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Unformatted text preview: -1-30 April, 2007Michael F. BrownCHEMISTRY 481 (Biophysical Chemistry)Problem Set 12 - STUDY GUIDETo be turned in by: NEVERBackground reading:Chapter 9Chapter 10Chapter 11.1–11.3, 11.7Chapter 13.1–13.3, 13.10Chapter 15Back of Chapter Problems related to the homework (optional):Problems 9.3–9.5, 9.7–9.8, 9.11–9.15, 9.17, 9.19–9.21Problems 10.12, 10.13, 10.18-10.20, 10.22, 10.26-10.28Problems 11.10, 11.11,11.13, 11.27, 11.17Problems 15.3, 15.5, 15.8, 15.11, 15.30Problem 1. One of the major applications of classical mechanics in biochemistry involvessolving Newton’s Laws of motion for macromolecules. This method is called moleculardynamics. It can be used to investigate the atomic motions of proteins, nucleic acids (DNA andRNA), and the lipids in membranes. Let us consider the molecular dynamics of membranelipids. The vibrations of the bonds joining the various atoms are modeled as a classicalharmonic oscillator.Consider a representative C–H bond of a lipid in a membrane bilayer. For simplicity, thebond vibrations are modeled in terms of the relative motion of the two atoms. In a center ofmass coordinate frame, the equation of motion is given by:d2xdt2+ω2x=Here ω=k/μand μis called the reduced mass, which is defined by:μ=mCmHmC+mHLet us assume that the force constant k= 450 N m–1for the case of a CH bond.a) What is the natural frequencyν0` / s–1of the harmonic oscillations of the C–H bonds?b) In one type of experiment, hydrogen (H) is replaced chemically by deuterium (D). Do youexpect the frequency of the bond oscillationsto increase, decrease, or remain unaltered uponsubstitution of D for H? Why?c) What is the natural frequencyν/ s–1of the C–D bond oscillations? (Assume the forceconstant kis the same as for a C–H bond.)-2-d) Calculate the forceneeded to produce vibrations of a C–H bond with an amplitude (A) of 10.0pm.Problem 2. The wavefunctioncorresponding to a hydrogenic 1sorbital is given byψ(r)=14πe−r/awhere the Bohr radius a= 52.9 pm. (Hint: the above wavefunction is not normalized, if youcannot normalize it then proceed to parts (b)–(e) without the normalization constant.)a) Find the normalizedwave function.b) Calculate the expectation valueof the radius, given by +r,.c) Calculate the expectation valueof the radius squared, given by +r2,.d) Calculate the varianceof the radius, defined by +r2,– +r,2.e) Calculate the root mean squareradius, defined by +r2,1/2.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 equationand state the eigenfunctionsand eigenvalues. Be sure todefine all symbols....
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