Chapter 5: Partition Function and Properties for Real Molecules

Transcription

1 Sprg 0 hrmdyamcs f a systm f dpdt partcls... h Partt fuct... Org f factr fr dtcal partcls...! wrkg th Hlmhltz fr rgy... 3 raslatal Partt Fuct: Partcl th bx wav fuct (atms)... 4 latg t thrmdyamc prprts... 5 Statstcal Mchacs f datmc mlculs... 7 bratal Partt fuct harmc apprxmat... 8 tatal Partt Fuct fr a datmc... h rg f th symmtry factr rtatal partt fuct... 3 Plyatmc Systms... 7 Chmcal acts ad Eulbrum... 0 Statstcal Mchacs... Cct t thrmdyamcs... Chaptr 5: Partt Fuct ad Prprts fr al Mlculs hrmdyamcs f a systm f dpdt partcls - glct tral dgrs f, partcular rtats, vbrats g. tms, rar gass - Wll lk at mlculs th gas phas, whch ar dlut ad at hgh tmpratur (dal gass) Quatum Hamlta H ˆ h( ) = tr- atmc tracts Slv h( ) ϕa( ) = εaϕa( ) Hˆ ( ϕa( ) ϕa( )... ϕ z( ) ) ( h( ) ϕa( ) ) ϕa( )... ϕz( ) ϕa( ) ( h( ) ϕb( ))... ϕz( ) + + ϕ ( ) ϕ ( ) h( ) ϕ ( ) = +... a b...( z ) ( εa εb... εz) ϕa( ) ϕb( )... ϕz( ) = prduct f sgl partcl wavfuct s gfuct - sum f - partcl g valus ttal rgs Chaptr 5:Partt Fucts ad Prprts f al Mlculs

2 Sprg 0 h Partt fuct Q= all stats abcd,,,... Eall / k ( ) a + b + c + d.... / k Q= ε ε ε ε a/ k b/ k c/ k ε ε ε =.. a b c Q=... a b c Sc vry atm/mlcul s th sam =... = Q= ( ) a b Whr s th umbr f partcls, s th partt fuct fr a sgl cmpt (mlcul). h vrall partt fuct Q f th systm s just th prduct f th dvdual partt fucts. Gral fatur: If Hamlta s a sum f trms wthut crss trms (tracts) partt fuct wll b a prduct f trms crrspdg t trms H frm wavfuct s prduct fuct t - hs s a bg smplfcat. Hwvr ths s ly partally crrct as t ly appls f th partcls wr dstgushabl frm athr. I may cass th partcls mult mlcul systms ar dstgushabl. Org f factr fr dtcal partcls! ll partcls atur shuld b vwd as thr bss r frms. Quatum mchacal wavfucts ar thr symmtrc r atsymmtrc udr trchag f partcl crdats Frms at symmtrc, Bss symmtrc If ψ ( ϕa( ) ϕb( ) ϕc( 3) ) = a b c I ur sum w cutd all prmutats f abc,, as dstct stats 3! ctrbuts But thr s ly fully atsymmtryc wavfuct abc (Slatr dtrmat) ad als ly symmtrc fuct. Fr 3 partcls dvd by 3! Fr partcls dvd by! Chaptr 5:Partt Fucts ad Prprts f al Mlculs

3 Sprg 0 If th umbr f avalabl stats >> th umbr f partcls th t s a vry gd apprxmat t smply dvd by! Q =! hs was d v fr classcal partt fucts, but th rass wr t clar (althugh [rrus] argumts wr mad) hs statstcs s kw as Bltzma Statstcs. h prcdur s t rgurusly crrct fr thr bss r frms g. ϕa = ϕb a= b, sam wavfuct Fr frms: wavfucts = 0 Fr bss: dffrt factr t cut symmtrc wavfucts 3 May mr stats tha # f partcls ( >> 0 ) Hwvr, Bltzma apprxmat s t always vald Eg. - Elctrs mtals - phts a lght surc - vry lght partcls (mr labrat dscuss latr ) I gral f w hav spcs B, tc. th partt fuct s gv by Q B = ( ) ( ) wrkg th Hlmhltz fr rgy B B!! Usg Strlg s apprxmat fr! klq B = l! = l = ( l ) = kl! ( l = k l! ) = l = k l l! k l = k = k = k= = 8.34 J/ mlk l = ( l l ) l = = (athr Strlg apprxmat) Chaptr 5:Partt Fucts ad Prprts f al Mlculs 3

