Phys 344 Ch 5 Lect 5 Feb 28 th,

Transcription

1 hys 44 Ch 5 Lect 5 Feb 8 th, Wed. /4 Fr. /6 Mn. /9 Wed. /11 Fr. / Dlute lutn 5.6 Checal Equlbru Revew Exa (C , 6.1 ltzann tatstcs nus: hys. r. hess 4p HW17: 7,76,8 HW18:8,84,86,88,89,91 HW19:,, 4, 6, ,87,9 HW14,15 HW16,17,18 Fr next te ake se pwerpnt t wrk thrugh ath nnunceents Frday Math Club 1:0 want yu guys t jn the t talk abut hw t hnr Davd. Last e s we have thrugh ut ths chapter, we ve lked t bbs Free Energy t deterne, when a syste s gt a chce, what chce t wll ake. We bserved that the bbs Free Energy was xed unxed x U x. Just t get a qualtatve feel, we dd just the easy parts shwed hw the frst tw ters depended n the xng ratn, x = /, nly waved ur hands abut hw the energy ter wuld be effected, and cpletely gnred the ressure-lue ter. tll that was enugh t understand se physcal phenena. Our task was t get a se-quanttatve understandng f hw xtures behave Hgeneus Mxtures vs. Dans We saw that, when t was energetcally favrable fr tw ppulatns t rean segregated, dans fred; hwever, thse dans weren t fr nn-zer teperature they theselves were slght xtures at lw teperatures, and re and re xed at hgher and hgher teperatures, untl se crtcal teperature s reached and the slutn beces hgeneus. Qualtatvely, ths akes ntutve sense; we argued t n derately quanttatve grunds. hase transtns We saw se very nterestng behavr. as-lqud. crss the -lqud transtn, dans als fred rch n ne substance and lqud rch n the ther. Lqud-ld. crss the lqud-sld transtn, there were even re nterestng pssbltes. If the sld preferred tw dfferent checal structures the sld tself wuld be n dans f dfferent checal rats, and thrugh eltng, yu get fur dfferent dans tw sld and tw lqud. 5.5 Dlute lutns It wuld be nce t get less qualtatve / re quanttatve. We can d that fr Dlute lutns, fr exaple, lghtly salted water. Defntns: lvent = the prary, r dnant substance. In ths exaple, the H 0

2 hys 44 Ch 5 Lect 5 Feb 8 th, 009 lute = the secndary, r nrty substance. he salt n ths exaple. he defnng crtera s that the slute s s dlute that yu really needn t wrry abut the a nteractng wth each ther r b cpetng fr states there are far re states avalable than there are slute partcles. hs s the exact sae crtera fr treatng a as a classcal deal the partcles never nteract and they dn t cpete fr states., as we try t thnk hw vares wth n a dlute slutn, we call upn the ( relatnshp fr deal ses. lvent and lute Checal tentals Our startng Equatns. Last te, we started fr U. hs te, we re gng t take xed unxed x x (Eq n 5.7 as ur startng pnt, because, fr an deal as, we actually have a quanttatve expressn fr the checal ptental., fr systes n whch the xed ppulatns are deal ses, we can get a very exact expressn; n cases n whch they re nly apprxately deal ses, we ll get apprxate expressns whch can stll be useful. We ll start wth a few ld, falar relatns: herdynac Identty fr : d d d (Eq n d 5. Whch eans bbs Free energy changes wth changng ppulatn as d d,. (Eq n 5.7 (whch s rather raculusly true k k ln ( n.6 fr an Ideal as h Oh, and f curse =k fr an Ideal as. etween these, we can put tgether relatnshps that explctly shw hw bbs Free Energy depends n the a xture (the nuber f slutes, the relatve nuber f slutes, the larty f slute Ideal lute and lvents. Let s explctly cnsder a sall ppulatn f slute n a large ppulatn f slvent, and fr cncreteness, we ll agng bth are deal ses. Fr a start, f the slutn s lvent, then the bbs Free Energy s sply (, slvent slvent nd the checal ptental fr a gven ateral can be lked up w, hw des change as we add slute?

3 hys 44 Ch 5 Lect 5 Feb 8 th, 009 d d slute, slute d slute slute slute d slute d slute d slute, we just need t fnd an expressn fr the slute checal ptental and ntegrate t. slute kln slute slute h k w, the lvent s by far the dnant, s t s far t say that t takes up st f the slvent space: = slvent k s k Makng ths substtutn sgnfcantly restrcts the range f quanttatve applcablty whle t s far t treat the slute as an deal, n any cases t wuldn t be far t treat the slvent as such. tll, the qualtatve results see rght. slute kln slvent slute k One way t separate ths ut s slute kln k slute h k slute h k kln slute slvent Observe that the frst ter s ly a functn f,, and slute (whch s a cnstant, we ll just call t f(,. slute slute f (, kln Eq n 5.70 slvent Even f we aren t dealng wth tw deal ses, we ght hpe fr ths basc functnal dependence. d slute slute, slvent f (, slute slute (, f (, f (, slvent kln slute slvent slute slute slute kln kln d f (, slute slute slvent slute slvent slute slute slute kln k k slute slvent slute k

4 hys 44 Ch 5 Lect 5 Feb 8 th, r. 7 w that we ve gt, we can fnd d slute k slvent = slvent (, (Equatn 5.69 d, slvent Unfrtunately, snce we re vergng n Chestry here, we re n a zne where Chestry vcabulary s cn, and rather than talkng abut the rat f slute/slvent, they phrase thngs n ters f Mlarty. Mlarty = les f slute / kg f slvent. slute avegadrnslute slute f (, kln f (, kln slvent f (, (, kln kln M avegadr slvent slvent slvent kln Ostc ressure Iagne a pl f 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 Osss. Why des t happen? In sple physcal ters, f there s ual pressure n nsde and utsde f the packet, that eans there s ual frce transferred acrss the packet wall va cllsns / crssngs f partcles n the tw sdes. Hwever, a sall fractn f thse cllsns n the nsde are due t slute, whch eans slghtly less f the cllsns are slvent than n the utsde s re slvent partcles ht and crss the wall fr the utsde t the nsde. Ostc ressure. get quanttatve, we can slve fr the pressure dfference that wuld stp the flw. he arguent fr the ulbru cndtn s akn t the arguents we d eplyed fr theral ulbru and echancal ulbru, nly ths te, we ll nze the whle syste s bbs Free Energy (rather than axzng ts entrpy. here wll be n transfer f partcles between the tw regns, and slutn, f the ttal bbs Free Energy s nzed wth respect t transfer f partcles, that s, f yu agne changng the nuber f slvent partcles n the sde f the ebrane by, d.sde,.e., the nuber n the slutn sde by -d.sde, then d d d d d sde slutn. sde sde slvent n slvent slute ttal sde slutn. sde 0 sde sde sde slvent dslvent dslvent d d slutn. sde slvent. sde slvent slvent slvent slvent sde dslvent d d ut these dervatves are just the checal ptentals f the slvent n the tw sdes sde slutn. sde (, (, slvent slvent slutn

5 hys 44 Ch 5 Lect 5 Feb 8 th, he value n the sde s just what we call, whereas the checal ptental n the slutn sde s slute k (, slutn s, slvent ( slute k (, (, slutn slvent w, f we a prr assue that the pressure n the slutn sde sn t very dfferent fr that n the sde, we can d a aylr seres expansn f the checal ptental at the slutn s pressure arund the -sde s pressure: r. 76 (, (, slutn hen we d have ut (, (, k slutn slutn slute slutn slvent, s 1 1 slvent slvent slvent slvent slutn slutn slute k slvent slute k slvent slute k slvent then slute k slutn Yu need t ncrease the pressure n the slutn sde such that the nuber f slvent cllsns wth the barrer ncreases t ual the nuber fr the slvent sde. hus ncreasng the nuber 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 ake se ntutve sense. he ttal pressures are the sae n bth sdes f the wall, but the cntrbutn f the slvent n the slutn sde s less than that n the sde by ths aunt. Meanwhle, the nuber f dffusns acrss the barrer, depends n the nuber f slvent cllsns wth the barrer n the exact sae way that the pressures d. Only ade t ths far, but there was a 15 n nterruptn at the begnnng f class, s I ay ake t further next te. lng and Freezng nts

6 hys 44 Ch 5 Lect 5 Feb 8 th, lng: In ulbru, bbs Free fr the tw sub-systes (the and the lqud are tgether nzed. s befre, ths eans that the crrespndng checal ptentals are ual. lqud slvent (, slvent (, he lqud slvent s checal ptental s slghtly reduced fr what t wuld be when lqud lqud :. k slvent (, slvent (, If we assue that the effect f addng the slvent s sall, then fr a gven eperature, the ulbru pressure s nly slghtly dfferent fr what t wuld be fr slvent,. Or, slarly, fr a gven ressure, the blng teperature s nly slghtly dfferent fr what t wuld be fr slvent,. Expand fr ressure lqud lqud, we say that. k slvent (, slvent (,. In turn, we say that the slvent checal ptental sn t far fr that fr ulbru,.e.,. lqud slvent (,. lqud slvent (,. lqud slvent ( slvent larly, slvent(, slvent(, ( Of curse, the ulbru checal ptentals fr and lqud are ual (that knd f defnes ulbru, s. lqud (, (, slvent slvent lqud, (, (,. lqud slvent. lqud slvent slvent. lqud slvent (, (, ( k slvent. lqud slvent k slvent ( (, slvent k ( slvent (, slvent ( s befre, the dervatves are sply / k ( ( lqud If yu assue that the vlue per partcle n the lqud phase s cnsderably saller than that n the phase, and the deal law hlds n phase, then k

7 hys 44 Ch 5 Lect 5 Feb 8 th, k k ( ( Expand fr eperature slar prcess results n k L where L s the latent heat f vaprzatn. lqud 1 We d 81 (they need fr 8 Cnsder freezng / eltng where the sld s lqud slvent sld slvent sld. slvent sld slvent (, (, (, (, lqud slvent sld slvent (. lqud slvent ut, at the ulbru teperature, sld. lqud. (, (, slvent sld slvent ( he dervatve s dervatve, s sld ( k sld slvent lqud slvent sld slvent lqud lqud ( ( ( sld slvent k lqud. slvent k k 1 slvent (, sld slvent lqud slvent ( k slarly fr the ther ( ( k sld lqud L fnally get the fr f 5.90, apprxate k as k r vce versa. r8