exam_1_key - -1 16 February 2007 Name Mz'deofe Era(A11...

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Unformatted text preview: -1- 16 February, 2007 Name Mz'deofe/ Era (A11 Social Security Number 146:; CHEMISTRY 481 (Biophysical Chemistry) Midterm Exam I Instructions: (I) Take it easy and relax!!! If there are any difficulties or questions just ask the Question Points instructor. (ii) This exam consists of 08 printed pages. Please make sure that your copy consists of this number of pages. (iii) Reference may be made only toa single formula sheet (two sides) (iv)Work out each of the following 05 problems. (v)Only one copy of this exam is to be taken per student and it is due immediately at the end of class. (w) Be sure to define all symbols used and indicate your answers clearly for maximum credit. A” answers should be expressed in SI units. (viii) Please see last page for physical r constants and conversion factors. Good [(2 Luck!!! “But you can't go through life applying Heisenberg’s Uncertainty Principle to everything." 02-16—07 course\chem481\07\exam\001\mw100 -2- Problem 1.‘ (20 points) For each of the wavefunctions indicated below, determine whether it is an eigenfunction of the linear momentum operator in, and explain briefly why. a) The function y(x) = em" b) The function y(x) = e‘ik" c) The function y(x) = em" + e‘i'“ /,. AW ope/”afar &per‘a2‘es on am agenquo/io‘q 710)];e/Jfle piva/ua“ of Me. eigenva/uc and #2 age/779» “‘1de: flyf : :72“ eJuW/y’an ope/“odor e—vq lue (6/24”) 4') F0!” fbe [z'nedr Mame» /VM ofg/vd‘or 3p}g_ -52? (afLAX) : -—L}$ (7“(3é) 67‘ka — X - 'kx = ’62,“? ei‘ gpfg : + k# e 1‘th ‘ I. aflkx is 624'; eijenj‘ardrfi'oh 07F ‘4‘734/6“ shag “enemy five pro «(u/dwfq 66:4st («gigs/7V4 he) Md #2 «5(7 ”A”; 94°04 aw 19) 424‘ ii <6“) = it (4A) {5“ Mg‘ , K : zgkA-a-L'I'X = ‘14? 42“"AK ' €-(,éx 75 am éléj‘e’77fM/JQé/oh 539(2 we 6/0/44.“ sz/“Mw’d‘ n- of 7%é el'yahwn/uc canal 7%6 eimfudaqon ( $983) 6) 47331 (e’ki’éx + Z-c-kx) -'-' ’5‘; (*ékkeflkx " c? 6.61%) 3 5. dx _ , J -Zk if § = (1mm) (g M -6 ,4) : +‘éé(€+£#x __ “.159 +z-kx —-c‘kx C/edréivé #2 x3 :73: am (beenfuwofi'cm slum (Am Jan /7( 06/6“; 4 am 5M(Z7‘0 elkem/a/IIZ) M Wavy/€24; fig 54 Me fuNcfi'On ' 372‘s. -3- Problem 2: (20 points) Consider a hydrogen atom in its ground state, for which the wave function (normalized) is: 1 —ra 1/)(r)=1/—3e I” we where a() = 52.9 pm is the Bohr radius. Find the root mean square distance (r2>“2 of the electron from the nucleus, in which ( > denotes the expectation value. -34- 2 75¢ expva/w‘rm VWof<fa>"/’ W057 (r3)— fzf’KOfi ra ZICr) J:- ( waru‘wvdrmy I: oormMzeJJ gfl = )fffé-r/“W” r; 24/40 r‘c/r snade 5/73 47.22%;- (7740: / ‘ 2F _ — / °° 41¢ ‘Rr “dr- ade d 7-?)f f f7;- 5/” f ¢ T , >(2/20)5’ —_)<—m9/0 '7)(¢/0 WS- . — 77/940 24-40) [_(_./ 1)] (177—0) __——_—> <ra)5’§j_ : #40 #7181 '2 4? (52.7fm) = 7/16 em 4- Problem 3: (20 points) In the process of vision, ll—cis—retinal is isomerized to all—trans—retinal. The structure of 11—cis— retinal is given below: Assume the free electron model is applicable and that the length of 11—cis—retinal is L = 1.2 nm. Calculate the wavelength of light that is absorbed by ll—cis—retinal. "44- 3-, 77 era 4%. .2z’Lefa a!) /z ban of; mmfcfiéfl._-_ a fafa€_af__ (fee/Me (/2~2 de/ocau’? 32¢ Q'S' ‘ _L;qflja;mz++_ .__.7___-_ _'___ 1 .+ “4" ' 1__ ( {aria/6 [/7 g 50K ,, __;_.fl-__. 1L : ree e/ea’ran ”yo/(1 _ _+___ ”L ‘ _T__$>_ IL - .. ' __ ___ ....... 3H. L. 4L ,,_ - .- -__ ___._._;__4L ._ _.__. 7L _ 7L 1L gg4ggmg.:__ * —~.--.——M= 67.x: _ _ ._/\_ _ _ __. -__-,_5;_;_-__8Lfl‘%_ '— __if‘.,-. .45 (QM/LL _ _ : gkcil (Olaf/)5 = 3 (7W0 x /o'3% )(iOOX/o 9mg") (/..:z,></0—7/72)‘2 (2'6 + f) (6. max /0"*’4 J5) A = $.6fx/O_7m 4M - MW fl—g—B'WflW—(‘WW — ( Qe (jg/0275 from flew? Mg/ Cz/[Ld‘a/fsfir‘f‘fag from d[/- trams QC (/— (is [some/ZS , 2 -5- Problem 4: (20 points) Infrared spectroscopy can be used to determine the types and amounts of secondary structure that is present in proteins, due to the influences of hydrogen bonding on the vibrational frequencies of the peptide bond. Assume a quantum mechanical harmonic oscillator model for the N—H stretching vibrations of the peptide bonds of a protein. a) Calculate the frequency of the infrared radiation absorbed by the N—H bonds in OL— helical regions of a protein in wavenumbers. Assume that the force constant is k = 640 N m“. b) Calculate the frequency of the infrared radiation absorbed by the N—H bonds in non-hydrogen bonded regions of a protein in wavenumbers. Assume that the force constant is k = 680 N m‘l. 0) Give a simple physical explanation for the influence of hydrogen bonding on the N—H stretching frequencies of proteins: . ~57: —- 4, FOP 4M W ”MG/V Awe/5 Mei/"V“?éy Ev :cngmw ;v=0»03/”' a) I \)k; Q /d/(e mfim] m <\dM4ss 777C szpa/wvb'm [De/weer: £240 my /€V€/§ is: = [Vr/+é)#w » (v+%,)¢7w 45 =: fig) NOW 72“” @My fair 64” 7‘rards/f/‘cms 4/ #0 bmv’c. 0507/426/ i: MW /'/d/\4 #11 e/eofra/wa7plué'g rdo/Jafiem; gI/kbf ;— 4505c;///w/23r by :— ZN (LET: : KW / Frequw7 of Ogcy/O‘fdf / :2 A '61 " i]— S/xfé‘a/ 47‘ Vale/32" : 2v % 777C K I 5*: J.; J; A 2/72. 9/71 V71 _/ 4) F0 ”'0‘ hoods/3522M g=g4o ””0 K . N _ //_ W” :2 3;: [6w Mia/(4672 m ”“24 2 £77 (2. 77g )0!) 3m 5*!) 3%9 : (3.3?3 X/Or W’l)(/02cm)—/ //77 9w: = 32%? an?” o N— h Wang. W é=520/V»," (71M$>A) F f 5’ [no (I; .) / 9 9 = W‘Ywfl” 277‘ 0?, 778X/agms") m 3 M c) 74g Farce, W5/4Af' is '/€ss. ”La/751145 flz u/‘éraj/me freyuency I 75 /CSS Wh /-h '67? Absence a?“ éydroim $006034?! 7774 7 kyJ/aign band/5 me ”0/“ as $7574 bé’C-mlse 766e, 'éydroyzry Z5 Shared be/weeg fwd 4/de )hsz‘ead of 567 caua/W/ éOnAoJ 1’0 *1 Sky/e Mom 9 @4273 ”zeal fly Whig/w; is /ess. -6— Problem 5: (20 points) Deuterium (2H) NMR spectroscopy is a powerful method for investigating the structural properties of nucleic acids such as DNA and RNA. The Hamiltonian is given by: H = Jr is = —yhBOiZ where A 121.0 = mill) The 2H nucleus has a spin of I = 1 and the selection rule for the allowed nuclear transitions is Am, = i 1. The value of the gyromagnetic ratio of the 2H nucleus is g = 4.11 x 107 rad 5‘1 Tesla“. Calculate the resonance frequency of an 2H nucleus in a magnetic field of B0 = 11.7 Tesla (T). ,7‘, 5, 71¢ resonahca fm7qenz? V is given iv‘MZEoé/fifreflmh? mafi/vmj bPM’s camsf‘w/ ‘45 :kv \«‘—- (cf/044104 7927“,“? 2,9 5” E #23) flmgz wz heel 7‘0 éa/o‘da/C 17?. fgflmt/m #60 )1”!an ”$7; 6r W #MIo-Mrv‘m W57 #f My af‘mfluc/eov may/76:42. MM” “V”: #9 W44 ”07%;. fieia’ ’l #7; = 97/2 (sway/3 900,53") 6 =;~-2/7v‘50f%¢= —3’%‘Eomfz/ = 52/ Q?“ 7., Eh: : ‘2’};30”? m: (’Z)0)+/) — Now a» Sig/com“ NA? ngmmkn qureswuj 5., = +2’72‘Ba E& :— o g (#1 5+, - -94§fso => AF: 15-1,— «0 —_ E0~E_/ = awe-go 4 a rd ; ad: 50 —— )ff V: I. E _— Mf'M/ni) 3 NS. 377. 0 QFrM/c/gde % -— 7 —; 2/ny - La__;_x_____/Os (274 5’ ”07%) —8— PH YSICAL CON STANTS‘I (ROUNDED TO FOUR SIGNIFICANT FIGURES) Qmmlil'v Sly/771ml Vulm' Speed of light in vacuum (- 2.998 X 10N m s' l (exact) Permeability of vacuum #0 477 X 10’7 N A‘2 (exact) % 12.57 x 1077 N A"2 Permittivity of vacuum 60 1074,02 (exact) .~. 8.854 x10“2 C2 N“ m-2 Planck constant I7 6.626 X 10'34 I s - 71/26 72 1.055 x 10‘34 J 5 Elementary charge e 1.602 X 10‘19 C Bohr magneton MB 9.274 x 10*2417'1 Nuclear magneton ,uN 5.051 X 10‘27 J T” Rydberg constant Rx 1.097 X 107 m‘] Bohr radius a0 5.292 x 10‘“ m Hartree energy Eh 4.360 X 10‘18 J Electron mass me 9.109 X 10731 kg Proton mass mp 1.673 X 10'27 kg Neutron mass mn 1.675 X 10‘27 kg Deuteron mass md 3.344 X 10‘27 kg Avogadro constant NA 6.022 x 1023 moi—1 Atomic mass constant mu 1.661 X 10‘27 kg Faraday constant F 9.649 X 104 C mol—1 Gas constant R 8.315 J K‘1 mol—1 8.315 x10—2 L bar K“ mor1 1.987 cal K'] mot“ 8.206 ><10—2 L atm K“ moi“ Boltzmann constant k 1.38] X 10‘23 J K” Acceleration due to gravity g 9.806 65 m s‘2 "The best values and their uncertainties are given in Appendix B. SOME NUMERICAL CONSTANTS AND CONVERSION FACTORS 77' = 3.141 592 65 2.54 cm inch—1 e = 2.718 281828 453.6 g 16—1 In x = logx/loge = 2.302 585 09logx 4.184] cal—l (exactly) 101325 N m—2 atm‘l 1.602 x10“19 J ev-1 105 N rn‘2 bar—1 10‘3 m3 L‘1 1.013 25 bar atm_l 133.32 Pa torr' ...
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