Nernst-Planck equation

Transcription

1 NenstPlan equaton The man poblem o the peous appoahes s that t s ey dult to estmate the ouplng between on luxes. n the NenstPlan appoxmaton t s assumed that l, ; Ths seems to mean that the luxes ae deoupled but due to the eletoneutalty ondton and the denton o elet uent densty,.4 thee s stong ouplng. Hene, wtng the on phenomenologal equaton and expandng the eletohemal potental, l m~ ln 3., l, 3. and ntodung the phenomenologal oeent, l we eah the NenstPlan equaton: ; / Aboe equaton apples n the Htto eeene ame o, n the absene o oneton, n the xed laboatoy oodnates.

2 thee s oneton n the system e.g. soluton low aoss a membane, we add the oespondng tem: 3.5 duson mgaton oneton on lux thus depends on thee atos: duson due to the onentaton gadent s law, mgaton eletophoess due to the potental gadent and oneton due to the solent low pumpng, stng. t has to be ealed that the oneton tem does not ognate om the gadent o the eletohemal potental but due to mehanal oes. t meely maes the onneton between the eeene ames. Solent low due to the gadent o the hemal potental o the solent.e. osmoss s possble but that eques an osmot membane that does not n pnple allow any solute low. Let s multply Eq. 3.5 by and sum oe all the spees: 3.6 The st obseaton s that the last tem s eo due to eletoneutalty,.e. oneton has no ontbuton to elet uent. Solng om aboe, t s obtaned m t t ln ln / 3.7

3 n Eq. 3.7 we hae used the ondutane o the soluton and the tanspot numbe o an on : t Eq. 3.7 has a ey poound message. Potental dop n the system has an eesble ontbuton, ohm loss,, and a eesble pat, duson potental. n the absene o onentaton gadents, we an wte the Ohm s law E and the dsspaton unton qohm E E that s nown as Joule heat. Although elet uent wee eo on luxes exst due to duson potental but the luxes o aton and anon spees must balane eah othe so that eletoneutalty s ept. Ohm potental dop always omes om an extenal soue and duson potental s an ntenal popety, asng om the deenes between the on mobltes see eq..88 n the textboo. Theeoe, duson potental usually s athe low, unless a system nludes ons wth sgnantly hgh moblty, suh as H o OH, o ey lage ons wth low moblty. As explaned n detal on p. 54 o the textboo, the deene between on mgaton and onduton s that the ome taes plae due to the total potental gadent, nludng both the ohm and duson potental, whle the latte omes only om the extenal soue uent. When we ay out an eletohemal expement we wsh to hae ontol oe the eletode potental but, n pate, the oltage we apply between the eletodes s patly onsumed n the ohm loss and duson potental. n the lght o Eq. 3.7 and 3.9 t s obous that neasng the soluton ondutty, we an edue these losses. These leads us to the onept o a suppotng eletolyte and a taeon to whh we etun late on

4 The analyss o a bnay system s useul n eealng some mpotant aspets o on tanspot. Let s onsde agan an eletolyte A u B u dssoatng ompletely to ons A and B. Hene, u and u whee s the eletolyte onentaton. Eletoneutalty eads u u. Let s wte the NenstPlan equatons o the both spees: u u u 3. u u u Summng aboe equatons ges u u u u Realng that / t s obtaned ate some algeba that u u u u u u u u u dentyng the ollowng goups o aables, u u ; u u u u u u u 3.3

5 u t ; t t u u we an wte the tanspot equaton as eale pesented: t u u ;, Eq. 3.3 s nown as the NenstHatley equaton, and s the duson oeent o the eletolyte. t shows that an on annot duse by tsel but t s oupled to a ounteon. Ths s tue also n a multomponent system although the equatons beome ey omplated. That s also the eason why sngle on mobltes annot be measued om the ondutane data o eletolytes home exese. nstead, loong at and the tanspot numbe t, that ae both measuable quanttes, t u u Example: The duson oeent o H s m /s, that o Cl.3 5 m /s and that o SO m /s. om the NenstHatley equaton, the duson oeent o HCl s m /s and that o H SO m /s. Ths shows that these anons slow down the tanse o H, whle H s aeleatng the tanse o the slowe anons. Ths s exatly the eet o duson potental, eleted n the alues o the eletolyte duson oeents. ue to nheent ouplng o on luxes a longange eletostat nteatons NenstPlan equaton wos ey nely, apat om ey onentated solutons. n ey onentated solutons onon nteatons beome moe mpotant, whh eques anothe appoah. 3.6

