Partition Functions and Ideal Gases
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1 Partitio Fuctios ad Idal Gass PFIG- You v lard about partitio fuctios ad som uss ow w ll xplor tm i mor dpt usig idal moatomic diatomic ad polyatomic gass! for w start rmmbr: Q( N ( N! N Wat ar N ad? W ow apply tis to t idal gas wr:. molculs ar idpdt.. umbr of stats gratly xcds t umbr of molculs (assumptio of low prssur.
2 Idal moatomic gass PFIG- Wr ca w put rgy ito a moatomic gas? ε ε ε atomic tras lc Oly ito traslatioal ad lctroic mods! total partitio fuctio is t product of t partitio fuctios from ac dgr of frdom: ( ( ( tras lc otal atomic partitio fuctio raslatioal atomic partitio fuctio Elctroic atomic partitio fuctio W ll cosidr bot sparatly
3 raslatios of Idal Gas: ( tras PFIG- Gral form of partitio fuctio: tras βε stats tras z Rcall from QM slids ε tras 8ma ( x c y b a z x x y z... So wat is tras? tras x βε x y z 8 y z x y z xp β ma ( x y z
4 PFIG-4 Lt s simplify tras ( 8 xp z y x z y x z y x z y x tras ma β βε c b a c b a 8 xp 8 xp 8 xp z y x z y x tras ma ma ma β β β 8 xp ( tras ma β Rcall: All tr sums ar t sam bcaus x y z av sam form! W ca simplify xprssio to:
5 tras is arly cotiuous PFIG-5 W d lik to solv tis xprssio but tr is o aalytical solutio for t sum! ( β xp 8 tras ma No fars tr is somtig w ca do! Sic traslatioal rgy lvls ar spacd vry clos togtr t sum is arly cotiuous fuctio ad w ca approximat t sum as a itgral wic w ca solv! β tras ( d xp 8 ma 0 Work t itgral Not limit cag oly way to solv but adds vry littl rror to rsult tras πmk ( / a
6 raslatioal rgy ε tras PFIG-6 Wit w ca calculat ay trmodyamic uatity!! I 7 (Z ots w sowd tis for t avrag rgy ε tras k l tras r tras πmk ( / ε tras k l / πmk / ε tras / k l k l ε tras k As w foud i Z ots! (Rcall: tis is pr atom.
7 Idal moatomic gas: ( lc PFIG-7 Nxt cosidr t lctroic cotributio to : lc Agai start from t gral form of but tis tim sum ovr lvls ratr ta stats: lc lvls g i Dgracy of lvl i βε i Ergy of lvl i W coos to st t lowst lctroic rgy stat at zro suc tat all igr rgy stats ar rlativ to t groud stat. ε 0 For moatomic gass!
8 a w simplify lc? PFIG-8 lc βε ( g g g βε (8.... trms ar gttig small rapidly lctroic rgy lvls ar spacd far apart ad trfor w typically oly d to cosidr t first trm or two i t sris Gral rul of tumb: At 00 K you oly d to kp trms wr ε < 0 cm - ( -βε > 0.008
9 A closr look at lctroic lvls PFIG-9 lc βε βε ( g g g... (8. Gral trds. Nobl gas atoms: ε 0 5 cm - (at 00K kp trm(s. Alkali mtal gas atoms: ε 0 4 cm - (at 00K kp trm(s. Halog gas atoms: ε 0 cm - (at 00K kp trm(s
10 Fial look at lc PFIG-0 I gral it is sufficit to kp oly t first two trms for lc lc ( g g βε Howvr you sould always kp i mid tat for vry ig tmpraturs (lik o t su or smallr valus of ε (lik i F tat additioal trms may cotribut. If you fid tat t scod trm is of rasoabl magitud (>% of first trm t you must cck to s tat t tird trm ca b glctd.
