6. Chemical Potential and the Grand Partition Function

Transcription

1 6. Chemcl Potetl d the Grd Prtto Fucto ome Mth Fcts (see ppedx E for detls) If F() s lytc fucto of stte vrles d such tht df d pd the t follows: F F p lso sce F p F we c coclude: p I other words cross dervtves re equl. hs s oe of my Mxwell reltos. hey re useful reltg prtl dervtves of stte fuctos sce oe my e eser to clculte or mesure th the other. lso f ( ) s lytc fucto of stte vrle d the t follows from the trple product rule: Dffusve Equlrum d the Chemcl Potetl Cosder the followg system where Σ d Σ re therml equlrum wth het th Σ t temperture d re seprted from ech other y permele memre so tht prtcles c dffuse ck d forth. ssume there re d prtcles Σ d Σ respectvely. Complete equlrum (therml d dffusve) wll e reched the totl etropy tot s mxmzed suject to the codtos tot

2 Cosder smll umer of prtcles d d smll mout d gog from Σ to Σ so tht d d d d d d d d d tot ce Σ d Σ re therml equlrum the d term. he codto for dffusve equlrum s the: the chemcl potetl :... where ll stte vrles re costt except. reservor s het th tht c exchge oth het d prtcles wth the system. reservor s lrge system so tht t c exchge het d prtcles wthout y chges ts d. reservor s chrcterzed y temperture d chemcl potetl (or potetls f there s more th oe kd of prtcle) Returg ow to sgle system wth vrle umer of prtcles ) (. ce s lytc fucto of stte vrles d : d d d d Eq. * pplyg the trple product rule ove whle keepg costt: p sce / d p Isertg these to Eq. * d d p d d. he solvg for d

3 3 d d pd d Eq ** mplyg hs s geerlzto of the frst lw of D geerlzed to clude trsfer of prtcles. ce t s dffcult to hold costt whe chges t my e eser to express terms of chge F : F mples df d d d Isertg Eq ** for d we ot df d pd d so F uppose Σ d Σ re ot dffusve equlrum d so hve dfferet chemcl potetls >. ssume d prtcles re trsferred from to where we do t specfy the sg of d. he the chge totl free eergy F F F s: F F df d d ce d d df ( d ) d ce F s mmum equlrum the chge df cused y d prtcles movg from to must e egtve d therefore d must e postve umer. I other words prtcles move from hgher chemcl potetl to lower chemcl potetl s expected We c lso express terms of the chge free ethlpy G cused y ddg prtcle. hs s useful sce t s eser to perform the prtcle ddto whle holdg p d costt rther th d. ce G F p p dg df pd dp d dp d hus

4 4 G p ote G ( p ) s extesve vrle (excludg y cotrutos from surfces.) he ulk cotruto to G s fucto of p d of whch oly s extesve. It follows: G hs c e see s follows. Let the sze of the system (d umer of prtcles) expd y fctor λ wth p d costt. ce G s extesve t must expd y the sme fctor or other words: λ G G( p λ ) ke the prtl dervtve of oth sdes wrt λ holdg p d costt: G G ( λ ) p For λ G G p Ifluece of Exterl Potetl We hve show ove tht the chemcl potetl s the sme for two systems Σ d Σ whch re therml d dffusve equlrum. However f there s some exterl potetl (grvttol electrcl etc ) whch chges the potetl eergy of ech prtcle Σ y mout E d tht the fluece of s dfferet for system Σ (.e. E E ) the the chemcl potetl (defed the sece of ) wll e dfferet Σ d Σ. I ths cse equlrum E E. o show ths cosder the free eergy Σ cludg E. F E df d p d d E d F ' E ; Eq. * mlrly F ' E I therml equlrum:

5 5 ' ' d thus E E If electrc feld s resposle for the eergy shft the ' s referred to s the electrochemcl potetl. (ote my texts the chemcl potetl s defed from Eq. * whch cludes the fluece of exterl potetl). Exmple: Electrochemcl Potetl Cosder two seprted metls wth work fuctos φ d φ respectvely where φ s the eergy requred to remove electro from the metl. hs s the dfferece chemcl potetls sde d outsde the metl e.g. φ v d φ v. ssume φ > φ. Oce they re rought to cotct electros wll flow from to (hgher chemcl potetl to lower potetl). I equlrum cotct voltge c develops etwee d so tht the electrochemcl potetl s equl o oth sdes of the terfce. kg to ccout the chrge of the electro s egtve ( q ). ( q ) ( q) or q( ) φ φ φ φ c q Erth s tmoshpere Cosder frst sgle prtcle the erth s grvttol feld where the grvttol potetl eergy: E mgz where z s heght ove se level. ssume the other cotrutos to the eergy re depedet of heght e.g. ketc eergy (temperture). he the cocl dstruto tells us the P ( dz Z exp[ mgz] dz so the prtcle desty: ( ( exp[ mgz] whch c e rewrtte mgz k l where s the prtcle desty t se level. Oe c use ths result to determe how the chemcl potetl for del gs depeds o desty.

