solution_4 - 06 February, 2006 Michael F. Brown CHEMISTRY...

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Unformatted text preview: 06 February, 2006 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 04 To be turned in by: Monday, 13 February 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, Chapter 12 Examples and exercises related to the homework (optional): Back of Chapter Exercises: 12.4, 12.10, 12.11, 112.15 Back of Chapter Problems: 12.3, 12.13 Problem 1. Nitric oxide is an important neurotransmitter molecule and is involved in many biological processes. Consider the following diatomic molecules together with their vibrational frequencies in wavenumbers: diatomic vibrational frequency ( 17/ cm’l) NO 1 904 CO 2 l 70 Assume that one can apply the harmonic oscillator model to the vibrational frequencies of the above molecules. a) Write the Schrodinger equation in terms of the reduced mass, and state the solutions for the energies. Be sure to define all symbols. b) Calculate the force constant for nitric oxide, NO: c) Calculate the force constant for carbon monoxide, CO: d) Rank the above molecules in terms of the stiffness of their chemical bonds and explain specifically the rationale for your choice: Problem 2. Let’s consider the case of a particle on a ring, which is important in NMR spectroscopy of biomolecules. The wave functions are given by: wmo) = [L em ’ 23’s Show that the above wave functions are eigenfunctions of the Hamiltonian and calculate the eigenvalues of the rotational energy. Problem 3. A particle on a ring is in a state described by the following wave function: w(¢) = a em) + b em (The above wave function is not necessarily normalized.) a) What is the normalized wave function? b) Calculate the expectation value of the angular momentum operator using the normalized wave function: 0) Calculate the expectation value of the kinetic energy operator describing the rotational motion: Problem 4. Cholesterol is an important constituent of biological membranes and is thought to be present in lipid “rafts” in complexes with proteins. As one example HIV is believed to utilize cholesterol rafts to gain entry into cells. Assume that cholesterol is labeled with deuterium (2H) and is studied in the gas phase using microwave spectroscopy. Assume also that the molecule rotates about a single axis and that the distance from the center of mass is r = 3 A. a) Calculate the probability density for the 2H atom located at a specific angle (1) (relative to space—fixed axes). Assume that the system is in an eigenstate of the Hamiltonian for rotation in a plane. b) Imagine that cholesterol is being studied in the gas phase. Calculate the wavelength of the microwave radiation that induces a transition between the first and second rotational energy levels of the cholesterol molecule: Problem 4. The familiar atomic orbitals involve the wavefunctions for a free particle on a spherical surface (V = 0), which are given by the famous spherical harmonics. Consider the spherical harmonic for l = 2 and m] = 0: Y;2)(6, 4)) = i (3c0326 — 1) 16% Show that the above is an eigenfunction of the Hamiltonian operator H and indicate the corresponding eigenvalue of the energy. 02-06-06 chem481\06\prob\004\mw100 _./_, /, 79 ctpr W h¢ernf¢ 034/7/Aa‘or' Made/{u fge (“gm/7",“ ofMJ/(Cufs We held 40 50”“ 5%r5'd/7'Jer’5 cywdiv'ch 740 467%}; W NaUZfUNol-I'Ong (6/2763741’” 05,005) 404 energies (eigenvalues). a) For f’he karma/77¢ as;7//Mf Mode/ %h< gaff/0794f XS: vcx) = {ikxl when k}: 7%: farc¢ Consfany‘. mcz Mm yank: fl? = E? if / Ix + ékxl’Wx) = Eva) 52’” 4X1 r {once cansylgnf (meg? 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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 University of Arizona- Tucson.

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solution_4 - 06 February, 2006 Michael F. Brown CHEMISTRY...

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