Problem_Set_3_Answers

Problem_Set_3_Answers - Chemistry 341 Physical Chemistry I...

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Unformatted text preview: Chemistry 341 Physical Chemistry I Fall 2011 Problem Set 3 Due: Friday September 23 Answer ANY FIVE (5) questions. 1. If the H2 molecule rotates in the plane of a crystalline surface (for a chemisorption situation), it can be approximated as a two-dimensional rigid rotor. For such a system calculate (in cm‘l) the lowest energy transition. Em '3 “ti 2 m: {ya-v9 :Qiila ta’.”3 1M2:_‘ \M'g:+l —-—-—- “11:0 2. Et—Efii; {c-x‘~co\‘i = fl 1. thia’la 1" )fl' . 22- 7Q,\QLL.P\M tam?» Talale- ‘3 ‘4 )‘l’ “$0 3F: (Lou? Cm x \53kQMrP.’q - 7. ((9.027. ac v5“ “we” - 3’: : (6.57.4, \Jaq 3A3 QfitCz-QQ% x Lowmfi‘) (8.3bq3uo'39’kc9 (7Q-‘qqub-‘zwu3t . (ham; 5.7—3 3 A! ‘ n In : (9.08% xm cm 7 mummy ‘ wwwsz ;. e- m Consider y; = Yf + Yl‘l . - (a) Is w an eigenfunction of the 2—D rigid rotor Hamiltonian? (b) Is 1/! an eigenfunction of the 3-D rigid rotor Hamiltonian? (0) What is the energy eigenvalue for each of these cases? La.) = 5515‘? Ewll/ m m-fl) titan“: Farm at?» Ciaméb we, image. J’- Am 1 {were -\- \ 7%: ’qugwfig’ 9-}: sign. “3% “be wig 3%? , Mt?“ ZACeeéa 2.. Lm‘gm‘é‘} 4- -L—- .. :55 1V 3}. awe Be sm‘etnw 3” E; $3..“th = ZAiw’é «we 5 twfi w 3% i DIE - CG W3“ “MA-D,ch -493- 11 sufie [3A.C°3§ 4. ‘ ("lvfll '1 we géanhwha m‘Q-mzle =1-2A3w7-6 -Afiq‘ 7.13:9ch ALMS _ 1. _‘ L 71/ _ .33; “Mae {\ 2M 6‘; shale ~ Egbll/ 4;,"- l__x__. \..;'?-e 15(__L_.. :E :1: Smsz w E mule Eb» EQ = Ezhfl’ ’I. Commevd': The 35 («A-or \MLS More enemy} +9....“ He, ab rofir Lemuse H" cam move :‘u. more tlirecHous. Several unstable species can be recognized spectroscopically by their pure rotational spectrum. Compute the rotational spacing 2B for the OH radical (Re =97 pm) and for the OD radical. Do you think these species can be differentiated experimentally? [Note that the rotational constant B is defined as h 87rch Where I is the moment of inertia] B: 2%": “l’gx -; 8312.21 7. ‘9‘ marl-R I": PR‘ bfifzc’f can . r: m‘mz (x1; W}, zi‘%\ (Mn-sz v: M‘Ml . _L (Mt‘l-Mz,‘ “A \i; (mo-z 329 3736351“ mm “D x teakgwéd ago-7 81$c37+ 69% Clint Uh “yum”? m4“) kt: LEORX \0'21 ice), - u . 2V5: (lo-(:16 1mg 3.43 but?- CZAQBX to“ c—m A") 0.97“! >1 \o’z-lkap (Q1 x today“ . ( kgm’uilw 3 7-5: 339,0 x lblcmd 29> : 57.930 OWL-l m. ‘ mm (bx ‘ r: M%M1. .L ._—.-—- n 7.5 "‘oU ( M$+Mt\ “A (52‘) 3 ‘2»: (2.ofll IM 7%? 3 (193%! Elm HOME; k; “4’.” (2.0th \O\ ve7+iSACm qu-t an) (Lou): to“ uni") ~27 Y‘ 1 .. ' . ‘° EX ' 29> (05) : Emu} ; Law “627 : egaq‘i; 2.15 (em ,4 ($3 2-0;“ x is“ keg, 2. Y>Leb\—. (5.52%st farmed”) 2?: (0D\: 20.02,. LWC' 7. (3. coh+muea\) 31.” Ge- l‘fi V0..\U.€.S a» ‘7. 7-7 LM—: "Pete 1‘ o+o_+£auo.‘ +v~kvxs~y+1¢w5 Shav-L‘vA b-e. a?" d¢+¢c+¢flp w avemwmb-cxs, Single-walled carbon nanotubes can be approximated by a particle-on—a— cylindrical surface model. Suppose the cylinder has a length E and radius a , with the z-axis along the cylinder. (a) Write the kinetic energy of the electron in terms of the length, 2, the radius a , and the radial angle 915 of the cylinder. (b) Combining ideas from the particle-in—a—box and 2-D rigid rotor models, show that the wavefunction can be written as I/jn = A Sin(n2[2) eim¢ . What are the allowed values of the quantum numbers n and m? (0) Write the energy expression in terms of m, n, I? , and a , and fundamental constants. For a nanotube of radius 550 pm and length 12,000 pm, compute the energy spacing between the two lowest energy levels. If there are 300 7r» electrons in the tube, calculate the spacing between the highest occupied and lowest unoccupied levels. What are their (m,n) quantum numbers? 3 ’X‘ 1. 12 Side View 4‘3? vim» at. L\3_\LMBLQV cl. Quzlilh Aer CM me View Hue M¢'\"'°h in “Re 26—d-$~ec:\-\,éu 05—1-9ch all 4. Pux—l-lee m or \D-box mm). +9.2 Merl-Law W fig Caerp- §>lqne LS- a¥ 0- ’va-b-lclz on a. fling}. 3.0 “W: ~fi A: .. Ali—L 3.: zme'évb“ 1mea7- 2.4,?— Cb} 37p "le ; agEcax M: ~15: the 4: aw 1mg Md amen . Atrawls Alf-this owl») Selw'bio‘vxs oi. ’H’Aa meii‘iafig Miner; {meld at. +9”. «Love Aenms are Se-l' +6 urns-bacwl-s figure Com‘\—.\nu£&> 2 = A4 (~le 11. a - euw vacluhiw ax. «harm—ahgfilw c5u$+b4~+ Ar gx‘ves ‘ mm W: A 2 q; 2 2. (:5 En M ‘3 "L ‘8‘ +— "kz m2 ' 8MP.” P-vmea‘Z e <0 ~ i‘ = .g—kaw‘sj gnu," 3 (<14on “1’3" has) (1-2. xxoflx 1047'»; 2 J <____9y_____..> 3-690"; 2:. ‘vD-lq 3‘ a .L . moozémgwg a me QMQZ . 1;} ‘ 13" K ’ 7. 2 '- 2wea. 8“ weq .. '2. .. (b-tozexwwa a) J" 3W7- Cq.\oq x to‘g‘kog (E'S‘o XLD-‘ZWAZ . < av Moo: 1 vi“: 1. 't‘ ‘— 0-126: e_V amen" ll ' E(»,w~\ = (6.0025 hi 4— 0426 1M1)e\( C423 New.) +Qjou\0.+\;rw 0—C- ‘ELM,\~L\ m WWW of. h Cg m and “590%HCKSKVU‘2‘ 41.9..” m>=\ —l~£._&+ w. :5 4- or — J we cow 4*“ up He. emery-5f lavzls 4-o Clo-:21, m: 23') re E (21,2) is He Mghefi' occmyneA level. ’11; wes+ bun) c.ch tack Lave} uvvgs‘aakcks +3 (“claim-:33. As a. ma} LLH’ AF: £05333 —E (12,23 13);: Cn-gozg—1.7a743eV AE: 0-oo3mi av C4.Cc>vr\-1vm aAZB E(n,m)leV — “II n Iil u Ii] Banana fl uuuuuu annuals: E “III n n