Chem 4502 Prof. Doreen Leopold 10/18/2017 Name (Please print) Quantum Chemistry and Spectroscopy Exam 2 (100 points, 50 minutes, 13 questions)

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1 Chm 5 Prof. Dor Lopold ANSWER KEY /8/7 Nm Pls prit Qutum Chmistry d Spctroscopy Em poits, 5 miuts, qustios Pls chck hr if you would prfr your grdd m to b rturd to you dirctly rthr th big icludd mog lphbtizd pils o th frot dsk. If so, pls pick up your grdd m from Dor bfor or ftr clss. You hv 5 miuts to do this m : - : AM. Pls us th bubbl sht isrtd i your m to rcord your swrs to th multipl choic qustios i Prt I. For ths multipl choic qustios i Prt I: w will grd oly your bubbl sht; w will ot grd th work writt o your m. pls bubbl i oly o swr pr qustio if mor th o swr is bubbld i, o crdit will b giv thr is o plty for icorrct swrs, so you should choos swr for vry qustio For th qustios i Prt II, w will grd th work writt o your m. For ths qustios, to rciv full crdit, you must show your work ot just th fil swr. Bfor th m bgis, pls bubbl i your m lst m first d your itrt ID your itrt ID is wht prcds "@um.du" i your U of M mil ddrss. Qustios r chos from th mtril covrd i th rdig, problms d lcturs up to Fri. th th of Octobr, icludig: Homwork d Problm St : Prticl-i--bo i Chptr, pp Probbility d Sttistics MthChptr B, pp. 6-7 Homwork 5: Hrmoic oscilltor Chptr 5, Sctios -7, pp Postults d pricipls of qutum mchics Chptr, pp. 5 - You my us o-progrmmbl, o-grphig clcultor. No ots r llowd. Thr is qutio sht o th lst pg of th m tht you c tr off. PLEASE DO NOT TURN THE PAGE UNTIL IT IS ANNOUNCED THAT YOU MAY BEGIN.

2 Aswrs to Multipl Choic Qustios Vrsios A d B hv th sm qustios i diffrt ordrs d som of th swrs r i diffrt ordrs lso. Vrsio A Vrsio B. G. G. C. B. A. D. D. A 5. D 5. C 6. C 6. A 7. H 7. H 8. F 8. E 9. F 9. F. B. C

3 Chm 5 Em /8/7 Nm Pls prit Totl grd to b filld i by grdr: poits out of Prt I. Multipl Choic qustios, 6 poits ch, 6 poits totl Pls bubbl i your m, itrt ID, d swrs o th bubbl sht providd, ug # pcil. Etr pcils r vilbl o th frot dsk. For th multipl choic qustios i Prt I, w will ot grd th work show o your m; w will grd oly th swrs tht you fill i o th bubbl sht. Pls bubbl i oly o swr pr qustio.. Th fr-lctro modl for th lctroic stts of lir cojugtd hydrocrbos uss th o-dimsiol prticl-i--bo pproimtio for th rltiv rgis of th d * molculr orbitls. For htri C 6 H 8, which hs si lctros, lt = th ovrll lgth of th molcul, d m = th mss of th lctro. I uits of h / 8 m, prdict th photo rgy rquird to cit lctro from th HOMO highst occupid molculr orbitl to th * LUMO lowst uoccupid molculr orbitl. A. B. C. D. E. 5 F. 6 G. 7 H. 9 I. 6 J. 5 Aswr: G. 7

4 #, cotiud For th systm to bsorb photo d udrgo this trsitio to th highr rgy stt, th photo rgy must qul th rgy diffrc btw th two stts of th systm. Hr, w pproimt this s th rgy diffrc btw th = d = molculr orbitls. I uits of h /8m, th rgy of th HOMO, which hs =, is = 9, d th rgy of th LUMO, with =, is = 6. So, th rgy diffrc for th systm is 6-9 = 7. This is lso th photo rgy rquird to citd this trsitio: E photo = E systm = 7 h / 8m. This is simplr vrsio of # o our Problm St. For similr problm o htri, s Homwork Chptr, problm 6. For th pplictio of this modl for butdi, s PIB lctur ots p. d tt pp Wh solvig th Schrödigr qutio for prticl i -dimsiol bo of lgth Å, whos wlls r t = -5 Å d = +5 Å, th qutiztio of rgy lvls will rsult from th rquirmt tht th wv fuctios: A. b ormlizd B. b zro t = d = Å C. b zro t = -5 Å d = +5 Å D. b zro t = - Å d = + Å E. hv mimum mplituds t th wlls Aswr: C. b zro t = -5 Å d = +5 Å Th id tht qutiztio of rgy lvls coms bout from th impositio of boudry coditios is importt grl rsult illustrtd by th prticl-i--bo PIB modl. For th PIB, w rquir tht th wv fuctio = t th wlls. S p. 8 d PIB lctur ots p..

5 . For prticl i -dimsiol bo of lgth with wlls t = d =, wht is th mig of this prssio? / d A. th vrg positio of th prticl B. th probbility tht th prticl will b foud btw = d = C. th most probbl positio of th prticl, for th stt with =. D. this itgrl is st qul to to dtrmi th vlu of th ormliztio tt E. th vrg vlu of th kitic rgy of th prticl Aswr: A. th vrg positio of th prticl S PIB lctur ots slid, Problm St #, d Homwork # problms i MthChptr B # d Chp. #.. As prdictd by th corrspodc pricipl, t vry high qutum umbrs, th prdictios of qutum mchics for prticulr typ of systm covrg towrd th corrspodig clssicl prdictios. Which of th followig is mpl of this covrgc for o-dimsiol prticl-i--bo systm t high qutum umbrs? A. th tulig probbility icrss from its lrgr vlu t low qutum umbrs B. th prticl is most likly to b foud r th wlls C. th prticl is most likly to b foud t th middl of th bo D. th prticl is bout qully likly to b foud ywhr i th bo E. th rgy spcigs btw succssiv stts tht is, btw stts whos vlus of diffr by o icrs Aswr: D. th prticl is bout qully likly to b foud ywhr i th bo A is icorrct bcus thr is o tulig i th PIB systm d lso if thr wr, th tulig probbility would dcrs, ot icrs, t high qutum umbrs, s i th hrmoic oscilltor. E is icorrct bcus it dos ot illustrt grmt with th corrspodc pricipl, c clssiclly, thr is o rgy qutiztio t ll. It is tru tht c th kitic rgy gos s, th spcigs btw djct stts icrs s icrss. Howvr, if o comprs this spcig to th totl rgy of th stt, th this rtio dos go to zro s icrss, i grmt with th clssicl prdictio of cotiuum of possibl kitic rgis. S PIB lctur ots pp. -, d th figur o p. 85 for =. 5

6 5. Assum tht two-dimsiol Hmiltoi oprtor is sprbl ito o-dimsiol trms, Ĥ,y = Ĥ + Ĥ y, whr th o-dimsiol solutios to th Schrödigr qutio r Ĥ = E d Ĥ y y = E y y. Which of th followig substitutios will llow us to solv th two-dimsiol Schrödigr qutio, Ĥ,y, y = E, y, y? A., y = + y d E, y = E + E y B., y = + y d E, y = E E y C., y = y d E, y = E E y D., y = y d E, y = E + E y Aswr: D., y = y d E, y = E + E y S PIB lctur ots pp. - for -dimsiol prticl-i--bo. 6. Which of th followig sttmts is/r tru cocrig th qutum mchicl hrmoic oscilltor modl for th vibrtiol motio of ditomic molcul? I. "" is th distc btw th two toms. II. For molcul i igstt with qutum umbr v of th Hmiltoi oprtor, th distc btw th two toms is wll-dfid. III. th most probbl vlu of i th groud stt is zro. A. I oly B. II oly C. III oly D. I d II oly E. I d III oly F. II d III oly G. I, II d III r ll tru H. No of ths sttmts I, II, III r tru. Aswr: C. III oly 6

7 I is fls; is thir displcmt from th quilibrium bod lgth so c b positiv or gtiv; s HO lctur ots p. 5. II is fls; thr is probbility distributio ssocitd with th distc btw th two toms, which c hv rg of vlus. Also, lthough th totl rgy th sum of th kitic d pottil rgis is wll-dfid, th pottil rgy lo, which dpds o, is ot wll-dfid. III is tru; s HO lctur ots pp. d From th ifrrd spctrum of H 5 Cl, o c dduc tht th vibrtiol "frqucy" of this molcul is 886 cm -. Wht is th forc tt of this molcul? Choos th closst swr. Not: th suprscript ms tht th mss of this chlori isotop is 5 mu. A. N/m B. 5 N/m C. N/m D. 5 N/m E. N/m F. N/m G. N/m H. 5 N/m I. 6 N/m J. 7 N/m Aswr: H. 5 N/m 7

8 8. Which of th followig sttmts is/r tru cocrig tulig? I. Tulig is prdictd i both th hrmoic oscilltor d th prticl-i--bo modls. II. I th hrmoic oscilltor modl, th probbility of tulig dcrss s th vibrtiol qutum umbr icrss. III. For th groud vibrtiol stt, ccordig to th hrmoic oscilltor pproimtio, th tulig probbility for HBr is grtr th for its dutrium isotop, DBr. A. I oly B. II oly C. III oly D. I d II oly E. I d III oly F. II d III oly G. I, II d III r ll tru H. No of ths sttmts I, II, III r tru. Aswr: F. II d III oly I is fls; th prticl-i--bo modl dos ot prdict tulig. Th wv fuctio must go to zro t th wlls tht dscrib th pottil rgy fuctio, which r modld s big ifiitly stp. II is tru; this is o mpl of th corrspodc pricipl s HO lctur ots p. 5. III is tru; th wv fuctio dcys mor rpidly for DBr du to its highr rducd mss µ, c th wv fuctio o th qutio sht gos s - /. Hr, = μk ½ /, d th forc tt, k, c b ssumd to b th sm for both HCl d DCl. S HO lctur ots, pgs d 9. Th rducd tulig probbility for th hvir isotop, dutrium, cotributs to th kitic isotop ffct, spcilly t low tmprturs. 8

9 9. I viw of th slctio ruls for vibrtiol spctroscopy i th hrmoic oscilltor pproimtio, which of th followig sttmts is/r tru? A. To bsorb ifrrd light d udrgo trsitio to highr rgy vibrtiol lvl, th molcul must hv prmt dipol momt. B. Th homouclr ditomic molcul O dos ot hv qutizd vibrtiol rgy lvls. C. HCl c bsorb photo d b citd from v = to v =. D. Isoltd rgo toms hv qutizd vibrtiol rgy lvls. E. Mor th o of th bov sttmts r tru. F. No of th bov sttmts r tru. Aswr: F. No of th bov sttmts r tru. To bsorb ifrrd light d udrgo trsitio to th t highr vibrtiol lvl or to mit photo d rl to lowr vibrtiol lvl, th dipol momt of th molcul must chg s it vibrts log tht vibrtiol mod. This is clld "dymic" dipol momt. Aothr slctio rul i th hrmoic oscilltor pproimtio is tht wh bsorbig or mittig photo, th molcul c oly mk trsitio btw djct vibrtiol rgy lvls v =. S HO lctur ots, pp Accordig to th postults of qutum mchics, which of th followig proprtis, if y, is/r ot cssry i ordr for fuctio to b ccptbl s qutum mchicl wv fuctio for prticulr systm? A. th fuctio must b ormlizbl B. th fuctio must b igfuctio of th kitic rgy oprtor C. th fuctio must mt th boudry coditios for tht systm D. both th fuctio d its first drivtiv must b fiit E. both th fuctio d its first drivtiv must b gl-vlud F. both th fuctio d its first drivtiv must b cotiuous G. All of th bov A - F r cssry proprtis Aswr: B. th fuctio must b igfuctio of th kitic rgy oprtor S lctur ots o postults, pp

