Unformatted text preview: 13 January, 2006 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 01 To be turned in by:Monday, 20 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 (12:3013:20 Tues, Koeffler 216; and 13:0013:50 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 Examples and exercises related to the homework (optional): Back of Chapter Exercises: Back of Chapter Problems: 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 CH 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:
d2 x + w2 x = 0 2 0 dt Here w 0 = k / m and m is called the reduced mass, which is defined by:
m / m C mH m C + mH Let us assume that the force constant k = 450 N m1 for the case of a CH bond. a) What is the natural frequency n0` / s1 of the harmonic oscillations of the CH 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? -1- c) What is the natural frequency n0 / s1 of the CD bond oscillations? (Assume the force constant k is the same as for a CH bond.) d) Calculate the force needed to produce vibrations of a CH bond with an amplitude (A) of 10.0 pm. Problem 2. Consider the following functions: k, kx, kx2, eikx , sin kx, cos kx, sinh kx, cosh kx, and exp(-kx2). [Hint: sinh kx (ekx ekx ) / 2 and cosh kx (ekx + ekx ) / 2.] a) Which of the above functions are eigenfunctions of the operator d/dx? b) What are the eigenvalues of d/dx corresponding to the eigenfunctions in part (a) ? c) Which of the above functions are eigenfunctions of the operator d2/dx2? d) What are the eigenvalues of d2/dx2 corresponding to the eigenfunctions in part (c) ? Problem 3. Being good sports let us consider the familiar (although mysterious!) hydrogen atom. The wavefunction corresponding to a hydrogenic 1s orbital is given by y (r) = 1 e - r/ a0 4p where the Bohr radius a0 = 52.9 pm. Note that the wavefunction is not necessarily normalized! a) Find the normalized wavefunction. b) Estimate the probability that an electron is in a volume t = 1.0 pm3 at the nucleus (r = 0). c) Estimate the probability that an electron is in a volume t = 1.0 pm3 in an arbitrary direction at the Bohr radius (r = a0). d) Estimate the probability that an electron is in a volume t = 1.0 pm3 at infinity (r = ). 01-13-06 chem481\06\prob\001\mw100 -2- ...
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