Physical and Interfacial Electrochemistry Lecture 4. Electrochemical Thermodynamics. Thermodynamics of electrochemical systems

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1 Physcal and Intrfacal Elctrochmstry 213 Lctur 4. Elctrochmcal Thrmodynamcs Modul JS CH334 MolcularThrmodynamcs and Kntcs Thrmodynamcs of lctrochmcal systms Thrmodynamcs, th scnc of possblts s of gnral utlty. Th wll stablshd mthods of thrmodynamcs may b radly appld to lctrochmcal s. W can radly comput thrmodynamc stat functons such as G, H and S for a chmcal racton by dtrmnng how th opn crcut potntal E vars wth soluton tmpratur. W can comput th thrmodynamc ffcncy of a ful provdd that G and H for th racton can b valuatd. W can also us masurmnts of qulbrum potntals to dtrmn th concntraton of a rdox actv substanc prsnt at th lctrod/soluton ntrfac. Ths s th bass for potntomtrc chmcal snsng. M Ox, Rd Rd',Ox' M ' - Anod Oxdaton - loss LHS + Cathod Rducton - gan RHS 1

2 Standard Elctrod Potntals Standard rducton potntal (E ) s th voltag assocatd wth a rducton racton at an lctrod whn all soluts ar 1 M and all gass ar at 1 atm. Rducton Racton H + (1 M) 2H 2 (1 atm) E = V Standard hydrogn lctrod (SHE) Masurmnt of standard rdox potntal E for th rdox coupl A(aq)/B(aq). lctron flow E rfrnc lctrod SHE P t H 2 n H 2 (g) A(aq) Pt ndcator lctrod H + (aq) B(aq) salt brdg tst rdox coupl E provds a quanttatv masur for th thrmodynamc tndncy of a rdox coupl to undrgo rducton or oxdaton. 2

3 Standard lctrod potntal E s for th racton as wrttn Th mor postv E th gratr th tndncy for th substanc to b rducd Th half- ractons ar rvrsbl Th sgn of E changs whn th racton s rvrsd Changng th stochomtrc coffcnts of a half- racton dos not chang th valu of E 19.3 W should rcall from our CH111 lctrochmstry lcturs that any combnaton of two rdox coupls may b usd to fabrcat a galvanc. Ths faclty can thn b usd to obtan usful thrmodynamc nformaton about a racton whch would b othrws dffcult to obtan. Hrn ls th usfulnss of lctrochmcal thrmodynamcs. Gvn any two rdox coupls A/B and P/Q w can radly us tabls of standard rducton potntals to dtrmn whch of th two coupls s prfrntally rducd. Onc ths s known th galvanc can b constructd, th nt potntal can b valuatd, and knowng ths usful thrmodynamc nformaton can b obtand for th racton. Th procdur s smpl to apply. On dtrmns th coupl wth th most postv standard rducton potntal (or th most postv qulbrum potntal E dtrmnd va th Nrnst quaton f th concntratons of th ractants dffr from 1 mol dm -3 ). Ths coupl wll undrgo rducton at th cathod. Th othr rdox coupl wll consquntly undrgo oxdaton at th anod. Ths nformaton can also b usd to dtrmn th drcton of lctron flow, for upon placng a load on th lctrons wll flow out of th anod bcaus of th occurrnc of a spontanous d-lctronaton (othrws known as oxdaton or lctron loss) racton, through th xtrnal crcut and nto th cathod causng a spontanous lctronaton (aka rducton or lctron gan) racton to occur. Hnc n a drvn th anod wll b th ngatv pol of th and th cathod th postv pol. Now accordng to th IUPAC convnton f th racton s spontanous th rsultant potntal wll b postv. W nsur that such s th cas by wrtng th cathod racton on th rhs, and th anod racton on th lhs of th dagram. Thn snc E,rhs s mor postv than E, lhs a postv potntal V wll b guarantd 3