10 Prt II. No-Multipl Choic Qustios qustios, poits totl For umricl qustios i Prt II, to rciv full crdit, pls show your work ot just th fil swr, d iclud th corrct uits with your swr.. 5 poits Cosidr prticl of mss m i o-dimsiol bo of lgth, with "wlls" t = d =. Assum tht th pottil rgy is zro isid th bo, d ifiit t th wlls d outsid th bo. Strtig with th Schrödigr qutio, solv it to obti th formuls for th wv fuctios d rgy lvls. To rciv full crdit, writ clrly d show ll of your stps. Ths stps should iclud writig th grl solutio to th diffrtil qutio, d clrly showig th rsults of pplyig pproprit boudry coditios. You do ot d to ormliz th wv fuctio. [Not: If w cot rd your drivtio or follow your rsoig, littl crdit will b giv.] poits: S. q with V = poit poits poits poits poits poits S PIB lctur ots pp. -5 d tt pp. 8-8.

11 . poits Cosidr plr molcul with cojugtd -systm which c b modld s -dimsiol "bo" with sids of lgth Å. If th molcul hs 6 lctros, clcult th wvlgth of light tht c b bsorbd to cit th molcul from its groud stt to its first citd lctroic stt. Pls show your work, d put your fil swr i th bo, icludig uits. poits for corrct dgrcis d cittio poits for corrct rgy spcig with corrct uits poits for corrct covrsio from rgy to wvlgth poits for corrct swr with corrct uits Mor grdig ots: M 6/ if rgy diffrc is ot qul to h /8m 8/ if swr i Jouls is corrct.8-9 J but is ot covrtd to wvlgth - if swr is corrct but uits r icorrct or omittd

12 . poits For molcul i its vibrtiol groud stt, ψ - / = / π ¼ whr = μk ½ / St up compltly th itgrl tht you would us to clcult th vrg pcttio vlu of th pottil rgy, ½ k. b Solv th corrct itgrl. Thr my b itgrl o th qutio sht tht could hlp you. c Wht is th molculs' totl vibrtiol rgy, d how dos its vrg pottil rgy compr to tht vlu? Aswr: Grdig: poits totl poit for limits o itgrl, pt for, poits for ½ k b 5 poits icl. pt for -^, poits for vlutig itgrl corrctly, pts for k / c poits icl. pts for corrct compriso, pts for substitutig k d o corrctly to compr vlus

13 List of Equtios, Costts & Covrsios c = m/s =.998 cm/s k B =.695 cm - / K h = 6.66 J s = h/π =.55 J s =.6 9 C V =.6 9 J corrspods to 866 cm - m = 9.9 kg m p =.67 7 kg mu =.66-7 kg H tom i = + i ν ~ = 9,678 / - / cm - E = - m r = ε h 8ε h π m Clssicl wv qutio u,t / = / u,t / t Norml mods of vibrtig strig of lgth : u,t = A ω t + π / Schrödigr qutio: / m d Ψ / d + VΨ = EΨ Momtum oprtor: = -i / P PIB ψ = / ½ π / E = h / 8m HO - / ψ = / π ¼ whr = μk ½ / E v = v+½ h o whr o = / π k/μ ½ d μ = m m / m +m ditomic

14 Equtios, cotiud: k k ik ik b b b b o o o B A F is solutio F k d F d C C F k B k A F is solutio F k d F d Equtios l Diffrti b b d b b d d gr positiv d d d d d d d d d d d d d d d C u u du C u du u : : it! 8 l