6 SteanMaxwell ton oeent appoah n onentated solutons the most natual appoah s to onsde the nteatons between all spees n the soluton. When a solute s tanseng though a soluton t olldes wth solent moleules and othe solutes, ausng ton to ts tanse. The tonal oe depends on the stength o the nteaton, haateed wth a ton oeent ', the numbe o ollsons, the numbe o patles le n the net gas theoy, expessed n mole atons, and the elate speed o the olldng patles. At steadystate, the dng oe o tanse s equal but opposte to the ton oes: dng oe tonal oe we onsde duson, the dng oe ould be the onentaton gadent, leadng to ' x 3.7 We an deelop uthe the let hand sde: ln m ' x m ' x 3.8 n ode to poeed, let s onsde st the smplest ase, a bnay soluton. om eq. 3.8, m 3.9 whee the mole aton and ae nluded n.

7 n the Htto eeene ame the solute lux s expessed as H m m x 3.9 The aboe equaton ntodues the SteanMaxwell duson oeent x ' 3. At modeately dlute solutons x and m ln /. Hene, H 3. At modeately dlute solutons the an and SteanMaxwell duson oeents thus ae equal, but de at hgh onentatons sgnantly. o a spheal patle the Stoes law ges the ton oeent as 6pR hn A, leadng to» BT 6pR hn 6pR h A 3. whee R s the patle adus and N A the Aogado s numbe. Although Stoes law was deed o maosop patles t wos supsngly well also to moleules and ons but not o ons n wate, see next page.

8 Valdty o Stoes law o and Cs n aous solents. l

9 Consdeng a bnay eletolyte soluton whee a salt dssoates nto u atons wth the hage numbe and u anons wth the hage numbe u u, the tanspot equatons beome ~ ~ m m 3.3 Also hee Onsage s epoal theoem apples,. Elet uent densty n Htto eeene s [ ] [ ] H H 3.4 Eq. 3.4 one agan shows that elet uent s ndependent o the eeene eloty hee and n the absene o uent n that ase the lux beomes staght om eq. 3.9, eplang. [ ] œ ß ø Œ º Ø g g m x x x ln ln ln H 3.5 The aboe equaton esembles the NenstPlan equaton wth the exepton o the atty oeent oeton. As the plot n the ollowng page shows, ths oeton tem an eah up to a. % sgnane n the ase o NaCl n wate. n pnple, ths oeton ould be mposed also n the NenstPlan equaton. ln / ln g ~ : m wth m

10 n the ase o pue onduton, n the absene o onentaton gadents, the equatons ae 3.6 Ths pa o equatons an be teated as ollows: 3.7 Subtatng eqs. 3.7 om eah othe ges note Onsage s theoem œ ß ø Œ º Ø 3.8 Solng n the om o Ohm s law, the ondutty s eahed as textboo eq..47, / / 3.9

11 n the pesene o both onentaton gadents and elet uent the textboo pesents the appopate equatons,.5.58, pp An nteestng example s on lquds,.e. solents that ae omed om atons and anons. n that ase the solent s also a hage ae and dssolng any eletolyte n an on lqud maes the system at least tenay, moe oten quatenay. The tanspot o an eletolyte n an on lqud s hadly desbed wth the NenstPlan equaton o s laws. Only when the onentaton o an eletolyte s ey low ompaed wth the on lqud these appoahes an be appled wth ae. n the ase o an on lqud alone, no onentaton gadents an exst, and the only possble tanspot poess s elet onduton: 3.3 Note that these equatons ae atually dental beause and. om ethe o these equatons t s obtaned: Hene, the only deene to the NenstPlan equaton s the SteanMaxwell duson oeent. Measung the ondutty o an on lqud the ton oeent and an be detemned.

12 we hae an on lqud A B and an eletolyte A Cl 3, the ollowng expesson an be deed ate lengthy algeba* t 3 3 M Ø m m ø t3 3 Œ 3 3 œ 3 3 M 3 M ß º whee M and ae the mola mass and mola olume o the eletolyte, espetely. Only at the lmt 3 «the tanspot equaton o Cl edue to the om o s law o NenstPlan wth elet uent, and 3 taes the om 3» The poblem o the SteanMaxwell omalsm s obous: the alues o the ton oeents ae meely unnown. Yet, t s the most appopate appoah as t onsdes all the nteatons n the soluton. t s lea that the teatment o multon systems beomes extemely dult. Theeoe, the NenstPlan equaton has eeed the mao ole n the study o on tanspot. SteanMaxwell appoah s used n modellng the tanspot o, e.g. mxtues o hydoabons n petohemal ndusty. * T. Vana et al. Eletohma Ata