CHAPTER 7 CHRONOPOTENTIOMETRY. In this technique the current flowing in the cell is instantaneously stepped from

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1 CHAPTE 7 CHONOPOTENTIOMETY In his echnique he curren flwing in he cell is insananeusly sepped frm zer sme finie value. The sluin is n sirred and a large ecess f suppring elecrlye is presen in he sluin; diffusin is he nly mass ransfer prcess be cnsidered. Elecrlysis a cnsan curren is cnduced wih he apparaus schemaically presened in Figure (). where P is a pwer supply whse upu curren remains cnsan regardless f he prcesses ccurring in he cell. The penial f he wrking elecrde E agains he reference elecrde E is recrded by means f insrumen V. Fr a simple reacin as described by Equain (), a chrnpenigram will ypically lk like he pl in Figure (). O + ne = () As he elecrlysis prceeds, here is a prgressive deplein f he elecrlyzed species a he surface f he wrking elecrde. As he curren pulse is applied here is an iniial sharp decrease in he penial as he duble layer capaciance is charged, unil a penial a which O is reduced is reached. There is hen a slw decrease in he penial deermined by he Nerns Equain, unil he surface cncenrain f O reaches essenially zer. The flu f O he surface is hen n lnger sufficien mainained he

2 applied curren, and he elecrde penial again falls mre sharply, unil a furher elecrde prcess ccurs. 7. Iniial and Bundary Cndiins Unless herwise saed, he fllwing cndiins are assumed be achieved: () he sluin is n sirred; () a large ecess f suppring elecrlye is presen in sluin, and he effec f migrain can be negleced; (3) cndiins f semi-infinie linear diffusin are achieved. Subsance O is reduced a a plane elecrde and he prduc f elecrlysis is sluble in sluin. Since he curren densiy is mainained cnsan during elecrlysis, he fllwing equain (, ) C i = nfd () can be wrien frm he definiin f he flu. Equain (), which is he firs bundary cndiin, can be wrien in he fllwing frm = 0

3 ( ) = λ =0, C (3) wih nfd i = λ (4) The secnd bundary cndiin is bained by epressing ha he sum f he flues fr subsance O and a he elecrde surface is equal zer. Thus ( ) ( ) 0,, 0 0 = + = = C D C D (5) The iniial cndiins can be seleced a priry, and generally ne can assume ha he cncenrain f subsance is equal zer befre elecrlysis and ha he cncenrain f subsance O is cnsan. Thus: () C (, 0) = 0, and C (, 0) = C. () The funcins C (, ) and C (, ) are bunded fr large values f. Thus, C (, ) C and C (, ) 0 fr Variain f The Cncenrains C (, ) and C (, ). The sluin f he abve bundary value prblem was repred by Karaglanf, wh calculaed he cncenrains f bh species. The cncenrains are ( ) + = 4 ep, D erfc D D C C λ π λ (6) ( ) = 4 ep, D erfc D D D D D C λ π λ (7) where he nains "erf" and "erfc" represen he errr inegral defined by he frmula: 3

4 λ erfλ = ep( z )dz π 0 (8) The funcin erf{ λ } is defined under he frm f a finie inegral, having zer as lwer limi and as upper limi f inegrain. Therefre, values f erf{ λ } are deermined nly by he variable λ, "z" being simply an auiliary variable. The variains f erf { λ } wih λ are shwn in Figure (3) fr values f λ cmprised beween 0 and. The errr funcin is zer when is argumen is equal zer, and he funcin appraches uniy when λ becmes sufficienly large. "erfc" is defined by he equain: erfc ( λ) erf ( λ) = (9) An eample values f C (, ) are pled agains in Figure (4) fr varius imes f elecrlysis and fr he fllwing daa: i =0 - A cm -, n=, D = 0-5 cm s -, C = 50-5 M /cm 3. All he curves f Figure (4) have he same slpe a =0, because he flu and cnsequenly he derivaive ( ), is cnsan a =0. C 4

5 7. Penial - Time Curves The penial is calculaed frm he Nerns equain, he cncenrains C (0, ) and C (0, ) being wrien frm (6) and (7). Thus T fd T C P E = E + ln + ln (0) nf f D nf P i P = π () nfd The sum f he firs w erms n he righ-hand f equain (0) is precisely he penial E / defined by equain: T i i E = E + ln () nf i d T f D ln E = E + (3) nf f D When a mercury elecrde is used, he penial E / is he plargraphic halfwave penial. Hence, equain (0) can be wrien as fllws 5

6 T C P E = E + ln (4) nf P The penial calculaed frm equain (4) is infinie when he numerar in he lgarihmic erm is equal zer, i.e. when he ime has he value τ defined by he fllwing relainship τ = C P (5) Acually he penial a ime τ increases ward mre cahdic values unil a new reacin ccurs a he elecrde: such a prcess can be he reducin f waer r he suppring elecrlye. By inrducing in Equain (4) he value f C epressed in erms f he ime τ defined by equain (5), ne bains he fllwing penial-ime relainship E = E T nf + ln τ (6) The abve equain has he same frm as he equain f a reversible plargraphic wave, he diffusin curren and he curren being replaced by τ and by /, respecively. Thus, he prperies f penial-ime curves can be deduced by simple ranspsiin f he hery f reversible plargraphic waves. The penial E / crrespnds a value f equal τ 4 as can be seen frm Figure (5). Equain (6) als shws ha a pl f he decimal lgarihm f he quaniy [( ) ] penial shuld yield a sraigh line whse reciprcal slpe is.3 τ versus T/nF. Lgarihmic pls are linear as prediced by equain (6), and he penials E / as shwn in Figure (6) are in gd agreemen wih he plargraphic half-wave penials. 6

7 Buler and Armsrng cined he erm "ransiin ime" designae he ime defined by equain (5). Accrding equains () and (5) he ransiin ime is nfc D τ = (7) i π The square r f he ransiin ime is prprinal he bulk cncenrain f subsance reacing a he elecrde and inversely prprinal he curren densiy i. Thus, ransiin imes can be grealy changed by variain f he curren densiy. The limis beween which he ransiin ime can be adjused are deermined by he eperimenal cndiins: () cnvecin shuld n inerfere wih diffusin. () he fracin f curren crrespnding he charging f he duble layer shuld remain negligible in cmparisn wih he al curren hrugh he cell. In pracice he ransiin ime shuld n eceed a few minues. Because f he charging f he 7

8 duble layer, ransiin imes shrer han ne millisecnd cann be measured wih reasnable precisin. Accrding equain (7), [called Sand equain (-3)], he prduc i τ shuld be independen f he curren densiy. Penial-Time Curves fr Tally Irreversible Prcesses: The rae f ally irreversible prcess is crrelaed he curren densiy by he equain: i nf = αn ( ) α FE k f, hc 0, ep (8) T The cmbinain f (6) and (8) yields i nf ( αnfe C P ) ep = k f, h (9) T 8

9 The ransiin ime τ is deermined by he cndiin C (0, ) = 0 as fr reversible prcesses. Hence, C = Pτ, and he crrespnding value f τ can be inrduced in equain (9). The equain fr he penial-ime curve is hus T T π D E = ln( τ ) ln (0) αnf αnf k f, h r in view f equain () T T E = ln( τ ) ln () αnf αnf τ An eample f penial ime curve is shwn in Figure (7) fr he reducin f idie. The penial shuld rise a ime =0 accrding equain (). I is seen frm equain () ha he shape f he penial-ime curve depends n he prduc α n, and ha he ransiin ime is independen f he kineic f he elecrchemical reacin. 9

10 The penial a ime zer depends n he parameers α n, k f, h and n he curren densiy. Penial-ime curves fr he ally irreversible prcesses may hus be shifed a cerain een in he penial scale by variain f he curren densiy. A pl f decimal lgarihm f {-(/τ ) / } versus penial is a sraigh line [Figure (8)] whse reciprcal value is.3t/α nf. Thus, α n is readily calculaed. The rae cnsan k is calculaed frm he penial a ime zer by applicain f equain (). f,h 7.3 Tw Cnsecuive Elecrchemical eacins Invlving Differen Subsances When w subsances O and O are reduced a differen penials, he penialime curve ehibis w disinc seps. The firs ransiin ime τ crrespnding he reducin f subsance O can be calculaed frm he reamen eplained abve. The secnd sep cann be deermined frm his simple reamen. As he elecrlysis 0

