Prelab Lecture Chmy 374 Thur., March 22, 2018 Edited 22mar18, 21mar18

Transcription

1 Prlab Lctur Cmy 374 Tur., Marc, 08 Editd mar8, mar8 LA REPORT:From t ClassicalTrmoISub-7.pdf andout: Was not a dry lab A partially complt spradst was postd on wb

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3 Not ruird 3

4 If solid is pur X Partial prssur is a ry common way of prssing concntration of gass. Dos NOT dpnd on wtr otr gass ar prsnt (if idal gas baior) 4

5 Iodin is a simpl molcul: W will NOW us t mass, bond lngt, ibrational fruncy, and at of sublimation and ntropy of t crystal to prdict K using statistical mcanics 5

6 . Vapor Prssur K p(- A 0 /RT) Finally, instad of uation (34), wic as bn mad compltly baffling by simplifying it to dat, w will us A A gas -A solid -RTlnQ gas RTQ solid E 0 0(sub) and ary t concntration (wic appars in trans oftn disguisd as t olum, V nrt/p I ), until w find t prssur tat maks A 0. Tat will b uilibrium, and tat will b t apor prssur A A A ap solid ap 0 0 A RT ln E0( sub) RT ln( ap) E0( sub) RT ln( solid ) solid { π mkt kt kt } p σ c 3/ 0 A RT E sub RT ν ib ln [( ) ( kt ) ] 0 ( ) ln( solid ) 0 6

7 A Cartoon: I (solid) I (gas) fw ibrational stats Stat ry,ry,ry,...many translational stats Stat Enrgy k T Euilibrium Constant ratio of aailabl stats 7

8 Qualitatily, wat will t uilibrium constant, (K N /N ) b for tis systm? 3 ustions: a. Gratr tan ( A E - T S is ngati)? b. Lss tan ( A E - T S is positi )? c. Can t tll? Stat Enrgy k T E 0 0 SR ln (aail. stats) partition function 8

9 Partition function numbr of aailabl stats How to gt t numbrs of stats??? Count tm Counting summing intgration Will b larg if nrgy ll spacing is small Tis is EASILY dtrmind by uantum mcanics: Wat is tru of ALL uations for nrgy lls in uantum mcanics?? Spacing mass confinmnt lngt m 9

10 Effct of nrgy ll spacing on partition function Spacing mass confinmnt lngt m Enrgy k T πmk Translational 0.0 m ~0 30 T 3/ V Rotational 0-0 m ~0 8π Ik T Vibrational 0 - m ~ 3N 5 j ν kt ib 0

11 Spacing mass confinmnt lngt m Eampl: Computing Q for an idal gas of monoatomic atoms Quantum mcanics gis t stats for a fr particl of mass m in a -dimnsional bo of lngt a n ε j, n,,3... 8ma Idal mans indpndnt particls, so tat t nrgy of a systm of N atoms will just b t sum of t indiidual nrgis: E j ε i, i i atom idntifir

12 Tr is noting spcial about. T nrgis for t y and z componnts of kintic nrgy ar indpndnt of tat for, so to gt t nrgy and partition functions for t 3 dimnsions can b obtaind simply from tos for dimnsion. E j E j E jy E jz and trans trans, trans,y trans,z ; (t probability of indpndnt nts is t product of t indiidual probabilitis.) Eac molcul dos its own ting. Trfor t systm Q can b writtn simply from t indiidual molcul translational partition n function: j /8ma E j / kt kt trans,d j j caus t lls ar so closly spacd, t uantum numbr, n j may b considrd a continuous ariabl (atoms ba almost classically), tus giing nj /8ma trans,d i.., t intgral of is a Gaussian function, bcaus t ariabl is suard. W us only alf, bcaus n is nr ngati. 0 k T dn

13 Gaussians T Gaussian function ( bll-sapd cur ) is so common in scinc, it is wort spnding tim gtting ry familiar wit it. Any Gaussian may b writtn as: In tis form, σ is t standard diation, and of cours as t sam units as. Tis Gaussian as σ.00 and looks lik: g() σ σ is t alf-widt at of t maimum. / Ara σ d ( π) σ.507 In gnral, t ara is always ry clos to ½ bas tims altitud, i..,t ara of a 3 triangl.

