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solution_set_12 - 30 April 2007 Michael F Brown CHEMISTRY...

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-1- 30 April, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 12 - STUDY GUIDE To be turned in by: NEVER Background reading: Chapter 9 Chapter 10 Chapter 11.1–11.3, 11.7 Chapter 13.1–13.3, 13.10 Chapter 15 Back of Chapter Problems related to the homework (optional): Problems 9.3–9.5, 9.7–9.8, 9.11–9.15, 9.17, 9.19–9.21 Problems 10.12, 10.13, 10.18-10.20, 10.22, 10.26-10.28 Problems 11.10, 11.11,11.13, 11.27, 11.17 Problems 15.3, 15.5, 15.8, 15.11, 15.30 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: d 2 x dt 2 + ω 0 2 x = 0 Here ω 0 = k / μ and μ is called the reduced mass , which is defined by: μ = m C m H m C + m H 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 ν 0` / 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 ν 0 / s –1 of the C–D bond oscillations? (Assume the force constant k is the same as for a C–H bond.)
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-2- d) Calculate the force needed to produce vibrations of a C–H bond with an amplitude ( A ) of 10.0 pm. Problem 2 . The wavefunction corresponding to a hydrogenic 1 s orbital is given by ψ ( r ) = 1 4 π e r / a 0 where the Bohr radius a 0 = 52.9 pm. ( Hint : the above wavefunction is not normalized, if you cannot normalize it then proceed to parts (b)–(e) without the normalization constant.) a) Find the normalized wave function. b) Calculate the expectation value of the radius, given by + r , . c) Calculate the expectation value of the radius squared, given by + r 2 , . d) Calculate the variance of the radius, defined by + r 2 , + r , 2 . e) Calculate the root mean square radius, defined by + r 2 , 1/2 . Problem 3 . An important pigment molecule found in plants is β -carotene: Assume that the electronic properties of β -carotene can be considered in terms of a particle-in-a- box , in which L = 2.0 nm. a) Write the Schrödinger equation and state the eigenfunctions and eigenvalues . Be sure to define all symbols. b) Given the free electron model, what is the wavelength of light absorbed by β -carotene? Problem 4 . A biochemist is interested in determining the amount of secondary structure (hydrogen bonding) in a newly discovered protein from human immunodeficiency virus (HIV).
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