Q Q N, V, e, Quantum Statistics for Ideal Gas. The Canonical Ensemble 10/12/2009. Physics 4362, Lecture #19. Dr. Peter Kroll

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1 Quantum Statistics fo Idal Gas Physics 436 Lctu #9 D. Pt Koll Assistant Pofsso Dpatmnt of Chmisty & Biochmisty Univsity of Txas Alington Will psnt a lctu ntitld: Squzing Matt and Pdicting w Compounds: Computational High Pssu Chmisty Thusday Octob 8 th :05 am Room S 6 Th lctu will b followd by a lunch/cption in S5 to allow him to answ qustions about gaduat pogams at UT- Alington. Th Canonical Ensmbl Ei Q Q V p i Q E i i

2 Quantum Idal Gas s s q q Bos-Einstin Statistics Paticls with intg valu of spin... qi... q j q j... qi... Fmi-Diac Statistics Paticls with half-intg valu of spin... q... q q... q... i j j i

3 E i i i n n i n s ln Q s BE statistics Planck Distibution n s s 3

4 FD statistics n s s BE statistics n s s Indpndnt idntical distinguishabl paticls Q q a i q Q q b E i... kt 4

5 Maxwll-Boltzmann Statistics Q Maxwll-Boltzmann Statistics n s s Gand Canonical Ensmbl i E i i 5

6 Gand Canonical Patition Function i ni Bos-Einstin statistics o limitation fo numb of paticls allowd p any quantum stat 0 n n 0 n 0 n 0 n n 6

7 Patition Function ln ln BE statistics Planck Distibution n s s BE statistics ln ln 7

8 Avag umb of Paticls ln V Pobability of occupancy of a givn singl paticl stat n s s s kt Fmi-Diac statistics o mo than on paticl is allowd p any quantum stat 8

9 Patition Function ln ln Avag umb of Paticls ln V Pobability of occupancy of a givn singl paticl stat n s s s kt 9

10 Classical Limit n s s kt Quantum Mchanics U m Tanslational Motion p h k m m m mv 0

11 On-dimnsional box n L n... h n n n... 8mL Th-dimnsional box h l m n n lm n... 8m a b c umb of quantum stats with ngy lss than ctain valu R l m n

12 Radius in Quantum Spac R 8mL 8mV h h 8mV R h 3 3 Volum in Quantum Spac G G 3 4 R mV 8m 3 8 h 6 h V Calculation of tanslational patition function l m n l m n kt kt g d h 3 kt 8mkTV 0

13 umb of quantum stats in th ngy intval d 3 3 dg d 8m 8m g V V d d 6 h 4 h 3 8m kt qt V d 4 h u mkt h 0 8mkT 8 V u du V 4 h 4 mkt h 3 V Patition Function qt Q! q t ln Q ln qt ln ln 3 mkt V ln Q ln h 3

14 Thmodynamic Functions of Monatomic Idal Gas 3 mkt V F kt ln Q kt ln h Pssu 3 ln Q mkt V p kt kt ln V T V h 3 mkt V kt kt 3 mkt V V h V h pv kt pv nrt Intnal Engy U kt kt T T h 3 ln Q mkt V ln V 3 mkt V 3 kt kt 3 mkt V T h h 4

15 Entopy 3 U F 3 mkt V S k k ln T h mkt k ln h 3 5 V 3 5 mkt kt k ln h p Chmical Potntial 3 G ln Q mkt V kt kt ln VT h 3 3 mkt V kt mkt V kt ln 3 h mkt V h h 3 3 mkt V mkt V kt ln kt kt kt ln h h 3 3 mkt kt mkt kt ln kt ln kt kt ln p h p h Polyatomic Idal Gas 5

16 Engy of th molcul t v Patition function fo a paticl qa V T aj kt j Indpndnt idntical indistinguishabl paticls Q q a Q i q b q! E i... kt 6

17 Maxwll-Boltzmann Statistics Th numb of quantum stats availabl to ach paticl must b lag compad to numb of paticls Evaluation of Patition Function q q q q q t v Tanslation Vibation Rotation Elctonic Excitation Tanslational Patition Function q t m m kt h 3 V 7

18 Elctonic patition function q j j D kt kt j kt q D Vibational motion u D f u f Oscillations x fx x mm m m f 8

19 Vibational Patition Function n n n 0... v n kt kt n kt q n0 n0 qv qv kt kt v T v T v k Rotational Motion j j j j 0... I Rotational Patition Function q j0 j j j j0 j j j j0 j kt j IkT T Ik 9

20 Rotational Patition Function j j T q j dj 0 j j T T d[ j j ] q 0 T Patition Function q Q! ln Q ln q ln q ln q ln q ln t v q t ln Q (ln ln qv ln q ln q ) Thmodynamic Functions of Diatomic Idal Gas 3 m m kt V IkT F kt ln Q kt ln kt ln h kt kt ln D kt ln 0

21 Pssu 3 ln Q mkt V p kt kt ln V T V h 3 mkt V kt kt 3 mkt V V h V h pv kt pv nrt Intnal Engy ln Q 3 U kt kt kt D kt T V Hat Capacity kt U 3 CV k k k T kt kt

22 Entopy 3 5 U F m m kt V S k ln T h IkT kt kln k kt kln kln kt Chmical Potntial 3 G ln Q m m kt V kt kt ln VT h IkT kt kt ln kt ln D kt ln Exampl Calculat th diffnc in Cv p mol fo H and D at 000K. You may assum that both spcis bhav as idal diatomic gass and you may nglct dissociation and lctonic xcitation of th molculs. vh 60K

23 Exampl Div an quation fo Cv th lctonic contibution to Cv. Vify that Cv appoachs to zo as tmpatu appoachs to zo o infinity. Vapo Pssu of a Solid Fluctuations of ns 3

24 Exampl Whn a paticl with spin ½ is placd in a magntic fild H its ngy lvl is split into H H and it has magntic momnt o along th diction of magntic fild spctivly. Suppos a systm consisting of such paticls is in a magntic fild H and kpt at tmpatu T. Find intnal ngy ntopy and th total magntic momnt of th systm with th hlp of canonical distibution. Idal Gas in a Gavitational Fild Exampl Consid a systm mad up of fou idntical paticls in a contain of volum V. Assum that ach paticl has availabl to it two ngy stats E and E E<E. Assum that th paticls can tak sam ngy lvls. a) Find q fo ach paticl b) Find Q fo th systm assuming th paticls to b distinguishabl c) Find Cv fo th systm and show that T 0 Cv 0; T Cv 0 4