch 3). 5MIT OpenCourseWarehttp://ocw.mit.edu 10.626 / 10.462 Electrochemical Energy Systems Spring 2011 For information about citing these materials or our Terms of Use, visit: