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2 6 Interfacal thermodynamc: Gbb equaton Luuk K. Koopal Chapter 6, Interfacal thermodynamc: Gbb equaton n Interface Scence, Second edton, 008, Wagenngen Unverty, Wagenngen, The Netherland. Avalable va: >publcaton, >Book / >Chapter Phycal Chemtry and Collod Scence P.O. Box EK Wagenngen The Netherland All rght reerved. No part of th publcaton may be reproduced, tored n a retreval ytem, or tranmtted, n any form or by any mean, wthout the pror permon n wrtng of Wagenngen Unverty, or a exprely permtted by law, or under term agreed wth the approprate reprographc rght organzaton. Enqure concernng reproducton outde the cope of the above hould be ent to the laboratory of Phycal Chemtry and Collod Scence, at the addre above. You mut not crculate th book n any other bndng or cover and you mut mpoe th ame condton on any acqurer. L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

3 6 Interfacal thermodynamc: Gbb equaton 91 6 Interfacal thermodynamc: Gbb equaton 6.1 Gbb conventon and Gbb dvdng plane Surface exce functon Gbb equaton Dervaton of the Gbb equaton and the poton of the Gbb urface Gbb-Duhem equaton and t combnaton wth the Gbb equaton Smple notaton of the Gbb equaton Central poton of the Gbb equaton Guggenhem conventon Bnary mxture and the Gbb equaton Expermental determnaton of Γ for a lqud nterface Ideally dlute mxture Compoton, urface exce Γ and relatve urface exce Γ Adorpton from a bnary mxture onto a old nterface Gbb equaton and urfactant adorpton on lqud nterface General behavor Ionc urfactant Thermodynamc contency Calculaton of the change n nterfacal entropy upon a change n compoton Surface charge effect on the adorpton of uncharged component Lteratuur Exerce 114 L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

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5 6 Interfacal thermodynamc: Gbb equaton 89 6 INTERFACIAL THERMODYNAMICS: GIBBS EQUATION 6.1 Gbb conventon and Gbb dvdng plane Equaton Secton 6 In equlbrum ytem compoed of a phae α (e.g. a lqud), and an mmcble phae β (e.g. a vapor) an nterface ext that can be characterzed wth t area and nterfacal tenon γ. Several thermodynamc nterpretaton of γ baed on dfferental equaton, ee Eq. (.9), (.18) and (.19), can be gven. In order to explore the thermodynamc properte of ytem wth an nterface further, a common procedure to plt up the entre ytem nto the bulk phae α and β and to conder the nterfacal properte a exce quantte compared to the two bulk phae. Modelng of the urface and nterfacal tenon alo baed on th approach. In th chapter th approach wll be worked out and the dervaton and ome applcaton of the Gbb equaton wll be gven. Fgure 6.1 Gbb conventon and Gbb dvdng plane poton A an example of the ue of nterfacal exce quantte the pure lqud/vapor equlbrum wll be condered frt, ee fgure 6.1. In general, the tranton from the lqud phae α to the vapor phae β wll occur n a regon of fnte thckne. In th regon gradual change occur n denty, molar energy, molar entropy and other ntenve thermodynamc functon. There no harp boundary of the lqud phae or a harp begn of the ga phae. However, t poble to thnk of a plane that eparate both phae. Th magnary plane of eparaton hould be tuated n the boundary layer between the two bulk phae α and β. Although a plane ha no volume ( V 0), t poble to attrbute all nterfacal properte to th dvdng plane. Th approach called the Gbb conventon and the plane telf the Gbb (dvdng) plane (or Gbb urface). It further aumed that at the Gbb plane dcontnute appear n denty, molar energy, molar entropy, molecular compoton, molecular nteracton and uch. Thu t aumed that the bulk properte of the lqud phae a well a thoe of the ga phae are homogenou up to the Gbb dvdng plane. For each extenve functon the dfference wth the bulk value gve an exce and th exce L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

6 6 Interfacal thermodynamc: Gbb equaton agned to the Gbb plane. Exce quantte that are derved on the ba of the Gbb conventon wll be ndcated by the upercrpt. Fgure 6.1 alo llutrate the concept of urface exce. The total molecular exce wth repect to the bulk phae, whch are thought to contnue to the arbtrarly choen Gbb plane, ndcated by the haded area. The natural poton of the Gbb plane for a lqud/vapour equlbrum of a pure ubtance that where the two haded area are equal, but oppote n gn. Wth th choce the molecular exce zero. A oon a the locaton of the Gbb plane ha been choen the volume of repectvely phae α and phae β, V α and V β, are determned. Becaue V = 0 thee volume have to make up the entre volume of the ytem: α β V = V + V (6.1) 6. Surface exce functon A mlar approach taken for more complcated ytem. Alo for oluton or mxture the poton of the Gbb plane can be fxed by gvng the urface exce of one of the component a certan value (motly = 0). Fxng the poton of the Gbb plane determne V α and V β. When the volume of α and β and ( V 0) have been determned, all other extenve properte, Y, of the ytem can be dvded over α, β and. Th done under the aumpton that Y ha a certan value n each of the bulk phae up to the Gbb plane plu a urface exce defned a: α β Y Y Y Y (6.) or, more explctly α α β β Y Y V y V y (6.3) where y α and y β α α β β are the dente Y / V and Y / V n each of the homogeneou phae α and β. Thee defnton are vald f Y contrbute n one or both of the bulk phae (α and β ), and n the nterface ( ). A change n a functon of tate can alo be dvded over α, β and n the ame way. For ntance, a change of the total nternal energy of the ytem du = TdS pdv + γ da + µ dn (.9) L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

7 6 Interfacal thermodynamc: Gbb equaton 91 can be plt up n a change n phae α : α α α α µ du = TdS pdv + dn (6.4) n phae β : β β β β µ du = TdS pdv + dn (6.5) and a change n the urface phae : γ µ du = TdS + da + dn (6.6) Wth th dvon t aumed that the entre ytem n equlbrum o that α β α β α β T = T = T = T, µ = µ = µ = µ and p = p = p. The lat condton mple that the nterface not curved. For curved nterface p α and p β dffer by an amount a large a the Laplace preure, ee ecton 4.. Equaton (6.4) to (6.6) can be ntegrated at contant T, p, µ and γ. For U α t found that: α α α α µ U = TS pv + n (6.7) For U β a mlar expreon apple. The exce nternal energy a a reult of the preence of the urface U, found to be γ µ U = TS + A+ n (6.8) For equlbrum procee the urface phae, that projected at the Gbb urface, may be condered a an ndependent peudo phae. For non-equlbrum procee th not the cae. The urface phae called a peudo phae becaue t cannot ext f there no α and/or β phae. Havng defned U, alo new functon of tate can be defned for the urface phae, jut a for the bulk phae (ee Ch. ). The urface exce enthalpy can be defned a: γ µ H U A= TS + n (6.9) where γ A take the place of the + pv term n the defnton of the bulk enthalpy. Wth the ad of (6.6) and (6.9) the functon dh can be ealy found L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

8 6 Interfacal thermodynamc: Gbb equaton dh = TdS Adγ + µ dn (6.10) The urface exce Helmholtz energy, F, and the urface exce Gbb energy, G, can now be derved from U en H n a mlar way a for the bulk phae. For F t follow: γ µ F U TS = A+ n (6.11) df = S dt + γ da+ µ dn (6.1) Equaton (6.1) can be ued for the decrpton of procee carred out at contant T and A. For procee at contant T and γ, G more utable: γ γ µ G H TS = U A TS = F A = n (6.13) dg = S dt Adγ + µ dn (6.14) A a conequence of the defnton preented above t follow that: α β U = U + U + U (6.15) α β H = H + H + H + γ A (6.16) α β F = F + F + F (6.17) α β G = G + G + G + γ A (6.18) It hould be remarked that γ A explctly appear n the expreon for H and for G. Th due to the defnton of H, ee Eq. (6.9), and the fact that γ A ha no bulk equvalent. Thu the general defnton (6.) and (6.3) are nvald for H and G. However, the defnton preented above do have the advantage that they have a cloe mlarty wth the defnton of the bulk expreon for the varou energe. The thermodynamc exce functon for flat urface are ummarzed n table 6.1 together wth the correpondng quantte for the homogeneou bulk phae. From the equaton of the ntegral exce quantte, U, H, F and G t follow, that F quantte: (6.11) provde a mple and general equaton for γ expreed n macrocopc L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

9 6 Interfacal thermodynamc: Gbb equaton 93 Table 6.1. Charactertc thermodynamc functon. State functon for a homogeneou bulk phae. U = TS pv + µ n H U + pv = TS + µ n F U TS = pv + µ n G H TS = µ n = du = TdS pdv + µ dn dh = TdS + Vdp + µ dn df = SdT pdv + µ dn dg = SdT + Vdp + µ dn = F + pv = U + pv TS State functon for a flat nterface: Gbb conventon ( V 0). γ µ U = TS + A+ n γ µ H U A= TS + n γ µ F U TS = A+ n µ G H TS = n = du = TdS + γ da+ µ dn dh = TdS Adγ + µ dn df = S dt + γ da+ µ dn dg = S dt Adγ + µ dn = F γa = U γa TS State functon for a flat nterfacal phae: Guggenhem conventon ( V = Aτ ). = + γ + µ U TS A n pv γ µ H U A+ pv = TS + n = γ + µ F U TS A n pv µ G H TS = n = du = TdS + γ da+ µ dn pdv dh = TdS Adγ + µ dn + V dp df = S dt + γ da+ µ dn pdv dg = S dt Adγ + µ dn + V dp = F γa + pv = U TS γa+ pv L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

10 6 Interfacal thermodynamc: Gbb equaton n F µ γ = = fa A µ Γ (6.19) where the urface exce quantte per unt area (areal quantte) are defned a: y a Y A When t qute clear that th an exce functon the upercrpt ometme left out. The urface exce n / A ndcated wth the ymbol (mol/m ) brefly ndcated a the (Gbb) adorpton and Γ. The above expreon emphaze that γ depend on the exce Helmholtz energy per unt area and on the amount of adorpton. Equaton (6.19) how alo that for pure ubtance Γ = 0 gve the natural poton of the nterface. In that cae γ equal to the exce Helmoltz energy per unt area. For mxture of lqud and oluton the tuaton more complcated: γ wll alway be alo determned by adorpton. 6.3 Gbb equaton Dervaton of the Gbb equaton and the poton of the Gbb urface The Gbb equaton gve the relatonhp between adorpton and change of the nterfacal tenon; t therefore the central law of nterface cence. The dervaton of Gbb law baed on the Gbb conventon and the Gbb plane or urface that eparate the two bulk phae. A already mentoned, the Gbb plane a urface wthout volume ( V 0) to whch all nterfacal exce quantte (ntegral en dfferental) are attrbuted. The magntude of the nterfacal exce quantte can be found by preentng the bulk phae α and β a homogenou up to the Gbb urface. The change n exce nternal energy, du, Eq. (6.6), wa derved n ecton 6.: du = TdS + γ da+ µ dn together wth the nternal energy, (6.8): U = TS + γ A+ µ n By takng the dervatve of (6.8) t follow that, n general, for a change du the followng relaton mut hold: d U = T d S + S d T + γd A + A dγ + µ d n + n dµ Combnaton of th equaton wth (6.6) lead to the elementary form of the Gbb equaton: L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

11 6 Interfacal thermodynamc: Gbb equaton 95 S dt + Adγ + n dµ = 0 Th can be wrtten n the better known notaton a dγ = dt Γ dµ (6.0) a wth S / A and Γ n / A, repectvely the urface exce entropy per unt area and the urface exce amount of mole of per unt area (the adorpton of ). Equaton (6.0) ugget that γ a functon of T and all µ '. Th however not the cae! To make th clear, we begn wth conderng a pure lqud n equlbrum wth t vapour. In th cae (6.0) goe over n a dγ = dt Γ dµ (6.1) At phae equlbrum of a pure ubtance, the ytem completely determned by keepng one parameter contant, e.g. the temperature. If T kept contant, then the vapour preure p contant and wth t G µ = µ alo contant. On the other hand f µ gven, then the varable p and T cannot be choen. Th mean that T and µ are not ndependent. Moreover, Γ cannot be determned wth Eq. (6.1), becaue ( γ / µ ) T mpoble: a change n µ accompaned by a change n T and the expreon requre T to be contant. The fact that Γ cannot be determned drectly related to the fact that Γ a functon of the poton of the Gbb plane. If the poton of the Gbb plane fxed, then Γ alo fxed (ee fgure 6.1). A mentoned before, the natural choce for a pure ubtance Γ = 0. Wth th choce Eq. (6.1) reduced to an equaton wth only ndependent varable: dγ = alo a a a dt, where the relatve urface entropy. The upercrpt expree that fxed through the choce Γ = Γ1 = 0. The value of calculated from γ ( T). Th reult ha already been hown n Ch.. Applcaton of (6.0) to a bnary ytem reult n: a can be unequvocally α dγ = dt Γ dµ Γ dµ 1 1 Jut a for a pure ubtance the µ ' and T cannot be vared ndependently, or equvalently, the Γ ' cannot be determned becaue the poton of the Gbb plane ha not been fxed. The problem can be ealy olved by fxng the Gbb plane and thu Γ1 and Γ. Once a value choen for one of the can be calculated. The choce of whch of the two Γ ', the poton of the Gbb plane fxed and the other Γ Γ ' a practcal one and not one of L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

12 6 Interfacal thermodynamc: Gbb equaton prncple. The fact that fxng one Γ alo fxng the other Γ llutrated n fgure 6.. The two fgure on the left hand de repreent the bnary A/W mxture. The denty profle near the nterface between the two phae (lqud and ga) are drawn ( ρ the denty and x the dtance to the Gbb plane). If the Gbb plane located by Γ W = 0 (lght hadng, fgure mot left), then there a relatvely large potve value for Γ A (dotted hadng). If the nterface fxed by choong Γ A = 0 (mddle fgure), then Γ W wll have a large negatve value. Thu fxng the poton of the Gbb plane determne the magntude of the Γ. For a oluton contanng a typcal urface actve ubtance that adorb from a very dlute oluton, the fgure at the rght apple. In th cae the mot logcal choce to take the olvent, W, a reference,.e. Γ W 0. The correpondng value of Γ A ndcated by the dotted hadng. Through the large exce of A n the nterfacal regon and the very low dente outde th regon, the exact poton of the Gbb plane hardly mportant for the value of Γ. However, for reaon of thermodynamc contency the Gbb plane tll ha to be located. A Poton Fgure 6.. The effect of the choce of the poton of the Gbb plane on the adorbed amount. The dente of W and A near the nterface are ndcated by the old lne. The dfferent hadng gve Γ W and Γ A. Now that t clear that one of the Γ ' ha to be fxed n order to apply the Gbb equaton, component 1 choen a reference. For the bnary ytem Eq. (6.0) can thu be wrtten a a dγ = dt Γ dµ (6.) The ndex mean that a a well a Γ are relatve to the value that ha been choen for Γ 1, e.g. Γ 1 = 0, whch fxe the poton of the Gbb plane. Γ the relatve urface exce of and (6.) reduce to a the relatve urface entropy (1 a reference). At contant temperature dγ = Γ dµ. At deal behavour, that mean dµ = RTdlnx, the value of Γ can now be found expermentally from the lope of the graph of γ a a functon of L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

13 6 Interfacal thermodynamc: Gbb equaton 97 ln x. For mult-component ytem the ame method ued and (6.0) become a T = dγ = d Γ dµ (6.3) In order to further llutrate the meanng of a and Γ, (6.3) can alo be derved n a dfferent way, ung the o-called Gbb-Duhem equaton. Th dervaton wll how clearly what the extra relatonhp between T, p and all µ and how, at the ame tme, the relatve exce quantte can be unequvocally defned Gbb-Duhem equaton and t combnaton wth the Gbb equaton The problem aocated wth the nterdependence of the varable n (6.0) can be dealt wth unambguouly by ue of the Gbb-Duhem equaton. Th equaton can be derved n a mlar way a the Gbb equaton. Startng pont for the dervaton are the energy change (6.4) and the energy (6.7) for each of the bulk phae, for example, phae α : and α α α α du = TdS pdv + µ dn (6.4) α α α α µ U = TS pv + n (6.7) Dfferentaton of (6.7) at contant T, p and µ gve α α α α α α α d U = T d S + S d T p d V V d p + µ d n + n dµ Combnaton wth (6.4) gve the Gbb-Duhem equaton for phae α : α α S dt V dp + n α dµ = 0 (6.4)a A mlar Gbb-Duhem relaton can be found for phae β : β β S dt V dp + n β dµ = 0 (6.4)b Both Gbb-Duhem equaton how that T, p and all µ ' cannot vary ndependently. The gnfcance of the Gbb-Duhem equaton can be elegantly llutrated for a bnary lqud mxture at contant T, p and neglectng the vapour phae. For thee condton Eq. (6.4) reduce to (ung mol fracton x = n / n ntead of n ' ): L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

14 6 Interfacal thermodynamc: Gbb equaton x dµ µ 1 = d 1 x Th equaton how very clear that the chemcal potental of one component of the mxture cannot change ndependently of the chemcal potental of the other component. Moreover, f the chemcal potental of component ncreae, then that of component 1 mut decreae and th decreae weghed by the rato of the mole fracton. Smlar relaton can be derved for the other partal molar quantte. For the preent purpoe we wll ue (6.4) to expre µ 1 a a functon of the other µ ' and T. Th can be done va elmnaton of dp from (6.4)a and (6.4)b α α β β ( / ) ( / ) S V S V α β c c dµ 1 = dt dµ α β α β c1 c (6.5) 1 = c1 c 1 where c = n / V the concentraton of n α or β. Th equaton how qute general how, at contant temperature, µ 1 change when a change occur n the chemcal potental of any other component. Th jut the knd of equaton needed to complement the elementary veron of the Gbb equaton (6.0). Subttuton of (6.5) n (6.0) gve the Gbb equaton wth only ndependent varable: where a T = dγ = d Γ dµ (6.3) a a the relatve urface entropy per unt area: a α α β β ( S / V ) ( S / V ) Γ 1 (6.6) α β c1 c 1 and Γ the relatve adorpton of : α β c c Γ Γ Γ1 α β c1 c 1 (6.7) The upercrpt mean that component 1 the reference. The thu found Gbb equaton dentcal to (6.3). Becaue thee equaton are made up of ndependent varable, the relatve urface exce quantte a en quantte are mlar to the earler mentoned Γ can be determned unequvocally. Both a and Γ acqured by fxng the poton of the Gbb plane (for example, by ettng Γ1 0 ). An advantage of the dervaton gven L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

15 6 Interfacal thermodynamc: Gbb equaton 99 here that expreon for a and Γ have been attaned. Yet, the ame expreon could have been derved f the dervaton baed on the frt way reaonng had been perted tll th level (ee Q.6.). Baed on the acqured expreon for a and Γ t can qute mply be hown that thee quantte are ndependent of the poton of the nterface (ee Q.6.1). Therefore, the value of Γ for Γ 1 = 0 exactly the ame a Γ for Γ1 0. The choce of Γ 1 = 0 the eaet and mot elegant choce for lqud nterface. It alo the mplet way of howng n a ketch uch a fgure 6. how large Γ = 0. The choce Γ 1 = 0, however, not necearly a good choce for old-flud nterface, becaue then the mot natural poton of the nterface the urface of the old ubtance and, n general, th doe not correpond wth Γ 1 = 0. The latter ue wll be condered n more detal n Ch. 10, ecton Smple notaton of the Gbb equaton In many crcumtance one encounter the Gbb equaton wrtten n a very mple notaton dγ = RTΓ dlnc (6.8) It hould be realzed that th mple equaton baed on a number of aumpton. Eq. (6.8) can only be derved from (6.3) f all µ j ' ( j 1 ) are contant; () T contant; (3) µ behave deally n order to atfy dµ = RTdlnc and (4) Γ a mplfed notaton for Γ and the Gbb plane located mplctly. In general, the latter two aumpton work out well when the adorpton occur from deally dlute oluton. In th cae ndeed dµ = RTdlnc and effect due to an mplct (unknown) locaton of the Gbb plane are mnor. The mplct aumpton on the locaton of the Gbb plane mut mean that the olvent choen a reference (component 1). The mole fracton of the dolved component of dlute oluton alway much maller than the mole fracton of the olvent ( x 1 1). Therefore, by neglectng the vapour phae the followng apple Γ Γ c Γ Γ x Γ Γ Γ Γ = 1 = 1 x 1 c1 x1 It can thu be een that the value of relatve exce Γ and the exce Γ how very lttle dfference. In fact th pcture of the adorpton correpond to the rght hand de fgure of fgure Central poton of the Gbb equaton L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

16 6 Interfacal thermodynamc: Gbb equaton The Gbb equaton can be appled to all type of nterface; t the lnk (at pt,, µ j con- tant) between the nterfacal tenon, γ (or the urface preure π ( = γ * γ), ee Ch. 8), the relatve adorpton of component, Γ, and the chemcal potental of, µ (or wth deal behavour ln x ). If the relaton between two of thee varable known, then the thrd can be calculated wth the help of the Gbb equaton. Th depcted chematcally n fgure 6.3. Fgure 6.3. Central poton of the Gbb equaton The relaton between Γ and ln x (or µ ) the adorpton otherm, that between γ and ln x the nterfacal tenon curve and that between γ (or π ) and Γ (or a = 1/ Γ ) the o-called urface preure curve. For all three relaton theoretcal equaton can alo be derved and thee equaton then have to be related to each other through the Gbb equaton. If th not the cae, then there a thermodynamc ncontency. The theoretcal relaton between γ (or π ) and Γ (or a = 1/ Γ ) are called urface or two-dmenonal (D) equaton of tate. A D-equaton of tate decrbe the tate of the urface ung urface exce quantte only. For lqud nterface (LG and LL) γ can ealy be meaured a a functon of the compoton of the oluton, x. The relatve adorpton, Γ, can now be calculated provded the condton have been choen properly and the relaton between the chemcal potental and the mole fracton known. The mot common tuaton n whch the analy doe not encounter problem when the adorpton take place from an deally dlute oluton. When a mall amount of a hghly urface actve component, whch noluble n the underlyng lqud, pread on that lqud, a o-called pread monolayer formed. Due to the preence of the monolayer the urface tenon of the lqud lowered wth π mn/m, takng the pure lqud a reference tate. When the area of the urface covered wth the monolayer vared at contant T and monolayer compoton, both the adorpton and the urface preure are vared (bear n mnd that Γ = 1/ a where a the avalable urface L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

17 6 Interfacal thermodynamc: Gbb equaton 101 area per mol and () Γ Γ wth an extremely dlute bulk phae). A thu obtaned π ( a) curve characterze the tate of the monolayer at that temperature and can be further analyzed ung a D-equaton of tate that can be derved wth the Gbb equaton. In Ch. 8 th wll be dcued n more detal. For SG and SL nterface γ not ealy meaurable, but n th cae the change n γ a a reult of adorpton can be calculated from the meaurement of adorpton onto S from repectvely the ga phae or the oluton phae. For ntance, the adorpton Γ of an deal ga onto S meaured, at contant T, a a functon of the relatve preure x ( = p/ p0 ). The urface preure π ( = γ γ ) a a reult of ga adorpton at a relatve ga preure x SG S SG can now be calculated ung the Gbb equaton: ( π ) SG T x= x x= 0 ( S ) = RT Γ dlnx (6.9) For ga adorpton S choen a reference component and becaue the ga phae hghly dlute (S) Γ Γ. Equaton (6.9) ndcate that π SG proportonal to the area under the otherm plotted a Γ (ln x). 6.5 Guggenhem conventon The Gbb conventon not the only way of determnng the nterfacal quantte. Le uual but n telf jut a good, the Guggenhem conventon. Wth th conventon the two mmcble phae are, formally, plt up nto a bulk phae α, a bulk phae β and an nterfacal layer wth thckne τ. The upercrpt ued to ndcate the Guggenhem conventon a oppoed to the upercrpt, for the Gbb conventon. In equlbrum the nterfacal layer may be condered to be a peudo phae. The nterfacal layer now o thck that all change n concentraton take place n the nterfacal layer. Once the thckne τ choen, t aumed that the two bulk phae are homogeneou up to the nterfacal layer. All dfference wth th tuaton are now made part of the nterfacal layer. The dffculte wth repect to the choce of the thckne of the nterfacal layer τ are comparable to thoe of the electon of the poton of the Gbb dvdng plane. Charactertc functon of tate can be derved for the urface phae n a mlar way a wth the Gbb conventon. The man functon are ummarzed n table 6.1. In th cae the volume of the nterfacal layer, V = τ A, alo play a role. In order to obtan relaton that can be ued n practce, alo n th conventon the Gbb-Duhem relaton are needed to avod nterdependence of the partal molar quantte. L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

18 6 Interfacal thermodynamc: Gbb equaton An advantage of ung the Guggenhem conventon ntead of the Gbb conventon that vualzaton of an nterfacal layer conceptually omewhat eaer than of the Gbb plane wthout volume. Becaue of th conceptual advantage the Guggenhem conventon wll be ued n Ch. 7, whch concerned wth the dervaton of expreon for the chemcal potental of the molecule n the urface layer. In th treatment t wll be aumed that the nterfacal layer a monolayer wth a contant thckne, τ m. In that cae V = Aτ m apple, and n procee where A kept contant, 6.6 Bnary mxture and the Gbb equaton V alo contant Compoton, urface exce Γ and relatve urface exce Γ To further elucdate the concept of urface exce, Γ, and relatve urface exce, Γ, a bnary mxture compoed of component 1 and condered. The compoton of the ytem depcted chematcally n fgure 6.4, where the mole fracton of component 1 and n the bulk phae and n the urface layer (nterfacal layer) are ndcated. The compoton of the bulk lqud phae and the nterfacal layer are both thought to be homogenou. The dente n the ga phae are o low that they can be afely neglected. N.B. To mplfy matter further the nterface can be condered a a monomolecular layer of molecule 1 and,.e. the urface layer a monolayer. Th can mplfy modelng of the urface layer. The mole fracton n the bulk lqud are defned a: n n n x x x = = and = = 1 1 n1+ n n n (6.30) where n 1 and n are the number of mole of 1 and n the bulk lqud and n the total number of mole n the bulk. The mole fracton n the urface layer are defned a: n1 n1 n 1 = = and = = 1 1 n1 + n nm nm θ θ θ (6.31) where 1 n and n are the number of mole of 1 and n the urface layer, and number of mole n that layer. n m the total In order to obtan the urface excee wth the Gbb conventon, the locaton of the Gbb plane ha to be choen. In fgure 6.4 two opton are preented, tuaton I and II. In tuaton I the Gbb plane located n the lqud phae jut under the monolayer and the Gbb L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

19 6 Interfacal thermodynamc: Gbb equaton 103 excee Γ1 and Γ can be decrbed a 1 I 1 m m ( Γ ) = θ n / A = (1 θ ) n / A (6.3) ( Γ ) = θ n / A (6.33) I m where A the area of the monolayer. poton Fgure 6.4. Compoton of the nterfacal phae (homogenou), the lqud phae and the ga phae n the cae of a bnary mxture where preferental adorpton of component occur on the LG nterface. The ga phae o dlute that the mole fracton of 1 and are neglgble. The mole fracton of n the lqud phae x, that n the urface layer θ. In tuaton I the Gbb dvdng plane tuated between the lqud phae and the urface layer, wherea n tuaton II t tuated between the urface layer and the ga phae. By combnng Eq. (6.3) and (6.33) t follow that 1 I I nm A I nm A 1 I ( Γ ) + ( Γ ) = / or ( Γ ) = ( / ) ( Γ ) (6.34) The two relatve Gbb excee Γ can now be calculated wth the help of (6.7): ( 1 1 ) I ( 1 ) I ( 1 ) x Γ = Γ Γ I = x 0 1 x nm θ x I = I 1 I = x1 A (1 x) ( Γ ) ( Γ ) ( Γ ) ( ) (6.35) Equaton (6.35) llutrate n a trkng way that the relatve exce Γ depend on both θ and x. In tuaton II, where the Gbb plane le between the ga phae and the urface layer, the L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

20 6 Interfacal thermodynamc: Gbb equaton followng apple: ( Γ ) = ( θ x ) n / A (6.36) 1 II 1 1 m ( Γ ) = ( θ x ) n / A (6.37) II m A expected, thee value are clearly dfferent from thoe that were obtaned above. From thee two expreon t follow that ( Γ ) + ( Γ ) = 0 or ( Γ ) = ( Γ ) (6.38) 1 II II 1 II II For the relatve urface excee t found n tuaton II that: ( 1 1 ) II ( 1 ) II ( 1 ) x Γ = Γ Γ II = x 0 1 x x n m θ x II = II 1 II = II + II = x1 x1 A (1 x) ( Γ ) ( Γ ) ( Γ ) ( Γ ) ( Γ ) ( ) (6.39) Comparng the reult of tuaton I wth thoe of tuaton II t can be een that ( Γ ) ( Γ ) I II, but that I Γ II ( Γ ) = ( ). From the latter t clear that the relatve urface exce ndependent of the poton of the Gbb plane, a t hould be Expermental determnaton of Γ for a lqud nterface (a) Perfect behavour Perfect behavour mean deal behavour of the chemcal potental over the entre mole fracton range. Th expected to happen when molecule 1 and are mlar n nature and approxmately equal n ze. For the expermental determnaton of the adorpton from a perfect bnary mxture the Gbb equaton can be appled to a ere of meaurement of the urface tenon a a functon of the mole fracton of component n the bulk lqud at T = contant: dγ dlnx T = RT Γ (6.40) where ue made of the relaton dµ = RTdlnx. If the relatve exce Γ determned expermentally, the mole fracton of component n the urface layer tll not known. The m adorbed amount per m, n θ / A, can be calculated wth (6.39), provded n / A, the total m L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

21 6 Interfacal thermodynamc: Gbb equaton 105 number of mole per m n de layer known. The value of n / A can be calculated f the value of the molar area a 1 and a are (about) the ame ( a a1 a) and that the adorpton take place n a monolayer. In order to obtan an etmate of a the molar denty of the component can be ued n combnaton wth a model for the packng of the molecule. (b) Non-perfect behavor Mot commonly mxture wll not how perfect behavor. In th cae the expreon for the chemcal potental not obvou and dµ RTdlnx, therefore (6.40) no longer vald. When the molecule are of mlar ze, regular behavour (ee Ch. 7) can be aumed and for th behavour the chemcal potental not only a functon of the mole fracton, but alo of the ntermolecular nteracton. When the molecule are not of mlar ze the tuaton even more complex. In th cae the actvte of the component n the mxture have to be known n order to apply the Gbb equaton. For ordnary mxture meaured actvte can be found n technologcal lterature. Unravelng of the ndvdual adorpton from the relatve Gbb exce can only be done by makng mplfyng aumpton. The fnal concluon that t rather complcated to analyze the adorpton from complex bnary mxture ung Gbb law. m Ideally dlute mxture Mot problem that may occur wth the analy of the adorpton behavour wth bnary mxture vanh f the adorpton of component from the mxture occur at extremely low mole fracton ( x << 1; θ >> x and x1 1). In th cae t can be aumed that the mxture behave a an deally dlute mxture of molecule n a ea of molecule 1 and dµ = RTdln x ( = RTdln c ). Th tuaton occur for ntance wth the adorpton of urfactant and trongly adorbng compact large molecule. Moreover, under th condton alo Γ m * Γ and n A a fully determned by the molar area of component. The latter can be obtaned from the adorpton aturaton level of component. For the olvent molecule that are expelled from the nterface 1 Γ1 1 0 Γ ( Γ = ), but th of no conequence for the analy. If neceary, the value of θ 1 can be mply found va θ = 1 θ. The mot gnfcant practcal cae of adorpton from dlute oluton that of 1 urfactant. In ecton 6.7 ome more attenton pad to th tuaton. For polymer oluton devaton from deal behavor often occur at uch low mole fracton that n practce the chemcal potental even n dlute oluton a complex functon of the L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

22 6 Interfacal thermodynamc: Gbb equaton compoton. A further problem wth polymer ample that they are polydpere. Th mean that a polymer oluton n fact a multcomponent mxture from whch electve adorpton may occur Adorpton from a bnary mxture onto a old nterface If n fgure 6.4 the ga phae exchanged for an noluble old phae, S, the pcture repreent that of adorpton from a bnary mxture onto a old ubtance. The mot natural poton of the nterface now tuaton II and Eq. (6.36) to (6.39) apply. Therefore, the adorpton Γ proportonal to ( θ x). A problem wth a old ubtance that γ SL cannot be drectly meaured. The change n γ SL reultng from the adorpton Γ can however, n prncple, be calculated ung Gbb law f Γ ( x ) determned eparately. How th can be done wll be dealt wth n ecton Gbb equaton and urfactant adorpton on lqud nterface General behavor One of the mot frequent ue of the Gbb equaton n the analy of the adorpton of non-onc and onc urfactant from dlute aqueou oluton on lqud nterface. Schematcally the effect of urfactant on the urface tenon of aqueou oluton hown n fgure 6.5. In the top part of the fgure a γ (ln c) curve (a) hown together wth the correpondng adorpton (urface exce) otherm (b). In the bottom part of fgure 6.5 the effect of the urfactant chan length on the urface tenon depcted for both a homologou ere of onc (c) and non-onc urfactant (d). A depcted n fgure 6.5a the urface tenon progrevely decreae wth ncreang urfactant concentraton (regon I). In regon II the decreae become approxmately lnear and at the tart of regon III the crtcal mcelle concentraton or CMC reached. At th concentraton the urfactant monomer aggregate nto mcelle that contan everal ten of monomer. For urfactant concentraton omewhat above the CMC (regon III) the monomer concentraton tay about contant and o doe the chemcal potental of the urfactant. L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

23 6 Interfacal thermodynamc: Gbb equaton 107 Fgure 6.5. Schematc llutraton of the effect of urfactant on the urface tenon of dlute aqueou oluton (a, c, d) and the adorpton otherm (b) that can be obtaned from fgure (a) through applcaton of the Gbb equaton. The knk n the urface tenon plot (a, c, d) the crtcal mcelle concentraton or CMC. Beyond the CMC the urfactant monomer concentraton about contant. In mot cae t afe to aume that up to the CMC the oluton deally dlute and dµ = RTdln c. Therefore, at contant temperature, t relatvely eay to calculate the ( w) relatve urface exce, Γ, from the dervatve of the γ (ln c) curve. The typcal reult depcted n fgure 6.5b. The adorpton frt ncreae but n regon II, where the lope of the γ (ln c) curve contant (wthn expermental error), t reache a plateau value. In regon III the monomer concentraton reman contant. Applcaton of the Gbb equaton n regon III not poble. However, t mot lkely that the adorpton tay at the level reached at the CMC Ionc urfactant When the Gbb equaton appled to onc urfactant the extra complcaton occur that L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

24 6 Interfacal thermodynamc: Gbb equaton the urfactant are docated. In th cae the expreon for the chemcal potental can be obtaned by conderng eparate on and takng nto account the electroneutralty of both the bulk oluton and the nterfacal layer, or by conderng neutral on combnaton n the form of alt. Both treatment hould lead to the ame reult. Here the treatment baed on onc component choen, becaue the expreon for the chemcal potental of the component mplet. The example are worked out for the frequently ued urfactant odum dodecylulphate (NaDS). Dolved n water (w) th urfactant completely docated both n the preence and abence of alt: + NaDS Na + DS Surfactant wthout alt In the abence of added alt (NaCl) and at contant temperature, the general Gbb equaton may be wrtten a: ( w) ( w) DS DS Na dγ = Γ dµ Γ dµ (6.41) Na where water taken a reference. Becaue the nterface ha to be electrcally neutral t follow that: ( NaDS ) ( w) ( w) ( w) DS = Na = Γ Γ Γ By neglectng onc actvty coeffcent (aumng they are contant) the followng apple: dµ DS = RTdlncDS and dµ Na = RTdlnc Na. Becaue of electrcal neutralty n the bulk t follow that cna = cds, o that dln cds =dlncna = dlncnads. Equaton (6.41) can therefore be wrtten a: ( w) NaDS dγ = RTΓ dlnc (6.4) NaDS Surfactant wth an exce of alt In th cae the general Gbb equaton hould be wrtten a: ( w) ( w) ( w) Na Na DS DS Cl Cl dγ = Γ dµ Γ dµ Γ dµ (6.43) Becaue NaCl preent n exce the followng apple: dµ Na 0 and dµ Cl = 0 and (6.43) L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

