Phys 344 Ch 5 Lect 4 Feb 28 th,

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1 hys 44 Ch 5 Lect 4 Feb 8 th, Wed /4 Fr /6 Mn /9 Wed /11 Fr / 1 55 Dlute Slutn 56 Chemcal Equlbrum Revew Exam (C 107 S 60, 61 Bltzmann Statstcs Bnus: hys Sr hess 4pm HW17: 7,76,8 HW18:8,84,86,88,89,91 HW19:,, 4, 6, ,87,9 HW14,15 HW16,17,18 Fr next tme make sme pwerpnt t wrk thrugh math nnuncements Frday Math Club 1:0 want yu guys t jn them t talk abut hw t hnr Davd Last me s we have thrugh ut ths chapter, we ve lked t Gbbs Free Energy t determne, when a system s gt a chce, what chce t wll make We bserved that the Gbbs Free Energy was Gmxed Gunmxed Smx + U mx + V Just t get a qualtatve feel, we dd just the easy parts shwed hw the frst tw terms depended n the mxng ratn, x B /, nly waved ur hands abut hw the energy term wuld be effected, and cmpletely gnred the ressure-vlume term Stll that was enugh t understand sme physcal phenmena Our task was t get a sem-quanttatve understandng f hw mxtures behave Hmgeneus Mxtures vs Dmans We saw that, when t was energetcally favrable fr tw ppulatns t reman segregated, dmans frmed; hwever, thse dmans weren t pure fr nn-zer temperature they themselves were slght mxtures at lw temperatures, and mre and mre mxed at hgher and hgher temperatures, untl sme crtcal temperature s reached and the slutn becmes hmgeneus Qualtatvely, ths makes ntutve sense; we argued t n mderately quanttatve grunds hase transtns We saw sme very nterestng behavr Gas-Lqud crss the -lqud transtn, dmans als frmed rch n ne substance and lqud rch n the ther Lqud-Sld crss the lqud-sld transtn, there were even mre nterestng pssbltes If the sld preferred tw dfferent chemcal structures the sld tself wuld be n dmans f dfferent chemcal rats, and thrugh meltng, yu get fur dfferent dmans tw sld and tw lqud 55 Dlute Slutns It wuld be nce t get less qualtatve / mre quanttatve We can d that fr Dlute Slutns, fr example, lghtly salted water Defntns: Slvent the prmary, r dmnant substance In ths example, the H 0

2 hys 44 Ch 5 Lect 4 Feb 8 th, 009 Slute the secndary, r mnrty substance he salt n ths example he defnng crtera s that the slute s s dlute that yu really needn t wrry abut them a nteractng wth each ther r b cmpetng fr states there are far mre states avalable than there are slute partcles hs s the exact same crtera fr treatng a as a classcal deal the partcles never nteract and they dn t cmpete fr states S, as we try t thnk hw G vares wth n a dlute slutn, we call upn the G( relatnshp fr deal ses Slvent and Slute Chemcal tentals Our startng Equatns Last tme, we started frm Gmxed Gunmxed Smx + U mx + V hs tme, we re gng t take G (Eq n 57 as ur startng pnt, because, fr an deal Gas, we actually have a quanttatve expressn fr the chemcal ptental S, fr systems n whch the mxed ppulatns are deal ses, we can get a very exact expressn; n cases n whch they re nly apprxmately deal ses, we ll get apprxmate expressns whch can stll be useful We ll start wth a few ld, famlar relatns: hermdynamc Identty fr G: dg Sd + Vd + (Eq n d 5 Whch means Gbbs Free energy changes wth changng ppulatn as dg d, G (Eq n 57 (whch s rather mraculusly true V πmk kln ( n 6 fr an Ideal Gas h Oh, and f curse Vk fr an Ideal Gas Between these, we can put tgether relatnshps that explctly shw hw Gbbs Free Energy depends n the a mxture (the number f slutes, the relatve number f slutes, the mlarty f slute Ideal Slute and Slvents Let s explctly cnsder a small ppulatn f slute n a large ppulatn f slvent, and fr cncreteness, we ll magng bth are deal ses Fr a start, f the slutn s pure Slvent, then the Gbbs Free Energy s smply G slvent(, slvent nd the chemcal ptental fr a gven materal can be lked up w, hw des G change as we add slute?

