Physcal & Intefacal Electochemsty 013 Lectue 3. Ion-on nteactons n electolyte solutons. Module JS CH3304 MoleculaThemodynamcs and Knetcs Ion-Ion Inteactons The themodynamc popetes of electolyte solutons exhbt maked devatons fom deal behavou. Ths s because stong electolytes ae completely dssocated nto ons and the devaton fom deal behavou s due to the opeaton of electcal nteactons between the ons: they ae long ange (they vay wth dstance as 1/) and ae essentally Coulombc. One should also consde shot ange ( -6 ) effects asng ethe fom ncomplete dssocaton o on assocaton. The latte facto becomes mpotant n non-aqueous solutons whee the delectc constant s low. Also n moe concentated onc solutons on/solvent nteactons become mpotant. Any theoy attemptng to pedct the equlbum themodynamc popetes of onc solutons must concentate on nteonc foces and be able to calculate the fee enegy change asng fom such nteactons. Intal State: No on/on nteactons G II Fnal State: on/on Inteactons opeatve G II = Gbbs enegy of on/on nteactons We model the ntal state of non nteactng ons as an assembly of dschaged sphees, wheeas the fnal state whee on/on nteactons opeate s modelled as an assembly of nteactng chaged sphees. Intal State: Assembly of dschaged Non-nteactng sphees W C G II Fnal State: Assembly of chaged nteactng sphees 1
oth catons and anons ae chaged up usng ths potocol. We wsh to solate the Gbbs enegy fo on/on nteactons due to one specfc on whch we label. Hence we wok n tems of chemcal potentals and ntoduce,i whch defnes the change n chemcal potental asng fom the nteacton between a specfc on and the assembly of all of the ons I. How do we calculate ths chemcal potental change? We assume that the efeence on s unchaged n an assembly of chaged ons. We then calculate the electostatc wok equed to chage up ths efeence on. W C, I Ths analyss s smla to that used n devng the on equaton. qz e I, A C A q0 NW N dq q q N A N A dq qdq q 0 4 0 4 0 q0 q N A q NA 4 0 4 0 0 q We can smplfy the latte expesson by notng The followng expesson fo the electostatc potental ( ). 4 0 Hence the change n chemcal potental s gven by the followng fundamental expesson. Intal state Dschaged efeence on Fnal State Chaged efeence on of adus. N A I, The poblem n essence : The appoach of Debye & Huckel (DH). Real System: Assembly of solvent Molecules & ons Solvated anon We note theefoe that the poblem n essence bols down to theoetcally evaluatng the electostatc potental ( ) at the suface of the efeence on usng the methods of classcal electostatcs. Hence the chemcal potental change due to the nteacton between a specfc efeence on and ts onc envonment essentally depends on the electostatc potental poduced at the suface of the efeence on by the est of the onc ensemble (assembly). We now dscuss the way n whch ths quantty s evaluated. The most smple appoach s due to that developed many yeas ago by Debye and Huckel (DH) (193). The man poblem s to quantfy the tme aveaged spatal dstbuton of ons aound the cental efeence on. The cental dea n the DH appoach s to eplace the stuctued solvent by a delectc contnuum as was done when we dscussed the on model. The net chage densty () due to all othe ons s epesented by an on cloud o onc atmosphee. Cental efeence on (assumed to be a caton) Solvent molecule Solvent eplaced by stuctueless Delectc medum of pemttvty Ionc atmosphee
We daw fom two dscplnes to quantfy the model. The fst s classcal electostatcs, and the second s statstcal mechancs. We also have a choce wth espect to on dmensons. () Pont chage model: most smple appoach, vald fo vey dlute solutons (slghtly polluted wate!!), esults n the fomulaton of the Debye-Huckel (DH) Lmtng Law. () Fnte on s model: moe ealstc, on s s quantfed by a s paamete a. () The on dstbuton about the cental efeence on s teated n tems of the Posson equaton of classcal electostatcs n tems of a contnuous chage densty (). Ths s only an appoxmaton snce n ealty one has a dscete dstbuton of counteons about the cental efeence on. Hence thee s a local excess of ons of opposte chage about a cental efeence on: ths s the onc atmosphee. Ionc atmosphee (due to tendency Of ons of opposte chage to cluste togethe) Dffuse chaged envonment Chage on onc atmosphee opposte to that on on Man assumptons of the DH Theoy 1. We assume that stong electolytes ae completely ond at all concentatons fo whch the theoy s vald. Ion assocaton poceses ae theefoe gnoed n the model.. Ions ae egaded as pont chages to a fst appoxmaton. Ths assumpton wll be vald fo vey dlute solutons. Fo moe concentated solutons the ons ae of fnte s (egaded as sphees of adus a) whch ae not subect to dstoton and whch possess a symmetc Coulomb feld. 3. Inteonc foces ae assumed to be Coulombc n fom: othe foces such as shot ange dspeson nteactons ae neglected. 4. The electcal potental enegy of an on s much smalle than ts themal enegy so we can state that q z e k T 5. The delectc constant of the solvent s unchanged by the pesence of the on. Hence delectc satuaton effects ae neglected. Hence the macoscopc delectc constant of the bulk solvent s used. Ths s a poo appoxmaton snce due to the effect of delectc satuaton effects, the effectve local delectc constant nea an on may be much less than ts bulk value. 3
Mathematcal development of DH Theoy. Chage densty In the followng analyss potentals and dstances wthn the electolyte ae consdeed elatve to a patcula on taken as efeence. We set () as the mean electostatc potental at a dstance fom the efeence on. Volume Element d - - + - - Electolyte Soluton Cental efeence on Solvent molecules Refeence on Medum of delectc Constant + medum Suoundng ons yeld excess Chage densty () We use classcal electostatcs to obtan An expesson temed the Posson equaton Whch seves to elate the electostatc potental () and the excess chage densty ( ) both evaluated at a pont dstant fom a cental efeence on as follows. () () 1 1 We note that the chage dstbuton about the cental efeence on s sphecally symmetc and so the potental s also sphecally symmetc. 0 0 8.854x10 CN m 78.5 (wate) 1 1 We now note that the excess chage densty s defned as the chage pe unt volume and we can eadly defne t n the followng manne. n () In ths latte expesson n () epesents the numbe densty (concentaton) of on labelled n the onc atmosphee dstant fom the cental on. 4
Now s whee Statstcal (Sadstcal??) Mechancs eas ts head and entes stage left. We can elate the numbe densty of on n the onc atmosphee to the bulk concentaton of the on whch we label n va the well known oltzmann Equaton. In the latte expesson the quantty U ( ) epesents the Potental of mean foce. Now f the cental efeence on s a caton, we can state that we have a local defct of catons and a local excess of anons n the onc atmosphee. Mathematcally ths can be stated by: n () n n n The opposte petans f the efeence on s an anon. We also assume that the potental of mean foce s gven by The followng smple expesson: U () U() n() n exp kt Ths assumpton s temed the cental appoxmaton. Now U elates to the tme Aveage foces between the ons. If on/on nteactons ae absent then U = 0. Hence n ( ) = n. If the nteacton foce s attactve then U < 0 and n ( ) > n, And thee s a local accumulaton of ons n excess of bulk values. In contast, f the nteacton foce s epulsve wth U > 0 then n ( ) < n, and thee s a local depleton of ons. If the cental on has a fnte s a then we can wte the followng. U a z U a 4 Futhemoe, as then () 0 0 Ths condton mples that electoneutalty petans the bulk soluton and that the excess chage densty s defned only on a local scale. Now comes the mpotant bt. We substtute the oltzmann equaton fom Statstcal Themodynamcs nto the Posson Equaton fom clacal electostatcs, and obtan The famous (o nfamous dependng on you vewpont) Posson-oltzmann (P) Equaton. 1 d d () 1 () n exp d d kt Ths non-lnea dffeental P equaton must be solved to obtan an expesson fo the electostatc potental ( ). 5
The P equaton s found n a geat many physcal stuatons petnent to electochemcal systems, most notably when one s dscussng the nteface between a plana electode and an electolyte soluton and when one consdes the nteface between a collodal patcle and ts suoundng envonment. In these mpotant stuatons one wll often have to attempt to solve the full non lnea equaton whch can be qute challengng. Howeve unde the pesent ccumstances we do not need to do ths and we can smplfy mattes consdeably by makng the mpotant assumpton that the soluton s so dlute that the ons wll aely be close togethe. Unde such ccumstances, the nteonc potental enegy wll be much less than the aveage themal enegy. Ths assumpton means that we can state the followng. 1 kt Now we can make the followng assumpton. When x s small we can assume that exp[-x] 1-x. exp 1 kt kt We now substtute ths appoxmaton nto the P equaton to obtan the followng. d n 1 1 1 1 n kt 1 d 1 n n d d k T kt In the latte we have ntoduced the electoneutlty Condton whch states that n 0 We also have ntoduced the DH paamete such that e zn kt 1 L D We wll show that the nvese kappa -1 = L D s an effectve measue of the onc atmosphee. We see that depends on the numbe densty of ons n the atmosphee and also note the followng quanttatve elatonshp: Nc A n N 3 Aom 10 Whee c denotes the mola concentaton of on and 0 denotes the densty of the solvent. We ecall the defnton of onc stength I as 1 1 cz mz Hence the DH paamete k s elated to the onc stength I as follows We now ntoduce the Debye Length L D as 1/ and wte that Ne A 0 kt I 1 LD kt 1 I 0Ne A Hence the Debye Length s a measue of the thckness of the onc atmosphee And we note that t deceases wth nceasng onc stength I. Fo an aqueous medum we note that 0 = 0.997 x 10-3 kgm -3, = 78.5, and assumng That T = 98 K, k = 1.38 x 10-3 JK -1, e = 1.60 x 10-19 C, N A = 6.03 x 10 3 mol -1 and 0 = 8.854 x 10-1 C N -1 m - we can calculate that the Debye Length can we wtten n the followng useful fomat. 10 1/ -1/ -1 L D 3.0510 mol kg m I/ molkg The followng esults ae obtaned fo a 1,1 electolyte at 98 K. m/mol 0.0001 0.001 0.01 0.10 1.0 kg -1 L D /nm 30.5 9.64 3.05 0.96 0.30 Hence as m 0, the onc atmosphee speads out and becomes moe dffuse. 6
We can calculate the total excess chage contaned n the onc atmosphee whch suounds the cental on as follows. z e exp 4 exp 4 Shell volume d We consde a sphecal shell of thckness d located At a dstance fom the cental efeence on. + d The chage n sphecal shell s gven by dq () d ()4 d Chage n shell volume dq () d Hence the total chage of the onc atmosphee s got by Integatng ove all the sphecal shells of value dq as follows. qcloud dq 4 ( ) d 4 expd exp d 4 0 0 0 0 The last ntegal may be solved va ntegaton by pats to obtan the fnal sgnfcant esult. q z eq cloud on Hence we have shown that the total chage on the suoundng on cloud s equal and opposte to that on the cental efeence on. Electoneutalty s theefoe satsfed. We now detemne the contbuton of the onc atmosphee to the electostatc potental at a pont dstant n the atmosphee fom the cental efeence on. on Atmosphee absent () () on cloud () () cloud on on () 4 () exp 4 cloud Ion plus atmosphee pesent Now fom the pncple of supeposton of potentals, the total potental at a dstance fom the cental efeence on s gven by the sum of the potental due to The cental on and that due to the on atmosphee. Hence we can make the followng analyss. Ion absent Hence the electostatc potental due to the onc cloud s gven by the followng expesson. cloud () exp 1 4 In the lmt of vey dlute solutons we can make the followng appoxmatons. L 1 1, D exp 1 exp 1 cloud Electostatc potental of Cloud n vey dlute soluton lmt. () 1 4 4 4L D 7