11 Fially w ca solv for Q! PFIG- For a moatomic idal gas w av: Q( N ( ( ( tras N! lc N wit tras πmk ( / lc βε ( g g
12 Fidig trmodyamic paramtrs U PFIG- U W ca ow calculat t avrag rgy E U k Plug i tras ad lc Nk So U l Q l Nk Nk U N ( Q( N ( tras ( lc( N! l U Nk Nk πmk or Ng / ε lc E l ( βε g g βε U molar rgy R tras lc Elctroic cotributio typically small (i.. gligibl
13 PFIG- Fidig trmodyamic paramtrs N d du Nk R d R d N Molar at capacity for a moatomic idal gas: ould also fid at capacity:
14 PFIG-4 Fidig trmodyamic paramtrs P lc tras N Nk Nk Q k P l( l l ( Nk g g mk Nk P l l / βε π Nk P R P! ( ( N N Q N ( ( ( lc tras Plug i tras ad lc Oly fuctio of So or molar prssur Look familiar??
15 Idal Moatomic Gas: A Summary Partitio Fuctio: From Q gt Ergy tras Q( N mk ( / ( N! N PFIG-5 π βε ( g g U Nk lc (molar U R Hat apacity Nk R Prssur P Nk P R
16 Addig complxity diatomic molculs PFIG-6 I additio to tras. ad lc. dgrs of frdom w d to cosidr:. Rotatios. ibratios Rigid Rotator Modl Harmoic Oscillator Modl ε ε ε ε diatomic tras ε lc otal Ergy z raslatioal Ergy Rotatioal Ergy ibratioal Ergy Elctroic Ergy (( (( c b a x
17 Diatomic Partitio Fuctio PFIG-7 Q. Wat will t form of t molcular diatomic partitio fuctio b giv: As. ε ε ε ε ε diatomic ( tras tras lc lc? Q. How will tis giv us t diatomic partitio fuctio? As. Q( N tras ( tras Now all w d to kow is t form of N! lc π ( m m k ( N lc ad. tras Start wit : is is t sam as i t moatomic cas but wit m m m! tras /
18 Diatomic Gass: lc PFIG-8 W dfi t zro of t lctroic rgy to b sparatd atoms at rst i tir groud lctroic rgy stats. Wit tis dfiitio ε D Ad lc ( g D k ( ε / k g / Not t sligt diffrc i lc btw moatomic ad diatomic gass! Figur 8.
19 PFIG-9 Diatomic ibratios ( v ε υ armoic oscillator approximatio is usd to dscrib ratios. Rcall from QM ots: ( 0 ( / 0 ( v v v v v stats υ β υ β υ β βε 0 x x ( υ β υ β υ β υ β v v ( / 0 / υ β υ β ( / v 0 ad lvls ar dgrat Plug ε ito summd ovr stats Usig rlatiosip: For < υ β x
20 ibratioal mpratur θ PFIG-0 It is commo to dfi a υ ratioal tmpratur Wat ar t uits of θ k? W ca writ i trms of θ ( βυ / βυ υ / k / ( /
21 Quotit Rul! ε ad PFIG- From t partitio fuctio w ca calculat t ratioal cotributio to t smbl avrag rgy: ε Nk l / ( / ε Ad Nk / d ε d
22 Wat is t diffrc i ts tr sts of data? PFIG- R / ( / lassical limit (at ig R Figurs 8. ad 8.6 θ
23 Fractio of populatio i stat v f v PFIG- f v βυ ( v ( / v / O is is also kow as? f(abl 8. f I f f f v > 0 f0
24 Diatomic atios PFIG-4 W us t rigid ator approximatio. (Wic is? ε J J ( J J 0... I g J J Plug ito... ( (J J 0 β J ( J / I Lik t ratioal w dfi a atioal tmpratur: bcoms: Ik ( J 0 (J J ( J /
25 Diatomic atios PFIG-5 is sris dos ot av a simpl closd form. Part of abl 8. Q. How did w gt aroud tis problm for ε tras? A. Ik if <<
26 ε ad << PFIG-6 At room tmpratur (or igr is a good approximatio for most molcular atios W will assum w ar i tis ig tmpratur limit to fid ε ad. Rotatioal rgy: ε Nk l Nk l Molar at capacity: d ε d A diatomic as dgrs of atioal frdom ac cotributs R/ to molar at capacity.