11 prceeds afer he ransiin ime τ, he penial f he plarizable elecrde adjuss iself a value a which subsance O is reduced. Subsance O cninues diffuse ward he elecrde a which i is immediaely reduced. As a resul he cnsan curren hrugh he cell is he sum f w cnribuins crrespnding he reducin f subsances O and O, respecively. The ransiin ime τ fr he secnd sep in he penial-ime curve is reached when he cncenrain f subsance O becmes equal zer a he elecrde surface. Iniial and bundary cndiins have be described derive he ransiin ime τ. In he wriing f hese cndiins i is cnvenien ake as rigin f he ransiin ime τ. The ime in he new scale will be represened by he symbl ', and he relainship beween and ' is ( τ ) {( ' = τ () The ransiin ime fr he secnd sep f he penial-ime curve τ is deermined by he cndiin (, τ ) 0 O 0 = C by he equain The quaniy n he righ-hand is prprinal π n FDO C τ + τ = (3) i ) } τ τ τ + (4) As a resul, he ransiin ime τ depends n he cncenrain f subsance O which is reduced a less cahdic penials. The rder f magniude f he increase in τ which resuls frm he cnribuin f O can be judged frm he paricular case in which

12 C C =, n = n, D O = D O ; (5) equain (3) yields fr such cndiins τ = 3τ. Sepwise Elecrde Prcesses-The Bundary Value Prblem: A subsance O is reduced in w seps invlving n and n elecrns, he reducin prduc being and. Penial-ime curves fr such prcesses ehibi w seps, when is reduced a markedly mre cahdic penials han subsance O. Afer he ransiin ime τ fr he firs sep subsance O cninues diffuse ward he elecrde a which i is direcly reduced subsance in a prcess invlving n + n elecrns. Furhermre, subsance prduced during he firs sep diffuses ward he elecrde a which is reduced in a prcess invlving n elecrns. The disribuin f subsance a he ransiin ime τ is given by equain (7) in which is made equal τ. The resuling epressin is he iniial cndiin fr he presen prblem. The bundary cndiin is bained by epressing ha he curren is he sum f w cnribuins crrespnding he reducin f subsances and O, respecively. Thus n D C (, ' ) ( n + n ) (, ' ) C D + = = 0 = 0 i F (6) Funcins C (, ) and C (, ) are bunded fr large values f. The fllwing equain fr he cncenrain f subsance a he elecrde surface is repred in he lieraure (4)

13 C i n + n ( 0, ' ) = τ ( τ + ) ' π nfd n (7) Transiin Time fr he Secnd Sep f he Penial-Time Curve: The ransiin ime τ is bained by equaing he righ-hand member in equain (7) zer. The resuling epressin can be wrien in he frm: C i n + n ( 0, ' ) = τ ( τ + ) ' π nfd n (8) which shws ha he relainship beween he ransiin imes τ and τ is remarkably simple. When n =n, he ransiin ime τ is equal 3τ. Eperimenal daa fr he reducin f ygen cnfirm he crrecness f he freding analysis. Penial-ime curves fr hese subsances are given in Figure (9). Oygen is reduced in w seps invlving w elecrns each, and cnsequenly τ = 3τ. Cahdic Prcess Fllwed by Andic Oidain: A subsance O is reduced and he direcin f he curren hrugh he elecrlyic cell is reversed a he ransiin ime τ crrespnding he reducin f O. Subsance is nw idized and a penial-ime curve is bserved fr his prcess. The cncenrain f subsance a he ransiin ime τ is epressed by equain (7) in which he ime is made equal τ. The resuling epressin is he iniial cndiin fr he presen bundary value prblem, since i is nw he cncenrain f subsance which is be calculaed. Iniial and bundary cndiins are he same as fr single elecrchemical reacin (see equain 4). 3

14 C (, ' ) = 0 = λ' i ' '= nfd λ (9) he inensiy i ' in he reidain prcess may n necessarily be adjused a he same value as in he reducin f O and. Hence he curren densiy i ' is inrduced in equain (9). The funcin C (, ' ) 0 fr The cncenrain f subsance during he re-idain is: C ( ' ) ( τ ) D + = θ π ep 4D θerfc, ( τ + ' ) D ( + ' ) D ' ( θ + λ' ) ep + ( θ + λ' ) erfc π 4D ' ' D (30) wih i = nfd θ (3) Variains f he cncenrain C (, ) wih disance frm he elecrde are shwn in Figure (0) fr he same daa as hse used in he cnsrucin f Figure (4) and 4

