H = d d q 1 d d q N d d p 1 d d p N exp

Transcription

1 8333: Sacal Mechanc I roblem Se # 7 Soluon Fall 3 Canoncal Enemble Non-harmonc Ga: The Hamlonan for a ga of N non neracng parcle n a d dmenonal box ha he form H A p a The paron funcon gven by ZN T d d q d d q N d d p d d p N exp d d qd d p exp βa p N β A p Ignorng hard core excluon each aom conrbue a d dmenonal volume o he negral over he paal degree of freedom and ZN T N d d p exp βa p N Obervng ha he negrand depend only on he magnude p p we can evaluae he negral n phercal coordnae ung d d p S d dpp d where S d denoe he urface area of a un phere n d dmenon a ZN T N Inroducng he varable x βa / p we have ZN T N S N d C N d N! N S d dpp d exp βap dn / A k B T N Sd A h d k B T N dxx d exp x dn/ where C denoe he numercal value of he negral We aume ha A and are boh real and pove Thee condon enure ha energy ncreae wh ncreang p The negral n fac equal o Cd dxx d exp x dx Γ d

2 and he paron funcon Z N! N dn/ Sd A Γ h d k B T N d b To calculae he preure and nernal energy noe ha he Helmholz free energy and ha F Fr calculang he preure: F F E T S k B T ln Z whle E ln Z T β k B T ln Z T Nk BT T Now calculang he nernal energy: E ln Z β β dn A ln d k B T Nk BT Noe ha for each degree of freedom wh energy A p we have he average value A p d k BT Th evaluae o 3 k BT for he 3 dmenonal deal ga c Now conder N daomc molecule wh H H where H A The expecaon value q q N N! dd q N! ealy calculaed by changng varable o p + p + K q q q q exp β H d d q d d p d d p N dd q d d q d d p d d p exp β H x q q and y q + q a noe ha he Jacoban of he ranformaon uny q q d d xd d y x exp βk x dd xd d y exp βk x d d x x exp βk x dd x exp βk x

3 Furher mplfyng he algebra by nroducng he varable z βk / x lead o q q βk / d d z z exp z dd z exp z d kbt K Here we have aumed ha he volume large enough o ha he range of negraon over he relave coordnae can be exended from o Alernavely noe ha for he degree of freedom x q q he energy K x Thu from par b we know ha e K x d kbt K x q q d kbt K And ye anoher way of calculang he expecaon value from q q β ln Z K Nd kbt K noe ha he relevan par of Z calculaed n par d below d For he deal ga he nernal energy depend only on emperaure T The ga n par c deal n he ene ha here are no molecule molecule neracon Therefore and Snce Nk B T C dq C dq de + d We now calculae he paron funcon Z N de + d C det zn 3 det det + + Nk B d d q d d q d d p d d p exp T T β H

4 where z d d q d d q d d p d d p exp β A d d q d d q exp βk q q Inroducng he varable x y and z a n par c Z N N! N N! N βk d/ z d exp z dz βk d/ Γ d N N N! βk dn/ βa Nd/ Now we can calculae he nernal energy a E ln Z β βa d/ Γ p p + A + K d d p exp βa d q q p N p d exp βap dp N d Nk BT + d Nk BT dnk B T From h reul he hea capace are obaned a C E T + d T Nk B + d + C E T dnk B + reulng n he rao + γ C C d/ + d/ + d/ + d/ + d + ******** Cure Sucepbly: The ne magnezaon of he N quanum pn M z µ m wh m a The Gbb paron funcon Z exp βb M exp βbµ {m } {m } 4 m m m expβµb m N

5 Thu we oban he ere Z exp βbµ + exp βbµ + + expβbµ + expβbµ N In general o evaluae a geomercal ere of he form ncreae he order of he ere by one and ubrac from he orgnal ere: S x + x + + x + x Sx x x + x + xs x x + S x x + x Noe ha he ame reul obaned wheher an neger or half neger quany Ung h expreon we ge N exp βbµ expβbµ + Z expβbµ N exp βbµ + / exp βbµ + / exp βbµ/ exp βbµ/ Subung n he proper rgonomerc deny b The Gbb free energy G E BM k B T ln Z N nh βµb + / Z nhβµb/ Nk B T lnnhβµb + / + Nk B T lnnhβµb/ Ung an approxmaon of nh θ for mall θ nh θ e θ e θ θ + θ3 + Oθ 5 3! for θ we fnd eng α βµb { G Nk B T ln α + + α α ln } + α + Oα 4 4

6 Ung he expanon ln + x x x / + x 3 /3 we fnd G Nk B T ln α + / 6 α/ + Oα 4 Nk B T ln + Nk B T α + 6 G Nµ B + + OB 4 6k B T c The magnec ucepbly χ M z / B obaned by nong ha he average magnezaon Thu χ M z B M z k B T ln Z B G B G B B Nµ + 3k B T whch obey Cure law χ c/t wh c Nµ + /3k B 3 Surfacan Adorpon: ******** a The paron funcon of a d-dmenonal deal ga gven by Nd { N d } Z d d d q N d!h dn d d p p exp β N d ε d + d m Nd e βn dε d N d! where λ h πmkb T The chemcal poenal calculaed from he Helmholz free energy a µ d F N k B T ln Z d T N d T ε d + k B T ln N d b The deny of parcle can alo be calculaed from he grand canoncal paron funcon whch for parcle n a d dmenonal pace Ξµ d T ZN d d T e βn dµ N d N d N d! Nd e βn dε d e βn dµ exp 6 e βµ ε d

7 The average number of parcle aborbed n he pace N d β µ ln Ξ β µ e βµ ε d e βµ εd We are nereed n he coexence of urfacan beween a d 3 dmenonal oluon and d dmenonal urface Dvdng he expreon for N 3 and N and akng no accoun ε ε 3 ε gve whch mple ha N N 3 Aλ eβε n N A nλeβε c I ha been uggeed ha a porou gel hould be regarded a fracal and he urfacan adorbed on urface reaed a a ga n d f dmenonal pace wh a non neger d f Ung he reul found n par b bu regardng he gel a a d f dmenonal conaner he adorbed parcle deny n gel nλ 3 d f exp βε 3 ε gel Thu by udyng he adorpon of parcle a a funcon of emperaure one can deermne he fracal dmenonaly d f of he urface The large conrbuon come from he dfference n energe If h leadng par accuraely deermned here a ubleadng dependence va λ 3 d f whch depend on d f ******** 7