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1、31Chapter3:IONICTRANSPORT1.Ionicfluxandcurrentdensity2.ValidityofOhmslawinelectrochemicalCells3.Transportnumberandconcentrationgradients4.Supportingelectrolyte5.Concentrationgradientsatelectrodesandtheirdependenceonthecurrentdensity6.Thelimitingcurrent32IONICTRANSPORTIonicfluxisdueto:Diffusion+Elect
2、ricMigration+Convection+.Nj=DjCjUjzjFCj+Cjv+.moles/seccm2Currentdensityisduetofluxofallchargecarryingspecies:i=zjFNj=FzjDjCjF2UjZj2Cj+FzjCjvHowever,becauseofelectroneutrality,zjCj=0,thelasttermisidenticallyzero,i=FzjDjCjComparewith(thedifferentialformof)Ohmslaw:i=Diffusion=DjCjMigration=UjZjFCjConve
3、ction=Cjv“Bulk”Boundarylayers+33Ionicfluxandthecurrentdensity:ThecurrentflowinelectrochemicalcellsisrelatedtotheionicfluxNj.Thelatteristypicallydescribedintermsofthreemajormodesoftransport:migrationofchargedionsdowntheelectricfield,diffusionofionsacrossaconcentrationgradientandtransportofionsduetobu
4、lkelectrolyteconvection.Consequently,thefluxofanionicspeciesiisgivenbyjjjjjjjNDCU FZ CC v Msec1cm21Electroneutralityispresentthroughoutthecellandisdescribedbyiiz C =0(eq./M)(M/cm3)=eq/cm32ThetotalcurrentdensityisdeterminedbymultiplyingthefluxofeachspeciesbyitschargeFZ;andsummingoverallionicspecies:i
5、=Fz NiiA/cm23SubstitutingEqs.1intoEq.3yields221iiiiii=F(u c)Fz Dc()iizFzCv4Becauseofelectroneutrality(Eq.2),thelasttermontherightisidenticallyzeroandcancels,givingtheexpressionforthetotalcurrentdensityintermsoftheconcentrationandpotentialfieldsinthecell:i=Fu c Fz Dc2iiiii()z125It becomes apparent th
6、at the current density is determined by both the potential andconcentrationgradients.ThevelocitytermisabsentinEq.5duetoelectroneutrality,however,convectionstillaffectstransportbycontrollingtheconcentrationfield.Tosolveforthecurrentdensity,onehastoknowthepotentialdistributioninthecell,andtheconcentra
7、tiondistributionofallspeciespresent,cj.Thelatterareaffectedbytheflow.However,asweshallsubsequentlysee,somesimplificationsareoftenpossible.Inanyevent,Eq.5,while giving the current density as a function of the potential and concentrationdistributions,doesnotprovidethenecessaryrelationshipsrequiredfors
8、olvingthelatter.Thesecomefromtheconstitutiverelationshipsdescribedbelow.34TheTransportParametersUandD:Twotransportparameterswereintroducedinthederivation:UandD.Theybothexpresstheabilityofaniontomoveundertheinfluenceofanappliedforce,hencewecanexpectthemtoberelated.Uisthemobility,i.e.,thevelocityofani
9、onduetoanappliedelectricfield.Disthediffusivityorthevelocityoftheionassociatedwithintermolecularcollisions.Bothparametersareexpectedtorelate(beinverselyproportional)tothesizeofthemoleculebeingtranslated,andbeaffectedalsobythenatureofthemedium(solvent)whichtypically(butnotalways)iswater.Accordingly,w
10、eshouldbeconsideringUi,jandDi,jwhereireferstothegivenionandjrepresentstheotherspeciesinthemedium.StokesEinsteinlawrelatesbothparameters:Di=RTUi 6wheretheindexjrepresentingthesolvent(water)hasbeenomitted.Wealsonotedbeforethattheequivalentionicconductivitywasrelatedtothemobilitythrough:i=F2ZiUi7Sincei
