STATISTICAL MECHANICS OF DIATOMIC GASES

Transcription

1 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) SAISICAL MECHAICS OF DIAOMIC GASES - Fo monatomic gas whos molculs hav th dgs of fdom of tanslatoy motion th intnal u 3 ngy and th spcific hat of on mol of gas a and cv R spctivly. V h ngy dos not dpnd upon th volum. - A molcul is a stabl compound of two lik o diffnt atoms. 3- h natu of focs that lads to th fomation of a molcul fom isolatd atoms is tatd in quantum mchanics. 4- h ngy of th molcul is mad up of: i. h tanslation kintic ngy of its cnt of mass Etans. ii. Engy of xcitation of th atomic lctons in th molcul E. iii. Engy of ation of th nucli along th axis joining thm E and iv. h ational kintic ngy du to th ational motion of th atoms about th cnt of mass of th molcul E. hus E Etans E E E h tanslational motion of a diatomic molcul is not quantizd. All th oth kinds of intnal motion of th molcul a quantizd i.. E E and E assum a disct sis of valus. Appoximations: I- Assum that th th kind of intnal motion of molcul a indpndnt of on anoth. II- h influnc of ational motion on ation can b nglctd. i.. th momnts of intial of molculs stay constant. 4 III- E V K i.. vy high tmpatu is quid to poduc a substantial numb of molculs in xcitd lctonic stats. his is why w a nglcting th contibution of th lctonic stats. IV- caus of 3 u R 3 4 E V to V E thn it is asy to find a fw ational and ation stats at oom tmpatu Vibational mods: Fo an oscillato with mass m and angula fquncy f th Hamiltonian in th dimnsions is: H( p q) p m q m Kintic ngy Potntial ngy and th ational ngy lvl is givn by: j ( j ) j ot hat: h ngis a qually spacd i.. j h gound stat has zo point ngy qual to / h stats a non-dgnat i.. gj = fo all j.

2 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) b h patition function: (us th gomtic sis summation ) b z (sinh a) a k v a v a ( a) a a a ot that: th valu of (has a units of tmpatu K) spaats btwn th quantum and classical gions Exampl: If you know that.9 V fo thn [ot: K 9 V.6 J J 336 K. k 3 J.38 k.38 ] 3 J So any tmpatu lss than o qual 336 K will b in quantum gion and any tmpatu gat than 336 K will b in classical gion. H.W. Calculat fo W shall also b concnd with th occupation numbs o with P / th faction of th H O and Cl. total numb of paticls with ngy j. h oltzmann distibution fo gi is j j j a a a ja Pj ( ) ( ) z h xponnt of th tm outsid th panthss can b wittn h h h j ( ) a j j ja j k k k k hus j j Pj ( ) A sktch of th last quation fo two tmpatus shows that th low th tmpatu th mo apidly th occupation numbs dcas with j (Figu 5.). At high tmpatus mo paticls populat th high ngy lvls. i i K h singl paticl patition function lads to a hat capacity of C / k / V U his hat capacity dcays xponntially as and tnds to R p mol as and tnds to zo as

3 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) Quantity Fomula Z z ( sinh a)? Patition function Hlmholtz f ngy Entopy Intnal ngy Pssu Vibational hat capacity k ( ) F k ln Z k ln(sinh a)? F k ln( ) F S k a coth a ln( a sinh a) V ln Z U w a V z ln P V U U C V V V k / / k S k ln( ) / k nr k / V nr / V k nr Substanc H (K) (K) k Ik O Cl Exampl: If you know that.9 V fo p mol at K. [ot: any calculat tmpatu lss than 336 K will b in quantum gion k c.38 ] 3 J Answ: Fom th abov tabl it was found that K ( ). Consquntly K will b in th 336 quantum gion. hn us a 3.36 W can hav / a J mol K c R a / a his is in xcllnt agmnt with th xpimntal valu c valu R 8.34 JK mol K 3.43 J mol K and away fom th classical 3

4 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) Rotational mods: Classically th kintic ngy of ation is: ( I) L K.E. I I I Using th lation m m thn th momnt of intia about an axis though th cnt of mass of a diatomic molcul is: mm I m m o m m o h ational ngy lvl is givn by: L l ( l ) I I l Wh I o is th ducd mass is th quilibium distanc btwn th nucli and is th momnt of intia of th molcul lativ to a ppndicula axis passing though th cnt of mass and l dtmining th angula momntum of th molcul lativ to th cnt of mass with its dgnacy gl (l ). h patition function: o l l l ( l ) / i ( ) ( ) l l l Ik z g l l A- Low tmpatus w just tak th fist two tms in th sis lading to l / z (l ) 3 l U k / 6 / CV 3k High tmpatus: At high tmpatu a vy lag numb of ational stats a occupid. Also th spacing of th ational lvls bcoms vy small compad with th thmal ngy and w may comput z by placing th summation by intgation. W also substitut l fo l+ and l fo l(l+). hn / / l l Z l classical dl l l dl which lads to intnal ngy of k and a hat capacity of R p mol du to ations Exampl: Calculat fo th diatomic nitogn. [ M 4.8 amu] Answ: h mass of a singl nitogn atom is m.35 kg th ducd mass is m/.63 kg and th momnt of intia of a molcul is I kg.976 m 4 kg.m I 4

5 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) h J s.88 K 8 Ik 47 3 J 8 4 kg.m.38 molcul.k Exampl: Calculat th patition function fo th hydogn molcul at 3 K givn that th momnt of 48 intia fo molcul is I 6.9 kg.m. Answ: 3 a) Using th classical xpssion: z classical wh h J s 64.3 K 8 Ik 48 3 J kg.m.38 molcul.k ot that 64.3 K 85.4 K so w hav to tat it using th quantum xpssion. ut w a going to tat it classically also to s th diffnc. i- Using th classical xpssion: ii- Using th quantum xpssion 3 z classical z (l ) l l ( l ) / on finds: ( ) l ( l ) / 64.3/ / 3 l z l Commnts: - h contibution du to ach ational lvl gos though a maximum at l = and thn dcay apidly bcoming ngligibl by l = 6. - his dcasing is mainly du to th facto l( l ) in th xponnt. 3- h diffnc btwn th classical and quantum sults is du placing a summation by an intgal whn quantization is still significant at this tmpatu Us th data of abl 5. to dtmin o th quilibium distanc btwn th nucli fo (a) an H molcul; (b) a CO molcul. Answ: hn k wh Ik and I and is th ducd mass k m m m m. a) Fo H m so mk 5

6 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) hn (amu)( J s kg/amu)(.38 3 J/K)(85.4 K) = m =.754 Å m m m m ( amu)(6 amu) amu 6 amu b) Fo CO amu hn J s 7 3 k (6.857 amu)(.66 kg/amu)( Summay h singl paticl patition function ot: Rgading E E E. C V h total hat capacity C C C C V V t V V E Etans E E E E E E E E tans m z z z z z V z z z h C C C C t V V t V V / 3 k k k / 3/ J/K)(.8 K) =. - m =. Å C Vwill b nglctd at low tmpatu bcaus th ngy diffnc / 3 k k k / Fo most diatomic molculs th spaation btwn th ational ngy lvls is much lss than k oom but th spaation btwn ational lvls is much gat than k oom. his mans that th hat capacity at oom tmpatu is typically 5 R / p mol: with 3 R / coming fom th tanslational hat capacity and th oth R fom th ations. At oom tmpatu th ational contibutions a small only coming into play at high tmpatu

7 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) Appndix Simpl Hamonic Oscillato Consid a on dimnsional hamonic oscillato with mass m and fquncy. a) Wit down th Hamiltonian. b) Us th ational ngy to calculat th classical patition function. E ( n ) h n c) Calculat th quantum patition function. Show that in th limit of found classically in pat (b). n h k this sult ducs to that d) Us th quantum mchanical patition function to calculat th intnal ngy ntopy and th hat capacity of a systm consisting of such oscillato as a function of tmpatu. Answ: a) In on dimnsion th Hamiltonian of th systm is givn by: ˆ pˆ x H ( p ) ˆ x qx m q x m b) classically th patition function fo singl oscillato is: sp x mw x p H dpxdqx m tans x h h Z Z Z dp dx m h m h Consquntly fo -oscillato systm w gt Z sp Z F k ln Z k ln H.W. pov th following thmodynamic quantitis k ln P S k P v V ln m m U k k (Equipatition thom) U C C k c) In quantum mchanics sinc th ngy lvls is E ( n ) h th patition function is givn by n 7

8 Pof. D. I. ass Phys54 7 -Ma-8 Diatomic_Gas (Ashly H. Cat chapt 5) Z a a En a na sp (sinh ) a a a n n a. a a a a thn Z sp In th limit a a is th sam sult as in pat b. d) Fo -indpndnt paticls Witing / k w could obtain th following sults: sp sinh Z Z a a a F k ln Z k ln sinh a k ln k ln a a ln a U ln Z coth a a a U CP Cv k a V a U F S k a coth a ln(sinh a) H.W. pov that in th limit of a th intnal ngy will b: U k Using th lation ln z. P V show that th quation of stat of a diatomic gas is th sam as that of a monatomic gas. Answ: ln Z P k. Wh Z ZtansZZ so ln Z ln Ztans ln Z ln Z V Z and Z do not dpnd on volum so 3 / mk Ztans V h ln Ztans P k V ln Z P k V 3 mk so ln Ztans lnv ln h lnv k k V V ln Z k V tans. So PV = k. which 8