27 Fractio i lvl J f J PFIG-7 f J (J J ( J / (J J ( J / is is also kow as? O Exampl: O at 00 K
28 Rotatioal Symmtry Numbr PFIG-8 Du to uatum mcaical cosidratios byod t scop of tis cours w d to add a symmtry umbr to t atioal partitio fuctio σ σ σ Htrouclar diatomic Homouclar diatomic Symmtry umbr
29 Diatomic: Put it all togtr PFIG-9 ( tras lc / / π ( m m k / σ ( g D / k
30 Idal Diatomic Gas: A Summary PFIG-0 Partitio Fuctio: From Q gt Q( N ( N! / / π ( m m k / σ ( g N D / k Ergy U R R R R / / N A D Hat apacity 5 R R / ( / Prssur: ik about your omwork wat would it b?
31 Polyatomic ibratios PFIG- Rmmbr for polyatomic ratios w divid t motios ito ormal mods ad xprss t coordiats as idpdt armoic oscillators ε α υ ( v Wat will t form of t partitio fuctio b? v 0 Wat is α for H O? ε Nk α / / Polyatomic ratioal rgy Hat capacity Nk α / ( /
32 Exampl: otributios to PFIG- Dgrat θ 954 K -5 4 ormal mods Nk 4 / ( / rtical bd Horizotal bd. 0.8 Symmtric strtc θ 890 K /R t 954 K t 890 K t 60 K / K Asymmtric strtc θ 60 K
33 Polyatomic Rotatios PFIG- Liar polyatomics: rsult is t sam as tat for a diatomic rigid ator. Noliar polyatomics: For ac of t dgrs of ratioal frdom (A ad w av a sparat momt of irtia ad a sparat atioal tmpratur I I A I σ σ σ OS O A I A I I A I A I I A
34 for polyatomic atios PFIG-4. From QM gt rgy lvls (rmmbr t uatios diffr for ac cas!:. Plug ito: for t sprical top: ( ε J J ( J J 0... I lvls J g J π ( σ π ( σ βε. Assum θ << & approximat sum as itgral: 4. Work t itgral / / π ( σ / A A / / g J J βε J ( djg J 0 / Sprical top Symmtric top Asymmtric top
35 Rotatioal cotributio to U ad PFIG-5 π ( σ π ( σ / / π ( σ / A A / / / Sprical top Symmtric top Asymmtric top U ε k l From calc U Not: s rsults ar du to our assumptio tat t distributio of lvls is cotiuous (θ <<.
36 PFIG-6 Summary: Polyatomic Gas k D g k M / 5 / / / ( ( σ π A D N R R U 5 / 5 ( 5 / / 5 R R Liar Polyatomic Noliar Polyatomic k D A g k M / 6 / / / / / ( ( σ π π A D N R R U 6 / ( 6 / / R R
37 ompariso wit Exprimt PFIG-7 For H O 90 K 560 K 560 K Poits ar xprimtal data ad t li is calculatd! ortical: Noliar polyatomic R R 6 / / ( Figur 8.7
38 Wat did w lar? PFIG-8 From statistical modls w ca lar about trmodyamic uatitis. W ca coct t microscopic ad t macroscopic troug Q. W ca fid Q troug t molcular partitio fuctio providd our systm follows oltzma statistics (i.. # rgy lvls gratr ta # of stats. From Q w ca calculat trmodyamic uatitis suc as U or. You av som isigt ito wy diffrt matrials av diffrt trmodyamics proprtis (.g. tik of ow uatum mcaical rgy lvls vary for diffrt matrials. Wat s xt? lassical trmodyamics ad t laws tat govr macroscopic trmodyamic uatitis.
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