11、wasaboutconstantformostions(i5070)duetotheiraboutuniformhydratedshell,exceptforprotons(i350)andhydroxylions(i200),weexpectthediffusivityformostionstobeaboutthesame.Weindeedfind:D105cm2/s8formostionsexceptfortprotonsandhydroxylions.Thisequivalency,coupledwiththesmallmagnitudeofDwasfirstrecognizedbyVe
12、njaminLevichandenabledhimtocharacterizetheRotatingDiskElectrode(RDE)asausefulandfairlyuniversaltoolforcharacterizingelectrolyticsolutions.35Afewquestions:Howcomethattheelectrolytevelocityappearsnottoaffectthecurrentdensity?DoesthismakesenseorcorrelatewithExperimentalevidence?Canwemeasureelectrolytec
13、onductivityusinganohmmeter?Whydowehaveconcentrationgradientsattheelectrodes?Howsignificantaretheconcentrationgradientsandwhatdotheydependon?Whichtermismoreimportant:FzjDjCjor?Cantherebeafunctioningelectrochemicalcell(i0)withnoconcentrationgradients?Whenaretheconcentrationgradientsestablished?Howlong
14、doesittakethemtobeestablishedortodecay?Howfardotheconcentrationgradientsextendintothesolution?Aretheconcentrationgradientsharmful,ordotheyhelpus?Cantheybeminimized?How?Howdoweplatefromanions?36The Transport Number jnjjjjjjjjMMjjCZCZiit1 11njjt (always)t+t0.5Binary electrolytetR 0 Well-supported elec
15、trolytes LVECu+SO4-Current is carried by all ions that are moving under the applied electrical field Define:The transport number of ionic species j=The fraction of the current that is carried by ionic species j in the absence of concentration gradient (i.e.,migration current in the well-mixed bulk).
16、373839CurrentandConcentrationGradientsThecurrentassociatedwithspeciesjisduetodiffusionandmigration:31aWecanwritedownthediffusionandmigrationcomponentsexplicitly32aWecanmakeuseofthetransportnumbertoexpress(onlyapproximately)themigrationcurrent33aSubstitutingthelasttermineq.32aintermsofthetransportnum
17、ber,34aWerecognizethatifweonlyhaveasinglereactionattheelectrode,e.g.,Or,inamoregeneralway,recognizingthattheremaybeastoichiometriccoefficient,sRforthereactant,35aweassignspeciesjasthereactant,recognizingthatifonlyonereactiontakesplace,theentirecurrentisassociatedwithit:iR=ij=i36aSubstitutingin34a:37
18、aSolvingfori,38aThenegativesigninEq.38indicatesthatthedirectionofthecurrentisdowntheconcentrationgradient.Byconvention,weassigncathodiccurrents(andoverpotentials)asnegativeandanodiccurrents(andoverpotentials)aspositive.Clearly,asthecurrentincreases(duetolargerpotentialdrivingforcefortheelectrodereac
19、tion),theconcentrationgradientmustincrease.Sincetheboundarylayerthicknessistypicallydeterminedbytheconvectiveflow,thisimpliesthattheconcentrationofthereactantnearthecathodedrops(andneartheanodeincreases).Rememberalsothatforwellsupportedelectrolytes,tR0,(whileforbinaryelectrolytestR0.5)DMjjjiii22jjjj
20、jjjiz D FCF z U C22MjjjjjF z U Citiijjjjjiz D FCt i2CueCu/zz n sRRs Mnes MRRRRiz D FCt i,11BERRRRRRRN Rz D F CCz D FCitt310CURRENTDENSITYATELECTRODESi=FZjDjCjCu+2eCu0nFDCRsR(1tR)DefinetheequivalentNernstboundarylayer,N:nFD(CBCE)sR(1tR)NNote:Byconvention,cathodiccurrentisnegative,anodicispositiveTheh
