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solution_2 - 23 January, 2006 Michael F. Brown CHEMISTRY...

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Unformatted text preview: 23 January, 2006 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 02 To be turned in by: Monday, 30 January Worked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the Lectures. You can get help with the homework problems in the Discussion Sections (12:30—13:20 Tues, Koeffler 216; and 13:00—13:50 Wed, Koeffler 216). The problem sets will be graded P+, P, or F and will be used to increase or decrease borderline grades. On all computational problems be sure to use SI units, and indicate your answer to the proper number of significant figures. For maximum credit show clearly how you obtained your answer, i.e. what numbers were combined to yield the final result. Background reading: Atkins & de Paula, Chapters 11 and 12 Examples and exercises related to the homework (optional): Back of Chapter Exercises: 11.8, 11.11, 11,14, 11.17, 11.19, 12.4, 12.5 Back of Chapter Problems: 11.4, 12.1 Problem 1. Being good sports let us consider the familiar (although mysterious!) hydrogen atom. The excited state wavefunction corresponding to a hydrogenic 2s orbital is given by m a 0 we): 1 [2—LJ6_r/2ao where the Bohr radius a0 = 52.9 pm. a) Find the normalized wavefunction. b) Estimate the probability that an electron is in a volume 1: = 1.0 pm3 at the nucleus (r = O). c) Estimate the probability that an electron is in a volume 1: = 1.0 pm3 in an arbitrary direction at the Bohr radius (r = a0). (1) Estimate the probability that an electron is in a volume 1: = 1.0 pm3 at infinity (r = 00). Problem 2. Consider a hydrogen atom in its ground state, for which the wave function (normalized) is: 1 e—r/a0 WU) = 3 Ilia 0 where a0 = 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. Problem 3. Let us consider two of the excited state wavefunctions for the hydrogen atom: a) Normalize the following wavefunction to unity: 2 _ I“ —r/ a 1P (r) = —— e 0 a b) Normalize the following wavefunction to unity: 1p (r) = rsinB cos (1,) e_r/2ao Problem 4. You are a theoretical Quantum Chemist interested in the electronic properties of conjugated polyenes. The ground-state wavefunction for a particle in a one-dimensional box of length L is: w (x) = 3 sin L L Assume that the box is 10.0 nm long. a) Calculate the probability that the particle is between x = 4.95 nm and 5.05 nm: b) Calculate the probability that the particle is between x = 1.95 nm and 2.05 nm: c) Calculate the probability that the particle is between x = 9.90 and 10.00 nm: d) Calculate the probability that the particle is in the right half of the box: e) Calculate the probability that the particle is in the central third of the box: 01 -23-06 chem48 1\06\prob\002\mw100 ,4, 75¢ for 6C 35 orb/3’41}; fl/nvan %: : _L 79. -1; 4/240 #(r) <3 do){ A) 7/9e anaeym {or harMa/fza/fan is; Aflf/Vcd/Qaft/ =1 KfidPMM'fa/fm WSW _ _,2 —r/o7~ L; AT/aiélé‘r)f(g 47,) (Q 2“) 6” whm g/ 1/ ’—“ rld/ fifty 66/ 6 d¢ 7" ’15 Wrew" z“0’ 57/er 5/44,” : 47r3dr Tb/(S f /' 1/6 S : i? : (fig/fjfl‘éfgar/Mo rid? 9Q _. ___/:_ —V40 9x6! : 0 (2 i)(?— 40>.e r f‘ :f(4__4-P+_L3)€'P/4o $00. 70‘ 4f gm» MW Mme 24/6/44 is of 7%; flw : 00 - / f Kné ‘4xe : Vi- (X:/‘)‘q: V40) 0 4mt/ For refereruce) /€/L m WW W 45%! 7hfl/m7 76/- 039; 3,4: -r / 3 foe/96 flag/r = ——2' : 26% 2 0/403 faD/‘gé-r/40/r: 34 : €404— 0 (//4o)4— d; 5" o l(//40); Than) 4 A ; 4—0402)- )4. 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W/amom. .7 (4:? _/0’0/ 2,4 /0 ‘4 _ J) PC7599; M99) = [MM—VJ. §/'/7(7/7/0'Oo)-§56375—00 _, ,7 _ »_7 Who /d-or 7 .5}; x 7/07? ‘79 7 ‘7 )1 7 .0. 22544 ...
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This note was uploaded on 04/02/2008 for the course CHEM 481 taught by Professor Brown during the Spring '06 term at Arizona.

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solution_2 - 23 January, 2006 Michael F. Brown CHEMISTRY...

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