25 6 Interfacal thermodynamc: Gbb equaton 109 change to: ( w) ( w) DS DS RT NaDS dγ = Γ dµ = Γ dlnc (6.44) NaDS Equaton (6.4) and (6.44) dffer by a factor becaue n the frt cae both the DS and the Na concentraton change and n the econd cae only the DS concentraton. Surfactant and alt preent n mlar amount The two tuaton n the prevou ecton are extreme cae. If the urfactant and alt concentraton are comparable, the adorpton can be analyzed when urface tenon meaurement are avalable at a range of contant alt concentraton. By way of llutraton fgure 6.6 how how the NaCl concentraton affect the urface tenon of the urfactant: odum dodecyl pyrdnum chlorde (DPC). It obvou that the alt concentraton change both the lope of the curve and the CMC. For the complete analy wth the Gbb equaton th et of curve requred plu the et of curve of γ v. ln c (alt). The latter can be obtaned by takng cro ecton of fgure 6.6 at contant urfactant concentraton. Content applcaton of the Gbb equaton wll gve not only ( w) DP ( w) Cl ( w) Na Γ but alo Γ and Γ. A mplfcaton of the Gbb analy poble when the adorpton of the urfactant coon neglected. In th cae one obtan (ee Q.6.3): ( w) urfactant dγ = prt Γ dlnc (6.45) urfactant where p = 1 + c c urfactant urfactant + c alt (6.46) wth urfactant = NaDS or DPC and alt = NaCl. When the urfactant caton and the alt caton are not the ame, the two equaton above alo apply a long a the co-on adorpton can be neglected. Th approxmaton hold bet for hghly charged urface,.e. at uffcently hgh urfactant on adorpton. L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

26 6 Interfacal thermodynamc: Gbb equaton Fgure 6.6. Set of γ v ln c curve for aqueou oluton of odum dodecyl pyrdnum chlorde (DPC) n the preence of dfferent concentraton of NaCl ( 0 mmol/l; 5 mmol/l; 0 mmol/l; Δ 100 mmol/l). 6.8 Thermodynamc contency A correct adorpton otherm equaton hould alway atfy the Gbb equaton and when th the cae the equaton thermodynamcally content. Thermodynamc contency play, for example, a role n compettve adorpton. Accordng to (6.3), the general Gbb equaton, the followng relaton hould hold for a mxture contanng component and j and a reference component 1: Γ dµ µ j Γ = j T, µ j T, µ (6.47) Th equaton can be ued for two purpoe. The change n relatve adorpton of component j a a functon of a change n the chemcal potental of component gve nformaton on the change n relatve adorpton of upon a change n the chemcal potental of j. () When an adorpton otherm equaton avalable that decrbe the compettve adorpton, then predcton obtaned wth th equaton hould atfy (6.47). If (6.47) not atfed, then the compettve adorpton otherm equaton that ued thermodynamcally ncontent (.e. ncorrect) and the predcton arng from that equaton hould be dregarded. 6.9 Calculaton of the change n nterfacal entropy upon a change n compoton That the Gbb equaton can lead to relaton that otherwe would be dffcult to envage L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

27 6 Interfacal thermodynamc: Gbb equaton 111 can be llutrated wth the followng example. The change n nterfacal entropy due to a change n the concentraton of component n a dlute mxture can be calculated, when the adorpton of component j from that mxture meaured a a functon of the temperature at contant compoton. The relaton between thee varable can be found by crodfferentaton n the general Gbb equaton (6.3): Γ a j = µ T T, µ j ' µ ' (6.48) The rght hand de not uually meaured, but gven a j / T c ' ( Γ ). The tranton of µ ' contant to c ' contant, when T not contant, can take place by makng ue of the extended chan rule for dfferentaton. Th not neceary for the left hand de, here T contant and for deal behavour µ j contant correpond wth c j contant. Although t farly complcated to carry out th applcaton of Gbb law, t neverthele how the power of the nterfacal thermodynamc Surface charge effect on the adorpton of uncharged component Smlarly a the prevou ecton, th ecton an llutraton of how the Gbb equaton can be ued to nterpret meaurement n an otherwe unexpected drecton. It alo an example of how the Gbb equaton can be ued for a complcated ytem wth many component. The example gve nght n the way the chargng of old by t charge determnng on n the preence of a background electrolyte can be decrbed and () that the urface charge may affect the adorpton of a non-onc component. The ytem that ha been tuded cont of an AgI-upenon n an aqueou oluton of KI, AgNO 3, KNO 3 (z) and butanol (b). It to be hown that quantty eparately) depend on the urface charge. The urface charge defned a ( w) b Γ (wthout meaurng th = F( Γ Γ ) (6.49) AgNO3 KI and can be obtaned expermentally by ttraton of an AgI dperon wth AgNO 3 or KI and meaurng the equlbrum pag (or pi) wth a et of Ag/AgI electrode and an approprate reference electrode. By ubtracton of the blank conumpton of AgNO 3 (or KI) from that n the preence of AgI the urface charge found. The relaton between ( = log c Ag ) the adorpton otherm for the charge determnng on (Ag + and I ). and pag L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