21、igherthecurrent,theloweristheinterfacialconcentration.CE=0definesthemaximumcurrentdensity:Thelimitingcurrent:nFDCBsR(1tR)NN=f(velocity,geometry,fluidproperties)tR=0inwellsupportedelectrolyte(e.g.;acid)tR=0.5inbinaryelectrolyte(e.g.;CuSO4only)i=i=CECBNCE=0iL=Thelimitingcurrentisthemaximalcurrentthatc
22、anbepassedforagivenreaction,irrespectiveofthevoltage.Itcanbeexceededonlywhensecondary,typicallyparasiticreactionsstarttakingplace.311IncreasingtheLimitingCurrent:Thelimitingcurrenthasamajorprocessdesignimplicationsincenoelectrochemicalreactioncanproceedatarateexceedingit,irrespectiveoftheappliedvolt
23、age.Theonlywaythelimitingcurrentcanbeexceededisthatonceitisreached,thevoltagewillswingtoahighervalue,eventuallykickingoffasecondary,typicallyparasiticreaction,e.g.,hydrogenevolution,resultinginadiminishedcurrentefficiency.Asnoted,thelimitingcurrentcanbeincreasedbyincreasingthebulkreactantconcentrati
24、on,however,wearelimitedbythesolubilitytoabout1M(=103M/cc).Wehavelittlecontroloverthediffusivity(typicallyincreasesslightlywithtemperature),andnocontrolovern,F,ands.Wecan,however,affectsignificantly,byenhancingtheconvectivevelocitypasttheelectrode.dependsmostlyonthevelocity,andalsoisafunctionofthegeo
25、metry(shapeoftheelectrode,andpositionalongit),andelectrolyteproperties(,andD).Beinginverselyproportionaltothevelocity,itbecomessmallerwithenhancedflow.Atlower,laminarflowrates,isinverselyproportionaltov0.33,however,athigher,turbulentflowrates,becomesinverselyproportionaltoapproximatelyv0.9.Ifnoconve
26、ctivevelocityispresent,increasesrapidlywithtime(weshallquantifythislater),andafterarelativelyshorttime(fewminutes)willshuttheprocessdownduetobecomingverylargeandiL,goingtozero.However,mostelectrochemicalsystems,willhavesomefreeconvection,thatwillkickinduetodensityvariationswithintheelectrolyte,andpr
27、eventfromdecayingtozero.Theeffectofsupportingelectrolyteonthelimitingcurrent:Withwellsupportedelectrolytes,i.e.,inthepresenceofanacid,base,orsalt,t0andwehave:,supBL wellportedrNnFDCisHowever,whenweremovethesupportingelectrolytewehavet0.5,giving:,sup2(0.5)BL binaryL wellportedrNnFDCiisAccordingly,byr
28、emovingthesupportingelectrolytewedoubleourmaximaltransportratesorhalfthemasstransportresistance.Whythendowetypicallyaddsupportingelectrolyte?Theanswerisrelatedtothefactthatbyaddingsupportingelectrolyteweincreasesignificantlytheelectrolyteconductivity,loweringtheohmicdropandimproving,asweshallseelate
29、r,thecurrentdistribution.Furthermore,mostelectrochemicalprocesses,particularlythoseinvolving312thedepositionofasolidphase,donotoperateclosetothemasstransportlimit(topreventdendriticdeposition),henceimprovingmasstransportisnotveryrelevant.Also,enhancingagitationandincreasingthereactantconcentrationar
30、etwoothermeansofimprovingmasstransport.Nonetheless,therearesituationswhenimprovingchemicallythemasstransportbyremovingthesupportingelectrolyteisuseful.Thereasonforthediminishedmasstransportinthepresenceofsupportingelectrolyteisthatduetothereducedmigrationinthepresenceofaweakelectricfield,henceessent
31、iallyallthereactantmustbebroughttotheelectrodebytransport(convectionanddiffusion)andvirtuallynonarriveviamigration.Thereisagreatsensitivitytothesupportingelectrolyte,andevenarelativelysmallamount,willreducetransport(andincreaseconductivity).313Theestablishmentofconcentrationgradientsinanelectrochemi
32、calcell:Considerfirstabinaryelectrolyte(CuSO4)inasimplecellconsistingoftwoparallel,infiniteelectrodes.AssumeamixedbulkwithaboundarylayerthicknessN.Furthermore,assumelinearconcentrationgradientwithinthethinboundarylayer.N NBeforeapplyingcurrent:iNCuMig=tri/nF0.5NCuNCu=i/nFThisdeficitof0.5NCucausesthe
33、cupricionconcentrationnexttothecathodetodrop,bringingaboutdiffusionalflux.CBC N314Whileweapplyacurrentdensityitothecathode,thecorrespondingfluxofcopperremovalbyplatingis:NCuTotal=i/nFSincemigrationflux,inthebinaryelectrolytebringsonlyNCuMig=tri/nF0.5i/nF0.5NCuTheremovalrateofcupricionsfromtheelectro
34、lytenexttothecathodeexceedstherateofreplenishmentbymigration,leadingtoadeficitandtolowerconcentrationnearthecathode.Thisloweredconcentrationbringsaboutdiffusionalfluxtowardsthecathode:NCuDif=D(dC/dx)|eD(CECB)/N0.5NCuTotal=0.5i/nFInitiallythisdiffusionalfluxiszero(initiallynoconc.gradients),however,a
35、stimepassesandthedeficitincreases,theconcentrationgradientsbuildupandthediffusionalfluxincreases.Oncethediffusionalfluxprovidestheentiredeficit,0.5NCu=0.5i/nFforthebinaryelectrolyte,iNCuMig=tri/nF0.5NCuNCu=i/nFNCuDiff=D(dC/dx)0.5NCuCBCEt315Theconcentrationgradientstabilizes.Thisestablishestheconcent
36、rationnearthecathode,Ce.:0(1)(1)RNEBcathodicRRNEBRs iCCby conventioninFDts iCCnFDtForthewellsupportedelectrolyte,theprocessissimilar,except,thattrisaboutzero.Accordingly,themigrationfluxofcopper,NCuMig=tri/nF0Isvanishinglysmall.Therefore,sincethecupricionsarestillremovedatafluxofNCu=i/nFandnocoppera
37、rrivestotheregionnexttothecathodebymigration,thedeficitbuildsupmuchfaster.Thediffusionalfluxbuildsupfaster,sinceitmustprovideallthecupricionsthatareremovedbyplating.Thediffusionalfluxwillbedoublethatforthebinaryelectrolyte,andtheconcentrationatthecathodewilltwicelower.Wehavenow:NCuDif=D(dC/dx)|eD(CB
38、CE)/N=NCuTotal=i/nFTheprocessesattheanode,areanalogous.Heretheamountofcopperdissolvedcantbecarriedoutjustbymigration.Thereforetheconcentrationbuildsupnexttotheanode,buildingupadiffusionalfluxawayfromtheanoderegion.316WhatabouttheSulfateconcentration?Considerfirstthecathoderegion.Clearly,sulfateisnot
39、removedorgeneratedbytheelectrodereaction,henceNSO4Total=0Sulfateiscarriedawayfromthecathodebymigration(duetotheelectricfield),since,unlikethecupricionitisnegativelycharged.NSO4Mig=tri/nF0.5i/nFAsaconsequence,adeficitofsulfatewillbegeneratednexttothecathode.Thiswilllowerthesulfateconcentrationnexttot
40、hecathode,resultingindiffusionalfluxofsulfatetowardsthecathode.NSO4Dif=D(dC/dx)|eD(CBCE)/N0.5i/nFSincethenetfluxofsulfateiszero,thesetwofluxesmustbeofoppositedirectionandequalinmagnitude.NSO4Dif=NSO4DifiNSO4Mig=tri/nF0.5NCuNSO4=0NSO4Diff=D(dC/dx)0.5NCuCBCEt317Thesulfateinterfacialconcentrationisiden
41、ticaltothisofthecupricionsandgivenby:(1)(1)RNRNEBBRRs is iCCCnFDtnFDtWefindthatthesulfateconcentrationprofileisidenticaltothisofthecupricions.Thiswasactuallyexpected,becauseduetoelectronetrality:44440022jjCuCuSOSOjCuSOCuCuCuSOz Cz CzCzCCCCz SulfateandProtonconcentrationsinthenonbinary(supported)elec
42、trolyte:Forthesulfateion:4444444444444444444444440000EBSOSOSOSOSOSOSOSOEBSOSOSOSOSOSOSOEBSOSOSOcathodicSOSOEBSOSOSOSOCCNDUzFCCCiNDtnFiCCtinFDiCCtnFD 318SincetSO4isnotequaltothatoftCuinthewellsupportedelectrolyte,theconcentrationprofileofthesulfatewillnolongerfollowthatofcopper.Electroneutralityismai
43、ntainedthroughtheprotonconcentration.Byanalogousderivationwefindthattheprotonconcentrationincreasesnearthecathode:44EBHHHHHHHHHEBSOSOHCuHHHHCCiNDU z FCFCCiiNDtFnF 0EBHHCuHHHHHHEBBCuHCuHHHHHHHHcathodicHHiiCCtFDnFDiiCCitCitiFDnFDnHowLongdoesittaketoestablishtheConcentrationGradients?Weshalllaterseetha
44、tthetimeconstantforthetransientprocessisgivenby25210/NtDDcmsNcm102103104ts100.11033107319Fluxesacrossthecell.Assumeabinaryelectrolyte,CuSO4:Cu+N NNMig0.5NN N NNDif0.5NNTotal=i/nFNDif=0Thereisafluxdeficitinbulk(Discontinuity)!N NNMig0.5NN N NNDif0.5NNTotal=i/nFNConv0.5NConvectioncarriesthereactantint
45、hebulk320Reviewquestions:Whydowehaveconcentrationgradientsattheelectrodes?Howsignificantaretheconcentrationgradientsandwhatdotheydependon?Whichtermismoreimportant:FzjDjCjor?Cantherebeafunctioningelectrochemicalcell(i0)withnoconcentrationgradients?Whenaretheconcentrationgradientsestablished?Howlongdo
46、esittakethemtobeestablishedortodecay?Howfardotheconcentrationgradientsextendintothesolution?Aretheconcentrationgradientsharmful,ordotheyhelpus?Cantheybeminimized?How?Canwedrawthedifferentfluxesacrossthecell?Whichspeciesactuallymovesinsidethecell?Howdoweplatefromanions?(platingofgoldfromKAu(CN)2)321E
47、xampleProblems:1.Twoparallelplatecopperelectrodesareimmersedin0.1MCuSO4electrolytesupportedby1MH2SO4.Eachelectrodeis4cm2inareaandtheinterelectrodegapis0.5cm.Convectionprovidesauniformboundarylayerthicknessof0.1mm.a.Assumingthattheonlysignificantpotentialdropisohmic(neglectingconcentrationvariationsa
48、ndkineticlimitations),estimatethevoltagerequiredtopassacurrentof40mA.(AssumecompletedissociationofH2SO4toH+andSO4=althoughthisisnotlikelytobetrue).b.Sketch,schematically,theconcentrationprofilesofallspeciespresent:Cu+,H+,SO4=.ForsimplicityassumeagaincompletedissociationofH2SO4c.Determinequantitative
49、lytheconcentrationofallspeciesattheelectrodes.d.Repeatpart(c)ifthecurrentdensityisraisedto200mA.e.Whatisthelimitingcurrentforthesystem?f.CanyouestimatetheconductivityofthesolutionandtheIRdrop?g.Repeatparts(e)and(f)iftheacidwascompletelyremoved.2.Goldisplatedoutofpotassiumgoldcyanidecomplexsolution(K
50、Au(CN)2)ofabout1troyouncepergallon,supportedby15g/Lmonosodiumphosphateand20g/Ldipotassiumphosphate.ThissolutionyieldsapHofaboutneutral(6.57.5).a.Consideringthatgoldistieddowntoanegativeionwhichdoesnotdissociatetoanyappreciableextent,howdoesthegoldgettothecathodeandplateout?b.Estimatethetransportnumb