4 Sprg 0 cta trms that d t scal larly wth, tractg partcls sm t tract! hs s a csuc f (at) symmtry rurmt f may- partcl wavfuct. Latr w wll s ~. kl = ~ l k wll b prprtal t, th d, as shuld b th cas. Smpl systm: - tractg atms ε = ε + ε + ε t α β γ : uclar wav fuct (ly uclar sp s mprtat) : lctrc (mprtat fr p shll atms) t : traslatal (ktc rgy) raslatal Partt Fuct: Partcl th bx wav fuct (atms) Lk at th uatum prblm - D systm, th xtraplat t 3D Df h d E mdx ψ = ψ s π ψ = x L h π πx πx s = E s m L L L h π h = = ml 8mL E,D h 8mL k d ψ π πx = s dx L L h = h π = =,,3... h Δ= ad x =Δ 8mL k t,d Δ = = x Sc th rgy spacgs ar small rlatv t k t s pssbl t us a tgral th plac f summat Gral tgral = ( ) = f( x ) dx= f ( x )( x x ) f x + x+ x =Δ ( + ) =Δ ( ) Δ= ( ) f x f x dx f x f( x) dx Δ ( ) Chaptr 5:Partt Fucts ad Prprts f al Mlculs 4

5 Sprg 0 Back t th partt fuct hs apprx s crrct f f ( x ) s smthly varyg = dx= t,d x x 0 Δ Δ (lk up th tgral) π 8mkL π L = π = = = Δ Δ 4 h 4 Λ t,d π Λ=Δ 4 h π = mπk whr Λ s calld th thrmal dbrgl wavlgth Lx t,d = Λ Mvg t 3D = t,3 D x, D y, D z,d L L L Λ Λ x y z t,3d = = 3 3 th mlcular partt fuct fr traslatal mt pprxmat s bst f partcl s havy, bx s larg classcal lmt (may rgy lvls, hgh dsty f stats) 3 3/ πmk πmk = = = α 3/ t,3d = =α 3 Λ = g (uclar) 3 Λ h h 3/ 3/ π mk df α = h 3/ (cstat) E / k = (sum vr stats) (lctrc, ly fr p- shll atms) = g α (sum vr rgy lvls) α E / k α E / E α : mlcular rgs, d t dpd!, latg t thrmdyamc prprts l =,3D = α 3/ α = l 3 = lα + l + l 3/ Chaptr 5:Partt Fucts ad Prprts f al Mlculs 5

6 Sprg 0 P = = P = (dal gas law!) 3/ d α S = = l d, d 3 = lα l l l d + + 3/ 3 α S = + l α = l + 3/ 3 U = 3 U = U 3 C = = 3/ α l = = 3 H = U + P 3 5 H = + = H 5 CP = = = l = kl 3/ α µ = = k l, 3/ k l 3/ α α = + k = k l k + k 3/ α µ = k l 3/ 3/ 5/ α k α k αk µ ( P,, ) = kl = kl = kl P P P α µ = µ = kl = l P 5/ α k P 0 G= µ = l P0 P 5/ 5/ k α k P Chaptr 5:Partt Fucts ad Prprts f al Mlculs 6

7 Sprg 0 5/ α k 0 G= l + l P P 0 P P0 P G= G0 + l G l P = P Chck frmula fr S cmpud t thrmdyamcs: 3/ 3 α S = + l 5/ 3 k α = + l P, cstat S l Δ = 3, cstat Δ S = l 5, P cstat Δ S = l P P, cstat S l P Δ = P t ll f dal gas thrmdyamcs fllws frm = kl partt fuct, ( ) Q t! = traslatal Statstcal Mchacs f datmc mlculs Mlculs hav: traslatal, rtatal, vbratal, lctrc ad uclar sp dgrs f frdm gd apprxmat: E = Et + Er + Ev + E + E hs s t ut tru, partcular thr culd b a cuplg btw rtatal ad vbratal mts (crtaly fr hghr lvls). = t r v (t ut tru as wll, cmplcats d ars) 3/ π Mk t s th sam as fr atms.. t = α, α =. Sam dal gas assumpts, h wrks bst at lw dsty, lght mlculs, ad wrks ly fr gass. t lw tmpraturs, gass cds r sldfy du t (wak) tracts. : typcally ly lctrc lvl ctrbuts. hs wuld b dffrt fr radcals r trplt stats. Ev th: smply accut fr dgracs. 3/ Chaptr 5:Partt Fucts ad Prprts f al Mlculs 7

