solution_set_1 - 16 January, 2007 Michael F. Brown...

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Unformatted text preview: 16 January, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 01 To be turned in by: Monday, 22 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 Tuesday, Koffler 216; and 13:00—13:50 Wednesday, Koffler 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: Silbey, Chapter 9 Back of Chapter Problems related to the homework (optional): Problems 9.11, 9.12 Problem 1. One of the major applications of classical mechanics in biochemistry involves solving Newton’s Laws of motion for macromolecules. This method is called molecular dynamics. It can be used to investigate the atomic motions of proteins, nucleic acids (DNA and RNA), and the lipids in membranes. Let us consider the molecular dynamics of membrane lipids. The vibrations of the bonds joining the various atoms are modeled as a classical harmonic oscillator. Consider a representative C—H bond of a lipid in a membrane bilayer. For simplicity, the bond vibrations are modeled in terms of the relative motion of the two atoms. In a center of mass coordinate frame, the equation of motion is given by: dzx F+ ng = 0 I Here too = qlk/u and ,u is called the reduced mass, which is defined by: mm _ c H ‘11,— mc+mH Let us assume that the force constant k = 450 N m‘1 for the case of a CH bond. a) What is the natural frequency vo/ s'1 of the harmonic oscillations of the C—H bonds? b) In one type of experiment, hydrogen (H) is replaced chemically by deuterium (D). Do you expect the frequency of the bond oscillations to increase, decrease, or remain unaltered upon substitution of D for H? Why? c) What is the natural frequency v0 / s‘1 of the C—D bond oscillations? (Assume the force constant k is the same as for a C—H bond.) (1) Calculate the force needed to produce vibrations of a C—H bond with an amplitude (A) of 10.0 pm. _1_ Problem 2. Physical chemists tell us that a typical chemical bond has a force constant in the range of k = 500 N rn‘l . Assume that we have a spring with the same force constant, and that a 1.00 pound weight is attached. How far will the spring stretch? Problem 3. You have landed a job in a new biotechnology company in Tucson and wish to consider the vibrations of a representative protein, say HIV protease. As a start, one can consider the classical harmonic oscillator model for the bond vibrations. Newton teaches us that the equation of motion is given by: dzx 37+ ng = 0 where wo = qlk m and u is the reduced mass of the two atoms comprising the chemical bond. a) What is the relative displacement x(t) of the atoms at which the velocity is greatest? b) What is the relative displacement x(t) of the atoms at which the acceleration is a maximum? [Hintz remember your calculus and denote the amplitude by A.] Problem 4. Consider the following functions: k, kx, kxz, eik" , sin kx, cos kx, sinh kx, cosh kx, and exp(—kx2). [Hintz sinh kx E (e’“ — e'k" ) / 2 and cosh [or E (6” + e‘” ) / 2.] a) Which of the above functions are eigenfunctions of the operator d/dx? b) What are the eigenvalues of d/a'x corresponding to the eigenfunctions in part (a) ? c) Which of the above functions are eigenfunctions of the operator dz/dxz? d) What are the eigenvalues of 51/2/de corresponding to the eigenfunctions in part (c) ? Problem 5. Being good sports let us consider the familiar (although mysterious!) hydrogen atom. The wavefunction corresponding to a hydrogenic ls orbital is given by 1 . w<r)=m where the Bohr radius (10 = 52.9 pm. Note that the wavefunction is not necessarily normalized! —r/a e o a) Find the normalized wavefunction. b) Estimate the probability that an electron is in a volume 17 = 1.0 pm3 at the nucleus (r = 0). 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). d) Estimate the probability that an electron is in a volume 1: = 1.0 pm3 at infinity (r = 00). 01-16—07 chem481\07\prob\001 l\rnw 100 .._ j__. .- -"r* “W V/r'v "*2 an: ' a7‘ #0 .5. v/7‘ éyads 5/; 12,5274 "1" "7667/6/36! 6431’”? Whit; as - ‘ ' 4/ W Airman/F5“ agc/‘fla/foqs 0/ W??? C—é’éon/S Z k L/x z‘arée (Anw‘dnf 1/, a Z, ¢4~~MI rafluceg/ M044. 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This homework help 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_set_1 - 16 January, 2007 Michael F. Brown...

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