15 fr he fllwing numerical values: i ' = i = 0 A cm, D =D =0-5 cm s -. The C (, ') versus curves f elecrlysis larger han ' = 0 ehibi a maimum. The cncenrain f a a sufficien disance frm he elecrde becmes slighly larger han he crrespnding iniial cncenrain a ime '=0; his resuls frm diffusin f subsance ward a regin f he sluin in which he cncenrain f is lwer han a he maimum f he C (, ') vs curve. The cncenrain f subsance O during reidain when he diffusin cefficiens D and D are equal is C (, ' ) C C (, ' ) 7.4 Transiin Time fr he e-idain Prcess = (3) The ransiin ime is deermined by he cndiin C (0, τ ')=0. By wriing (3) fr =0 and slving fr he ransiin ime τ ' fr he re-idain prcess ne bains θ τ ' = (33) ( θ + λ' ) θ When θ = λ ', when he curren densiies i and simple frm i ' are equal, equain () akes very 5

16 τ '= 3τ (34) which shws ha he ransiin ime fr he re-idain prcess is equal ne hird f he ransiin ime fr he iniial cahdic prcess, he curren densiy being he same in bh prcesses. An eample f penial-ime curve is given in Figure (). 6

17 EXPEIMENTAL equired equipmen and supplies P wrking elecrde, A=0.5 cm. 0-3 M 3 [ ( CN ) ] Fe = 0-3 M. 6 [KCl] = M PA Mdel 35 Three cmparmen elecrchemical cell. Objecives: The bjecives f his eperimen are by using chrnpenimery and chrnpenimery wih curren reversal deermine: () he dependence f he ransiin ime n he bulk cncenrain f elecracive species reacing a he elecrde and () he dependence f he ransiin ime n applied curren densiy. The elecrchemical cell emplyed fr hese sudies shuld be cnveninal hreecmparmen design wih cnac beween he wrking elecrde cmparmen and he reference elecrde via a Luggin prbe. The chrnpenimeric eperimens shuld 3 [ ] be carried u using sandard calmel elecrde (SCE) in () 0-3 M ( CN ) Fe and M KCl. Daa Analysis: Values f () he ransiin imes as a funcin f he applied curren densiies fr 3 [ ] cnsan cncenrains f ( CN ) Fe and () fr ransiin imes as a funcin f 6 6 7

18 3 [ ] differen cncenrains f ( CN ) Fe a cnsan curren densiies and (3) penial vs. 6 lg / τ were bained by Ppv and Laiinen (unpublished resuls) and are given in Table (), Table () and Table (3). Table. Values f Transiin Time fr Differen Cahdic Currens Obained in 0-3 M Fe(CN) M KCl Cncenrain Curren Densiy Transiin Time elaive Sandard 0-3 M µ A/cm 3 Measured, (s) Deviain % % % % % % Table. Values f Transiin Time Obained fr Differen Cncenrains f Fe(CN) 6-3 a Cnsan Cahdic Curren Densiy f 50 A/cm. [Fe(CN) 6-3 ] Applied Curren Transiin Time elaive Sandard 0-3 M µ A/cm 3 Measured, (s) Deviain % % % % 8

19 Penial (mv) [SCE] τ Table 3. Penial vs lg τ τ lg τ, (s) Pl i τ vs i; iτ vs /i; i τ /C vs C; τ vs C Cmpue (a) he diffusin cefficien f he elecracive species in M KCl. (b) he number f elecrns invlved in he prcess (c) Discuss he lgarihmic pl fr reversible elecrde prcesses (d) cmpare he "n" value bained frm he slpe f he lgarihmic pl fr reversible elecrde prcesses and "n" value bained frm Sand Equain. 9

20 EFEENCES. B. N. Ppv and H. A. Laiinen, J. Elecrchem. Sc., 7, 4, 48, (970).. B. N. Ppv and H. A. Laiinen, J. Elecrchem. Sc., 0, 0, 346, (973). 3.. Cvekvic, B. N. Ppv and H. A. Laiinen, J. Elecrchem. Sc.,,, 66, (975). 4. Bris B. Damaskin, "The Principles f Curren Mehds fr he Sudy f Elecrchemical eacins, Edir, Gleb Maamnv, McGraw-Hill Bk C., New Yrk, (968) 0

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