Partition Functions for independent and distinguishable particles
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1 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur /0.0/.77 vs. q for dsngushabl vs ndsngushabl syss Drvaon of hrodynac Proprs fro :, S,, µ, P xapls Paron Funcons for ndpndn and dsngushabl parcls W wan o gnralz for dsngushabl and ndsngushabl parcls. L s ak asr on ourslvs by consdrng only ndpndn subsyss,.., h parcls do no nrac. hn h nrgy of h whol sys, wrn as, ε + can b splfd bcaus h nd r (ε nracon ) s 0. hs allows us o say: ( ) ε nracon q whol sys. Dsngushabl parcls (and ndpndn), ndpndn of ach ohr, and labld: ε,, a ε,, b ε + ε xp a q xp ε and q xp ε a b xp ( ε + ε ) b a b ε xp caus h sus ar ndpndn of ach ohr a ε ε xp q xp Gnralz for N ndpndn and dsngushabl parcls: b N q q ε xp
2 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur /0.0/.77. Indsngushabl parcls (and ndpndn) Now: no and labls! ε + ε whr,,,,,. xp xp ( ε + ε ) Now: canno facor ou of su du o ndsngushably: can spara sus WHY? parcl ε 0 parcl ε 67 Can b dsngushd fro h suaon whr parcl ε 67 parcl ε 0 So ovrcounng s prsn. Dvd by! In gnral, for N parcls, dvd by N! q! N q N! Drvng hrodynac Proprs usng ll hrodynac quans can b calculad fro h paron funcon h olzann facor and paron funcon ar h wo os poran quans for akng sascal chancal calculaons. If w hav a odl for a aral for whch w can calcula h paron funcon, w know vryhng hr s o know abou h hrodynacs of ha odl. Now w wll rla our favor hrodynac proprs o q, h paron funcon. hs s our k bwn h croscopc and acroscopc dscrpons. sng h convnn duy varabl /k b o splfy hngs.
3 Lcur /0.0/.77 3 Drvng nrgy, p s rck so hn Subsung hs no <> > < Drvng S: p p k S s p 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld
4 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur /0.0/.77 S k Spl Σ up S k + s r: + nd r: s So nd r s Cobng boh rs: S k + W can do hs for svral ohr proprs! Hlholz Fr nrgy, S k + 4
5 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur /0.0/.77 Chcal ponal, µ µ N, V µ N, V Prssur, P P V, N P V Now hav all h hrodynac proprs as a funcon of, h paron funcon. W can us hs n a coupl xapls. n pplcaon xapl: Vsualzng h coplx sas of a DN olcul L s consdr h unwndng of a suprhlx of DN as an xapl of usng h Gbbs fr nrgy o dscrb h populaon of sas. Closd suprhlcal DN can b unwound by ran wh DNs, whch ncks h DN. h brak n on chan allows h doubl hlx o ws rlav o s axs and rlax h suprcog n rspons o hral flucuaons. h DN can b hald by ran wh lgas o sal h nck. Whn nckd, h DN wll achv an qulbru whr so of h DN s coplly unwound, so has on rgh-handd ws, so has on lf-handd ws, so has wo rgh-handd wss, and so on. Whn h lgas s addd o frz h flucuang DN by fxng h nck, h collcon of DN olculs s capurd n an qulbru dsrbuon of dffrn confguraons. W can us h conncon bwn h probably of confguraons and h fr nrgy o prdc hs dsrbuon. (snbrg and Crohrs), N Iag rovd du o copyrgh rasons. Plas s: Fgur 4-0 n snbrg, Davd S., and Donald M. Crohrs. Physcal chsry: wh applcaons o h lf scncs. Mnlo Park, C: nan/cungs, 979. ISN: X. 5
6 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur /0.0/.77 h frozn collcon of DN olculs wh dffrn dgrs of suprhlcy can b sparad by gl lcrophorss o allow analyss of h rlav concnraons of ach spcs: Iag rovd du o copyrgh rasons. Plas s: Fgur 4-(a) n snbrg, Davd S., and Donald M. Crohrs. Physcal chsry: wh applcaons o h lf scncs. Mnlo Park, C: nan/cungs, 979. ISN: X. h gl lcrophorss char shows a clar sparaon of unqu DN spcs, occurrng a dffrn concnraons as a funcon of hr suprhlcy. (h y-axs rprsns concnraon, whl h x- axs rprsns dsanc along h gl.) h paks hav bn dnod wh valus of, asurng h nubr of suprhlcal wss n h DN prsn n ach pak: ε rlaxd crcular DN, ε+ on lf-handd suprhlcal ws, ε- on rgh-handd suprhlcal ws, c. How can w prdc h rlav concnraons obsrvd abov? s wh all sascal chancs calculaons, w sar wh a odl: Hr, w wan a odl for how h fr nrgy vars wh ws n h DN suprhlx. W wll sar fro a vry spl odl for h wsng nrgy of h DN col ( and show ha corrcly prdcs h obsrvd dsrbuon of wss). W ar all falar wh h spl ar funcon known as Hook s law whch dscrbs h rlaonshp bwn h rsorng forc on a sprng and h dsplacn of h sprng: F -kx, whr k s h sprng consan. wsng DN s no a spl sprng, bu can b hough of as a orsonal sprng- a col wh a rsorng forc whn a orqu s appld. o rnd you, a orqu ( Τ) rsuls whn a forc acs n a radal annr hrough an axs r: Τ F r oh h forc F and radus r ar vcors. nalogous o h spl ar sprng, a orsonal sprng fls a orqu whch s ar o h appld ws: Τ orsonal_ sprng θ Hr k s a orsonal sprng consan and θ s h angl of h ws. If w assu h sprng can only undrgo ngral nubrs of wss, hn w could rwr hs as: Τ orsonal_ sprng k θ ws τ Whr θ ws s sply h angl for on ws of h sprng, and τ s h oal nubr of wss (θ θ ws τ). Jus as h forc on a ar sprng s rlad o a chang n ponal nrgy F kx V x, w can rla h orqu on our DN orsonal sprng o a chang n s fr nrgy: 6
7 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur /0.0/.77 Τ orsonal_ sprng k θ ws τ G τ G τ k θ ws τ τ G τ In h quaon abov, w cobn h consans no on sffnss parar ( k θ ws ) o splfy h xprsson. W ar usng fr nrgy rahr han chancal ponal nrgy hr bcaus hs olcular sys (h wsng DN col) has nrnal dgrs of frdo (.g., bonds aong h DN srands) ha could also b affcd by suprcog. If w ask wha s h fr nrgy of on parcular DN olcul ha has so nubr of wss τ, w hav: G τ τ s h nubr of suprhlcal urns; ngav for rgh-hand urns, posv for lf-hand urns. sng our k bwn h fr nrgy and h probably of obsrvng a sa wh a parcular nrgy w hav for h wsng DN: P τ R all nrgs n τ n R o rla hs o our asurd quany (concnraon of spcs I, proporonal o h pak n our gl lcrophorss xprn), w sply rcognz: c c o P Whr c o s h oal concnraon of DN. h prsnc of h squard r n h xponn ans hs dsrbuon has a Gaussan shap (h sa rsul w dscussd las lcur- xcp for hs spl odl, h nr probably dsrbuon s Gaussan, no us nar h pak of h dsrbuon). Fng h asurd concnraon daa wh a Gaussan curv, w fnd h hory prdcs h obsrvd dsrbuon of suprhlcs vry wll: 7
8 0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur /0.0/.77 Iags rovd du o copyrgh rasons. Plas s: Fgur 4- n snbrg, Davd S., and Donald M. Crohrs. Physcal chsry: wh applcaons o h lf scncs. Mnlo Park, C: nan/cungs, 979. ISN: X. 8
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