6 6 Let e the chemcl potetl t se level. I equlrum the totl grvttolchemcl (grvchemcl) potetl (eergy) s depedet of heght: ' ( z ) gz ( k ( l s fucto of z the desty drops expoetlly the chemcl potetl drops lerly d the grvttol potetl rses lerly to keep the grvchemcl potetl costt. ote s fucto of desty ( (wth costt) the chemcl potetl must logrthmclly s the desty (. We wll soo see tht must lwys e egtve for del gs d just ecomes less egtve t hgher destes. I the preset cse there s competto etwee the grvttol potetl whch teds to cocetrte t the gs t se level d the chemcl potetl whch teds to de-cocetrte the toms. hs tur c e trced ck to tedecy to mxmze etropy y dstrutg the gs over lrger volume. Grd Prtto Fucto Cosder ope system Σ coupled to reservor Σ t temperture d chemcl potetl. ssume Σ hs sgle stte wth eergy E f t s ether occuped wth oe prtcle or s uoccuped. here s o etropy Σ depedet of whether t s occuped or ot. e.g. If Σ were H tom E would e 3.6e. he reservor d thus the totl system wll hve smller etropy whe the stte Σ s occuped (f E s postve). I prtculr t wll e reduced y mout E E ( ) ( ) where the secod equl sg follows the deftos for d. I prtculr d

7 7 hus whe prtcle leves the reservor the chge etropy hs two cotrutos oe from the prtcle umer decresg d secod from the eergy the reservor decresg y mout E. Geerlzg to stte tht c hold prtcles (ote here s ot the desty ut occupto umer) : E( ) where d of the lrge reservor re depedet of ut E () s geerl otrvl fucto of. If the prtcles were o-terctg the E ( ) E. However the cse of the H tom E( ) 3. 6e. Wht s E () for the H tom (egtve o)? he prolty tht the stte wth prtcles d eergy E () versus the prolty tht the stte s uoccuped: Ω exp[ / / k ] occ occ E( ) exp[ / k ] exp exp[ ( E( ))] Ωuocc exp[ uocc / / k ] k s the cocl dstruto ths must hold true for mcrosystem wth multple eergy levels d occupces. he oly wy for ths to hold true s tht the prolty for the system to cot prtcles mcrostte wth eergy E must e proportol to the G s fctor exp[ ( E )] : P( E ) Ξ exp[ ( E )] where the ormlzto costt s the sum over ll possle G s fctors Ξ exp[ ( )] E hs s clled grd prtto fucto whch s geerlzto of the prtto fucto to systems cotct wth reservor where oth eergy d prtcle umer c vry. he prolty tht the system hs prtcles s oted y summg over ll eerges wth prtcles: P( ) P( ) Ξ exp[ ( E )] Ξ exp[ ] Ξ α Z exp[ E ] where α exp[ ] s clled the ctvty d Z s the prtto fucto for prtcles. Ξ exp[ ( E )] k Ξ Ξ lξ lα α Ξ Ξ α

8 8 tomc Iozto Cosder tom wth ozto eergy I cotct wth reservor of electros t temperture d chemcl potetl. We choose the zero of eergy of the tom to e the ozed stte so tht whe the electro s o the tom E I. If we gore electro sp d excted stte there re oly two possle mcrosttes of the system correspodg to E d E I. he grd prtto fucto: Ξ α exp[ I ] wth me occupcy α Ξ α exp[ I ] Ξ α α exp[ I ] exp[ ( I )] thus the stte wll hve 5% occupcy / whe I. he chemcl potetl (or ctvty) s fucto of the prtcle desty d temperture. It coveet to express t terms of the qutum cocetrto q ( q Chpter 6) 3/ mk q ( ) π where we hve dded the effect of sp whch creses the sgle prtcle prtto fucto y fctor of. I the lmt << q the ctvty s smply: α / or equvletly: q k l q We wll show ths s true for del gs soo ut ote t s cosstet wth the desty vrto we foud for gs grvttol feld where: ( ( α( ( k l k lα k l k l 9 3 Cosder the surfce of the su where the electro desty 6 m 7 d 64 K or k. 55e. t ths gorg electro sp.5 m 8 [ ] 9. e 9 6 k lα.55e l.55e l ote s egtve 7.5 for o degeerte gses where << q ut creses wth umer desty. I the cse of L where I 5. 4e whch s cosderly less th the occupto prolty s much less th. q exp[( ) /.55] thus eve though k << I ozto s erly complete. How do you expl tht? lso so fr we hve eglected tht the electro s /. How does ths ffect Ξ d

9 9 α? Fluctutos he me squred fluctuto occupcy: E Ξ Ξ Ξ )] ( exp[ Ξ Ξ Ξ Ξ thus f s sestve to fluctutos wll e lrge. ummry: