Quadrupole terms in the Maxwell equations: Born energy, partial molar volume and entropy of ions. Debye-Hückel theory in a quadrupolarizable medium

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1 5 uaduole tems n the Maxwell equatons: Bon enegy, atal mola volume and entoy of ons. ebye-hückel theoy n a quaduolazable medum Radom I. Slavchov,* a Tzanko I. Ivanov b a eatment of Physcal Chemsty, Faculty of Chemsty and Phamacy, Sofa Unvesty, 1164 Sofa, Bulgaa b Theoetcal Physcs eatment, Faculty of Physcs, Sofa Unvesty, 1164 Sofa, Bulgaa * E-mal:fh@chem.un-sofa.bg 0 A new equaton of state elatng the macoscoc quaduole moment densty to the gadent of the feld E n an sotoc flud s deved: = (E U E/), whee the quaduolazablty s ootonal to the squaed molecula quaduole moment. Usng ths equaton of state, a genealzed exesson fo the Bon enegy of an on dssolved n quaduola solvent s obtaned. It tuns out that the otental and the enegy of a ont chage n a quaduola medum ae fnte. Fom the obtaned Bon enegy, the atal mola volume and the atal mola entoy of a dssolved on follow. Both ae comaed to exemental data fo a lage numbe of smle ons n aqueous solutons. Fom the comason the value of the quaduola length L s detemned, L = ( /) 1/ = 1- Å. Futhe, the extended ebye-hückel model s genealzed to ons n a quaduola solvent. If quaduole tems ae allowed n the macoscoc Coulomb law, they esult n suesson of the gadent of the electc feld. In esult, the electc double laye s slghtly exanded. The actvty coeffcents obtaned wthn ths model nvolve thee chaactestc lengths: ebye length, on adus and quaduola length L. Comason to exemental data shows that mnmal dstance between ons s equal to the sum of the bae on ad; the concet fo on hydaton as an obstacle fo ons to come nto contact s not needed fo the undestandng of the exemental data Intoducton The macoscoc Posson equaton of electostatcs combnes the statc macoscoc Coulomb and Amee laws, = (); (1) E =, () wth a lnea deendence of the electc dslacement feld on 0 the electc feld ntensty E 0 E+ P = 0 E+ P E = E. () Hee s the fee chage numbe densty; s the electostatc otental; 0 + P = 0 s the absolute delectc emttvty, 0 s the vacuum emttvty, s the elatve emttvty of the medum, P s the macoscoc olazablty of the medum. Fo a homogeneous medum ( = 0) the Posson equaton fo follows fom Eqs (1)-(): ( ). (4) Fo conductng meda, one must ovde also an equaton of state fo the deendence (). A common assumton s that the chages ae dstbuted accodng to the Boltzmann dstbuton ( ) ec ex( e / T ), (5) whee e = ez s the absolute chage of the th on, e s the electon chage, Z s the elatve onc chage, C = C el s the local concentaton of the th on, stands fo ts stochometc numbe, C el s the electolyte concentaton, T[J] = k B T[K] s the themodynamc temeatue. Insetng Eq (5) nto Eq (4), one obtans the Posson-Boltzmann equaton, wdely used n hyscal chemsty and collod scence. Numeous basc concets such as ebye-hückel double laye1,, Gouy model fo chaged nteface, 4, aves adsoton model fo onc sufactant adsoton5, 6, the electostatc dsjonng essue n LVO theoy7-9, electoknetc -otental, etc., ae meely a consequence of Eqs (4)-(5). It has been ealy ecognzed that both Posson and Boltzmann equatons (4)-(5) have sevee lmtatons. The devaton of Eq (4) nvolves a multole exanson of the local otental u to dole tems,.e., t neglects the quaduole moment densty Eq () s stctly vald fo lnea meda 14. The Boltzmann dstbuton (5) s only a fst aoxmaton vald fo deal soluton -17 ; othe extenal otentals excet e often ase In ode to make Eqs ()-(5) alcable to eal systems, numeous coectons have been oosed, to ont a few: () Coectons to the Boltzmann dstbuton (5) by ntoducton of vaous addtonal nteacton otentals, ethe fo on-on non-electostatc nteacton -16 o vaous on-suface nteactons 18-0 ; () Coectons fo the macoscoc natue of the equaton, nvolvng exlct molecula teatment of the fst neghbo nteactons 1,14, o othe dsceteness effects -4 ; () Coecton fo the delectc satuaton,.e., the deendence of on the electc feld ntensty 5,6,14 E; (v) coectons elated to the nhomogenety of the medum ( 0; e.g., Refs. 7,8); (v) coelaton effects,17 and non-local electostatc effects 9-, etc. Evey majo coecton of Eqs (4)-(5) have been an metus fo econsdeaton of the basc concets followng fom the Posson-Boltzmann equaton. Whle most studes n hyscal chemsty ctczed manly the Boltzmann at of Posson-Boltzmann equaton, seveal studes of otcal henomena -6 attacked the Posson at of t. It was demonstated that the quaduola tems n the macoscoc Coulomb law (1) become qute sgnfcant n cases whee hgh gadents of E ae esent. In such cases, quaduola tem n the dslacement feld need to be ntoduced 6 : = 0 E + P ½. (6) Hee, s the macoscoc densty of the quaduole moment 85 tenso. Note that the coeffcent n font of deends on the choce of defnton of quaduole moment (fo convenence, a devaton of Eq (6) and the defntons of the nvolved quanttes ae gven n Sulementay nfomaton A ). The substtuton of

2 Eq (6) nto Eq (1) yelds a genealzaton of the Posson equaton (4), 1 E, (7) whch oens a vast feld fo analyss of the effect of the 5 quaduole moments of the molecules comosng a medum on the oetes of chaged atcles n such medum. It have been ecently demonstated that quaduole tems n can lay a ole n solvent-solute nteacton 1,1. The coecton fo wll be motant f the solvent molecules ossess lage quaduole moment such s the case of wate 7 and many othes, ncludng non-ola meda of low dole moment but hgh quaduole moment such as lqud CO, fluoocabons etc. 1,8. The uose of ou study s to analyze the consequences of the new tem n Maxwell equaton fo seveal basc oblems of hyscal chemsty of electolyte solutons and collod chemsty. Eq (7) s lagely unknown to hyscal chemsts and vtually has neve been used n collod scence. Thee ae thee easons fo ths neglgence. Fst, Eq (7) s useless wthout an equaton of state fo. Thee ae seveal exstng studes of ths consttutve 0 elaton 9-4,1,1,6 5 0 but all ae scacely analyzed. Theefoe, n Secton, we wll deve a new equaton of state as smle as ossble, showng that s a lnea functon of E U E/, wth a scala coeffcent of ootonalty the quaduolazablty (hee U s the unt tenso). The second eason fo Eq (7) to be unknown n the collod feld