Statistical Thermodynamics Essential Concepts. (Boltzmann Population, Partition Functions, Entropy, Enthalpy, Free Energy) - lecture 5 -

Transcription

1 Statstcal Thrmodyamcs sstal Cocpts (Boltzma Populato, Partto Fuctos, tropy, thalpy, Fr rgy) - lctur 5 -

2 uatum mchacs of atoms ad molculs STATISTICAL MCHANICS ulbrum Proprts: Thrmodyamcs MACROSCOPIC Proprts Tm-dpdt bhavor: Chmcal ktcs

3 Mcroscopc world Macroscopc world uatum stats ad rgs of a molcul Proprts of a molcul: gomtry multpol momts ozato pottal lctro affty spctroscopc proprts vbratos/rotatos tc. Thrmodyamc stats ad rgs of a hug smbl of molculs (matral) Proprts of a matral: phas (bolg/frzg) tmpratur prssur hat (thalpy) ordr/dsordr (tropy) fr rgy magtc suscptblty optcal actvty tc. Proprts of a molcul ( Wavfucto ) STATISTICAL THRMODYNAMICS Proprts of a mattr: ( Partto fucto ) 3

4 Stat fuctos ad thr rlatos kow from classcal thrmodyamcs U H U + pv H A U TS G H TS A G A + pv G S, U, H, A, G stat fuctos caot b masurd drctly S, U, H, A, G stat-fucto chags ca b masurd drctly dothrmc/xothrmc racto ( trms of H) drgoc/xrgoc racto ( trms of G) 4

5 Dftos of systms statstcal thrmodyamcs {N, V, } Mcrocaocal systm (solatd systm) {N, V, T} Caocal systm (closd systm) {N, µ, T} Grad-caocal systm (op systm) 5

6 Caocal 4 Toward th prcpls of statstcal thrmodyamcs - Boltzma dstrbuto Collcto of thought systms 3 3 uatum stats of systms ad thr rgs:,, 3, umbr of systms havg. 3 3 Th: 4 tot tot Mcrocacal W{ N} tot Thr xsts th most probabl cofgurato wth th maxmal wght.. 6

7 Toward th prcpls of statstcal thrmodyamcs - Boltzma dstrbuto Havg ths uatos: tot tot W max { N} tot )Strlg approxmato M ) Mthod of Lagrag multplrs tot Partto Fucto > M Partto Fucto for a classcal systm: ddp ( p ), 7

8 8 U / / /, l l U T T N V Partto fucto ad tral rgy U N V T U, l

9 Partto fucto ad tropy du d + d d rv TdS l + l From Boltzma dstrbuto: [ ] TdS d l d l 0 d ds d k S k l l Itgrato 9

10 0 Partto fucto ad tropy k S l k S / l + V N T T k S, l l I N,V, smbl W /p

11 Partto fucto ad fr rgy Sc Th, A U TS A l Sc Th, G G A + pv l l V V N, T Lcturs 8/9 o how to valuat A or G by mas of Molcular Dyamcs (MD) ad Mot Carlo (MC) tchus whr tratd classcally: ( p ), ddp

12 Molcular partto fucto For dal gas (gas of o-tractg atoms/molculs): N N N th umbr of partcls (molculs) uclar lctroc vbrato rotato traslato tral ε k ε 0 g k k * k g k ε k

13 Molcular partto fucto N N ( ) ε g N N Nε 0 k k U 0 k ZPV # mods U + 0 N lc hν l l Itral rgy at 0K l tras A U A 0 tras + l N l A rot + A vb + A l * tras +... l rot l vb l l

14 4 Traslatoal partto fucto rgy of a atom/molcul th 3D box (Soluto of th Schrodgr uato wth th ft pottal wll): c b a m h z y x ( ) ( ) ( ) z z y y x x c m h b m h a m h tras ) ( 8 ) ( 8 ) ( 8 / 0 C d x C C x x x π V h m tras 3/ π 3D box RT N U tras 3 3 RT N H tras 5 5

15 Vbratoal partto fucto Vbratoal rgy of a molcul th 3N-6 harmoc pottal (Soluto of th Schrodgr uato): Icludd to U 0 3N-6 wll vb #mods l + hν l vb 3N 6 l hv l / 3N 6 l hvl / hvl / ( ) hvl / hvl / ( ) vb 3N 6 l hvl / hv l / 5

16 Vbratoal partto fucto ad vbratoal tropy S k l + T l T N, V S vb Nk l vb + T l T vb N, V vb 3N 6 l hvl / S vb Small rror low fr. v l R 3N 6 l hvl l hvl / ( ) Larg rror S vl [ ] hν / 0 l S vb To ovrcom (partally) a problm by cosdrg vbratoal low-frucy dgrs of frdom as fr or hdrd rotors 6

17 Rotatoal partto fucto S for rotatoal moto of datomcs rot h J ( J 8π I + ) g rot J + rot 8π I σ h Polyatomcs: U rot H rot 3 N 3 RT rot π σ / 8π h 3/ ( I I I ) / A B C 7

18 Gbbs fr rgy of a racto ( a mplct soluto) - smplst approxmato G RT l + pv RT RT l Vald for dal gas approx Idal-gas harmoc-oscllator/rgd rotor approxmato: G [U 0 + RT RTl] gp + solv lc + ZPV + RT RTl vb rot tras + solv Low-frucy mod ssus Solvato rgy dscussd Lct. 6 Harmoc oscllator Approx. fals G lc + ZPV RTl vb rot tras + solv 8

19 Th rlato btw K ad th partto fucto R K P R R tras R rot R vb P GR P U 0 RT R ( P R l l ) U RT 0 l > G RT P U RT R P / 0 / K R 9

20 (Gbbs/Hlmholz) fr rgy of a racto ( a xplct soluto) G A f pv 0 ( p ), ddp A l A l ( p, ) ( p, ) ( p, ) ddp ddp A ( p, ) l P(, p) ddp Probablty of bg a partcular pot of a phas spac A A P A Mor lcturs 8/9 R l M P M P / l M R M R / 0