14 In t intgral n /8ma kt trans, dn 0 Ara undr cur. Wat is making tis intgral LARGE? i.., wat maks t ponnt small? Larg mass and larg spac to run around gi small nrgy ll spacing. Dos t Gaussian a larg or small standard diation? σ 4ma k T/ So, σ (4ma k T/ ) /, and t intgral undr t wol cur is (π) / (4ma k T/ ) /, and alf tat bcaus t intgral starts at 0 gis: trans,d (πma k T/ ) /, wr a is t lngt of t bo. caus t motion in t 3 dimnsions ar indpndnt, t 3 D partition function is simply trans ( trans,d ) 3 (πma k T/ ) 3/ (πmk T/ ) 3/ V, wr V a a y a z. 4

15 Rotational partition function rotation J( J ) /8π µ r E / /8 j kt kt π µ r kt ( ) j j kt /8π µ r J de J /8 π µ r /8π µ r but d J(J ) ( ) kt kt kt rotation d [ ] r r kt /8π µ r J kt 0 /8 π µ 0 /8π µ 8π µ r kt kt Wy will tis intgral b ry, ry, ry muc smallr tan t translational on? T mass is sam confinmnt lngt r t bond lngt for I.67 Angstroms 5

16 Vibrational Partition Function Potntial Enrgy D ond lngt D 0 E of dissociation ½ ν ib Us Harmonic Osc. T is not ig ε ( ½ )ν ib, wit 0,,,3, ib So, ib gos to as T gos to 0 ( ν ) k T if zro is at bottom Coosing zro of nrgy is arbitrary It is NEATER to coos zro of nrgy ν k T if zro is at 0 at 0 instad of bottom of wll ib 6

17 Nglcting zro point E gis sam numbr of aailabl stats ε ( )ν ib, wit 0,,,3, Now nrgy dos not includ zro point ibrational nrgy wic dos not a anyting to do wit tmpratur ib ( ib call ν k T ib ib, 0,,, 3... βν βν ν k T ) βν ( ) 7

18 )... ( ib )... ( βν ib k T ib ν 8

19 Concpt of ibrational tmpratur ib ν k T Obiously, ν/k as t units of tmpratur; it is t tmpratur suc tat t oltzmann factor - A common sortand is to dfin Θ ib ν/k ib Θib T 9

20 Putting it all togtr: For any molcul: trans rot ib lc For a linar polyatomic molcul: 3/ 3N 5 πmk T T Θ σθrot j Wat is σ??? Tis is NOT standard diation; it is t SYMMETRY NUMER For a omonuclar diatomic σ, bcaus t total wafunction must itr b symmtric or anti-symmtric to intrcang of two idntical particls. Rcall tat cang of idntical lctrons ruirs t wafunction to cang sign. Tis lads to t Pauli clusion principl for all frmions,.g., lctrons. Iodin is a frmion bcaus it as 53 protons and 74 nutrons, an odd numbr of frmions. caus t rotational functions wit n J ar symmtric, ty will not b occupid. A good approimation is to diid by bcaus rougly alf of t 0 rotational stats cannot b occupid. ib T i g lc, i E k T lc, i

21 . Vapor Prssur K p(- A 0 /RT) Finally, instad of uation (34), wic as bn mad compltly baffling by simplifying it to dat, w will us A A gas -A solid -RTlnQ gas RTQ solid E 0 0(sub) and ary t concntration (wic appars in trans oftn disguisd as t olum, V nrt/p I ), until w find t prssur tat maks A 0. Tat will b uilibrium, and tat will b t apor prssur A A A ap solid ap 0 0 A RT ln E0( sub) RT ln( ap) E0( sub) RT ln( solid ) solid { π mkt kt kt } p σ c 3/ 0 A RT E sub RT ν ib ln [( ) ( kt ) ] 0 ( ) ln( solid ) 0