log 10 [ j/(a cm-2) ]

Transcription

1 CHEM465/865, 6-3, Lecture 18, Oct. 16, 6 Mass Transport Tafel plot: reduction of Mn(IV) to Mn(III) at Pt in 7.5 M H SO 4 at 98 K, various concentrations. log 1 [ /(A cm-) ] E E [V] vs. SHE -η = E -E

2 Corresponds to the following reaction scheme (cathodic process, reduction), η < : We have seen, how the rate of the reaction is influenced y E and y surface concentrations of involved species. Consider expression for the encountered current: C C I C, s = = nfk c exp A C ( 1 α ) F( E E ) RT Note: sign of current and E E <

3 Tafel-plot: electroanalytical tool to determine (cathode) exchange current density C, =.3RT Tafel-slope C ( 1 α ) C (cathode) transfer coefficient α. F Current depends exponentially on E. However, current cannot grow unlimitedly with E. Progress of a reaction is accompanied y concentration variations toward interior of solution affects a region that grows in thickness with time!

4 Three forms of mass transport: diffusion: nonuniform concentrations + entropic forces, acting to smooth the uneven distriutions convection: action of a force on the solution (pump, gas flow in pressure gradient, hydraulic permeation, gravity, etc.). E.g. convection under laminar flow conditions: c t = v x c x migration: electrostatic effect, voltage variation in solution, difficult to calculate for real solutions due to ion solvation effects try to avoid migration y adding supporting, inert electrolyte which levels potential variations in solution. Current density in solution with conductivity σ : Ohm S = σ ϕ x

5 Overall: Flux of electrical current at electrode involves electrode kinetics, and 3-dimensional diffusion, convection and migration consistent treatment of all these effects within one system is impossile, in experiment as well as in theory! understand conditions and electrode geometries under which mass transport limitations can e avoided Diffusion-limitations Transport of matter y random molecular motion (maximum entropy striving for uniform distriutions). Two limiting cases (as usual, realistic situations are etween these limits): rate of reaction >> rate of diffusion rate of reaction << rate of diffusion Which case did we consider so far???

6 Now: Let s focus on diffusion Diffusion normal to electrode surface (x-direction) Fick s first law (1855): Fick s second law (1855): c Jdiff = D x c t c = D x Fick s second law: permits prediction of variation of concentration of different species as a function of position and time (need initial and oundary conditions) Important analogy: heat conduction Can e solved with all sorts of oundary and initial condition. Usually, the solution involves Laplace transforms. (for details: see J. Crank, The Mathematics of Diffusion, nd Edition, 1975, Oxford University Press and H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press 1959).

7 Evaluate: How does diffusion affect current measured at an electrode? Semiempirical treatment Comparison with full analytical solution Potential sweep experiment Potential step experiment (using potentiostat) E t E

8 1. Current limited entirely y diffusion surface concentration: s c c 1 3 x/µm Every reactant molecule that arrives at interface reacts immediately In which region of the Tafel-plot are we now? Region, where current does not depend on E any more!

9 How deep does the disturance penetrate into the solution? mean free path or diffusion layer thickness: e.g. ygen in solution: δ πdt D.55 1 cm s What are the currents that diffusion can sustain? c δ 5 δ 9 µm after 1s 1 3 s c c c c diff = nfd nfd = nfd x δ δ x/µm Insert δ: diff = nfc D π t Cottrell-equation

10 Diffusion-limited current decreases with time! The longer a measurement takes, the more severe are the diffusion limitations. Reactant supplied from more distant regions. How do concentration profiles and ( ) diff t look like? c E E 1-3 s 1 - s (no current) 1-1 s δ c E E >> (diffusion limited current) n = 1, F = 96485Cmol, c = 1 mol l, D =.55 1 α =.5, = 1 5 Acm 5 cm s, 1 3 x/µm 3 diff / A cm - 1 diff = nfc D π t t/s

11 . Kinetic limitations and mass transport: MIXED KINETICS! Finite surface concentration c s > and alance of fluxes oundary condition at electrode surface: rate of consumption (kinetics) = rate of supply (transport) C = diff C ( 1 α ) FE E s nfk c exp = RT again use simple semiempirical treatment insert δ and solve f or s s kc exp = D s kc c s c -nfd x E E c c E E exp = c c δ s ( ) x= D π t s c E E s c = and = nfk c exp E E π t 1+ k exp D

12 and, thus E E exp = = C * *, where nfk c E E π t 1+ k exp D With definitions of diff and * this can e rewritten E E exp E E 1+ exp * diff, C * * = with diff = nfc * D π t (Cottrell) Be aware that diff is a function of time! Two important limiting cases are reproduced Kinetic limitations more severe, << * * diff : C = * exp E E Tafel-equation Diffusion limitations more severe, >> * * diff : C D π t = nfc Cottrell-equation Two handles to steer etween the limiting cases: t and E

13 Show results for n = 1, F = 96485Cmol, c = 1 mol l, 5 D =.55 1 cm s, α =.5, * 5 = 1 Acm k 7 = 1 cm s, RT = 5.7 mv = (98 K) F C Reactant distriution and ( ) t for fixed E E =.7 V. 1-4 s s 1 - s 1-1 s 8 t= 6 η =.7 V c.5 / A cm x/cm..5.1 t/s

14 Consider: measurement of current-voltage relationship (potential-step experiment). How do you perform the measurement? t 1 t t 3 E-E t/s After which time do you record the current? Assume: each point a new step from equilirium. The measurement time is a property of the equipment (How many charges do you have to collect to reach appropriate accuracy current measurement).

15 Tafel-plots recorded with distinct measurement times: log 1 ( / Acm - ) - kinetics diffusion t = 1-4 s t = 1-3 s t = 1 - s t = 1-1 s t = 1 s less sensitive equipment E-E η /V / V

16 Compare with full solution Diffusion equation: Initial condition (t = ): c t c c = D x = c everywhere Boundary conditions: E E c x : c c k c x D x s = a nd x = : e p = - x= c s - where k E E = Kc, K = exp x D x= Solution: straightforward mathematics (use Laplace transform) can e found in H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press 1959.

17 Variation of concentration with x and t (, ) c x t x x = 1 erfc exp ( Kx K Dt ) erfc K Dt, c Dt 4Dt Concentration at x = : ( =, ) c x t c = exp Current density: exp ( K Dt) erfc( K Dt) ( ) erfc( ) * K Dt K Dt = exp E E Compare the two solutions: good correspondence! s 1-3 s 1 - s 1-1 s c / mol cm x / cm

18 8 t= 6 / A cm - 4 exact apprimation..5.1 t/s Apprimate solution is sufficient for a asic understanding. The semiempirical apprimation works pretty well.

19 Consider potential sweep experiment: E E = At diffusion.1 / Acm -.5 onset of reduction sweep rate E-E = 1 mv/s * t E-E / V How to confine concentration variations to a thin region and not let them ecome limiting? Control transport: vigorous stirring (e.g. rotating disc electrode) and supporting electrolyte good transport due to convection and migration Control electrode geometry: microelectrodes

20 Diffusion Overpotential In a voltammetric experiment, the electrode potential E is controlled with a potentiostat (chronoamperometric measurement). The current density is determined y the following relation E E exp * = * E E 1+ exp * diff. C (, current through ce ll) This can e easily rewritten as (-η relation) = η exp η. 1+ exp * diff Now assume that the experiment is performed under current control (chronopotentiometric measurement). What is the value of η corresponding to a fixed value of? Solve for η

21 η η 1+ exp = * exp diff = 1 * exp η diff Take ln on oth sides and collect all -dependent terms on right hand side: η = ln ln 1 = * ηk + η diff First term: usual Tafel-equation in asence of mass diff transport limitations reaction overpotential: η K Second term: overpotential due to mass transport limitations diffusion overpotyential: η diff

22 .4 kinetics, η k = 1-7 Acm - overpotentials. diff = 1.1 Acm - diffusion, η diff / Acm - E cell =1.3 V - η k - η diff / V / Acm -