Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

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1 Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y NernstPlnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly neutrl system (liquid or solid), inluding diffusion oeffiient, diffusion flux, nd ompre these prmeters with those we previously developed for the neutrl systems of toms or moleules, iming to see how the eletrostti potentil estlished with the ions ffet the diffusion kinetis. Further understnding of the hrmony diffusion of ions nd the ounter ions (ions) for neutrl system of : how to dedue the NernstPlnk eqution, nd understnd the oupled (in hrmony) diffusion oeffiient,. Lol internl eletril field, s defined s E=/, n e uilt up if the diffusion oeffiient of nd ions re different, tht is. In lst leture: We lerned how to dedue the diffusion flux for ions, speifilly the ions nd ions for neutrl system, whih n e, for exmple, slt like CCl (=1, =) dissolved in medium (e.g., wter), where it dissoites into free ions: one tion C, nd two nion Cl , diffusing in the queous medium. The proess of dissoition of is usully referred s ioniztion, s written s where nd re the vlenes of ion (tion) nd ion (nion) respetively. Note tht >0 ut <0, nd =. The potentil round n individul ion is now onsisted of oth the regulr hemil potentil (s mrked s µ) nd the eletril potentil (s mrked s Φ) uilt up y the eletril hrge; nd this omined potentil is usully referred s eletrohemil potentil s mrked s η. Here for the nd ions, we hve: η = µ e Φ nd η = µ e Φ iffusion of (tully nd ions) in the medium is regrded s oupled diffusion of wherein numer of tions move in hrmony with numer of nions, so s to mintin eletroneutrlity within the system (in other words, no eletrostti potentil reted, otherwise the free energy will inrese). For the oupled (hrmony) diffusion s ssumed ove, we hve derived the diffusion flux for nd ions: 1
2 C dη C = = e k T k T C = k T (1) C dη C = = e k T k T C = k T () Where the nd re the diffusion oeffiient of nd ion, respetively, nd C is the onentrtion of the dissolved slt. From the two equtions ove, we hve =, or = 0, s indeed onsistent with the oupled diffusion flux of nd ions s disussed ove, where the diffusion of is in hrmony with (or neutrlized y) the diffusion of. lso onsidering the eletroneutrlity of, we hve =, where >0, <0 then we hve, = defining = = This implies tht when ions nd ions move in hrmony, it is s if one moleule moves. Now let s understnd this hrmony through the following tretment. Relling the ssumption we mde for the ioniztion equilirium, whih remins during the diffusion: Tht mens, µ µ = µ Where µ is the hemil potentil of. Note tht is eletrilly neutrl.
3 ifferentiting the ove eqution, = (3) gin, onsidering = We n hve = α nd = α, whereα is positive integer. For exmple: for slts like KCl, CCl, α =1; for slts like MgSO 4, α =. Thus, timing the oth side of Eq. (3) withα, we hve = α α α Or, d d d = α (4) The lefthnd side is the term in prenthesis in equtions (1) nd (). Sustituting Eq. (4) into Eqs. (1) nd (), we hve C = α k T Or, C α = k T (5) The sme wy, we hve, C α = k T (6) now C α = { } k T (7) With Then, = α C α = { } k T 3
4 The sme wy, C α = { } K T (8) Now, from Eqs. (7) nd (8), we hve indeed seen tht = =  gin, when ions nd ions move in hrmony, it is s if one moleule moves. so, 1 α = C { } K T (9) Let s write = α (10) The ove is known s the NernstPlnk eqution. is the diffusion oeffiient of in the medium under onsidertion. It is pprently relted to the individul ioni diffusion oeffiients, n thus e regrded s prmeter refleting the oupled diffusion of nd ions. Tking s defined in Eq. (10), the diffusion flux for in generl n e given y = C { } (11) k T onentrtion moility fore Where C = C, = α, µ = µ µ, s defined ove, respetively. Considering two extreme onditions: if >>, α = = if >>, α = = 4
5 Let us now return to the eqution for the potentil grdient d Φ given y the eqution we developed in lst leture, = e (1) For dilute solution of s ssumed ove, we hve µ = µ ktln C = µ ktln C 0 0 µ = µ ktln C = µ ktln C 0 0 The ove ssumption is not ompletely true, even for dilute solutions. However, we will use it for simpliity. Then, kt d = C kt d = C So, Eq. (1) n e rewritten s = e ( ) kt d = e C Rememer = α, = α, so, = α, = α Sustituting oth nd into the ove eqution, we hve α( ) k T 1 d = e C The lol eletril field (E) is given y α( ) k T 1 d E = = e C This is lol internl field, not mesurle. If =, there is no internl eletril field. 5
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