8 Sprg 0 Uxpctd cmplcat: strg cuplg btw uclar ad rtatal wavfuct (athr mafstat f (at)symmtry). r rtat + uclar shuld b tratd tgthr Paul prcpl fr ucl. t: th factr was als rsult f rurg (at)symmtrc wavfucts. hs! aspct wll b dscussd latr. Smpl rsults ar btad wh vbrats ad rtats ca b tratd sparatly ( cuplg) ad harmc scllatr s usd fr vbrats. bratal Partt fuct harmc apprxmat Eharm = kx x= 0 (Harmc apprxmat) whr ( ) ˆ h d H µ dx mm whr µ = m + m = + kx (rducd mass) E = + ω h = 0,,... ad k ω = µ ( k s th frc cstat ad dpds th mlcul) (ly d k ad µ ad yu wll fd all th rgs) hs wuld b dscussd a uatum mchacs class Fdg th partt fuct Ergy lvls harmc scllatr: E hω / k E / k hω / k = = hω+ h ω = 0,,... ( hω / k ) ( hω / k) = St ( h ) y= ω/k = 0,,.. ( h ω k ) ( h ω k ) ( h ω k ) = = y / / / s ( hω / ) = y k ( hω/ k ) ( hω/ k 3 ) (...) = + y+ y + y + = y = 0,,... = 0,, y = ( + y+ y + y ) = (kw math rlat) y Chaptr 5:Partt Fucts ad Prprts f al Mlculs 8

9 Sprg 0 ( hω / k ) = y ( hω / k ) = ( hω / k ) partt fuct wth zr pt rgy cludd Df vbratal tmpratur, ω = h k ( / ) v = ( / ) J / JK = K (klvs) s s dmslss Fr sm mlculs, th tral rtat ca rag frm rlatvly small smthg lk C- H strtchg h ω 300cm h ω 00cm, t Kwg k = 0.695cm K, hω / / k hω / k ( ) ( / ) = dd fr a dct ppulat ω = h ca rag frm K k, sc rags frm K, a larg valu s Ergy scal s t cvt f w wat t csdr mxturs f mlculs, as w hav chs th zr f rgy as th bttm f th wll. Mr cvt: chs zr f rgy as th rgy f dsscatd atms sam rgy scal fr all mlculs Ev = D + hω+ h ω = D + hω 0 sults w bta frm partt fuct ar dpdt f th vrall shft rgy hω / k D hω / k = D / k Chaptr 5:Partt Fucts ad Prprts f al Mlculs 9

10 Sprg 0 D0 h vbratal partt fuct bcms / k = ( / ) 0 / s larg (lw ) = D / k If ( ) 0 s small (hgh ) D / k = If ( / ) hrmdyamc valus = ( ) D0 / k / ( l D ( )) 0 / k l / / D 0 kl( ) = k = + (hgh lmt) / k / S = = kl( ) / If w multply ths by th w gt / k S = kl + v / ( ) U = + S = D + 0 / k ( ) ( ) ( ) / / / ( ) / µ = D0 kl( ) = +, hs s all xact, fr harmc scllatr Csdr th larg lmt f bttr D 0 / k = D0 / k v / / s larg x /, x / % = kl = D0 kl S% =+ kl + k U% = % + S% = D0 + k = D0 + C = k = Chaptr 5:Partt Fucts ad Prprts f al Mlculs 0

