ON HE UREN DENSY ND OVERENSON SGNS. HE SE OF HE NERFE BEWEEN WO MMSBLE ELEROLYE SOLUONS N OPEN RU ONDON. Mhalcuc and S. Lupu abstract: For a spontaneous electrode reacton the entropy producton anhe current densty across the electrodc nterface anhe overtenson under whch the electrode reacton occurs are ntmately lned. From ths relatonshp, consderng only the sgn conventon for the anodc and cathodc overtenson, the sgns of the anodc and cathodc current denstes, n a galvanc cell and n an electrolyss cell, could be deduced. No other conventon, except the completely accepted overtenson conventon anhe physcal conventon for current, s made. he charge transfer through the ES could be treated by electrodc concepts as an electrode reacton (at sngle- or mult-electrode). ntroducton n two prevous papers [1,2] we have treatehe case of a charge transfer occurrng at a metal/electrolyte nterface whch behaves ether as a sngle electrode [1] or as a multelectrode [2] when the charge transport reacton taes place spontaneously. n ths paper we are dealng wth the nterface between wo mmscble Electrolyte Solutons (ES) case n whch two ons of opposte charge are transferred spontaneously across the ES. Even the onc charge transfer across the ES may be treated as an electrode reacton [3]: obvously, the electron transfer across the ES resembles more wth the usual electrode reacton tang place to the M/S electrodc nterface.hese onc charge transfers are modelled as a d-electrode reacton formng the so-called d-electrode of transfer [4,5]. mong dfferent electrodc nterface the ES s of maor mportance especally n understandng the bologcal membrane electrochemstry. t an ES one could meet dfferent cases of onc charge transfer: sngle-electrode of transfer for a sngle onc speces [4,5], d-electrode of transfer [4,5] for two onc speces (of the same or opposte charge) or mult-electrode for at least three onc charges transferred across the ES. Depart. of Physcal hemstry, Faculty of hemstry, Unversty of Bucharest, Blvd. Elsabeta 4-12, RO- 70346, Bucharest, Romana Depart. of nalytcal hemstry, Faculty of ndustral hemstry, Poltehnca Unversty of Bucharest, alea Grvte 132, RO-78122, Bucharest, Romana nalele Unverst dn Bucuret hme, nul X (sere nou), vol. -, pag. 227 236 opyrght nalele Unverst dn Bucuret
228. MHLU S. LUPU heoretcal One consders a two-phase system formed by two oncally conductng phases, denoted here by S 1 (n the left-sde) and S 2 (n the rght-sde) wth an nterphase regon developed between (havng dfferent specal propertes from the two phases n drect contact but determned by these two formng phases). hs nterphase regon s the so-called electrodc nterface at whch an electrode reacton taes place [6]. he electrode reacton means the transfer of some electrcally charged speces (usually catons and anons for an ES) between the phase S 1, whose nner (Galvan) electrcal potental s, anhe phase S 2, whose nner (Galvan) electrcal potental s S 2 S 1. ang nto account that ths onc charge transfer, between the two phases of well-defned electrcal potental, s usually accompaned by a change n the chemcal composton of at least one of the two formng phases, then at the electrodc nterface an electrode reacton taes place. Obvously the charge transfer tself wth no precedng and/or followng chemcal reacton could be seen as an electrode reacton [4,5,7-10]. Between the two phases of the system, through ths specal nd of electrodc nterface, a flow of onc speces, postvely and negatvely charged, occurs; obvously, ths charge flow s accompaned by a mass transfer too (the ons havng both mass and charge). But the system as a whole s closed and, therefore, the spontaneous exchange of charge and matter nsde the system, s the cause of the entropy producton. n ts progress from the ntal state to the fnal state of equlbrum, the system moves by a steady-state route because of the condton of open crcut (or a ero-polaraton control). Let us assume that the electrode reacton occurrng at the ES s an onc charge transfer of a caton and of an anon whch cross the nterphase regon from the more concentrated electrolyte soluton (let be S 1 ) to the more dluted electrolyte soluton (let be S 2 ). Obvously, the passage of the caton from the left to the rght drecton (forward drecton) anhe passage of the anon from the rght to the left drecton (bacward drecton) are electrcally equvalent because that means a transport of a postve quantty of electrcty from the left to the rght. On the contrary, the passage of the caton from the rght to the left drecton (bacward drecton) anhe passage of the anon from the left to the rght drecton (forward drecton) are also electrcally equvalent because the former means a transport of a negatve quantty of electrcty from the left to the rght anhe latter a transport of a negatve quantty of electrcty from rght the to the left. n fact, for ths chosen nterface both the caton anhe anon pass the nterface from the left to the rght, each of two carryng, at the begnnng, the same quantty of electrcty because of the open crcut condton. Durng the steady-state progress of mass and charge transfer, the passage of the two ons could occur n both drectons because the bacwarransfer becomes more and more ntense (the forwarransfer becomng less and less ntense (untl the two processes, forward and bacward, balance each other)). n order to reach the equlbrum state the caton anhe anon wll pass across the ES n the forward drecton under open crcut condton ( 0 or ; ) each ndvdual onc charge transfer beng out of ts own equlbrum state (for the E E, rev value and for for, rev the E E, rev value). For ths reason, the passage of the caton n, rev the forward drecton beng an oxdaton occurs under an anodc overtenson assstance and
ON HE URREN DENSY ND OVERENSON SGNS 229 the passage of the anon n the forward drecton beng a reducton occurs under a cathodc overtenson assstance. So the equvalent electrode reacton mght be wrtten n the followng form: (1) where the stochometrc coeffcents for the reactants (1 ) and are negatve( 0 ) and for the products and are postve( 0 ). Of course, the prncple of charge conservaton s respected beng expressed by the equaton (here 1): 0 (2) By ntroducng the degree of advancement of the electrode reacton d [11,12], one can wrte the followng relatonshps for the components n the electrode reacton: d d n d n (3) d d n d n (3 ) descrbng the connecton wth the stochometrc coeffcent anhe mole number of each consumed (for and ) or produced ( and ) speces. s t s nown, the electrode reacton rate s gven by the rato between the ncrement of the degree of advancement, d, anhe nterval of the tme, dt, n whch ths ncrement occurs: r d (4) dt Expressng the rates wth respect to and, on one hand, and and, on the other hand, one can wrte: d n d n d n d n r, r, r, r (5) or, f one taes account to the equaton (4), one obtan: r d d d d,,, d r d r t t r (6) obvously, the followng relatonshp exsts between the ndvdual rates: r r r (7) r r r (7 )
230. MHLU S. LUPU n the electrode netcs the rate of the electrode reacton s expressed as current densty: Fr (8) therefore, nsertng, n turn, each ndvdual equaton from the equaton (6) nto the equaton (8) one gets: d F ; d F ; and consequently, the elementary charge beng: d F (8 ) d F (8 ) d q (9) the elementary charges assocated wth each current densty are gven by: d q d q F d ; d q F d (9 ) F d ; d q F d (9 ) s mentoned above, the electrodc nterface endowed wth an electrode reacton could be seen as a two-phase system and a charge transfer between the two phases whch determnes chemcal composton changes at least n one of the two phases. From a thermodynamc vewpont, one can assume that the nternal energy equaton (the so-called Gbbs equaton) extended wth the electrcal term: du d S p dv d n d q (10) s stll vald, where counts both for the reactants and and for the products and. herefore, by splttng the sum sgns, one can rewrte the equaton (10) n the followng form: du d S p dv d n d n where counts for the reactants d q d q S S and n whch the nner (Galvan) electrcal potental s (11) (they belong to the S electrolyte soluton ), and for the products and (they belong to the S electrolyte soluton n whch the nner (Galvan) electrcal potental s ). nsertng the equatons (3) and (9, 9, 9 and 9 ) nto the equaton S (11) one gets a d dependence of nternal energy: S
ON HE URREN DENSY ND OVERENSON SGNS 231 du d S p d V ( d d ) ( F d F d ) S S Rewrtng the equaton (12), by gatherng all the terms referrng to the same speces, one gets: du d S p d V ( F )d ( F )d S S ang nto account the meanng of the electrochemcal potental [13],, whch conssts of the chemcal potental,, contrbuton anhe electrcal wor, F, contrbuton F ) the equaton (13) can be wrtten n a shorter form: ( or n another form: (12) (13) du d S p dv d d (14) du d S p d V ( d d ) (15) from whch t s possble to obtan the shortest form that follows: du d S p dv d (16) n electrochemcal affnty term :,, G G (17) he electrochemcal affnty s defned as follows (.e., n the same manner as the chemcal affnty s defned): G G ; p, G p, G (as concerns the others quanttes thers meanngs are the usual ones: G ). (18) G, One nows the relatonshp between electrochemcal Gbbs energy and chemcal Gbbs energy:
232. MHLU S. LUPU G G nfe (19) where E s the electrode potental of the electrode endowed wth the electrode reacton (1):, rev ( S S ) 0 E (20) S S E when there s only transferable (20 ) E when there s only transferable (20 ), rev ( S S ) 0 Usng the correspondng meanngs n affnty terms of Gbbs energes, gven n the equatons (18), the equaton (19) can be wrtten n the form that governs the progress of the electrode reacton: nfe (21) (here n 1 because t was assumehat only 1 F of postve electrcty was transported across the electrodc nterface whch s the ES from the left to the rght, f not every t must be multpled by n). n the electrochemcal equlbrum state ( 0, E Erev ) the electrochemcal affnty s ero: 0, 0, 0 nfe,rev (22) where, 0, 0 nfe,rev 0 (22 ) E rev s the reversble (equlbrum) electrode potental: E rev ( S S ) 0 f the electrode reacton s not at the equlbrum state ( 0, electrochemcal affnty s postve: 0, 0, 0 nfe, rev E Erev (20 ) ) the (22 ) (22 ) 0, 0, 0 nfe, rev ombnng the equatons (22) and (22 ) anang nto account that: (23), 0, 0 (23 ), 0, 0 (because they depend on the same chemcal potentals) one gets: nf( E E ) (24 ), 0, rev
ON HE URREN DENSY ND OVERENSON SGNS 233 anhen by usng the overtenson noton: nf( E E ) (24 ), 0, rev, 0, 0 nf nf (25 ) (25 ) whch, for our dscusson, shows that the reducton (cathodc) electrode reacton () s spontaneously havng negatve overtenson anherefore postve electrochemcal affnty. From equaton (16) one gets the total entropy change: 1 p d S du dv d (26) as a sum [14] of the entropy change d e S wth the exteror anhe entropy change d S due to the spontaneous electrode reacton occurrng at the electrode nterface: d S d S d S (27) e where the electrochemcal affnty, n open crcut condton, could be wrtten n the followng manner:, 0 (28), 0 ang nto account the expresson for d e S : 1 p de S du dv (29) one gets the expresson for d S as: d S d (30) d S d S d S d d (30 ) showng that the electrochemcal affnty of the spontaneous electrode reacton s responsble for the entropy producton n the two-phase system. he rate of entropy change/ncrease s gven by the equaton:
234. MHLU S. LUPU d S d S d S d d d (31) where plays the role of a thermodynamc force closely relateo the electrode reacton rate. he electrochemcal affnty anhe electrode reacton rate must have the same sgn. ombnng the equaton (30) wth the equaton (25) one obtan frstly: d S d S d S nf( d d ) dt nf d dt because one can wrte (consderng the physcal conventon on the current, the drecton of s the same as that of the movng postve charge): (31) anhen: nf d (32) e dt nf d (32 ) e d S 0 (33) 0; 0 0 for 0; 0 0 for an equaton that offers the crteron for the current densty sgn for an electrode reacton occurrng spontaneously at the ES when the two-phase system goes from the ntal sate to the equlbrum fnal state: the current densty must be taen wth the same sgn as the overtenson n an spontaneous electrode reacton. herefore, n such a system where the entropy producton s postve, f 0 (.e., an anodc overtenson) then 0 (.e., the (33)
ON HE URREN DENSY ND OVERENSON SGNS 235 partal anodc current densty s also postve). onsequently, f 0 (.e., a cathodc overtenson) then 0 (.e., the partal cathodc current densty s also negatve). On the contrary, n a drven two-phase system (an electrolyss cell) where the entropy producton s not postve, f 0 (.e., an anodc overtenson) then 0 (.e., the partal anodc current densty s negatve). onsequently, f 0 (.e., a cathodc overtenson) then 0 (.e., the partal cathodc current densty s postve). For a multelectrode (endowed wth n dfferent electrode reactons tang place smultaneously at the ES) a general equaton, rememberng the multelectrode case from the metal/electrolyte nterface [2,15,16], could be obtaned: d S n 1 descrbng the spontaneous entropy producton as a sum of entropy changes of each electrode reacton component of the multelectrode. 