4 A P B Q E E RHS E LHS E cathod E anod lctron flow Anod Oxdaton M - P(aq) V A(aq) + M Cathod Rducton E,anod Q(aq) B(aq) E,cathod A n B Cathod PQm Anod E E( T ) G, H, S, salt brdg rdox coupl ma np mb nq Q aa m n B Q R m n aa A P E Rlaton btwn thrmodynamcs of racton and obsrvd potntal Whn a spontanous racton taks plac n a Galvanc, lctrons ar dpostd n on lctrod (th st of oxdaton or anod) and collctd from anothr (th st of rducton or cathod), and so thr s a nt flow of currnt whch can b usd to do lctrcal work W. From thrmodynamcs w not that maxmum lctrcal work don at constant tmpratur and prssur W s qual to th chang n Gbbs fr nrgy G for th nt racton. W us basc physcs to valuat th lctrcal work W don n movng n mol Elctrons through a potntal dffrnc of E. W q E E 1 lctron : W N E 1 mol lctrons : A n mol lctrons : G W n N E n F E A W n N A E G W 4

5 A n B E A, B P m Q E P Q ma mn mb np nm nq, W assum that E A,B s mor postv than E P,Q and so s assgnd as th cathod racton. W subtract th two ractons to obtan th followng rsult. ma np mb nq ma nq mb np To procd w subtract th corrspondng thrmodynamc stat functons In ths cas G,,,,,, G mg ng m nfe mfe nmf E E nmfe A B P Q A B P Q A B P Q Not that nm dnots th numbr of lctrons transfrrd pr mol of racton as wrttn. ma np mb nq A n B Cathod m n abaq Nt Cll Racton QR m n P Q m Anod a a A P Th stablshmnt of qulbrum dos not mply th cssaton of rdox actvty at th ntrfac. Th condton of qulbrum mpls an qualty n th lctrochmcal potntals of th transfrrng spcs n th two phass and n th stablshmnt of a compnsatng two way flow of charg across th ntrfac rsultng n a dfnt qulbrum potntal dffrnc or E. A sngl qulbrum potntal dffrnc may not b masurd. Instad a potntal s masurd btwn two lctrods (a tst or ndcator lctrod and a rfrnc lctrod). Ths s a potntomtrc masurmnt. Th potntal of th ndcator lctrod s rlatd to th actvts of on or mor of th componnts of th tst soluton and t thrfor dtrmns th ovrall qulbrum potntal E. Undr dal crcumstancs, th rspons of th ndcator lctrod to changs n analyt spcs actvty at th ndcator lctrod/soluton ntrfac should b rapd, rvrsbl and govrnd by th Nrnst quaton. Th ET racton nvolvng th analyt spcs should b kntcally facl and th rato of th analyt/product concntraton should dpnd on th ntrfacal potntal dffrnc va th Nrnst quaton. 5

6 Th potntomtrc masurmnt. Potntal radng Dvc (DVM) In a potntomtrc masurmnt two lctrods ar usd. Ths consst of th ndcator or snsng lctrod, and a rfrnc lctrod. A Elctroanalytcal masurmnts rlatng potntal to analyt B concntraton rly on th rspons of on lctrod only (th ndcator lctrod). Rfrnc Indcator Th othr lctrod, th rfrnc lctrod lctrod lctrod s ndpndnt of th Soluton contanng soluton composton and provds analyt spcs A stabl constant potntal. Th opn crcut potntal s masurd usng a potntal masurng dvc such as a potntomtr, a hgh mpdanc voltamtr or an lctromtr. Equlbrum condton btwn phass: chmcal potntal. It s wll known from basc chmcal thrmodynamcs that f two phass and wth a common unchargd spcs, ar brought togthr, thn th tndncy of spcs to pass from phas to phas wll b dtrmnd by th dffrnc n th chmcal Potntal btwn th two phass. Th condton for qulbrum s Th standard thrmodynamc dfnton of th chmcal potntal s: G n nk, P, T Altrnatvly w can vw th chmcal potntal of a spcs n a phas as a masur of th work that must b don for th rvrsbl transfr of on mol of unchargd spcs from th gasous stat of unt fugacty (th rfrnc stat) nto th bulk of phas. In lctrochmstry w dal wth chargd spcs and chargd phass. Phas ~ Phas 6

7 Elctrochmcal Actvty W consdr th work don W n transfrrng a spcs from th ntror of a standard phas to th ntror of th phas of ntrst. W also assum that th spcs has a charg q = z. Standard Phas W Dstnaton Phas Th lctrochmcal actvty can b dfnd n th followng mannr. q zf W a axp axp xp kt B RT kt B a a If two phass and contan a spcs wth dffrnt lctrochmcal actvts such that th lctrochmcal actvty of spcs n phas s gratr than that of phas thn thr s a tndncy for spcs to lav phas and ntr phas. Th drvng forc for th transport of spcs s th dffrnc n lctrochmcal actvty btwn th two phass. In th lattr xprsson a rprsnts th actvty of spcs. Now from th dfnton of lctrochmcal actvty zf aaxp RT zf aaxp RT Hnc a a zf xp a a RT a zf xp a RT W can follow th lad of Lws and ntroduc th dffrnc n lctrochmcal potntal as follows. W can mmdatly dduc a rlatonshp btwn th lctrochmcal potntal dffrnc and th rato of lctrochmcal actvts btwn two phass and va th followng rlatonshps. a a RT ln RT ln zf a a zf Hnc th dffrnc n lctrochmcal potntal s splt up nto two dstnct componnts. Frst, th dffrnc n chmcal potntal and scond th dffrnc n lctrcal potntal. Hnc w not that th lctrochmcal potntal s dfnd as th work rqurd to transfr 1 mol of chargd spcs from nfnty n vacuum nto a matral phas. Ths work conssts of thr sparat trms. Th frst consttuts a chmcal trm whch ncluds all short rang ntractons btwn spcs (such as an on) and ts nvronmnt (on/dpol ntracton, on/nducd dpol ntractons, dsprson forcs tc). Ths consttuts th chmcal potntal trm. Th scond consttuts an lctrostatc trm lnkd to th crossng of th layr of orntd ntrfacal dpols (z F). Th thrd consttuts an lctrostatc trm lnkd to th charg of th phas (z F). Th outr potntal y s th work don n brngng a tst charg from nfnty up to a pont outsd a phas whr th nflunc of short rang mag forcs can b nglctd. Th surfac potntal c dfns th work don to brng a tst charg across th surfac layr of orntd dpols at th ntrfac. Hnc th nnr Galvan potntal f s thn dfnd as th work don to brng th tst charg from nfnty to th nsd of th phas n quston and so w dfn: 7

8 Elctrod q M Elctrolyt Soluton q S zf zf RT ln a z F Vacuum Chargd Intrfac Dsmantl ntrfac. Rmov all xcss charg & orntd dpol layrs Vacuum = = Dpol layr across mtal No dpol layr Vacuum Unchargd mtal Charg lctrod Charg soluton Unchargd soluton No dpol layr Orntd dpol Layr on soluton Chargd lctrod wth Dpol layr q M q S Chargd soluton wth Dpol layr Elctrochmcal Potntal In lctrochmstry w dal wth chargd spcs and chargd phass, and on ntroducs th da of th lctrochmcal potntal whch s dfnd as th work xpndd n transfrrng on mol of chargd spcs from a gvn rfrnc stat at nfnty nto th bulk of an lctrcally chargd phas. G n nk, P, T ~ G Elctrochmcal potntal Elctrochmcal Gbbs nrgy It s somtms usful to sparat th lctrochmcal potntal nto chmcal and lctrcal componnts as follows. zf q Spcs valnc Chmcal potntal Galvan lctrcal potntal Spcs charg Equlbrum nvolvng chargd spcs transfr btwn two adacnt phass s attand whn no dffrnc xsts btwn th lctrochmcal potntals of that spcs n th two phass. 8

9 Rgorous Analyss of Elctrochmcal Equlbrum E F Enrgy Rfrnc Vacuum Lvl Flld lctronc nrgy lvls Bfor Contact E F Ox Rd Whn two phass com nto contact th lctrochmcal potntals of th spcs n ach phas quat. For qulbrum at mtal/soluton ntrfac th lctrochmcal potntal of th lctron n both phass quat. Mtal Rfrnc Vacuum Lvl Aftr Contact Va procss of Charg transfr Enrgy E F Mtal Flld lctronc nrgy lvls Ox Rd Th Nrnst Equaton W consdr th followng ET racton. Ox At qulbrum zox n W not th followng Also Rd n zox zrd zo zr zrd n O R F O OzOF zf R R R RT ln a RT ln a O O O R R R Hnc ln ln O RT ao n zof RT a z F R R R Smplfyng w gt ar RO ln R O a n RT z z F O ar RO ln a n RT nf O Hnc R O RT a R ln F n n a Also F At qulbrum R O RT a R ln n n ao Smplfyng w gt O F O R n RT ao ln nf nf a R RT a O ln nf a R ORn nf Ths s th Nrnst Equaton. 9

10 Rvw of Thrmodynamcs W rcall that th Gbbs nrgy G s usd To dtrmn whthr a chmcal racton Procds spontanously or not. W consdr th gas phas racton A(g) B (g). W lt dnot th xtnt of racton. Clarly < < 1. Whn = w hav pur A and whn = 1 w hav 1 mol A dstroyd and 1 mol B formd. Also dn A = -d and dn B = + d whr n dnots th quantty (mol) of matral usd up or formd. By dfnton th chang n Gbbs nrgy dg at constant T and P s rlatd to th chmcal potntal as follows: dg AdnA BdnB Ad Bd B Ad Furthrmor (1) In th lattr th symbol (1) r G = racton Gbbs fr nrgy Snc vars wth composton Thn so also dos r G. G dg PT, d (2) Hnc w gt from qn. 1 and 2 G PT, G r B A (3) 1

11 If A > B thn A B s spontanous and r Gs ngatv. If A > B thn B A s spontanous and r G s postv. If A = m B thn r G = and chmcal qulbrum has bn achvd. Snc A and B ar dal gass thn w wrt pa A A RT ln p pb B B RT ln p pb pa B A B A RT ln RT ln r p p p B B A RT ln pa G RTln Q r R Q R = racton quotnt K = Equlbrum constant QR G RTln K RTln QR RTln K G G RTln Q r r R At qulbrum Q R = K and r G= G RT r ln K Gbbs nrgy and chmcal qulbrum. G Racton not spontanous In forward drcton P R G Equlbrum Q=K Q larg, Q>K [P]>>[R] G postv Q small, Q<K [P]<<[R] G ngatv G G R P Racton spontanous In forward drcton Standard stat Q=1 lnq= ln Q G G RT ln Q 11

12 Th xprsson ust drvd for th spcal cas A B can also b drvd mor gnrally. If w st n as th stochomtrc coffcnt of spcs (ngatv for ractants and postv for products, w can drv th followng. G dg d d PT, G G r PT, G RT ln a r rg RT ln a rgrtln a G G RTln Q r r R W can rlat chmcal potntal to actvty a as follows. RTln a In th lattr w hav ntroducd th followng dfntons. Q a R ln a Agan at qulbrum G f Standard fr nrgy oof formaton of spcs. v ln a rg QR K rg RTln K K a G G r f Potntomtrc Masurmnts W now mnton th practcalts of conductng a potntomtrc masurmnt. A two lctrod lctrochmcal s usd. Ths conssts of a rfrnc lctrod and an ndcator lctrod. Th obct of thxrcsstomakamasurmnt of th qulbrum potntal wthout drawng any sgnfcant currnt snc w not that th qulbrum potntal s dfnd as E = ( --> ) whr Df dnots th Galvan potntal dffrnc masurd btwn th trmnals.ths obctv s achvd thr usng a null dtctng potntomtr or a hgh mpdanc voltmtr. 12

13 Th scond masurmnt protocol nvolvs us of an lctromtr. Th lattr s basd on a voltag followr crcut. A voltag followr mploys an opratonal amplfr. Th amplfr has two nput trmnals calld th summng pont S (or nvrtng nput) and th followr nput F (or non-nvrtng nput). Not that th postv or ngatv sgns at th nput trmnals do not rflct th nput voltag polarty but rathr th non nvrtng and nvrtng natur rspctvly of th nputs. Now th fundamntal proprty of th opratonal amplfr s that th output voltag V o s th nvrtd, amplfd voltag dffrnc V whr V = V - -V + dnots th voltag of th nvrtng nput wth rspct to th non nvrtng nput. Hnc w can wrt that V o AV AV AV In th lattr A dnots th opn loop gan of th amplfr. Although dally A should b nfnt t wll typcally b 1 5. V o AV AV o V V o V 1 1 V A Also dally th amplfr should xhbt an nfnt nput mpdanc so that thy can accpt nput voltags wthout drawng any currnt from th voltag sourc. Ths s why w can masur a voltag wthout any prturbaton. In practc th nput mpdanc wll b larg but fnt (typcally 1 6 ). An dal amplfr should also b abl to supply any dsrd currnt to a load. Th output mpdanc should b zro. In practc amplfrs can supply currnts n th ma rang although hghr currnt output can also b achvd. Kpng ths commnts n mnd w can now dscuss th opraton of th voltag followr crcut prsntd In ths confguraton th ntr output voltag s rturnd to th nvrtng nput.. If V rprsnts th nput voltag thn w s from th analyss outlnd across that th crcut s calld a voltag followr bcaus th output voltag s th sam as th nput voltag. Th crcut offrs a vry hgh nput mpdanc and a vry low output mpdanc and can thrfor b usd to masuravoltagwthoutprturbngth voltag sgnfcantly. Th Nrnst quaton. Th potntal dvlopd by a Galvanc dpnds on th composton of th. From thrmodynamcs th Gbbs nrgy chang for a chmcal racton G vars wth composton of th racton mxtur n a wll dfnd mannr. W us th rlatonshp btwn G and E to obtan th Nrnst quaton. rg nfe rg nfe nfe nfe RT ln Q RT ln nf E E Q R R T = 298K G G RTln Q r r R RT/F = 25.7 mv At T = 298 K.592 E E log Q R n Nrnst qn. holds for sngl rdox coupls and nt ractons. QR Racton quotnt products ractants 13

14 Zn( s) Cu E 2 ( aq) Zn E 2.59 log 2 ( aq) Cu( s) 2 Zn 2 Cu Equlbrum G E Q K W R Dad Q 1 W larg Q 1 E E As oprats 2 2 [ Zn ] [ Cu ] Q R E Q Zn Cu 2 2 Q 1 W small Dtrmnaton of thrmodynamc paramtrs from E vs tmpratur data. Masurmnt of th zro currnt potntal E as a functon of tmpratur T nabls thrmodynamc quantts such as th racton nthalpy H and racton ntropy S to b valuatd for a racton. Gbbs-Hlmholtz qn. G r rh rgt T P r G nfe E a b E T rh nfe T nfe T E nfe nft T P E rh nfet T P 2 T T ct P T a, b and c tc ar constants, whch can b postv or ngatv. T s a rfrnc tmpratur (298K) Tmpratur coffcnt of zro currnt potntal obtand from xprmntal E=E(T) data. Typcal valus l n rang VK -1 14

15 Onc H and G ar known thn S may b valuatd. rg rh TrS rh rg rs T 1 E rs nfenft nfe T T P E rs nf T Elctrochmcal masurmnts of potntal conductd undr condtons of zro currnt flow as a functon of tmpratur provd a sophstcatd mthod of dtrmnng usful thrmodynamc quantts. P Fundamntals of potntomtrc masurmnt : th Nrnst Equaton. Th potntal of th ndcator lctrod s rlatd to th actvts of on or mor of th componnts of lctron flow th tst soluton and t thrfor dtrmns th ovrall E qulbrum potntal E. Undr dal rfrnc crcumstancs, lctrod th rspons of SHE th ndcator lctrod to changs n analyt Pt spcs actvty at th ndcator lctrod/ soluton ntrfac should b rapd, rvrsbl and govrnd by th Nrnst quaton. H 2 n H 2 (g) H + (aq) A(aq) B(aq) salt brdg rdox coupl Th ET racton nvolvng th analyt spcs should b kntcally facl and th rato of th analyt/product concntraton should dpnd on th ntrfacal potntal dffrnc va th Nrnst quaton. Pt ndcator lctrod tst analyt 15

16 Hydrogn/oxygn ful Rmmbr CH111 Elctrochmstry: GnFE q, E E E q, q, C q, A Ballard PEM Ful Cll. 16

17 max max Effcncy of ful (I). GnFEq, Effcncy =work output/hat nput Eq, Eq, C Eq, A For lctrochmcal or cold combuston: Hat nput ntalphy chang H for racton Work output Gbbs nrgy chang G for racton rg rh TrS TrS 1 rh rh rh nfeq, H r Usually G H r 1 r H 2 (g) + ½ O 2 (g) H 2 O (l) G = kcal mol -1 ; H = kcal mol -1 n = 2, E q = V, =.83. Effcncy of ful (II). For all ral ful systms trmnal potntal E dos not qual th qulbrum valu E q, but wll b lss than t. Furthrmor E wll dcras n valu wth ncrasng currnt drawn from th ful. nfe nf E q ral H Ths occurs bcaus of: Slownss of on or mor ntrmdat stps of ractons occurrng at on or both lctrods. Slownss of mass transport procsss thr ractants to, or products from, th lctrods. Ohmc losss through th lctrolyt. ( ) H Sum of all ovrpotntal Losss. 17

18 Nrnst Equaton nvolvng both an lctron and proton transfr. W consdr th followng racton whch nvolvs both th transfr of m protons and n lctrons. Ths s a stuaton oftn found n bochmcal and organc ractons. A mh n B Th Nrnst quaton for ths typ of proton/lctron transfr qulbrum s gvn by th followng xprsson. E RT a B E ln m nf aaah Ths xprsson can b radly smplfd to th followng form E 2.33RT a B m RT E E ln 2.33 ph nf aa n F RTm SN 2.33 F n ph Ida: Potntomtrc masurmnts can gv rs to accurat ph masurmnts (spcally mtal oxd lctrods Lyons TEECE Group Rsarch 212/213). Hnc w prdct that a Nrnst plot of opn crcut or qulbrum Potntal vrsus soluton ph should b lnar Wth a Nrnst slop S N whos valu s drctly rlatd to th rdox stochomtry of th rdox racton through th m/n rato. Whn m = n thn w prdct that S N = - 2/33RT/F Whch s clos to 6 mv pr unt chang n ph at 298 K. Mmbran Potntal Snc w ar consdrng chargd spcs w dfn qulbrum In trms of qualty of lctrochmcal potntals. ( ) zf zf a a a M + M + a M + X - M + X - zf zf zf Now RT ln a RT ln a Hnc ln ln a RT ln a RT a zf RT a RT a ln zf zf a s th lctrc potntal ncssary To prvnt qualzaton of onc actvts by Dfuson across th mmbran. Nt - v Of cours a smlar rsult for th mmbran potntal can b obtand by qualzng th rato of lctrochmcal actvts and notng that th followng prtans. Mmbran prmabl only to M + on. Nt + v Now f w assum that W gt th fnal xprsson for th mmbran potntal RT a M ln zf a a zf a a xp 1 a RT RT a M ln zf a 18