11 Sprg 0 Classcal Eupartt thrm: Fr vry ( P ) x ad tral rgy s pr ml ctrbut t Evry vbratal ml ctrbuts t U, ad t x Hamlta, th ctrbut t C s C at hgh trmpratur tatal Partt Fuct fr a datmc Us th s calld rgd rtr apprxmat, glct cuplg btw rtats ad vbrats (small rrr) h Quatum Mchacal Hamlta fr rtats ˆ ˆ L mm H = µ = µ m + m ˆL : a agular mmtum pratr dpdg θϕ,, : truclar dstac t: ˆL s th sam pratr that shws up th H- atm rbtal s s,p,d,f fucts m ( θϕ) = h ( + ) ( θϕ) ˆ m l, l, Ly l l y ml = l... + l l = 0 s 0 l = p -, 0, + l = d -, -, 0, +, + l = 3 f - 3, -, -, 0, +, +, +3 Kw sluts fr rgy gvalus h EJ = J ( J + ) = BJ ( J + ) J = 0,,... µ g = J + Has dgracy ( ) m Stats: (, ) J J Y θϕ m J, J... J, J = + ( J + ) hc Ergy lvls ar ft xprssd cm hυ = = hck% λ EJ = BJ %( J + ) = h 8π cµ ( cm ) µ = I (mmt f rta) cvt cvrs: cm tatal partt fuct J = 0,,.. ( ) BJ ( J / ) k = J + + t: w sum vr rgy lvls J, ad d t xplctly clud dgracs Usg a math prgram, ca carry ut th sum xplctly (g. u utl J max = 00) Chaptr 5:Partt Fucts ad Prprts f al Mlculs

12 Sprg 0 I practc, th hgh tmpratur lmt rplacs th sum by a tgral ( ) Bx %( x+ / ) k % x+ dx 0 Substtut ( ) y= x x+ = x + x ( ) dy = x + dx By/ k k By/ k k % dy % % % % % = = = 0 B% 0 B% k = % B = whr B % %, k % ar uts f cm B = tatal tmpratur ( Klvs) k hs frmula s crrct fr htruclar datmcs wth havy masss lk CO, but fr hmuclar cas, lk H t s ff by a factr f. Crrctg fr ths k = = σb σ whr σ s th symmtry factr ; σ = fr htruclar, σ = fr hmuclar udrstad th symmtry factr has t tak uclar sp t csdrat. It s a csuc f th Paul prcpl fr ucl. It s cmparabl t th Bltzma factr! Q =.! I th xt lctur I wll dscuss th rtatal partt fuct fr H, D ad HD stps. hs wll gv us a bttr da f th rg f th mystrus σ. hrmdyamcs hgh tmpratur lmt: Ctrbuts du t rtatal dgr f frdm = l σ S = = l +, σ U = + S = C, = µ = = l σ, Chaptr 5:Partt Fucts ad Prprts f al Mlculs

13 Sprg 0 Prbablty t fd mlculs rgy lvl P( E J ) ( J + ) J( J+ ) / P( E ) = J ( J ) σ J( J + ) / + W ca als plt th prbablty t fd a partcular stat ( frm J + ) ψ, m J J hs paks at th grud stat, whch always has th hghst prbablty h rg f th symmtry factr rtatal partt fuct ucl ca b bss (csstg f v umbr f frms) r frms (csstg f dd umbr f frms). hs charactr s rflctd by uclar sp: Bss wll hav tgr sp, Frms hav half tgr sp. ucl ar dscrbd by Quatum Mchacal wav fucts ad thy by fudamtal symmtrs f atur - ψ s symmtrc udr trchag f dtcal bss - ψ s at symmtrc udr trchag f dtcal frms Csdr a systm csstg f ucl datmcs uclar wavfuct has bth a spatal ad a sp part. Fcus frst H - atm, sp ½, αβ, fucts. uclar sp fucts: Chaptr 5:Partt Fucts ad Prprts f al Mlculs 3

14 Sprg 0 α( ) α ( ) rthhydrg α( ) β( ) β( ) α( ) = trplt β( ) β ( ) + Parahydrg α( ) β( ) β( ) α( ) = sglt + symmtrc atsymmtrc uclar sp fucts ar vrtually dgrat (v prsc f a magtc fld) Fr us th symmtry f th sp- gfucts ar mst mprtat. Csdr ucl f gral sp I Symmtrc ( mm + mm ) ( I + )( I + ) = ( I + )( I + ) (symmtrc fucts) 3 I = = 3 tsymmtrc ( mm mm ) (cmpar symmtrc ( ) ( I + )( I) = I( I + ) (fucts) I = = (s abv) + ad atsymmtrc ( ), cludg dagal) Symmtry f sp fuct udr prmutat s udrstd. What abut spatal part f uclar wav fuct? r r Csdr th uclar crdats, + = r cm (fr dtcal ucl) r r = sθcsϕ sθsϕ csθ r r = uclar wavfuct: r ψ ψ ψ θ, ϕ ( ) ( ) ( ) t cm Chaptr 5:Partt Fucts ad Prprts f al Mlculs 4

