N J of oscillators in the three lowest quantum

Transcription

1 . a) Calculat th fractinal numbr f scillatrs in th thr lwst quantum stats (j,,,) fr fr and Sl: ( ) ( ) ( ) ( ) ( ) fr usth sam apprach fr fr j fr frm q. b)

2 .) a) Fr a systm f lcalizd distinguishabl scillatrs, Bltzmann statistics applis. Shw that th ntrpy S is givn by Νj S Κ Ν Ν Accrding t Bltzmann statistics S n g W Ν! Π! S Κ w w Ν! g w! w! () Ν g! Apply strling apprximatin w Ν Ν Ν Νj w Ν Ν Ν w w Ν j Ν Ν Ν Νj Νj ( Ν Ν ) Ν Ν Ν w Ν Ν Put this valu f w in q () Ν S Κ Ν Ν n Ν S Κ Ν Ν Ν!! Νj b) Substitut th Bltzmann distributin in th prvius rsult t shw that S Νκ Z Τ Prvius rsult is n Ν S Κ Ν Ν W nw that g κτ z

3 Put th valu f Ν in q and Ν g K S ΚΝ z z S K K g S K z g g S K z K K S K z K S K z S K z S K z K c) sing th xprssin drivd in th txt fr and, Prv that S K whr. Examin th bhaviur f S and apprachs zr K Frm th prvius rsult S K z () Put valu in q () K S _ K z K K S K z b pag 8 ()

4 W nw that z z Put this valu f z in q () K S S K K K S

5 .) cnsidr diatmic mlculs at a tmpratur a) Find th numbr in ach f th thr lwst ratinal stats sl: b) Find th ratinal nrgy f th systm. 6

6 .6) s th data f abl. t dtrmin r, th quilibrium distanc btwn th nucli a) An H mlcul b) A CO mlcul Ans: Charactristics mpratur f H 8. Charactristics mpratur f CO.8 h π K.8 K a) Equilibrium Distanc btwn ucli fr H? W nw I µ Whr I mmnt f inrtia µ Rducd mass S mm I m m () m m m m Calculatin f I W nw that ( H ) I I 6.66 (.8 ) I.7 gm Put th valu f I in q () and m ( H ).67 7

7 mm I m m.7 7 (.67 ) 7 7 ( ) ( H ) 7. m b) Equilibrium Distanc btwn C-O ucli? 6 Mass f C (atm) m. 6 Mass f O (atm) m.66 Calculating f I I K (. ).8 I Put this valu in q () part (a) mm I m m ( CO). m

8 .7) Cnsidr a diatmic gas nar rm tmpratur. Shw that th ntrpy is Sl: πm S h 7 Fr diatmic mlculs S S At rm tmpratur S S xcit trans S S S xcit, ( dsnt cntribut!) Fr translatinal mtin, th mlculs ar tratd as nn-distinguishabl assmbly Z V πm h ( thr dg r f frdm ) ( Z ) V πm h Fr atinal mtin (thy ar distinguishabl in trms f intic nrgy)

9 S Z Z ( fr distinguis habl assmbly ) du t hm nuclar V πm S systm h 7 V πm h mlcul.8) fr a ilml f nitrgn ( ) at standard tmpratur and prssur, cmputr a) h intrnal nrgy ; b) th Hlmhltz functin F; and c) th ntrpy S slv this prblm, n nds t nw and bth can b fund frm r th txtb and.9 fr r

10 a) ml ml bcaus r trans b) Hlmzltz Functin F, can b calculatd thrugh th partitin functin

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12 F F F Z.m F Z t F t πm V h...9 F F. ( Z ) ( nn distinguishabl particls) Z ( Distinguishabl particls such as & atin) ( Z ) 9 ( 7.86 ) ( 7.) ( ).9 F t Z 98.6 Z 98. Z.6.6. F Z Z fr F t 6 π F t Z ( (. ) ).87 7

13 c) s F F S 7 (.87 ) ) sing th rlatin P Z V diatmic gas is th sam as that f a mnatmic gas. Fr diatmic mlculs, h partitin functin Z Z trans πm V h Z Z trans Z shw that th quatin f stat f a Z ( h sam as mnatmic cas) ( indpndnt f vlumv ) Z ( indpndnt f vlumv ) Ztrans P V thrfr it is th sam as mnatmic gas systm!.) Calculat th spcific hat capacity at cnstant vlum fr hydrgn H at s Equatin. C V 6 nr ( ) 6 6 ( ) (..69).6