s that t s a fouth-ode equaton wth esect to, and eques the use of new bounday condtons. Seemngly, these new condtons have been deved only ecently 9,. We wll evew ths oblem n Secton. Fnally, the thd obstacle to use Eq (7) s that t nvolves a new aamete of unknown value. We wll gve n ths ae both theoetcal estmaton and values detemned fom ndeendent sets of exemental data fo ons n wate. In Sectons 4 and 5, Eq (7) s used to envestgate the most basc concets n the hyscal chemsty of electolyte solutons Bon enegy and ebye-hückel dffuse onc atmoshee. It wll be shown that the coecton n Eq (7) n fact leads to esults whch have no counteat n the fame of Posson equaton (4), notably, fnte electostatc otental and enegy of a ont chage n quaduolazable medum (smla esult was obtaned n Ref. 1). In Sectons 4 and 5, we comae ou esults fo both the Bon enegy and the ebye-hückel dffuse atmoshee n quaduola medum to exemental data, whch allows us to detemne the value of quaduolazablty of wate.. Equatons of state fo the quaduole moment densty The oblem fo the consttutve elaton between and the feld gadent E has been addessed seveal tmes,6,9-4,1,1. Usng as a statng ont the aoach of Jeon and Km 1, we wll be able to obtan a new smle equaton of state whch elates to the feld gadent E and the molecula oetes of the solvent. Consde an deal gas consstng of molecules ossessng a sold quaduole moment tenso q 0 (fo the sake of smlcty, the molecule s assumed non-olazable and wth no dole moment). Snce q 0 s symmetcal and taceless, by a sutable choce of the coodnate system t can be dagonalzed 4 and n the geneal case, ts dagonal fom s: qxx 0 0 qxx qyy qzz q0 0 qyy 0 U. (8) 0 0 q zz The tem (q xx +q yy +q zz )/U ensues that the tace of q 0 s 0 and can be added because the feld ceated by a quaduole n vacuum s nvaant wth esect to the oeaton of exchangng the quaduole stength q 0 wth q 0 + XU whee X s any scala 11,44. The molecule s constantly otatng. An abtay otaton changes the quaduole moment tenso fom q 0 to q. Fo a otaton at abtay Eulean angles, and, the Eule matx E s gven by: cos cos sn cos sn sn sn sn cos cos sn cos E cos sn sn sn. cos sn sn cos cos cos cos cos sn sn cos sn cos (9) The elaton between the tenso q fo a andomly oented molecule and the tenso q 0 s q (,, ) E E q. () j k jl 0 kl In the absence of a gadent of the electc feld the aveage value of q s 0. In an extenal electc feld gadent E, the molecule tends to oentate tself n ode to mnmze ts electc enegy, gven by the exesson (Eq 4.17 of Jackson 11 ): 1 uel q : E. (11) The oentaton of the molecule must follow the Boltzmann dstbuton whch can be lneazed n the case of u el /T << 1: cn ex( uel / T ) cn 1 uel / T. (1) Hee, c n s a nomalzng coeffcent calculated as n π π π el. (1) c 1/ 1 u / T snddd 1/ 8π The aveage quaduole moment q of a molecule can be calculated dectly usng Eqs (8)-(1): π π π q E E q q sn d d d U /. (14) Hee, the molecula quaduolazablty q was ntoduced, elated to the dagonal comonents of q 0 as follows: q : q / T q 0 0 q q q q q q q q q / T. () xx yy zz xx yy xx zz yy zz Eq () was obtaned e.g. n Ref. 1. The macoscoc densty of the quaduole moment n a gas acted uon by a feld gadent E s the gas concentaton C tmes q, Eq (14): E U E /. (16) Hee, the macoscoc quaduolazablty s gven by C Cq : q / T. (17) q 0 0 Ou consttutve elaton Eq (16) s a dect consequence of the geneal fom (8) of the molecula sold quaduole and the lneazed Boltzmann dstbuton (1). Note that accodng to Eq (16) s taceless 11, n contast to Eq.5 of Jeon and Km 1. Refs. 4 and contan some dscusson n favo of the choce. Howeve, n Sulementay nfomaton A, we esent

3 5 0 5 aguments that the use of tenso wth non-zeo tace s ncomatble wth Eqs (6)-(7). Eq.4 of Chtanvs 1 ostulates an equaton of state n whch only the U E tem of ou Eq (16) s esent,.e., accodng to hm, has only dagonal elements and a non-zeo tace. In geneal, deends not only on the feld gadent but also on the feld E tself, and, on the othe hand, electc feld gadent E can nduce non-zeo dole moment 9,6. Fo an deal gas of sold doles wthn the lnea aoxmaton fo, ths s not the case. Ths can be shown by a dect calculaton analogous to the devaton of Eq (16): f the molecule has dole moment 0 and quaduole moment q 0, then n extenal feld E and feld gadent E ts enegy s 11 : 1 uel E q : E. (18) Usng ths exesson nstead of Eq (11), one can calculate the aveage dole and quaduole moments. Ths calculaton yelds fo agan Eq (16), because the tems ootonal to E cancel each othe. Calculaton of the macoscoc olazaton P gves the classcal esult 14 : P = P E, / P C C0 T, (19) whee and P ae the molecula and the macoscoc olazabltes. The devaton above s stctly vald fo a gas of sold multoles. It can be eadly genealzed to nclude molecula olazabltes and quaduolazabltes 1. Ths yelds nstead of Eq (17) the exesson q : q /, (0) q q0 0 0 T whee s the aveage ntnsc (atomc + electonc) molecula q0 quaduolazablty (Eq 4.5 of Jeon and Km 1 ). Eq (0) can be 0 comaed to the well-known fomula fo the olazablty 14 / 0 0 T, (1) whee s the aveage ntnsc molecula olazablty. In 0 addton, n the case of lquds one can ntoduce a Clausus- Mossott tye of elaton fo the local gadent E to the macoscoc quaduole moment densty and a eacton feld (smla to the elaton between local feld and aveage macoscoc olazaton 14 P). The local feld s nvestgated n Refs. 1 and 1. We shall not attemt such a genealzaton n ou study and n what follows we wll assume that the equaton of state (16) s vald fo sotoc fluds, ovded that E and E ae not too lage (n ode Eq (1) to be alcable). Fo dense fluds, Eq (17) fo wll be nvald but t stll must gve the coect ode of magntude of the quaduolazablty. Usng the values of the quaduole moment of wate fom Ref. 7: q xx = Cm, q yy = Cm and q zz = Cm (a facto of / fo the dffeent defntons of q 0 used hee and n Ref. 7 must be accounted fo), we can calculate the value = 1-0 Fm fom Eq (17). Both and the q0 Clausus-Mossott effect ncease. Fo comason, the exemental value fo the olazablty of wate s P = 0 = F/m, whch s about tmes hghe than the one calculated though the estmaton / P C0 T. By analogy, we can assume that s seveal tmes lage than the value followng fom Eq (17). Let us now estmate the essue and temeatue devatves of. Assumng that the molecula quaduolazablty q s ndeendent on, fom Eq (17) t follows that C T, () C T T whee T s the sothemal comessblty. Snce Eq (17) s aoxmate, the esult Eq () also gves only an estmate of /. Fo the temeatue devatve of (sutably made dmensonless by a facto of T/ ), we use Eq (0) fo the deendence of the molecula quaduolazablty on temeatue and the elaton = C q to obtan: T T C T q T C T q T v q0 : q0 / T T. () q : q / T q0 0 0 Hee v = C 1 ( C/ T) s the coeffcent of themal exanson. Fo wate 46, T v = To estmate the second tem, we assume that q0 q0 : q 0 / T (wate has hgh quaduole moment q 0 and t s a had molecule of low ntnsc olazablty and ehas low 0 ). In ths lmt, the second q0 tem n Eq () s about 1, much lage n absolute value than T. Theefoe, we can wte aoxmately that v T T 1. (4). Bounday condtons fo the genealzed Posson equaton Wthn the quaduola aoxmaton, the Coulomb-Amee law (7) s of fouth ode wth esect to snce uon substtutng Eq (16) n Eq (7) one obtans: 1 E E U E / ( ). (5) In a homogeneous medum, ths equaton smlfes to 4 ( ) E E L. (6) Hee, we have ntoduced the quaduola length L defned wth the elaton: L /. (7) Fom the estmaton of n the end of the evous Secton, we can say that L = ( /) 1/ > 0.Å, ehas seveal tmes lage. Eqs (6)-(7) ae of the same fom as those of Chtanvs 1, wth the only dffeence that he obtaned dffeent numecal coeffcent n Eq (7). We ae manly concened wth shecal symmety n ths study, whee Eq (6) eads: 1 d E L d d 1 d E ( ). (8) d d d d 95 We wll need an exlct exesson fo and ; the gadent and the dvegence of E n shecal coodnates ae gven by: d E / d 0 0 de E E 0 E / 0 ; E. (9) 0 0 E / d Then, fom Eq (16) one obtans 0 0 de E d, 0 0 1

4 d E de E e, (0) d d whee e s a unt vecto, collnea wth the adus-vecto. The bounday condtons of Eq (7) have been deved ecently by Gaham and Raab 9, usng the sngula dstbutons aoach of Bedeaux et al n the case of a flat bounday suface of an ansotoc medum wth abtay equaton of state; altenatve devaton, agan fo flat bounday, was gven n Ref.. Followng the aoach of Gaham and Raab, we wll deduce hee the bounday condtons of Eq (7) at a shecal suface dvdng two sotoc hases. Fst, we wte the sngula dstbutons of E, and : E η E η E ; η η ; (1) S η η δ. Hee, X + and X denote the coesondng hyscal quanttes fo the hase stuated at > R and < R, esectvely; S s the suface chage densty; the notatons ± and stand fo the Heavsde functon and the ac -functon: η η( R) ; η η( R ) ; δ δ( R ). () To obtan the necessay bounday condtons, we nset Eqs (1) nto Eq (7) and use the educblty of ±, and ts devatve 1 = d/d. We need fst to calculate E and :, whee E and ae gven by Eq (1): E η E η E δ E E ; 1 : η : δe δ e δ ( ) 1 η : δe δ e δ ( ). () Usng Eqs (), we can wte Eq (7) n the fom 1 η E 1 η E S d E e e d δ d E e e d R 1 δ 1 ( R) ( R) 0. (4) Fo the devaton of Eqs ()-(4), we have used the oetes of the sngula functons: + = e ; = e ; = e 1 ; 1 () = 1 (R) d /d =R. ecomoston of Eq (4) yelds, fst, the bulk equatons fo the two hases (the coeffcents of ± n Eq (4)): 1 E. () Settng the facto multlyng n Eq (4) to 0, we obtan a genealzaton of the Gauss law fo the quaduola meda: d E e e d R d S E e e. (6) d R The last tem of Eq (4), ootonal to 1, esults n a new bounday condton, whch balances the quaduole moment denstes on the two sdes of the shecal suface: ( R) ( R) 0. (7) We now substtute Eqs (0) nto Eqs (6) and (7) and obtan the exlct fom of the bounday condtons. Eq (6) eads: d E de E de E E d d d R d E de E de E E d d d R S. (8) The exlct fom of Eq (7) s: de E de E 0 ; (9) d d R R Subtactng Eq (8) and Eq (9), we obtan the elaton: S, () whch s fomally equvalent to the classcal Gauss law, but one must kee n mnd that nvolves hghe devatves of the feld E, cf. Eq (6). 4. Effect of the quaduolazablty of a medum on the Bon eneges, atal mola volumes and entoes of dssolved ons In ths at of ou study the geneal equaton (8) of electostatcs n quaduola meda at shecal symmety and ts bounday condtons (8)-(9) deved n the evous sectons wll be used to solve seveal basc electostatc oblems of hgh sgnfcance to the hyscal chemsty of electolyte solutons Pont chage n an nsulato We solve Eq (8) wth = e (). The geneal soluton of the equaton s: 1 Е k 1 k 1 ex k 1 ex. L L L L (41) In ode to detemne the thee ntegaton constants k 1, k and k, we need to mose thee condtons on E. The fst one s to eque E to tend to a fnte value as (ths gves k = 0). The second condton s that the asymtotc behavo of E at s unaffected by the esence of quaduoles, that s, the feld of a ont chage at tends to q/4. Ths condton yelds k 1 = q/4 (the same esult can be obtaned by the Gauss law). Thee s one fnal condton needed to detemne k. Ou assumton s to eque that E tends to somethng fnte as 0.e. thee s no sngulaty of E at 0, whch yelds k = k 1. Eq (8) has thus a fnte soluton, whch s: e 1 Е 1 1 ex. (4) 4π L L Integaton of ths esult gves the followng fomula fo the electostatc otental: e 1 ex( / L ) 85. (4) 4π The otental n = 0 s also fnte, and ts value at = 0 s 0 = e /4L. The on has, theefoe, a fnte enegy: uel e0 / e / 8π L. (44) Ths s n maked contast to the case of on n vacuum whee the 90 otental s dvegng as 1/ and the electostatc self-enegy of a

5 ont chage s nfnte (Fg. 1). Fo a ont chage n wate at T = 5 C, f L = Å, we obtan 0 = 9 mv and u el =.6 T. Eq.8 of Chtanvs 1 has the same fom as Eq (4) (but hs elaton between L and s dffeent). Eq (4) can be comaed also to 5 Eq.48 of Jeon and Km 1, who obtaned a dvegent otental snce they used anothe consttutve elaton fo and mled dffeent condtons on the solutons to detemne the ntegaton constants. In ode to cooboate ou non-classcal choce fo the fnte feld condton, a dffeent aoach wll be esented n the followng Secton 4. to deve the same esult (4), by lacng the chage nto a shecal ty of adus R (at = R the bounday condtons deved n Secton wll be aled) and then takng the lmt R 0 of the esultng otental. 0 5 Fg. 1. Electostatc otental of a ont chage n a quaduola medum vs. the dstance fom the ont chage n wate, Eq (4), fo vaous quaduola lengths L. In a quaduola medum, the ont chage has fnte otental at = 0. Sold lne: L = Å; dashed lne: L = 1 Å; dashdotted lne: L = 0 (the classcal soluton). 4.. Ion of fnte sze n an nsulato Thee ae vaous models of an on of fnte sze n a soluton, whch yeld the same exesson fo the Bon enegy 49,8,. The smlest model assumes that the on s a ont chage stuated nto a ty,.e., n an emty shee of emttvty 0 and adus R ; the emty shee s located n a medum of delectc emttvty. Ths model neglects the detaled chage dstbuton n the on and can be genealzed n vaous ways 1,1,17,1,,7-,. Hee, n ode to kee the ctue smle, we wll hold on to the emty shee model, only addng nto account the quaduolazablty 0 of the medum. Smla oblem (an entty of cetan chage dstbuton laced nto a ty n a medum wth ntnsc quaduolazablty) was consdeed by Chtanvs 1 and Jeon and Km 1 usng dffeent equaton of state and a dffeent set of bounday condtons. We fomulate the oblem wth the followng equatons: () Insde the shee (suesct ), at < R, thee ae no chages, bound o fee, aat fom the cental on of chage e : 0 E e δ( ). () () Outsde the shee (no suesct), at > R, Eq (8) s vald wth = 0: 1 d E L d d 1 d E 0. (46) d d d d () The bounday condtons at = R ae gven by Eqs (8)- (9): 0 d E de E E E L 0 ; (47) d d R de E 0 ; (48) d R The soluton of Eqs ()-(48) n tems of the electostatc otental s: e 1 e 1 e L R at < R ; 4π 4π R 4π L L R R 0 0 (49) e 1 ex ( ) / L R L 1 at > R ; () 4π L L R R we used also the condton = at = R. As R 0, Eq () fo smlfes to Eq (4) fo a ont chage, whch justfes the assumton fo fnte E and at 0 whch was used n Secton 4.1 to deve Eq (4) that s, the esults obtaned n Secton 4.1 can be obtaned altenatvely by takng the lmt fom Eq () wthout makng use of such a non-classcal condton. The self-enegy u el of the on s detemned by the otental = e /4 0, ceated by the olazed medum and actng uon the on. It s obtaned fom Eq (49) as 1 e 1 1 L R R uel eδ 8πR 0 L L R R. (51) It can be comaed to Eq.1 of Chtanvs 1, who used nstead of ou Eq (48) a condton fo contnuty of de /d at = R, wthout dscusson. Jeon and Km 1 used anothe condton fo non-oscllatoy soluton, also wth no good justfcaton. When L 0, Eq (51) smlfes to the famla exesson fo the Bon enegy 49, : e 1 1 ubon. (5) 8πR 0 It follows fom Eq (51) that the Bon enegy of small ons s moe stongly affected fom the quaduolazablty of the medum. The obtaned esult (51) fo the effect of the quaduola length L on the self-enegy u el s not easy to test dectly due to the fact that, at least fo wate, the 1/ 0 tem s by fa lage than the second tem n the backets of Eq (51), whch nvolves L. To test Eq (51) we wll elmnate the tem 1/ 0 by dffeentatng u el ethe wth esect to o to T n Sectons 4.4 and 0. Po to ths, we need fst to dscuss the elaton of R to the cystallogahc adus R of the on. Followng Latme, Ptze and Slansky, we assume that R,c = R + L c and R,a = R + L a (5) fo catons and anons, esectvely. Latme, Ptze and Slansky assumed that the lengths R + L c and R + L a ae measues of the dstance between an on and the dole closest to t. Rashn and Hong 51 agued that R + L c s, n fact, the covalent adus of the caton. The length L c s the same fo all catons, and L a s the same fo all anons; both ae of the ode of the wate effectve adus. Latme, Ptze and Slansky obtaned L c = 0.85 Å and L a = 0.1 Å, usng data fo the fee eneges fo hydaton 0 of few monovalent ons (L c > L a because L a must be about the dstance of the anon to a oton and L c must be about the dstance of a caton and an oxygen atom). We wll detemne these values moe ecsely n the followng secton, usng a lage set of data fo Ion fee enegy of hydaton The chemcal otental of an on n dlute aqueous soluton s: T ln C / C. (54) 0 0 The standad molaty-based chemcal otental 0

6 (coesondng to a standad concentaton C 0 = 1 M) eflects the state of a sngle on n the soluton, ncludng the effect of the on s feld on the molecules of the solvent n the vcnty of the on. Fo ths enegy one can wte,17,,5 : nta ( T ) uel πr gv T ln C0. () The fst tem n Eq (), nta, s elated to the ntamolecula state of the on tself. Fo smle ons, t s assumed that ths tem s the same n gaseous state and n any solvent. The second tem, u el, stands fo the electostatc on-wate nteacton; we wll use fo u el the genealzed exesson fo the Bon enegy of the on, Eq (51), nvolvng the quaduola length L. The thd tem, v 0, s the mechanc wok fo ntoducng an on of adus R nto a medum at essue ; g V s a ackng facto standng fo the fact that the eal volume v 0 occued by the on s not a shee (of volume 4/ R ) but athe a olyhedon. Fo examle, f the ackng of the solvent molecules aound the on s dodecahedal, then g V = 1.; fo a cubc ackng, t s 1.6; fo vey lage ons, g V s close to 1. Snce fo vey small ons the tem v 0 s unmotant, and we ae not gong to analyze data fo lage ones 0 due to the moe comlcated stuctue of the standad otental 0, we wll assume that g V s aoxmately 1. fo all ons studed below. The fouth tem, TlnC 0, ognates fom the choce of the standad state. Othe contbutons to Eq (), such as the enegy fo ty fomaton 5 and vaous secfc nteactons, 5 ae hee neglected fo smlcty. Ths makes Eq () nalcable to lage ons. Snce the electostatc effects we ae nvestgatng ae sgnfcant fo small ons only, ths s not a dawback, but all data fo mola hydaton eneges and atal mola volumes fo ons lage than. Å wll be neglected below. Hydohobc 0 effect s esecally motant fo the atal mola entoy 5, theefoe, only data fo ons smalle than. Å wll be taken nto account. The full lst of data-onts s gven n the Sulementay nfomaton C. The standad mola fee enegy of hydaton of an on 0 = 0 G 0 s the enegy fo tansfe of 1 mol ons fom a hyothetcal deal gas at standad essue 0 to a hyothetcal deal 1M soluton 54. The exesson fo 0 follows fom Eq () fo 0 and an analogous exesson fo G 0 : TC 0 Δ0 T ln πr gv Z e 1 1 L R R 8πR 0 L L R R. (56) The fst tem stands fo the choce of standad states n the aqueous soluton (hyothetcal deal soluton wth concentaton C 0 = 1M = 00 N A m -, N A Avogado s numbe) and n the gas state (deal gas of ons at standad essue 0 = 1 Pa). The second tem stands fo the mechancal wok fo ntoducng an on of volume 4/ R nto the aqueous soluton. Ths tem s neglgble comaed to the othe ones n Eq (56). The thd tem s the electostatc enegy fo tansfeng an on fom a gas hase to an aqueous soluton. The electostatc enegy n the gas (the 1/ 0 tem n the backets) s about 0 tmes lage than the esectve enegy n the soluton (the tem ootonal to 1/). Theefoe, ou exesson (56) fo 0 yelds essentally the same esults fo the value of 0 as those of Latme, Ptze and Slansky who dd not accounted fo L. Nevetheless, snce we wll use a moe extended set of themodynamc data fo ons, we eeated the calculatons by fttng the data fo 0 taken fom Refs. 54 and 5. ata-onts fo lage ons as well as cetan ons of hgh olazablty o dole moments wee neglected (cf. Sulementay nfomaton C fo the lst). The met functon s defned as: Δ ( Lc, La, L ) 4 Δ 0,th( Lc, La, L ;Z, R ) Δ 0,ex (Z, R ) Z1, (57) N f whee 0,th s the edcted value accodng to Eq (56) of an on of valence Z and bae adus R ; 0,ex s the esectve exemental value; N=86 s the numbe of data onts and f s the numbe of fee aametes used n the otmzaton ocedue. Ths met functon s almost ndeendent on L. The esults fom the mnmzaton of wth esect to L c and L a ae gven n Table 1 fo vaous values of L. They ae n good ageement wth the esults of Latme, Ptze and Slansky, L c = 0.85 Å and L a = 0.1 Å, and ae almost ndeendent on L. The comason of Eq (56) wth exemental data s llustated n Fg.. Table 1. Values of L c and L a obtaned fom the ft of the exemental data fo the hydaton eneges of vaous ons wth the theoetcal exesson, Eq (56), at thee dffeent values of L. 85 f L c [Å] L a [Å] L [Å] [kj/mol] 0.84± ± ± ± ± ± Fg.. Mola hydaton enegy 0 [kj/mol] vs. bae on ad R [Å]. ata fo catons (blue) and anons (ed) of vaous valence (ccles monovalent, cosses dvalent, stas tvalent, squaes tetavalent). The lnes ae dawn accodng to the theoetcal edcton Eq (56), wth L c = 0.84 Å and L a = 0. Å, as obtaned fom the mnmzaton of, Eq (57), at fxed value of L (L = 1Å). Although Eq (56) s n satsfactoy ageement wth the exemental data, one must kee n mnd t s an ovesmlfed model of an on n a medum. The stongest assumton used n ts devaton s that the contnual model neglects the dscete natue of the solvent-on nteactons. The exesson fo the Bon

7 enegy was coected by many authos n ode to take an exlct account fo the dscete stuctue of matte (cf. Chate 5.7 of Ref. 17 fo a summay). The homogenety condton = 0 has been ctczed e.g. by Abe 8 ; the effect of delectc satuaton 5 has been analyzed by Ladle and Pegs 6. Nonlocal electostatc theoy was aled to the self-enegy oblem by Baslevsky and Pasons 1, (note that the esence of E n the defnton (6) of the dslacement feld makes the electostatc oblems n quaduola meda nonlocal 1 ). All these effects contbute to the value of 0, but these coectons wll be neglected n the dscussons below. In addton, the valdty of Eq (16) fo lquds s a hyothess only. Theefoe, the comason of Eq (56) and ts devatves (Eqs (58) and (66) below) wth the exemental data should be consdeed wth cauton Ion atal molecula volume The models fo the atal molecula volume v of ons ae evewed n Ref.. We wll consde only the values of v at nfnte dlutons. The atal molecula volume v of the on n aqueous soluton s calculated by takng the devatve of 0 + TlnC /C 0, cf. Eqs (54)-(), wth esect to : 4 Z e R v πr gv T T 8π R 0 Z e L R 8π L L R R Z e L L R L. (58) 8π L L R R Hee T = - lnv w / s the comessblty of wate (the tem T T s elatvely small and s usually neglected); v w s wate s mola volume. In the thd tem ootonal to R /, we neglected 1/ n comason wth 1/ 0. If L = 0, ou exesson (58) smlfes to the famla fomula fo v followng fom the Bon enegy (5): 4 Z e R Z e v πr gv T T 8π 0R 8π R. (59) The exesson (58) edcts the lmtng atal molecula volume of an on at nfnte dlutons as a functon of the on cystallogahc adus R. Whle the hydaton enegy 0 s vtually ndeendent of L, the atal mola volume s senstve to the value of L, whch allows us to use Eq (58) to detemne L fom the exemental data. We use the data fo catons and anons of vaous valence assembled by Macus 54, neglectng ons of comlex stuctue and lage R, cf. Sulementay nfomaton C. The met functon of the otmzaton ocedue s defned as: Lc L L a v gv, L,,, 4 Lc L L a v,th Z, R ; gv, L,,, v,ex(z, R ) Z1 ; () N f the total numbe of data-onts used n the otmzaton s N = 97. Fst we need to estmate all aametes n Eq (58). The deendence () was detemned by the dect measuements 56 and allows the calculaton of / ; we use the followng value : Pa. (61) The value of the comessblty s T = Pa -1 and t coesonds to a atal mola volume N A T T = 1.1 ml/mol. The values of 1/ / and T ae vey close to each othe snce s almost lnea functon of the wate concentaton comae to Eq () fo. Usng Eqs (61) and () we can estmate L / : 1 L Pa. (6) L Ths value s qute small and theefoe the last tem n the backets of Eq (58) lays lttle ole fo the atal mola volume v of an on n wate and can even be neglected. Fo the geomety facto g V we can use the value 1., cf. the dscusson followng Eq (). In all cases, we used the values L c = 0.84 Å and L a = 0. Å obtaned n the evous secton, cf. Table 1, when calculatng R = R + L c,a. The value of R / s n geneal dffeent fo catons and anons, R / = L c / o L a /. We tested aganst the exemental data two smlfyng assumtons egadng the two devatves L a,c / n ode to decease the numbe of fee aametes. The fst ossblty nvestgated s that they ae equal, L / L /. (6) c a The second one s that the quantty 1/L a,c L a,c / s the same fo both catons and anons: 1 Lc 1 La. (64) Lc La We tested both assumtons and Eq (64) was found to be n much bette ageement wth the exemental data, cf. Table. We tested vaous combnatons of fxed and fee aametes fo the otmzaton ocedue (Table ), n ode to test the senstvty of v to the aametes and the assumed aoxmatons fo g V, R /, L /. In summay, the esults ae: () v has a shallow mnmum and the uncetanty of the values of the fttng aametes s hgh. Ths s llustated n Fg. S1 n Sulementay mateal B. () The assumton Eq (64) yeld lowe dseson than Eq (6) (comae ows c and d n Table ). If both L c / and L c / ae used as fee aametes, we obtan values whch agee wthn the uncetanty wth Eq (64) (cf. ows b, g and 4). The devatve L -1 c L c / s found to be negatve and has a value of the ode of Pa. Ths value can be comaed to the essue deendence of the dstance L w between two wate molecules; snce v w ~ L w, 1 vw 1 Lw Lw. () v L L w w w Theefoe, L -1 w L w / = T / = Pa. Thus L -1 w L w / s one ode of magntude lage than L -1 c L c /, whch suggests that the stuctue of the hydaton shell of an on s fa moe ncomessble than the stuctue of wate tself. () We tested whethe the assumed value g V = 1. of the ackng facto gves good esults by allowng g V to be a fee aamete. Ths yelded a bette dseson and a slghtly hghe value: g V ~ (cf. ows a, f and 4 n Table ). Ths suggests that the ackng of the hydaton shell aound the on s less dense than dodecahedal (g V = 1.) but dense than cubc (g V = 1.6). (v) We tested whethe the aoxmate value L -1 L / = Pa -1, (6), yelds good esults by consdeng t as a fee aamete. Unfotunately, ths etuned almost the same dseson and unealstc values of L -1 L / and L, snce v s almost ndeendent on ths quantty. Theefoe, we consde the esults n ows e and h - n Table nadequate and use L 1 L / = Pa -1. (v) When one accounts fo the effect of L on v, ths yelds only slghtly lowe dseson (comae ows a and f o b and g n

8 Table ). L affects the data fo the smallest ons only (L +, Be +, Al + ) and t exlans why the atal mola volumes ae moe ostve than the ones edcted fom the classcal model wth L = 0. Fo examle, the atal mola volume calculated fo Al + (R 5 = 0.5Å) wth the aametes n ow f s - ml/mol, and f one sets L = 0, the esult wll be -85 ml/mol. The exemental value s ml/mol. The value of L obtaned fom the vaous vaants of the otmzaton ocedue vaes between 1 and Å. Fom the estmaton of n Secton, we can edct that L s few tmes lage than 0.Å. Stll, a dffeence of one ode of magntude between the value of L estmated fom Eqs (17) and (7) and the exemental one s unexected. Nevetheless, the value of L wll be confmed wth ndeendent data fo the atal mola entoy of vaous ons and data fo the actvty coeffcent n the followng two sectons. The comason between Eq (58) and the exemental data s llustated n Fg. (aametes fom ow c ). Table. Results fom the otmzaton of v, Eq (), wth esect to vaous aametes. Blue felds ndcate fxed values of the esectve 0 aametes. 5 0 L 1 1 L c 1 L f L L L L c a g v [Å] V [Pa -1 ] [Pa -1 ] [Pa -1 ] [ml/mol] 1 1 a Eq (64) b c Eq (64) d ~ Eq (6) e f Eq (64) g h Eq (64) Fg.. Patal mola volumes v [ml/mol] of ons of vaous valences n nfntely dluted aqueous solutons as functons of the onc cystallogahc ad R. Sold ccles: monovalent ons; cosses: dvalent; stas: tvalent; boxes: tetavalent; blue and ed catons and anons. ata assembled by Macus 54. Lnes: Eq (58) wth Z = 1,,, 4 and R = R + L c o R + L a; the values fo the aametes wee obtaned fom the otmzaton of v, Eq (), wth esect to two fttng aametes, L = 1.1 Å and L -1 c L c/ = Pa -1, fo all 8 cuves. a 4.5. Standad entoy of hydaton The atal mola entoy s of the on n wate can be calculated by takng mnus the devatve of 0 + TlnC /C 0, cf. Eqs (54)-(), wth esect to T. The mola entoy fo hydaton, s, s calculated analogously as ( 0 + TlnC /C 0 Tln/ 0 )/ T, cf. Eq (56) fo 0. The standad mola entoy fo hydaton s 0 s obtaned,54 by settng = 0 and C = C 0 n s. Usng Eq (56), one fnds the followng exesson fo (the dmensonless) s 0 : 0 1 Δ 0 ln TC 1 e v s T R 8πR T 0 0 e L R 8π L L R R T e L L R L. (66) 8π L L R R T In Eq (66), all quanttes excet fo L, L / T and R / T = L c,a / T ae known. The hydaton entoy s 0 has been measued wth easonable accuacy fo a lage numbe of ons 54. The exemental data assembled by Macus 54 can be used to obtan a second estmaton of L fom an ndeendent set of data (besdes the atal volumes), by comang Eq (66) to them. To do so, we defne the met functon: Lc L L a s L,,, T T T 4 Lc L L a Δs0,th Z, R; L,,, Δ s0,ex(z, R ) Z1 T T T. N f (67) ata fo N = 68 ons of valence Z = 1 4 ae analyzed (cf. Sulementay mateal C). The followng values ae used fo the aametes n Eqs (66)- (67). Fo the temeatue deendence of, we use the exemental data fo (T) fom Refs. and 46, whch gves: T 1.. (68) T Fo the coeffcent of themal exanson, we take 46 T v = 0.076, coesondng to entoy of k B N A = 0.6 J/Kmol. We can also estmate L / T fom Eqs (7), (4) and (68): T L T T (69) L T T T Snce ths value s qute small, the tem ootonal to L / T wll have nsgnfcant contbuton to the value of s 0. The devatve R / T has dffeent values fo catons and anons, L c / T and L a / T esectvely. In ode to decease the numbe of fee aametes, we tested agan two ossble aoxmatons, a fst one that L c / T = L a / T (whch was found to be n dsageement wth the exemental data), and a second one, that 1 Lc 1 La ; () Lc T La T comae to the assumtons (6)-(64) fo L c / and L a /. We agan attemted vaous combnatons of fxed and fee aametes n the otmzaton ocedue (Table ), n ode to analyze the senstvty of s to R / T, L / T and L. The esults ae: () s has a shallow mnmum and does not allow fo a vey ecse detemnaton of the aametes nvolved. The deendence of s on L and T/L c L c / T (cf. ow d of Table ) s

9 5 llustated n Fg. S n Sulementay mateal B. () The exemental data agee well wth Eq (). If both L c / T and L a / T ae used as fee aametes, close values of T/L c,a L c,a / T ae obtaned, cf. ows c and f n Table. The value of T/L c L c / T s ostve and has a value of the ode of 0.0. Ths value can be comaed to the temeatue deendence of the dstance L w between two wate molecules; fom the elaton 1 vw Lw (71) v T L T w w we fnd that T/L w L w / T = T = 0.. Smlaly to the essue deendence, cf. Eq. (), wate exanson coeffcent s hghe by an ode of magntude comaed to the esectve deendence of R on T. Ths s anothe oof that the stuctue of wate s moe lable than the stuctue of the hydaton shell of an on. () The aoxmate value T/L L / T = 0.18 yeld esults whch does not dffe n comason to the model wth neglected L / T (ows b and d ). If L / T s left as a fee aamete, the otmzaton ocedue etuns the same dseson but unealstc lage negatve values of L / T (ow e ). Theefoe, we consde 0 the esults n ow e unealstc and we use the value T/L L / T = In fact, T/L L / T can be safely neglected. (v) The effect of L on s 0, and esectvely, on the dseson s, s not vey lage. It manly affects s 0 of vey small ons, by 5 deceasng the absolute value of the entoes by -0%. Fo examle, the entoy of Al + calculated fom Eq (66) wth the aametes gven n ow d n Table fo L and L c / T s -597 J/Kmol, whle wth L = 0 t s -678 J/Kmol (the exemental s 0 of Al + s J/Kmol). The esults fo the value of L does 0 not deend stongly on the otmzaton ocedue, and gve values n the ange L = Å, somewhat smalle but of the same ode as the edcton fom the atal mola volume data. The value of coesondng to L = 0.Å (ow d ) s = L = 1-0 Fm, o about 1 tmes hghe than the edcton fom the deal gas fomula (17). Fg. 4. Negatve entoes of hydaton s 0 [J/Kmol] fo mono, d, t and tetavalent ons as functons of the cystallogahc onc ad R [Å]. ata assembled by Macus 54. Lnes: Eq (66) wth two fttng aametes: L = 0.8 Å and T/L c L c/ T = 0.05, obtaned fom the mnmzaton of s, Eq (67). Sold ccles: monovalent ons; cosses: dvalent; stas: tvalent; boxes: tetavalent; blue and ed catons and anons. v Table. Results fom the otmzaton of s, Eq (67), wth esect to vaous aametes. Blue felds ndcate fxed values of the esectve aametes. f T L T Lc T La L [Å] L T Lc T La T s [J/molK] a Eq () 5.5 b Eq () 5.4 c d Eq () 5.4 e Eq () 5. f uaduolazablty n the ebye-hückel theoy The model of ebye-hückel1 of the electc double laye of a dssolved on s a basc concet n electolyte chemsty,17. It has been coected at least as many tmes as the Posson-Boltzmann equaton (4), but to ou knowledge the coectons nvolved ethe the Boltzmann dstbuton o the homogenety condton = 0. We ae modfyng the Posson equaton tself by accountng fo the quaduole tem :, Eq (7), and n ths Secton the effect of ths tem on the stuctue of the onc atmoshee aound an on s nvestgated Pont chage n conductng meda We solve Eq (8) wth beng gven by the sum of a ont chage and Boltzmann-dstbuted fee chages, Eq (5): ( ) e δ( ) e C ex( e / T ). (7) j j j Followng ebye and Hückel1, we exand the Boltzmann dstbuton n sees at low otentals, ( ) eδ( ) / L ; (7) hee the ebye length L s defned wth the exesson L T e C. (74) / j j The soluton of Eq (8) wth gven by Eq (7) s: e l l ex( / l) ex( / l), () 4π l l whee we used agan the condton (0) < to detemne one ntegaton constant, cf. Sectons In Eq (), we ntoduced he chaactestc lengths l and l defned as l L 1 4 L / L 1/, 1/ / l L L L. (76) The nvese elatons defnng L and L wth l and l ae: L l l ; L l l. (77) In dlute solutons, L >> L and both l and l ae eal; l s about equal to L and l s about equal to L, whch s the eason fo the choce of ndces. At a cetan ctcal value of the ebye length, L = L (f L = Å, ths coesond to onc stength of 0.6 M), the lengths l and l become equal (Fg. 5). At hghe onc stengths and smalle L, the lengths become comlex conjugates,.e., the otental () whle dmnshng wth dstance exhbts an oscllatoy behavo. When L < L, Eq () can be eesented as: e l Re l Im Re Im ex l sn l, (78) 4π lrelim lre lim lre lim whee l Re = Rel and l Im = Iml ; t s easy to show that l Im =

10 5 (L L L ) 1/ / and l Re = (L L +L ) 1/ /. We wll leave the deee analyss fo a futue ae, snce the oscllatons of the otental ae elatvely unmotant fo ou cuent oblem. Fg. 5. mensonless chaactestc lengths l /L and l /L as functons of the dmensonless ebye length L /L, Eqs (76). Red sold lne: Re(l /L ); ed dashed lne: Re(l /L ); blue lnes: Im(l /L ) and Im(l /L ). The otental () s fnte and ts value at = 0 s e l l 0 4πl l l l e 4π L L / L L / L 1/ 1/. (79) The esectve enegy u el = e 0 / of the on n the medum s e l l uel 8πl l l l e 8π L L / L L / L 1/ 1/. () Ths exesson can be exanded n sees when L to obtan the lmtng law fo the enegy u el n dlute solutons: L e e e L uel... (81) 8π L 8π L 16π L The fst tem n Eq (81) s the quaduola self-enegy of a ont chage, cf. Eq (44). The esence of the dffuse electc double laye deceases ths enegy wth the ebye-hückel enegy H : 0 H T ln H = e /8L. (8) Eq (8) s the well-known lmtng ebye-hückel law; H s the actvty coeffcent of an on n dluted electolyte soluton. The thd tem n Eq (81) eesents the leadng coecton of the lmtng ebye-hückel law fo a soluton of ont chages n a 5 quaduola medum. It s ostve, whch means that the lmtng law undeestmates the actvty coeffcent H at hgh concentaton, whch s ndeed the case. The coecton fo L s of the same ode (C 1 el ) as the coecton fo the fnte on sze R, theefoe, n ode to comae the edcted effect of the 0 quaduolazablty on the actvty coeffcent wth the exemental data, we need to genealze the extended ebye- Hückel model (the genealzaton of the lmtng ebye-hückel model fo shecal nstead of ont chages). 5.. Fnte-sze chage n conductng meda As was the case wth the Bon enegy, the extended ebye- Hückel model can also be deved by vaous models of the on and ts ty. The smlest one whch yelds the coect esults assumes nstead of Eq (7) the chage densty: eδ( ), R, ( ) (8) / L, R. Ths equaton eflects that the ons fom the ebye atmoshee cannot aoach the cental on to a dstance smalle than R, whch must be the sum of the two cystallogahc ad R + and R of the anon and the caton: R = R + + R. (84) Eq (8) neglects the fact that two ons of the same sgn can aoach each othe to dstances R + o R - dffeent fom R; ths s elatvely unmotant snce the eulsve electostatc foce decease the co-on concentaton n the vcnty of the cental on to values close to 0. Also, the model assumes that s the same nsde the ty and n the soluton. If Eq (8) s substtuted nto the Posson equaton (4), and ths equaton s solved, t wll gve fo the actvty coeffcent the famous esult, known as the extended ebye-hückel theoy1: e T ln H-R. (85) 8π L R Ths equaton has much wde alcablty than the lmtng law (8). Howeve, nstead of usng Eq (84), the dstance R s always used as a fttng aamete and ts value s geneally hghe than Eq (84) would edct as onted out by Isaelashvll, to obtan ageement wth measued solublty and othe themodynamc data, t has been found necessay to coect the cystal lattce ad of ons by nceasng them by 0.0 to 0. nm when the ons ae n wate. The hyothess that we nvestgate hee s that ths dsceancy eflects the neglected effect of the quaduola tems n Posson equaton. Fg. 6. Electostatc otental of Na + on n quaduola medum (L = Å) contanng fee chages (NaF of concentaton C el = 0.01 and 1 M). The mnmal dstance between Na + and F - s R =. Å (the sum of the cystallogahc ad of Na + and F - ). The soluton fo at C el = 1 M s oscllatng-decayng and has shallow extema (the fst s = mv at = 4.7 Å). To ntoduce n the extended ebye-hückel model, we substtute Eq (8) nto the genealzaton (8) of the Posson equaton of electostatcs (no jum of o occus at = R).

11 The geneal soluton of ths oblem s: = l l l l ex 1 ex 1 A0 A A, R; ex( / l ex( / ) ) l B B, R. (86) The fve ntegaton constants A 0, A +, A -, B and B ae detemned by fou condtons fo contnuty of and ts devatves (fst, second and thd) at = R and the electoneutalty condton: / 4π d 0. (87) R e L The soluton of these 5 condtons s tval but lengthy. The soluton s gven n Sulementay nfomaton (executable Male 17 code). The fnal esult fo accodng to Eq (86) s llustated n Fg. 6 at two concentatons of the electolyte wth L = Å and R =. Å. Fom Eq (86) and the values of A 0, A +, A -, B and B, one can calculate the otental 0 = (=0) actng uon the cental on. It s smly elated to the actvty coeffcent: e e e T ln 0 F( L, R, L ) 4π L, (88) 8π L whee the self-enegy e /8L n the absence of othe ons s subtacted, Eq (44). Hee, we have ntoduced the functon F: 1 d d l d 1 e 1e l l l l 1 F 4l l 1 d d l d l 1 e l 1e l d 1 1 e l l l l d l l d l l 1 ; l 1 1 d l 1 l d l l 1 e l 1 1 l 1 l d l l 1 l 1 (89) whee the dmensonless lengths l and d ae defned as l = l /L and d = R/L. In dluted solutons, Eq (88) can be exanded n sees and the esult s: T ln l l l L e R 4L ex( R / L ) L ex( R / L ) 1, 8π L L (90) whch can be used at low electolyte concentaton. Howeve, n the most nteestng fo ou consdeaton egon of concentatons, C el ~1M, whee the effect of L s most motant, Eqs (88)-(89) must be used wthout smlfcatons. The comason of Eq (88) to the exemental data fo the actvty coeffcent allows the detemnaton of the sngle unknown aamete n t, the quaduola length L, ovded that ou model s sutable fo the electolyte unde nvestgaton. The effect of L s commeasuable wth seveal othe effects, e.g., the 1 secfc dseson on-on nteactons, coelaton effects etc.,17. In the case of small ons, dect electostatc nteacton wll eval ove these effects. NaF s esecally sutable fo the comason wth Eq (88) snce F - s the smallest anon, t has low olazablty and mnmzes the non-electostatc tems that ae not ncluded n the Boltzmann dstbuton (5) used n ou devaton. Besdes, Na + and F - have smla ad, whch mnmzes the eo fom the aoxmaton that the double laye stats fom R + + R - (nstead of usng lengths, R + + R - fo counte-ons and R + o R - fo co-ons). The exemental data fo the mean actvty coeffcent ± of NaF s well-descbed by the fomula,57 : A Cm lg 1 B C m C m, (91) whee A s the ebye-hückel coeffcent (-0.58 kg 1/ /mol 1/ at 5 ), and the sem-emcal aametes B and b have the values 57 B = 1.8 kg 1/ /mol 1/ and = kg/mol. Eq (91) s vald u to molal concentaton C m = 1 mol/kg. Snce the atal mola volume of NaF s vey small 54, the elaton between molaty and molalty s smly C el [mol/l] = C m m, whee m = kg/l. Fg. 7. Mean on actvty coeffcent ± vs. molalty C m. Comason between the exemental nteolaton fomula Eq (91) (geen dotted lne) of the NaF data 57 and the genealzed ebye-hückel model whch takes nto account quaduolazablty, Eqs (88)-(89), wth R = R Na + R F =. Å followng fom the onc cystallogahc ad 54 and quaduola length L =.1 Å obtaned as a fttng aamete. Fo comason, the lmtng ebye-hückel model (R = L = 0), Eq (8), and the extended ebye-hückel equaton, Eq (85), wth R =. Å ae gven. The comason between the exemental deendence (91) fo NaF and Eqs (88)-(89) s shown n Fg. 7. To detemne the best value of L, we otmzed the met functon 1 mol/kg m m m, (9) 0 ( L ) ln ( C ; L ) ln ( C ) dc wth ± beng the exemental mean actvty coeffcent gven by Eq (91) and s the theoetcal actvty coeffcent fom Eqs (88)- (89). The best value of L s.11±0.06 Å, n good ageement wth the values found fom the atal volume data and lage than those followng fom the entoy data. The ageement between theoy and exement s excellent (Fg. 7). Fo comason, the extended ebye-hückel model s also shown n Fg. 7, wth R =. Å as edcted by Eq (84) (nstead of usng t as a fttng aamete). The lmtng ebye-hückel law (L = R = 0) s also gven fo comason. Note that the effect of L s qute sgnfcant t s of the same ode as the effect of R.

12 6. Conclusons Ou wok nvestgates the effects of the quaduole moment of the molecules n a medum on the oetes of chaged atcles dssolved n ths medum, usng a macoscoc aoach based on 5 the quaduola Coulomb-Amee law (6), genealzng the classcal Posson equaton of electostatcs. () We deved a new equaton of state, Eq (16), elatng the macoscoc densty of quaduole moment and the feld gadent E n gas of quaduoles. The tenso has zeo tace, unlke the one used n Refs. 1 and 1. Ou consttutve elaton nvolves a sngle scala coeffcent, the quaduolazablty, whch was estmated to be = 1-0 Fm o few tmes lage. () We deved the bounday condtons needed fo the fouthode quaduola Coulomb-Amee law (6) at a shecal suface between two meda of dffeent delectc emttvty and quaduolazablty, Eqs. (8)-(9). () The otental of a ont chage n quaduola medum s fnte even at = 0, cf. Eq (4). Ths unexected esult was obtaned evously by Chtanvs 1 wth anothe consttutve elaton fo. (v) The classcal model fo a dssolved on as a chage n a ty was genealzed fo the case of quaduola medum. It was shown that the quaduolazablty of wate affects sgnfcantly the themodynamc oetes atal mola volume v and entoy fo hydaton s 0 of small ons n aqueous soluton. Fom ths effect and the exemental themodynamc data fo v and s 0 fom Ref. 54, the value of the quaduola length, L = ( /) 1/ = 1- Å, could be estmated. (v) The ebye-hückel model fo the dffuse onc atmoshee of an on was genealzed by ncludng the quaduolazablty of the medum n t. Comason wth data fo the actvty coeffcent ± of NaF allowed ndeendent detemnaton of the value of L, whch yelded agan L ~Å. The mnmal dstance of aoach R between ons n the extended ebye-hückel model must not be coected fom the exected value, R = R + + R -, whee R + and R - ae the cystallogahc ad. (v) The ode of magntude of and L obtaned fom these sets of exemental data (v, s 0 and ± ) comaes well wth the ode edcted by othe authos 1,1. (v) The essue and temeatue devatves of wee estmated theoetcally, cf. Eqs ()-(). The estmated values of / and / T show that the effect fom these devatves on the atal mola volume and entoy, Eqs (58) and (66), of the dssolved on s neglgble. The essue and temeatue devatves of the adus of the ty R aound an on wee estmated fom exemental data. Fom the values t can be concluded that the stuctue of the hydaton shell of an on s about tmes stffe than the stuctue of wate. Although the esults obtaned hee ae encouagng, one must not foget that ou aoach uses some stong aoxmatons. Fst, the consttutve elaton Eq (16) s stctly vald fo dluted deal gas only. The assumton that the equaton of state kees the same fom n dense lqud needs addtonal justfcaton. Also, ou model fo the dssolved on (ont chage n a ty) s clealy an ovesmlfcaton, as dscussed n Secton 4. and 5.. Nevetheless, we obtan self-consstent esults and we have enough oof to asset that quaduolazablty has measuable effect on many themodynamc chaactestcs of the dssolved ons (v, s 0 and ± ). The esults obtaned hee fo the equaton of state fo, the bounday condton fo the genealzed Maxwell equatons of electostatcs and the value of the quaduolazablty of wate wll be used n the followng study of ths sees fo the analyss of seveal motant oblems of collod scence. Acknowledgements Ths wok was funded by the Bulgaan Natonal Scence Fund Gant VU 0/1 fom 0. R. Slavchov s gateful to the FP7 oject BeyondEveest. Consultatons wth Pof. Ala Tadje ae gatefully acknowledged. Notes and efeences See sulemental mateal at [URL wll be nseted by AIP] fo: A. evaton of the macoscoc Maxwell equatons of electostatcs wth account fo the quaduole moment. B. Analyss of the met functon of the otmzaton oblems fo the atal molecula volumes and entoes of ons n aqueous soluton C. 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