15 Sprg 0 r r th r ad ar uaffctd cm r r r r P = If Hwvr: ( ) ( ) = sθcsϕ sθsϕ csθ r r Itrchagg ucl s uvalt t θ π θ cs π θ = cs θ ( ) ( ) s( π θ) = s( θ) + cs( ϕ+ π) = cs( ϕ) s( ϕ+ π) = s( ϕ) ϕ ϕ π Hc trchagg r ad r s uvalt t chagg θ π θ ϕ ϕ + π m l ( m, + ) = Yl ( θϕ, ) m Y ( θϕ, ) Y π θ π ϕ + l v l dd rasfrmats ar uvalt t x x y y (vrs) z z s, d, g fucts ar v udr vrs p, f fucts ar dd udr vrs J = 0,,4.. v udr J =,3,5.. dd udr tsymmtrc uclar wavfucts (frms): trplt φ ψ θ, ϕ sp Ev s glt sp Odd ( ) Or φ ψ ( θ, ϕ) Symmtrc uclar wavfucts (bss): trplt sp Ev s glt sp Odd l J Odd J h ly allwd cmbats fr H Ev (, ) φ ψ θ ϕ Or φ ψ ( θ, ϕ) ( H s frm) J Ev J Ovrall symmtrc fr D Odd ( D s bs) Chaptr 5:Partt Fucts ad Prprts f al Mlculs 5

16 Sprg 0 h rstrct t thr vrall symmtrc wavfucts r vrall atsymmtrc wavfucts amuts t a cuplg btw rtatal ad uclar partt fuct. Hc thy shuld b tratd tgthr Fr H (frms, atsymmtrc) I = ( ) ( ) ( ) H BJ J / k BJ ( J / ) k = J=,3,5.. J= 0,,4.. J J Fr D (bss, symmtrc) I = ( ) ( ) ( ) D BJ J / k BJ ( J / ) k = J= 0,,4.. J=,3,5.. J J mmbr umbr f sp stats: H( I = /) D( I = ) Odd I( I + ) 3 Ev ( I + )( I + ) 3 6 tal ( I + ) Fr HD symmtry rurmt, g = 3 = 6(dgracs) J = 0,,.. ( ) BJ ( J / ) k HD = 6 J + + I gral fr hmuclar datmcs wth sp I I + I + Odd + I I + Frm: ( )( ) ( )( ) Bss: ( )( ) Odd ( )( ) I I + + I + I + Ev Ev Fr htruclar datmcs: ttal = I + I + Fr sp I ad sp I B ( )( ) B Hw ds ths rduc t th symmtry factr σ fr rtatal wavfuct? a) ( ) b) + = I + = v dd Prf f b): ( + ) v dd ttal uclar J BJ ( J / ) k + J = J = 0,,4 k = 0,,.. ( 4 ) Bk( k )/ k = k + + ybk ( / ) k dy = 0 B Smlarly: BJ ( J / ) k ( J + ) + J k J =,3,5 k y 4k k = +, dy = 8k + = +, k = 0,,.. Chaptr 5:Partt Fucts ad Prprts f al Mlculs 6

17 Sprg 0 J =,3,5 ( )( ) ( ( k + ) + ) + + B k k / k = ( + )( + ), dy = ( k + ) + ( k + ) y k k ybk ( / ) k dy = 0 B Sam hgh tmpratur lmt hs aalyss s ut vlvd, W wll d a smulat Matlab t clarfy Plyatmc Systms Csdr systm wth atms 3 crdats. 3 cllctv crdats dscrb th vrall traslat f ctr f mass.if w chs t ptmz th ulbrum gmtry w ca dtfy 3 cllctv crdats that dscrb rgd rtat ( fr lar mlculs). 3 6cllctv crdats rma that dscrb tral vbrats ( 3 5 fr lar mlcul) Slv lctrc Schrdgr uat fr fxd uclar pst r H ψ, r = E r r ψ, r ({ }) { } ( ) ({ }) ( { }) 3 dmsal pttal rgy surfac (PES) E = 0 xtrm PES j, =, 3 Dffrt smrs: dffrt mma PES rast Stat: Saddl pts PES (Max drct, m all thrs) aylr srs f pttal rgy surfac arud mmum r r r E r r E r r r r E = E + + r r r r ( ) ( ) ( ) ( ) ( ) E Extrmum = 0 Mass- wghtd Hssa j = j = Chaptr 5:Partt Fucts ad Prprts f al Mlculs 7

