6. Chemical Potential and the Grand Partition Function

Transcription

1 6. Chemcl Potentl nd the Grnd Prtton Functon ome Mth Fcts (see ppendx E for detls) If F() s n nlytc functon of stte vrles nd such tht df d pd then t follows: F F p lso snce F p F we cn conclude: p In other words cross dervtves re equl. hs s one of mny Mxwell reltons. hey re useful n reltng prtl dervtves of stte functons snce one my e eser to clculte or mesure thn the other. lso f ( ) s n nlytc functon of stte vrle nd then t follows from the trple product rule: Dffusve Equlrum nd the Chemcl Potentl Consder the followng system where Σ nd Σ re n therml equlrum wth het th Σ t temperture nd re seprted from ech other y permele memrne so tht prtcles cn dffuse ck nd forth. ssume there re nd prtcles n Σ nd Σ respectvely. Complete equlrum (therml nd dffusve) wll e reched the totl entropy tot s mxmzed suject to the condtons tot

2 Consder smll numer of prtcles d nd smll mount d gong from Σ to Σ so tht d d d d d d d d d tot nce Σ nd Σ re n therml equlrum the nd term. he condton for dffusve equlrum s then: the chemcl potentl :... µ where ll stte vrles re constnt except. reservor s het th tht cn exchnge oth het nd prtcles wth the system. reservor s lrge system so tht t cn exchnge het nd prtcles wthout ny chnges n ts nd µ. reservor s chrcterzed y temperture nd chemcl potentl (or potentls f there s more thn one knd of prtcle) Returnng now to sngle system wth vrle numer of prtcles ) (. nce s n nlytc functon of stte vrles nd : d d d d Eq. * pplyng the trple product rule ove whle keepng constnt: p snce / nd p Insertng these nto Eq. * d d p d d µ. hen solvng for d

3 3 d d pd µ d Eqn ** mplyng µ hs s generlzton of the frst lw of D generlzed to nclude trnsfer of prtcles. nce t s dffcult to hold constnt when chnges t my e eser to express µ n terms of chnge n F : F mples df d d d Insertng Eqn ** for d we otn df d pd µ d so µ F uppose Σ nd Σ re not n dffusve equlrum nd so hve dfferent chemcl potentls µ > µ. ssume d prtcles re trnsferred from to where we don t specfy the sgn of d. hen the chnge n totl free energy F F F s: F F df d d nce d d df ( µ µ d ) d nce F s mnmum n equlrum the chnge df cused y d prtcles movng from to must e negtve nd therefore d must e postve numer. In other words prtcles move from hgher chemcl potentl to lower chemcl potentl s expected We cn lso express µ n terms of the chnge n free enthlpy G cused y ddng prtcle. hs s useful snce t s eser to perform the prtcle ddton whle holdng p nd constnt rther thn nd. nce G F p p dg df pd dp d dp µ d hus

4 4 µ G p ote G ( p ) s n extensve vrle (excludng ny contrutons from surfces.) he ulk contruton to G s functon of p nd of whch only s extensve. It follows: G µ hs cn e seen s follows. Let the sze of the system (nd numer of prtcles) expnd y fctor λ wth p nd constnt. nce G s extensve t must expnd y the sme fctor or n other words: λ G G( p λ ) ke the prtl dervtve of oth sdes wrt λ holdng p nd constnt: G G ( λ ) p For λ G G µ p Influence of Externl Potentl We hve shown ove tht the chemcl potentl µ s the sme for two systems Σ nd Σ whch re n therml nd dffusve equlrum. However f there s some externl potentl (grvttonl electrcl etc ) whch chnges the potentl energy of ech prtcle n Σ y n mount E nd tht the nfluence of s dfferent for system Σ (.e. E E ) then the chemcl potentl (defned n the sence of ) wll e dfferent n Σ nd Σ. In ths cse n equlrum µ E µ E. o show ths consder the free energy n Σ ncludng E. F E df d p d µ d E d µ F µ ' E ; Eq. * mlrly µ F ' µ E In therml equlrum:

5 5 µ ' µ ' nd thus µ E µ E If n electrc feld s responsle for the energy shft then µ ' s referred to s the electrochemcl potentl. (ote n mny texts the chemcl potentl s defned from Eq. * whch ncludes the nfluence of n externl potentl). Exmple: Electrochemcl Potentl Consder two seprted metls wth work functons φ nd φ respectvely where φ s the energy requred to remove n electron from the metl. hs s the dfference n chemcl potentls nsde nd outsde the metl e.g. φ µ v µ nd φ µ v µ. ssume φ > φ. Once they re rought nto contct electrons wll flow from to (hgher chemcl potentl to lower potentl). In equlrum contct voltge c develops etween nd so tht the electrochemcl potentl s equl on oth sdes of the nterfce. kng nto ccount the chrge of the electron s negtve ( q ). µ ( q ) µ ( q) or q( ) µ µ φ φ φ φ c q Erth s tmoshpere Consder frst sngle prtcle n the erth s grvttonl feld where the grvttonl potentl energy: E mgz where z s heght ove se level. ssume the other contrutons to the energy re ndependent of heght e.g. knetc energy (temperture). hen the cnoncl dstruton tells us the P ( dz Z exp[ βmgz] dz so the prtcle densty: ( ( exp[ βmgz] whch cn e rewrtten mgz k ln where s the prtcle densty t se level. One cn use ths result to determne how the chemcl potentl for n del gs depends on densty.