3 hys 44 Ch 5 Lect 4 Feb 8 th, 009 dg d G slute, G G + slute d slute slute slute d slute dg slute d slute S, we just need t fnd an expressn fr the slute chemcal ptental and ntegrate t V πm slutek slute kln slute h w, the Slvent s by far the dmnant, s t s far t say that t takes up mst f the slvent space: V slvent k s V k Makng ths substtutn sgnfcantly restrcts the range f quanttatve applcablty whle t s far t treat the slute as an deal, n many cases t wuldn t be far t treat the slvent as such Stll, the qualtatve results seem rght slvent k πm slutek slute kln slute h One way t separate ths ut s k πm slutek slute slute kln + kln h slvent Observe that the frst term s purely a functn f,, and m slute (whch s a cnstant, we ll just call t f(, slute slute f (, + kln Eq n 570 slvent Even f we aren t dealng wth tw deal ses, we mght hpe fr ths basc functnal dependence G G G G G G slute d f (, + k ln slute slute f (, + slute [ ] d slvent slute slute slutek ln [ ] slutek slvent slute k ln [ ] k G G + slute f (, + slute slvent S, G (, + f (, + kln slvent slvent slute slute slute slute [ ] k slvent slute

4 hys 44 Ch 5 Lect 4 Feb 8 th, r 7 w that we ve gt G, we can fnd dg slutek slvent slvent(, (Equatn 569 d, slvent Unfrtunately, snce we re vergng n Chemstry here, we re n a zne where Chemstry vcabulary s cmmn, and rather than talkng abut the rat f slute/slvent, they phrase thngs n terms f Mlarty Mlarty mles f slute / kg f slvent slute f (, + kln[ m ] f (, + kln (, + kln avegadr [ M ] slvent slute slvent slvent f (, + kln + kln n slvent slute m slvent avegadr Osmtc ressure Imagne a pl f pure slvent, say, water Yu put n a packet f slute, say, salt n the slvent he packet walls allw slvent but nt slute (water but nt salt t pass What happens s the packet swells as t sucks n water hs s Osmss Why des t happen? In smple physcal terms, f there s ual pressure n nsde and utsde f the packet, that means there s ual frce transferred acrss the packet wall va cllsns / crssngs f partcles n the tw sdes Hwever, a small fractn f thse cllsns n the nsde are due t slute, whch means slghtly less f the cllsns are slvent than n the utsde s mre slvent partcles ht and crss the wall frm the utsde t the nsde Osmtc ressure get quanttatve, we can slve fr the pressure dfference that wuld stp the flw he argument fr the ulbrum cndtn s akn t the arguments we d emplyed fr thermal ulbrum and mechancal ulbrum, nly ths tme, we ll mnmze the whle system s Gbbs Free Energy (rather than maxmzng ts entrpy here wll be n transfer f partcles between the tw regns, pure and slutn, f the ttal Gbbs Free Energy s mnmzed wth respect t transfer f partcles, that s, f yu magne changng the number f slvent partcles n the pure sde f the membrane by, d puresde, e, the number n the slutn sde by -d puresde, then dg dg ttal puresde dgslutnsde + 0 puresde puresde puresde dslvent dslvent dslvent dgpuresde dgslutnsde dgslutnsde puresde puresde slvent sde dslvent dslvent dslvent But these dervatves are just the chemcal ptentals f the slvent n the tw sdes puresde slutnsde (, (, slvent pure slvent slutn slvent m n slute slvent m slvent

5 hys 44 Ch 5 Lect 4 Feb 8 th, r 76 he value n the pure sde s just what we call (, pure whereas the chemcal ptental n the slutn sde s slutek (, slutn s, slvent slutek (, pure (, slutn slvent w, f we a prr assume that the pressure n the slutn sde sn t very dfferent frm that n the pure sde, we can d a aylr seres expansn f the pure chemcal ptental at the slutn s pressure arund the pure-sde s pressure: (, slutn (, pure + ( slutn pure hen we d have slutek (, pure (, pure + ( slutn pure ( slvent slutek slutn pure slvent G 1 G 1 But, s V then slvent slvent slvent slutek ( slutn pure V slvent ( ( slutn slutn pure pure slute k slvent slute k slvent V Yu need t ncrease the pressure n the slutn sde such that the number f slvent cllsns wth the barrer ncreases t ual the number frm the slvent sde hus ncreasng the number f cllsns als ncreases the pressure he bk pnts ut that ths s essentally the pressure that the slute exerts n the wall He dscurages thnkng f t t lterally that way; hwever, t make sme ntutve sense he ttal pressures are the same n bth sdes f the wall, but the cntrbutn f the slvent n the slutn sde s less than that n the pure sde by ths amunt Meanwhle, the number f dffusns acrss the barrer, depends n the number f slvent cllsns wth the barrer n the exact same way that the pressures d Only made t ths far, but there was a 15 mn nterruptn at the begnnng f class, s I may make t further next tme Blng and Freezng nts

6 hys 44 Ch 5 Lect 4 Feb 8 th, Blng: In ulbrum, Gbbs Free fr the tw sub-systems (the and the lqud are tgether mnmzed s befre, ths means that the crrespndng chemcal ptentals are ual lqud slvent (, slvent(, he lqud slvent s chemcal ptental s slghtly reduced frm what t wuld be when lqud lqud Bk pure: slvent(, slvent (, If we assume that the effect f addng the slvent s small, then fr a gven emperature, the ulbrum pressure s nly slghtly dfferent frm what t wuld be fr pure slvent, Or, smlarly, fr a gven ressure, the blng temperature s nly slghtly dfferent frm what t wuld be fr pure slvent, Expand fr ressure S, we say that lqud lqud B slvent(, slvent (, In turn, we say that the pure slvent chemcal ptental sn t far frm that fr ulbrum, e, lqud lqud lqud slvent slvent (, slvent (, + ( slvent Smlarly, slvent(, slvent(, + ( Of curse, the pure ulbrum chemcal ptentals fr and lqud are ual (that knd f defnes ulbrum, s lqud (, (, slvent slvent lqud S, (, (, lqud slvent lqud slvent slvent slvent lqud slvent Bk (, (, lqud slvent + B k ( slvent (, slvent B k ( s befre, the dervatves are smply V/ ( k slvent (, slvent + V Bk V ( ( lqud If yu assume that the vlume per partcle n the lqud phase s cnsderably smaller than that n the phase, and the deal law hlds V n phase, then k (

7 hys 44 Ch 5 Lect 4 Feb 8 th, Bk k ( B B ( 1 Expand fr emperature k smlar prcess results n + 1 B L where L S S s the latent heat f vaprzatn ( lqud We d 81 (they need fr 8 Cnsder freezng / meltng where the sld s pure lqud sld slvent slvent sld slvent lqud lqud Bk (, slvent(, slvent (, sld lqud sld slvent lqud slvent B k slvent (, + ( slvent (, + ( But, at the pure ulbrum temperature, lqud sld (, (, slvent slvent sld slvent ( lqud slvent Bk ( he dervatve s sld sld slvent Gslvent 1 dervatve, s S S sld lqud Bk ( ( Bk [ S S ] sld lqud [ S S ] ( slvent S sld slvent smlarly fr the ther ( ( Bk sld lqud L fnally get the frm f 590, apprxmate k as k r vce versa r8