We now wsh to evaluate the total potental at the suface of the efeence on (assumed to be a pont chage). 1 1 () 4 LD z ( 0),0 4 LD Now we fnally ecall that the change n chemcal potental due to on/on nteactons s gven by The followng expesson. N A NA I,,0 (1) 8 LD Ths s the fundamental theoetcal esult whch we eque. We now elate ths esult to the Themodynamcs of electolyte solutons. Fo an deal soluton contanng non-nteactng solute we can wte the followng expesson fo the Chemcal potental. 0 ( deal) RT lnm In contast fo a eal soluton contanng nteactng solute patcles we ntoduce actvtes athe than molaltes whee we note that a = m. 0 0 ( eal) RT lna RT ln m Now we note that I, ( eal) ( deal) Hence, RT ln I () We can now of couse get a handle on the meanng of the onc actvty coeffcent as t elates to the change n actvty of onc solvaton fom compang eqn.1 and eqn.. ln N z e z e A 1 1 LD LD 8 0 RT 8 0 kt (3) log S DHLL Az We now ecall that the nvese Debye Length s elated To the Ionc stength I of the soluton accodng to the followng elatonshp. 1 0Ne A 1/ L D I (4) 0 kt Fom eqn.3 and eqn.4 we deve the elatonshp between the onc actvty coeffcent abd the squae oot of the Ionc Stength whch s the fundamental esult of the Debye-Huckel Theoy vald fo vey dlute electolyte solutons, and s temed the DH Lmtng Law. 1/ log Az I (5) 1.8 10 A.3038 0NAe kt T 3 6 3/ 3/ 0 Fo an aqueous soluton ( =78.5) at T = 98 K the DH constant A = 0.51 mol -1/ kg 1/. The DH lmtng law expessed n eqn.5 wll be vald only fo vey dlute solutons whee m < 10-3 mol kg -1. One apdly obseves maked devatons fom ths law as the soluton becomes moe complcated. Now we have deved an expesson fo the actvty of a sngle on. Such a quantty cannot be measued expementally. Instead we can measue the mean onc actvty coeffcent, and one can show that the followng holds. 1/ log Az z I AFV 8
The valence facto F V = z + z - can be evaluated as follows fo dffeent electolyte types. Electolyte Type Example Ionc valence F V (1,1) NaCl z + = 1, z - = -1 1 (1,) MgCl z + =, z - = -1 (1,3) LaCl 3 z + = 3, z - = -1 3 (,) MgSO 4 z + =, z - = - 4 (,3) Fe (SO 4 ) 3 z + = 3, z - = - 6 The DH lmtng Law holds only fo the most dlute solutons (slghtly polluted wate. The model must be modfed n a numbe of espects f ageement between theoy And expement s to be obtaned at hghe concentatons. Extenson to basc DH Model The most smple extenson s that of fnte on s. The pont chage pctue of a efeence on s only vald fo vey dlute solutons. If we wsh to descbe moe concentated solutons, then we must assume that the efeence on s of a fnte s. We theefoe ntoduce a fnte on s paamete a. The on s paamete cannot : () e less than the sum of the cystallogaphc ad of the ons () Moe than the sum of the hydated ad of the ons () Is most pobably less than the sum of the solvated ad because the solvaton shells may be cushed. Hence the paamete a should be temed the mean dstance of closest appoach. Hence fo a fnte s on, the onc atmosphee stats at a dstance a fom the cente of the efeence on. Ion L D a Atmosphee 9
As pevously noted the pont chage pctue s vald only fo vey dlute solutons but becomes less so as concentaton s nceased. We theefoe ntoduce the on s paamete a the mean dstance of closest appoach. We agan may wte the followng expessons. In the latte A s a constant of ntegaton Whch must be evaluated fom the bounday condtons. Agan, the chage caed by an nfntesmally thn shell of soluton located at a dstance fom the cental on s gven by the followng expesson. dq 4 d A A exp exp Now fo an on of fnte s, the onc atmosphee begns at a dstance a fom the cente of the efeence on. Also the total chage caed by all the sphecal shells between = a and = s obtaned by summaton ove all sphecal shells. Ths chage s balanced by the equal and opposte chage of the cental efeence on (the electoneutalty condton). Hence mathematcally we can make the followng asseton. 4 d Ths expesson may be solved va ntegaton by pat to obtan the a followng esult. 1 a 4 A exp d 4 A exp a a exp a Hence the electostatc potental at a dstance Fom the efeence on s gven by the followng expesson. exp exp a a () 4 1 A 4 1 a We agan use the supeposton pncple to obtan an expesson fo cloud as follows. () () cloud on on() 4 expaexp cloud () 1 4 1a Snce no ons fom the atmosphee can appoach the cental on moe closely than = a, then the potental that exsts at the ste of the cental efeence on due to the onc atmosphee s obtaned by settng = a n th expesson ust deved above to obtan the followng esult. 1 1 1 ( a) 1 1 1 4a1a 4 a 4 1a Now fo dlute dolutons a <<1 and the expesson ust deved decomposes to that altady Obtaned fo the pont chage model as ndeed t should. 10
We now consde the Guntelbeg-Mulle chagng pocess. The efeence on s ntoduced nto the soluton n an unchaged state. One then nceases the chage gadually to ts fnal value gven by z e. The wok done n ths pocess s now evaluated. The electostatc wok equed pe on W c to ncease the chage on on fom q = 0 to q = z e can be calculated va the followng pocedue. q z e z e W ( a) dq 1 q dq C 4 1 a q 0 0 1 q 4 1 a WC 8 1a 0 Hence the change n chemcal potental due to the nteacton between the cental efeence on and the assocated onc atmosphee s gven by the followng expesson below. N 8 1 a A I, NW A C Ths s the fundamental theoetcal esult fo fnte s ons. As befoe we can state the followng elatonshps. I, RT ln N A ln 8RT 1a 8k T 1a Hence we may obtan the extended DH equaton fo fnte s ons as follows. Whee we have noted the followng Az log 1 a 0 3 6 0NAe 1 1.810 A 3/ 3/ 3/.30380k T T Ne A 0 kt Ne 5.010 11 A 0 1/ 0 kt T Fo wate whee = 78.5 at T = 98 K we note that A = 0.51 kg 1/ mol -1/ And = 3.8 x 10 9 mol -1/ kg 1/ m -1 11
Relatonshp between Ionc Stength I and molalty m, I = km X - X - X 3- X 4- M + 1 3 6 10 M + 3 4 15 1 M 3+ 6 15 9 4 M 4+ 10 1 4 16 Relatonshp between actvty coeffcents on vaous scales. f mole facton scale (x ) molal scale (m ) y mola scale (c ) f 10.001MS m 0.001 S / 0 f y c M M 0.001cM 0 m0 y 0 m0 y 10.001mM c c y M M 0 S mola mass of salt mola mass of solvent = desnty of soluton = densty of solvent = = stochometc onzaton facto Fnally the extended DH equaton may be wtten n tems of the mean onc actvty coeffcent n the followng manne. Az z AF log V 1a 1a A easonable ft fo the on s paamete s 0.4 nm. The a facto s best teated as an adustable paamete used to obtan a best ft between expementally obtaned and theoetcally calculated actvty coeffcents. Note that fo hgh concentatons of some electolytes the paamete a attans negatve values whch of couse ae not physcally easonable fo concentated onc solutons. The extended DH equaton s geneally vald up to I 0.05 mol kg -1. One gets even moe complex behavou as the electolyte soluton ecomes moe concentated. Ths ponts out that the extended DH model s stll too smplstc an appoach. A bette ft follows the expesson pesented below, Whee C s an adustable paamete. Az z log C 1a Hydaton effects by Robnson and Stokes have poven to be useful. We quckly ente ealm of cuent eseach when concentated solutons ae consdeed. 1