18 Sprg 0 E H = M M j j j r r = (t accut fr uclar masss uclar ktc rgy trm) Dagalz H j 6 (5) gvalus ar 0: crrspd t vrall traslat, vrall rtat. 3 6 ( 3 5) gvalus ε f Hssa crrspd t rmal md. By dagalzg th Hssa th vbratal prblm s rducd t 3 6dpdt harmc scllatr prblms d ( ) + ε χ( ) = E χ( ) d ε k Df ω = (aalg f ω =, wth m = ) m ( ) E = + ω h If all ε, ω > 0 th statary pts s mmum. If prcsly f th ε s gatv, r, ω s magary, th structur s trast stat bratal frucs ad rmal mds ar btad frm Hssa. tats: Pst f mmum: α, j α = xyz,,, j=,,... (umbr f ucl) rα cm = mk α, k k m Mmt f rta tsr: j ( α )( β ) ( ) j j cm j cm δ γ j j cm I = m + m αβ, α, β, αβ, γ, j= j γ I αβ, 3 x 3 matrx Dagalzg matrx I ylds 3 gvalus I, IB, I C - Sphrcal p I = IB = IC - Symmtrc p I = IB > IC (prlat cgar) r I > IB = IC (blat dsk) - symmtrc p I > IB > IC tatal gvalus spctrum ca b calculatd purly frm I, IB, I C. h varus cass ar smwhat cmplcatd Hgh mpratur partt fuct always has a smpl frm (usd practc) Chaptr 5:Partt Fucts ad Prprts f al Mlculs 8

19 Sprg 0 ( ) / / π = σ B C hs frmula always wrks, xcpt fr lar mlculs, whr uss rtatal dgrs f frdm) / = h I = µ I =, σ = h I ( Symmtry Factrσ : # f pur rtats th pt grup f th mlcul, kw frm grup thry. (# f rtats that map mlcul t tslf). HO σ = CH 4 σ = 3 4 = H 3 σ = 3 CH 6 6 σ = Ovrall partt fuct fr plyatmc mlcul: d = raslatal: bratal: v = t v 3/ = 3/ π Mk t α α hω / k 3 6 = hω / k = h M = m O factr fr ach vbratal md, cludg zrpt frucy j j tatal: π = σ B C = h tc. I uclar Sp = ( + ) I α I α : uclar magtc mmt fr uclus α α Elctrc: D / k r E / k Δ D : atmzat rgy, bttm f wll sparatd atms May uatts ca b calculatd accuratly frm ctmprary lctrc structur calculats. Gmtrs ad vbratal frucs ar farly accurat (but wth harmc apprxmats). tmzat rgs/ract rgs wuld b th hardst t bta accuratly. h harmc apprxmat s pr fr flppy mlculs. hs s dffcult t crrct. Othr vry lw frucs f vbrats, tral rtat fr xampl tha CH3 CH3 Pttal alg trsal md Chaptr 5:Partt Fucts ad Prprts f al Mlculs 9

20 Sprg 0 ( ϕ) ( ϕ) = cs 3 : barrr hght hs ca b cludd (glct md- cuplg). hr ar prblms fr flppy mlculs thugh. Chmcal acts ad Eulbrum Csdr racts gas phas a) hrmdyamcs: Prttyp ract: a + bb Ä cc + dd abcd:,,, stchmtrc cffcts BCD:,,, chmcal spcs actats Prducts Wrt t th frm cc + dd a bb = 0 υ = 0 υ > 0fr prducts υ < 0 fr ractats Sc w wll csdr ulbrum, ractats vs prducts s a arbtrary chc t chmcal ulbrum Δ = 0 ad υµ = 0 G ract = + µ µ P = stadard prssur, ( ) l ( P / P ) P = partal prssur f spcs Fr dal gass: P = xp x : ml fract f spcs µ = µ + l P/ P + l x ( ) ( ) µ (, ) l = P+ x (altratv xprss ml fracts) Δ G = υµ = 0 ract ( ) l ( / ) = υµ + P P υ ( ) l ( / ) = υµ + P P υ Chaptr 5:Partt Fucts ad Prprts f al Mlculs 0