I! I?! “.m- n nmm I! E El I! I! I! n u I! I! n —--— a 2.3505 2.476 28 3.4842 w 0 4 4.3660 5. SAB Problem 9.32, p. 345. v 9.32 Use information in Example 9.21 to calculate the fre— quency and wavenumber of the radiation required to take the H35Cl molecule from J = 1 to J = 2. (6,4,2; M63“ 3A— 3 9.1V“ (2. a 9‘4 x lo‘“ kc) m1) 3):. —»\ A. V; a.) :. \.170 X \o pl b. fizc-A M Ms” c 1‘2. _‘ I.?."lo><\° A. w a ~l z-‘Mg xioi l :3: 42.795 CWC‘I M :4: -( ham-LA: 3 1) 6. SAB Problem 9.33, p. 345. ® 9.33 Use the Schrodinger equation for a rigid rotor in three dimensions to calculate the rotational energy when the wavefunction is given by the spherical harmonic 1/10. What is the magnitude of the angular momentum? (m\ Raw. 3M5 Tate 01.2, Pam, we. have LIL \‘(c : Acogg uIA-E. A (3191:» t ’11». 3cer' A‘.VL%QJF eobucd'tgk w Xsea 3kg uh“. CLN‘L, P531] IE“ 4w:— (smeane‘ EV“ 11 sine Be 3% Sun-"4 “34.1. ‘ J‘A 4— 2. SL-stnzefl : Pr? 5 a, 0 Couture +5 E =11? gilt—HQ 9LT. E = it. (9304' '3 “Man i=1 7—1— Eat} TL". s. r-esu.\-\- meant-gag SA—pb gqbn‘qnqu’Pfi331) or Bin mugs , p.391. g 2 g g é % g E E E % ma“ L / Ha my,“ CM prom 3% new Quota) Pew, -=. Lita—q fl; For the diatomic molecule 1111350 the rotational constant B is 10.59 cm‘l. Determine for 1HE‘SCZ : (a) the internuclear distance and (b) the wavenumber for the transition J=2 —> J=3. [Note that the rotational constant is defined as B _ h _ 87rch Where I is the moment of inertia] v: M‘ “Aa _ L (Md—M13 “A -3 - y—z (Loo-182 o3? (34.958 $52—2qu x11: 1: mule,‘ (100'? $525 ($7 +?>'~l~%24 $92. 7243 ‘l Ce-oza x toumu-Q” km: {4921 x 1611 k3: 1mm '1 ' ’P‘ : ’9‘ El 81"ch é-Q‘zc 3: $3in (R : ant 04911 x at“ up(zmtzxmmwg‘d(Io-sch) .(szi‘jy‘ J 'I (R =- 1.335 x V3.2" } 1m l-Z‘J‘S x lu‘mws u inc/«,4. xv m x, Csmymre lb cRe, : \27.‘-N.S rm. “‘kiv‘gvx {w $1M“: Tabla last, Pym. (‘7.Cuw‘nnme (b3 ' E; = Bat; SKI—kn ' AS 2 E3497, Ar: = .9”; : RC; = 9:0”. {acumzm} zgmc 3:“; Obtain the allowed energy levels of a particle on a ring by considering the condition of standing waves that do not disappear by destructive interference along With the de Broglie relation. [This question is similar to question 8 in Problem Set 1.] ° QVL%D\,\ ox“ WLo mend-Lem LET-".2 PR C‘W rte-Ha: QPPe'S'tJre Sfyfi corres‘pend +0 éP?oS'l:\"e done oil—Loves ofi +Vo~ve\- ' (Legroglt'e, vglofi‘loh 7v:- {L C93 3? ' Concxt’l‘s'on aFuv $41“);an waves we— Cud' +39. right} 4_____~_.__.l_ 14—“ 2TH? ~44 $+oendiivt§ waves ui“ ne+ lav. de§rrfil€£¢£ 'L g} destrucsrtve \In‘t-eragevvi e q—fl‘bermam?‘ Cvacles around a Y‘ukfi, '2} QDK'FInQQMs around +2.2 Tl’ng‘ NOQ‘A. be in ‘quse Kg-L-QY— fubsecbde-vd' ro+oCchs around +6ie vino) -.’. In) =2'TTRI w’HrQ h=0,'\,,-. (33 “04%: hr—O is Pcfis‘flrfile be muse ‘lfiis Cow-ESFOVU‘95 ‘i‘o : 0L1 wkfck Is PcSSCHe beams; as. mm 00 an? Vo+a~+10hs meg ‘PGSS’IHE. ' “ow WG. Cambine. misfits mf‘fl: Y in ' F: “a :y‘t can-han A 5 -+ 'k 1%- Lg..- h R 1 +14 NH n=6 \Z —- -— )3 I L3: \M F) wd‘fi W} :thl)LZ’ , ' Mm “FR-a Quart»va E wh‘10k '\$ 0.“ hinefi‘fg energy), E = .va2: L Mr"- M} - ZR — 1—7“; 1 ' 41-23- " 11 9. One of the eigenfunctions for a particle moving on a ring in the (X,y)—p1ane is y/ = A 343'” . a. Determine the normalization constant A. . Is 1/1 an eigenfimction of I; ? If so What is the eigenvalue? 0. Is the particle rotating clockwise or counterclockwise in this state? Explain. 5—754 nremrawv!-vammmawfimwmmawwmuwmmmmWMQXWWWW 41 um. “I.” ,7 . W, mmflmklm'fqfiwu-‘m/«wvtvluvm/N is in ét ganguvxc‘htm 5L L3 \«Mlfl .U, ‘14: *‘ew $Qf¥t0l£ rat—«Jees cluakwi$¢ \obki'vmk chmnn aw 49.72 {no.3 Cor Coum+ifclokkwt51 l‘aolkwmy talent} Pos’n’u’xle‘ ; ascxsX I+Kgn +3; “wwwlux Meme,“ LLMA t 99““ be nick? ’Hixe wgaéod-{ve ~73 axis. 10. The particle on a ring is a usefill model for the motion of electrons around the porphine ring Porphine (free base form) the conjugated macrocycle that forms the structural basis of the heme group and the chlorophylls. We may treat the group as a circular ring of radius 440 pm, with 22 fl-electrons in the conjugated system moving along the perimeter of the ring. We take it that each of the allowed levels is occupied by 2 electrons. Calculate: a. the energy of an electron in the HOMO (highest occupied molecular orbital) level, b. the angular momentum of an electron in the HOMO level, 0. the wavelength of radiation absorbed which causes a transition between the HOMO and LUMO (lowest unoccupied molecular orbital) levels. "12:45 A'b— + WJ=+S 4+“ “417% _Ory___ —+i/‘_ .441... + m£=-\ + ———!fi,~_ m£:+\ 4+— m2“, 1 7_ . Ema: :n'zm’Ra- w-Q W‘Hi Mez°2+l>tZJH Q, E : “a1 GHQ-L 4. .. -3 gTZW‘RL A E+ 1‘- CGIG‘ZE >He'z'f’q 3A3?" (153 , (kg mail “5 8171‘ Cq.luQXle‘3‘ lag] (uqu wo‘nm)?’ J 3 E1; = 7vg% X \e'mfi L16: its (G.Q'ZL¢XLO.%L‘ 3 A) 17“ L33: 57:73. \< my“ 3,4, (\o . Cou+xnueA 3 'L A: 8W2 (23°13 x wanna-P) (qnoq x 363‘h$\(qLID>C16\Zm _ < '5 > 0") (Lb 2t; x 10'3"" FAQ ‘25 MEI x: g-V'thoq -. 91% NM ...
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Problem_Set_3_Answers - Chemistry 341 Physical Chemistry I...

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