0 (34) onclusons he equaton (34) governs the progress of an electrode reacton occurrng spontaneously at an electrodc nterface. hs s the case of an electrode reacton tang place spontaneously nto a two-phase system contanng an ES (a two-phase system whch s not ntally at equlbrum). Durng ts occurrence, ths electrode reacton has a postve entropy producton. For the case when only two ons of dfferent sgns pass across the ES ether n closed or open crcut condton the electrode reacton s, n fact, a d-electrode reacton. hs charge transfer of two ons was called d-electrode of transfer. herefore, n closed crcut condton, for an anodc occurrence of the d-electrode reacton, a postve overtenson generates an anodc current of the caton electrode reacton (whch s postve) and a negatve overtenson generates a cathodc current of the anon electrode reacton (whch s negatve). For a cathodc occurrence of the d-electrode reacton a negatve overtenson generates a cathodc current of the caton electrode reacton (whch s negatve) and a postve overtenson generates an anodc current of the anon electrode reacton (whch s postve). Even n open crcut condton, f, for example, the concentraton of the electrolyte s greater n the left-sde phase, namely S 1, than n the rght-sde phase, namely S 2, the ndvdual electrode reacton for the caton occurs under anodc overtenson whle the ndvdual electrode reacton for the anon occurs under cathodc overtenson, the overall current beng ero (the charge transfer occurng from the left to the rght). For an electrode reacton that does not tae place spontaneously at an electrodc nterface, the entropy producton s not postve. o tae place, the electrode reacton needs to be helped by an external energy contrbuton. hs s the case of an electrode reacton occurrng at an electrodc nterface belongng to a two-phase system n whch a charge transfer has to be drven. For these nonspontaneous electrode reactons the product of overtenson by current densty s always negatve. s a consequence, for an anodc
236. MHLU S. LUPU occurrence, the postve overtenson generates an anodc current that s negatve, whle for a cathodc occurrence the negatve overtenson generates a cathodc current, whch s postve. herefore, the usual classfcaton of electrode reactons as reversble (fast) and rreversble (slow), whose crteron s the charge transfer rate, has nothng to do wth the terms of reversble and rreversble processes consdered by thermodynamcs. Such fast and/or slow electrode reactons can occur at the electrode nterface nto a self-drven cell as spontaneous electrode reacton (havng a postve entropy producton) as well as nto an externally drven cell as nonspontaneous electrode reacton (for whch the entropy producton s not postve). oncludng, the anodc current s postve for a spontaneous progress of a two-phase system and negatve for a nonspontaneous progress of a two-phase system and, conversely, the cathodc current s negatve for the former system and postve for the latter system. REFERENES 1. Mhalcuc,., Lupu, S. nalele Unverst Bucuret, X (sera nou), vol., 2002, 127-134. 2. Mhalcuc,., Lupu, S. nalele Unverst Bucuret, X (sera nou), vol. -, 2003. 3. Mhalcuc,., Boncocat, N., Borda, M., Spataru,. (1998) Rev. Roum. hm., 43, (11), 1003-1009. 4. Mhalcuc,. (1998) Rev. Roum. hm., 43, (12), 1113-1120; (1999) Rev. Roum. hm., 44, (1), 19-23; (1999) Rev. Roum. hm., 44 (7), 635-642; (1999) Rev. Roum. hm., 44 (8), 781-787. 5. Mhalcuc,. (1998) Rev. hm. (Bucuret), 49, (9), 632-637, (1998); Rev. hm. (Bucuret), 49, (10), 708-712, 1998; (2000) Rev. hm. (Bucuret), 51, (5), 359-364, (2000); (2002) Rev. hm. (Bucuret), 5, (), -, 2002, n press 6. Boncocat, N. (1996) Electrochme aplca, Daca Europa-Nova, moara. 7. Gavach,. (1973) J. hm. Phys, 70(10), 1478. 8. Melroy, O. R., Buc, R. P.(1982) J. Electroanal. hem., 136, 19. 9. Koryta, J. (1979) Electrochmca cta, 24, 293-300; (1984) Electrochmca cta, 29 (4),445-452; (1988) Electrochmca cta, 33 (2), 189-197. 10. Kauch,. (1995) Electrochmca cta, 40 (18), 2999-3003. 11. de Donder, h. (1928, 1931, 1934) L affnté, Gauther-Vllars, Pars, de Donder, h. (1936) L affnté, Gauther-Vllars, Pars. 12. de Donder, h., van Rysselberghe, P. (1936) ffnty, Stanford Unversty Press, Stanford. 13. Guggenhem, E.. (1949) hermodynamcs, North Holland Publshng ompany. 14. Prgogne,. (1967) ntroducton to themodynamcs of rreversble processes, nterscence Publshers, New Yor London Sydney a dvson of John Wley & Sons. 15. Zaharov, E. M. (1964) Zhurn. F.Khm., 38, 2950. 16. Gudell, R. (1970) rans. Faraday Soc., 66, 1185-1193, 1194-202.