21 Sprg 0 Df ulbrum cstat ( / ) I practc w ca calculat ( ) K = P P υ p υµ ( ) = l Kp υµ ( ) µ frm QM ad Stat- Mch frst prcpl thry f chmcal ulbrum Fr prvus xampl: a + bb Ä cc + dd K = c ( PC / P) ( PD / P) ( P / P ) ( P / P ) p a b B Oft t s asr t wrk wth mlfracts P = xp ( / ) υ ( / ) υ P K = P P = x P P υ x = ( / ) P P υ υ Δ x ( P/ P) υ υ υ = ( / ) Δ = ( ) υ = ( / ) K = K P P υ P x K x K P P x p d Δυ frm stat mch Δ = Statstcal Mchacs Chmcal pttal = Gbbs Fr rgy G = + P = + k (dal gas) = k l + k! = k l + k ( l ) + k = k l + k l ( ) ( ) = k l / = l / t = v t 3/ h = α M α M = π Mk k P = k = P 5/ t αm k = P 3/ Chaptr 5:Partt Fucts ad Prprts f al Mlculs

22 Sprg 0 = D / k v h / k ω (datmc),, : ca all b calculatd fr ach spcs chmcal racts t dtrms th prssur dpdc 5/ t αm k P l l l = + P P 5/ α k, l M t= trs Δ Gract P Cct t thrmdyamcs = K ( ) l K υµ p = υ l υ = l = l υ p = υ t, υ υ υ υ ( v ) ( ) ( ) ( ) = = K K K K K ach factr s a rat f crrspdg s t, v Lt us aalyz dffrt factrs: - uclar factr: asst, sc th umbr f ucl ds t chag btw ractats ad prducts ad thr ds uclar sp prducts = K ractats = (always) - rtatal factrs: Just has t b calculatd fr ach mlcul, fr atms = Fr datmcs us = (gd ugh) σ Plyatmcs: π = σ B C 3/ Δυ - tmpratur dpdc f (f atms/datmcs) υ Chaptr 5:Partt Fucts ad Prprts f al Mlculs

23 Sprg 0 Δυ 5/ k - traslatal factr: P Fr bth rtatal ad traslat factrs υ α υ υ x Δυ Δ = dpdc rflcts trpc ctrbuts Δ υ > 0 mr spcs prduct sd cras trpy (# f stats) up ract - vbratal + lctrc factr: hs factrs ar umrcally mst mprtat. Lt us try t udrstad hw th trms rgat. Lt us fr dftss csdr a ccrt ract HC CH PES alg ract crdat W wuld mak a harmc scllatr mdl fr ractat ad prducts (3 rmal mds ach). h chag lctrc rgy s dtrmd frm th dffrc rgy at th bttms f th wll. h zrpt vbrats rgy fr ach spcs wuld b =,...3. Sum vr rmal mds E zp ( ) = h ω. Hr I gral rgy dffrc fr ract ca b wrtt as sum vr lctrc rgs at th rspctv mma ad sum vr zr pt frucs Δ E =Δ E +Δ E zp = υe + h ω υ ( ) E/ k K = Δ v, ( ) : dcats th ctrbut du t th grud stats f varus spcs Chaptr 5:Partt Fucts ad Prprts f al Mlculs 3

24 Sprg 0 d Fr ach spcs thr wuld b addt th factr vx, = / k h. hs factr s ω vry cls t uty (). I thk t wuld b clarst t wrt th ctrbut as K K K zp v K Kzp = = Δ E / k Ezp / k Δ Δ E = υ E Δ Ezp = h ω υ, Kv = ω / k, h : labl fr spcs : lvl fr rmal md hs clarly dcats th rg f th varus trms K = K K K K K p zp t v hs dcats th mprtac f varus factrs K >> K > K ~ K > K υ zp t v K, K ctrbut t a xptal factr zp Ek / Δ ad ths abslutly dmats th ulbrum cstat. Othr factrs dpds pwr f. Prssur dpdc drvs frm P traslatal partt fuct l. W hav rachd th sstal usag f statstcal P mchacs chmstry. Wrthwhl t lk at xampls Chaptr 5:Partt Fucts ad Prprts f al Mlculs 4