28 6 Interfacal thermodynamc: Gbb equaton Fgure 6.7 depct the relaton between and pag for AgI at contant T n the abence of butanol and n the preence of four butanol concentraton. (Smlar relaton have been found for other adorbent and neutral molecule). From the reult t appear that by addng butanol the lope of the curve decreae. Th ha to have a drect relaton wth the butanol adorpton. A marked phenomenon the common nterecton pont at not correpond wth the pont of zero charge. m pag, whch doe Fgure 6.7. Surface charge of AgI at varou concentraton of butanol. To analyze the problem, the general Gbb equaton (6.3) wrtten out ung neutral component and T = contant: ( w) ( w) ( w) ( w) ( w) AgI AgI AgNO3 AgNO3 KI KI z z b b d γ = Γ d µ Γ d µ Γ d µ Γ d µ Γ d µ Becaue of the equlbrum AgNO3 + KI Ä AgI + KNO3 t follow that dµ + dµ AgNO3 = dµ AgI + dµ z. Now one varable can be elmnated: µ KI choen. Further dµ AgI = 0 becaue AgI preent a a old ubtance and T contant. By takng all th nto account and ung the defnton of the urface charge (6.49) t follow that ( w) w ( KI ) ( ) ( w) dγ = dµ AgNO Γ d d 3 z + Γ µ z Γ b µ b F Cro-dfferentaton between the frt and the thrd term at contant µ z gve: 1 F ( w) Γ b = µ b 3 3,, µ T AgNO µ AgNO µ z µ b, µ z, T KI 0 Becaue T contant (and thu alo the µ ' ), µ z, µ b and µ AgNO 3 contant equvalent to L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

29 6 Interfacal thermodynamc: Gbb equaton 113 c, c repectvely pag contant. Becaue of the equlbrum AgNO3 Ä Ag + NO3 t z b further apple that dµ = dµ + dµ, where AgNO3 Ag NO3 NO3 + dµ 0 becaue µ z ( KNO 3) contant (exce KNO 3). For dµ Ag t can be wrtten RTdlncAg =.30RTd pag. After ubttuton t follow that: ( w) Γ b = µ b pag, cz, T c b, cz, T F RT pag By ntegraton over the pag tartng from the common nterecton pont (m) one obtan Γ ( w) ( w) m b Γb pag m µ b pag pag, c z, T.30RT ( pag) ( pag ) = dpag F (6.50) The left hand de of (6.50) gve the butanol adorpton at a gven pag mnu that at (common nterecton pont). In the rght hand de m pag µ b can agan be replaced by RTdlnc b. The dfferental quotent can be read from fgure 6.7 after whch graphcal ntegraton can follow. For the moment only the gn mportant a th wll how what the meanng of the m pag : a) rght of m pag : µ b T, pag, cz > 0, dpag > 0, thu R. de of (6.50) < 0. b) left of m pag : µ b T, pag, cz < 0, dpag < 0, thu R. de of (6.50) < 0. Concluon: Γ ( pag ) at t maxmum n the common nterecton pont. Wthout b meaurng Γ ( pag ) the preent analy how that the adorpton of neutral butanol b depend on the urface charge, () how trong the dependence and (3) that the common nterecton pont n fgure 6.7 the pont of maxmum butanol adorpton. The phycal explanaton of a maxmum n the adorpton of neutral molecule onto a charged urface that water dpole are exchanged for alphatc egment wthout a dpole moment. The orentaton of the adorbed water dpole depend on the urface charge and the exchange of thee dpole eaet when they are randomly orented. Apparently th the cae for AgI at a certan charge ( 8 mc/m ) and not at the pont of zero charge. L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

30 6 Interfacal thermodynamc: Gbb equaton 6.11 Lteratuur F.C. Goodrch, The thermodynamc of flud nterface n Surface and Collod Scence, Vol. 1, E. Matjevc Ed., Wley-Intercence, New York, 1969, p. 1. E.A. Guggenhem, Thermodynamc, 3rd ed. North-Holland Publhng Company, Amterdam, 1957, p. 46. (Guggenhem conventon). 6.1 Exerce Q6.1 In the fgure below the nterfacal regon of a bnary ytem compoed of component 1 and drawn. Show that the relatve adorpton Γ ndependent of the poton of the Gbb plane. Th can be done by locatng the Gbb plane frt at poton x and to wrte down the expreon for Γ ( x) and () to hft the Gbb plane over a dtance λ to poton y, ee the fgure, and to how that Γ ( y) baed on Γ 1 ( y) and Γ ( y ) the ame a Γ ( x). Q6. Show that Eq. (a) hold for a heterogeneou ytem compoed of a phae α and a phae β eparated wth an nterface wth an area A. where α β α α Γ AΓ A = ( c c )( V V ) (a) Γ the relatve adorpton baed on Γ 1 = 0, and the correpondng α β dvon of the total volume V = V + V, Γ the adorpton wth the Gbb plane n an arbtrary poton wth V + V = V. Show that applcaton of Eq. (a) to component 1 and lead to α β c c Γ = Γ Γ1 α β c1 c 1 α β L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.

31 6 Interfacal thermodynamc: Gbb equaton 115 Q6.3 Derve Eq. (6.45) and the correpondng expreon for p, Eq. (6.46), for a ytem compoed of NaDS and NaCl. Do the ame for a ytem compoed of NaDS and KCl. Indcate clearly the aumpton that hould be made n order to arrve at (6.45) Q6.4. In an attempt to derve the Gbb equaton n an alternatve way, the followng equaton of tate defned obtan by cro-dfferentaton n th equaton that Z G µ n. Derve the expreon for dz and n j γ = A µ pt,, µ ' j pt,, µ j' I t allowed to make the ubttuton j ( j ) pt,, µ ' Gbb conventon requred? Γ = n A? I n th cae the Q6.5 Wth an electro-capllary meter the nterfacal tenon between mercury and an aqueou electrolyte oluton can be meaured a a functon of the appled potental, E, acro the nterface. For uch a ytem the followng equaton of tate apple dg= SdT + Vdp+ EdQ+ γ da+ µ dn where Q the total charge at the nterface. The term EdQ the electrcal contrbuton to the Gbb energy of the ytem. The urface charge denty can be Q A and the dfferental electrcal capacty a defned a ( ),, ', C = ( E) pt,, n '. pt n E Show that the electro-capllary curve,.e. the functon γ ( E), a parabola when the capacty C a contant. Show alo that the pont of zero charge correpond to the top of the capllary curve. L.K. Koopal, Interface Scence, 008, Wagenngen Unverty, Wagenngen, The Netherland.