Q Q N, V, e, Quantum Statistics for Ideal Gas and Black Body Radiation. The Canonical Ensemble

Transcription

1 Quantum Statistics fo Idal Gas and Black Body Radiation Physics 436 Lctu #0 Th Canonical Ensmbl Ei Q Q N V p i 1 Q E i i Bos-Einstin Statistics Paticls with intg valu of spin... qi... q j q j... qi... 1

2 Fmi-Diac Statistics Paticls with half-intg valu of spin... q... q q... q... i j j i E N i i i n n i n s 1 ln Q s

3 BE statistics Planck Distibution n s 1 s 1 FD statistics n s 1 s 1 BE statistics n s 1 s 1 3

4 Maxwll-Boltzmann Statistics n s s Gand Canonical Ensmbl i E N i i BE Planck Patition Function 1 ln ln

5 BE statistics 1 ln ln FD Patition Function 1 ln ln 1 Classical Limit n s 1 s kt 1 5

6 Maxwll-Boltzmann Statistics Th numb of quantum stats availabl to ach paticl must b lag compad to numb of paticls Quantum Mchanics U m Tanslational Motion 1 p h k m m m mv 6

7 On-dimnsional box n L n 1... h n n n mL Th-dimnsional box h l m n n lm n m a b c Numb of quantum stats with ngy lss than ctain valu R l m n 7

8 Radius in Quantum Spac R 8mL 8mV h h 8mV R h Volum in Quantum Spac G G 3 4 R mV 8m 3 8 h 6 h V Calculation of tanslational patition function l m n1 l m n kt kt g d h 3 kt 8mkTV 0 1 8

9 Numb of quantum stats in th ngy intval d 3 3 dg d 8m 8m g V V d d 6 h 4 h 1 3 8m 1 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 N qt Q N! q t ln Q N ln qt N ln N N N ln N 3 mkt V ln Q Nln h N 9

10 Thmodynamic Functions of Monatomic Idal Gas 3 mkt V F kt ln Q NkT ln h N Pssu 3 ln Q mkt V p kt NkT ln V T N V h N 3 1 mkt V NkT NkT 3 mkt V V h N V h N pv NkT pv nrt Intnal Engy U kt NkT T T h N 3 ln Q mkt V ln V N 3 1 mkt V 3 NkT NkT 3 mkt V T h N h N 10

11 Entopy 3 U F 3 mkt V S Nk Nk ln T h N mkt Nk ln h 3 5 V N 3 5 mkt kt Nk ln h p Chmical Potntial 3 G ln Q mkt V kt kt N ln N N VT N h N 3 3 mkt V ktn mkt V kt ln 3 h N mkt V N h N h N 3 3 mkt V mkt V kt ln kt kt kt ln h N h N 3 3 mkt kt mkt kt ln kt ln kt kt ln p h p h Polyatomic Idal Gas 11

12 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 N q N! E i... kt 1

13 Evaluation of Patition Function q q q q q t v Tanslation Vibation Rotation Elctonic Excitation Tanslational Patition Function q t m1 m kt h 3 V Elctonic patition function q j j D kt kt 1 1 j kt q D 13

14 Vibational motion 1 u D f u f Oscillations x fx x mm 1 m m 1 f 1 Vibational Patition Function 1 n n n v n kt kt n kt q n0 n0 qv 1 qv 1 kt kt v T v T v k 14

15 Rotational Motion j j j 1 j I Rotational Patition Function q j0 j j j 1 j0 j j1 j j0 j kt j 1 IkT T 1 Ik Rotational Patition Function j j1 T q j 1 dj 0 j j1 T T d[ j j 1 ] q 0 T 1 15

16 Patition Function N q Q N! ln Q N ln q N ln q N ln q N ln q N ln N N t v q t ln Q N(ln ln qv ln q ln q ) N Thmodynamic Functions of Diatomic Idal Gas 3 m1 m kt V IkT F kt ln Q NkT ln NkT ln h N N kt NkT ln 1 ND NkT ln 1 Pssu 3 ln Q mkt V p kt NkT ln V T N V h N 3 1 mkt V NkT NkT 3 mkt V V h N V h N pv NkT pv nrt 16

17 Intnal Engy ln Q 3 N N U kt NkT NkT ND kt T VN 1 Hat Capacity kt U 3 CV Nk Nk Nk T kt kt 1 Entopy 3 5 U F m1 m kt V S Nk ln T h N IkT kt Nkln Nk kt Nkln 1 Nkln kt

18 Chmical Potntial 3 G ln Q m1 m kt V kt kt ln N N VT h N IkT kt kt ln kt ln 1 D kt ln 1 Black Body Poblm Photon Gas Gas of f photons Unity spin 18

19 Consvation of Engy nh n i i mh m j j Chmical Equilibium nm 0 ( nm) 0 0 Statistical Ensmbl Gand Canonical Ensmbl Systm of fixd VT 19

20 Standing Wavs a n Th numb of allowd fquncy stats 1 dimnsion a i 3 dimnsions ix a iy a i a z cos cos cos 1 3 Quantum Numb Spac R i i i x y z 4a cos 1 cos cos 3 4a 0

21 Th total numb of wavlngths bigg o qual R a 4 a G Th total numb of stats with fquncis lss o qual a 1 G V c Th total numb of stats with ngy lss o qual 1 V G 3 3 c 3 1

22 Boltzmann Statistics? Is th numb of quantum stats availabl to ach paticl lag compad to th numb of paticls? Th numb of mods in givn fquncy ang d d 1 V 1 g G V 3 3 d d 3 c c 3 Patition Function n 1 ln ln 1 n n 1 n 1

23 Summation ov Engy lvls h 1 ln g ln 1 c 1 akt Intgation vs. Summation 1 ln g ln 1 d 0 V 1 ln 3 1 d c V 1 ln 3 1 d c V V x dx x 3 c 1 3 c 1 1 d V x V Vk T x dx hc 1 0 c 15 c Thmodynamic Functions of Photon Gas 3

24 Pssu ln Vk T k T p kt kt V T V 45 c 45 c Intnal Engy and Stfan- Boltzmann Law ln Vk T Vk T U kt kt 3 3 T V T 45 c 15 c 1 p u 3 Hat Capacity C V E Vk T 4 Vk T 3 3 T V T 15 c 15 c

25 Entopy ln Vk T Vk T 4 Vk T S kt k ln T 15 c 45 c 45 c Avag Numb of Paticls N ln ln kt V V T n 1 1 n n dx 3 x 1 c N g V V 1 x d c Vk T 3 c 3 3 5

26 Th Spctal Engy Distibution 3 3 Vk T kt N d c kt / kt 0 N 3 3 Vk T 3 3 kt / kt c 1 kt E kt N kt E 3 3 Vk T kt 3 3 / kt c 1 3 Planck s Distibution Win s displacmnt law kt 1max kt max 1 6

27 Win s fomula at low tmpatu 3 3 E Vk T kt 3 3 /kt c 3 Rayligh-Jans fomula at high tmpatu 3 3 E Vk T 3 3 c kt Exampl Find intnsity of adiation adsobd by absolutly black body which is in thmal quilibium with suounding photon gas. 7

28 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 610K 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 8

29 Fluctuations of ns 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 N 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 9

30 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 E1 and E E1<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 30