6 6 Let µ e the chemcl potentl t se level. In equlrum the totl grvttonlchemcl (grv-chemcl) potentl s ndependent of heght: µ ' µ ( z ) µgz µ µ ( µ k ( ln s functon of z the densty drops exponentlly the chemcl potentl drops lnerly nd the grvttonl potentl rses lnerly to keep the grv-chemcl potentl constnt. ote s functon of densty ( (wth constnt) the chemcl potentl must logrthmclly s the densty (. We wll soon see tht µ must lwys e negtve for n del gs nd just ecomes less negtve t hgher denstes. In the present cse there s competton etween the grvttonl potentl whch tends to concentrte t the gs t se level nd the chemcl potentl whch tends to de-concentrte the toms. hs n turn cn e trced ck to tendency to mxmze entropy y dstrutng the gs over lrger volume. Grnd Prtton Functon Consder n open system Σ coupled to reservor Σ t temperture nd chemcl potentl µ. ssume Σ hs sngle stte wth energy E f t s ether occuped wth one prtcle or s unoccuped. here s no entropy n Σ ndependent of whether t s occuped or not. e.g. If Σ were H tom E would e 3.6e. he reservor nd thus the totl system wll hve smller entropy when the stte n Σ s occuped (f E s postve). In prtculr t wll e reduced y n mount E µ E ( ) ( ) where the second equl sgn follows the defntons for µ nd. In prtculr µ nd

7 7 hus when prtcle leves the reservor the chnge n entropy hs two contrutons one from the prtcle numer decresng nd second from the energy n the reservor decresng y n mount E. Generlzng to stte tht cn hold n prtcles (note n here s not the densty ut n occupton numer) : nµ E( n) where µ nd of the lrge reservor re ndependent of n ut E (n) s n generl nontrvl functon of n. If the prtcles were non-nterctng then E ( n) ne. However n the cse of the H tom E( ) 3. 6e. Wht s E () for the H tom (negtve on)? he prolty tht the stte wth n prtcles nd energy E (n) versus the prolty tht the stte s unoccuped: Ω exp[ / / k ] occ occ nµ E( n) exp[ / k ] exp exp[ ( n E( n))] Ωunocc exp[ unocc / / k ] k β µ s n the cnoncl dstruton ths must hold true for mcrosystem wth multple energy levels nd occupnces. he only wy for ths to hold true s tht the prolty for the system to contn n prtcles n mcrostte wth energy E must e proportonl to the G s fctor exp[ β ( nµ E )] : P( n E ) Ξ exp[ β ( nµ E )] where the normlzton constnt s the sum over ll possle G s fctors Ξ exp[ β ( nµ )] n E hs s clled grnd prtton functon whch s generlzton of the prtton functon to systems n contct wth reservor where oth energy nd prtcle numer cn vry. he prolty tht the system hs n prtcles s otned y summng over ll energes wth n prtcles: P( n) Ξ exp[ β ( nµ E )] Ξ exp[ βnµ ] exp[ βe ] P( n) Ξ n α Z n where α exp[βµ ] s clled the ctvty nd Zn s the prtton functon for n prtcles.

8 8 n Ξ n n exp[ β ( nµ E )] k Ξ Ξ µ ln Ξ lnα α Ξ Ξ α Where we hve used: α exp[ βµ ] α βα µ µ α α αnd µ ln[ α] β tomc Ionzton Consder n tom wth onzton energy I n contct wth reservor of electrons t temperture nd chemcl potentl µ. We choose the zero of energy of the tom to e the onzed stte so tht when the electron s on the tom E I. If we gnore electron spn nd excted stte there re only two possle mcrosttes of the system correspondng to n E nd n E I. he grnd prtton functon: Ξ α exp[ βi ] wth men occupncy α Ξ α exp[ βi ] n Ξ α α exp[ βi ] exp[ β ( µ I )] thus the stte wll hve 5% occupncy n / when µ I. he chemcl potentl (or ctvty) s functon of the prtcle densty nd temperture. It convenent to express t n terms of the quntum concentrton q ( n q n Chpter 6) 3/ mk q ( ) π where we hve dded the effect of spn whch ncreses the sngle prtcle prtton functon y fctor of. In the lmt << q the ctvty s smply: α / or equvlently: q µ k ln q We wll show ths s true for n del gs soon ut note t s consstent wth the densty vrton we found for gs n grvttonl feld where: ( ( α( µ ( µ k ln k lnα k ln k ln

9 9 9 Consder the surfce of the sun where the electron densty 6 m nd 64 K or k. 55e. t ths gnorng electron spn 8 [ ] 9. e q m 9 6 µ k lnα.55e ln.55e ln ote µ s negtve 7.5 for non-degenerte gses where << ut ncreses wth numer densty. In the cse q of L where I 5. 4e whch s consderly less thn µ the occupton prolty s much less thn. n 4 8 exp[( ) /.55] thus even though k << I onzton s nerly complete. How do you expln tht? lso so fr we hve neglected tht the electron s /. How does ths ffect Ξ nd α? Fluctutons n n he men squred fluctuton n occupncy: n n n n Ξ n β Ξ Ξ n exp[ β ( nµ E )] β Ξ n µ Ξ µ n β Ξ µ Ξ µ n β µ β µ n thus f µ s nsenstve to n fluctutons n n wll e lrge. ummry: