Download Chemistry 453 March 17, 2008 Enter answers in a Blue Book Final

Chemistry 453
March 17, 2008
Enter answers in a Blue Book
Final Examination
Useful Constants:
• 1 Joule=1J=1 Nt-m=1 kg-m2/s2=107erg…where 1 erg=1g-cm2/s2.
-1
-1
• Universal Gas Constant R=8.31 J K mole
-23
• Boltzmann’s Constant=k=R/NA=1.38x10 J/K, where NA is Avagadro’s Number.
-34
• Planck’s constant h=6.626x10 J-sec.
-18
• Rydberg constant =2.18x10 J
-1
-1
• 1 poise =1P=1gm cm sec =0.1 kg m-1 s-1
•
•
-8
-10
1 Angstrom=10 cm=10 m.
me=mass of the electron=9.1x10-31 kg
Note: Parts 1-3 deal with material that we have covered since the midterm exam.
Parts 4-6 are cumulative and deal with material covered during the entire course.
Part 1 (28 points) Answer FOUR out of the following six questions.
Question 1.1 Explain the terms T1 and T2 as they are applied in Nuclear Magnetic
Resonance Spectroscopy. In each case relate T1 and T2 to expressions for the time
dependence of transverse (Mx and My) or longitudinal (Mz) magnetization.
Question 1.2 Define the term FRET as it is used in the spectroscopy of fluorescent
molecules. How can FRET give structural information on biological polymers? What are
the complicating factors?
Question 1.3 What is a spin echo, as the term is used in Nuclear Magnetic Resonance?
How is a spin echo produced?
Question 1.4 Define the partition function. What does the partition function quantify and
how does its value change with temperature? What is its value in the high temperature
limit? Explain.
Question 4.5 The laws of motion predict that the acceleration of an object falling in a
gravitational field is independent of its mass, assuming the presence of a gravitational
force only. To prove this Galileo dropped two objects with different weights from the top
of the Leaning Tower of Pisa. Although the experiment largely supported Galileo’s
position, in fact the weights reached the ground at slightly different times. Explain why
this might have happened and what additional force might be responsible for this result.
Question 1.6 Define the term random walk. Give two examples of physical systems that
that can be described using the random walk model.
Part 2 (36 points) Work THREE out of the following FIVE problems.
Problem 2.1 Calculate the rotational partition function of HBr. Assume the length of the
H-Br bond is 1.21x10-10m and T=1000K.
Problem 2.2 Calculate the frictional coefficient of a rod-like macromolecule which is
modeled as a stiff chain of 20 beads. Assume each bead has a diameter of 20Å. Assume
the viscosity of the solvent is 0.01P.
Problem 2.3 Calculate the difference in energy between the “spin up” and “spin down”
states for a 13C nucleus in a magnetic field of 11.75T. The magnetogyric ratio of 13C is
6.73x107 rad T-1 s-1. Calculate also the Larmor frequency.
Problem 2.4 Consider the two distributions of 7 particles among five energy levels.
Distribution 1
Distribution 2
The values of the level energies are given in terms of multiples of ∆E, the spacing
between the levels. Calculate the average energy for the two distributions assuming
∆E=1.38x10-20J. Which distribution is more probable? Justify your answer quantitatively.
2.5 Calculate the sedimentation coefficient for the DNA 200,000 base pairs in length at
T=293K. Assume each base pair weighs on average 660 gm/mole. Assume the specific
volume of the DNA is 0.51mL/gm. Assume the density of water is 1gm/mL. Give your
answer in units of Svedbergs. Announced in class: treat DNA as random coil. Assume
A =3.4Å and η =0.01P.
Part 3 (36points) Perform ONE out of the two multi-step calculations.
Problem 3.1) Assume 1 mole of He gas at 1000K.
a) Calculate the single particle translational partition function for He. Assume
V=1m3
b) Calculate the entropy for 1 mole of He.
c) Using your results from parts a and b, calculate the Gibbs energy G and the
enthalpy H for He at T=1000K
Problem 3.2. The following data are available for human hemoglobin at T=293K:
• Molecular weight=64,500 gm/mole
• Specific volume= V2 =0.75 mL/gm
• The Perrin factor, which is ratio of f, the frictional coefficient of hydrated
hemoglobin, to f0, the frictional coefficient of unhydrated hemoglobin is
f
= 1.16
f0
(a) Calculate the radius of hemoglobin, assuming it is an unhydrated sphere.
(b) Calculate radius of hydrated hemoglobin. Assume hydrated hemoglobin is also
spherical. Assume the solvent viscosity is η=0.01P.
(c) Calculate the mass of water molecules bound per gram of hemoglobin. Assume
the density of water is 1gm/mL. How many water molecules hydrate each
hemoglobin molecule?
Part 4 (28 points) Answer FOUR out of the following SIX questions. Limit
discussion to less than 200 words. Use equations where helpful or required, but
detailed calculations are not necessary.
Question 4.1 State the Franck-Condon Principle. Make a sketch of an electronic
transition from a ground state singlet state to an excited state that demonstrates this
principle.
Question 4.2 State the Correspondence Principle of Quantum mechanics. How do the
wave functions of the quantum mechanical linear harmonic oscillator demonstrate this
principle?
Question 4.3 In one of our homework sets, it was shown that in a two energy level
system, the number of particles in the higher energy level cannot by thermal means be
made larger than the number of particles in the lower energy level. Explain how in
Nuclear Magnetic Resonance Spectroscopy, the number of nuclear spins in the higher
energy “spin down” level can be made to exceed the number of nuclear spins in the lower
energy “spin up” level.
Question 4.4 Define the term saturation as it applies to the spectroscopy of a two energy
level system.
Question 4.5. What is the difference between a system when it is at equilibrium and
when it is at steady state? Reference your answer to the case of transport processes.
Question 4.6: The N particle canonical partition function Q is related to the single
qN
particle partition function q by Q =
. Explain the factor N!. When is it valid to use this
N!
factor?
Part 5: (36 points) Perform THREE of the following FIVE calculations.
Problem 5.1 For the purpose of calculating electronic transition frequencies, assume a
delocalized electron in butadiene can be treated as a particle in a one dimensional box. If
the wavelength of radiation required to promote an electron from the n=2 to n=3 energy
level is 2.20x10-7 m, calculate the length of butadiene.
Problem 5.2 A spherical protein has a radius in solution of R=3.55nm. Calculate the
coefficient of rotational diffusion, assuming T=300K and the viscosity is 0.01P. Assume
stick boundary conditions.
Problem 5.3 Calculate the wavelength of radiation that would be required to cause a
transition from the J=1 to the J=2 rotational state of H35Cl. H35Cl can be treated as a
quantum mechanical rigid rotor with a moment of inertia I = 2.74 × 10−47 kgm 2
Problem 5.4. Calculate the translational partition function for H2 at T=1000K. Assume
V=1m3
Problem 5.5 Calculate the average squared displacement x 2 and the average squared
momentum p 2 for the n=0 state for an oscillator of reduced mass 10-26 kg and a spring
constant of 1600 Nt/m.
Part 6 (36 points) Perform ONE out of the TWO multi-step calculations
Problem 6.1 For carbon monoxide CO the vibrational temperature is ΘV =
hν
= 3103K .
kB
Assume T=1000K
a) Calculate the vibrational partition function, assuming a quantum mechanical
harmonic oscillator model. Compare this expression to the high temperature limit
expression.
b) Again using the quantum mechanical harmonic oscillator model, calculate the
vibrational energy at T=1000K. Assume 1 mole of CO molecules.
c) Calculate the hear capacity at constant volume CV for CO at T=1000K
Solution: The heat capacity has the same definition for both conventions:
Problem 6.2. For a diatomic species, A-B, the vibration of the molecule may be modeled
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as the motion of a simple harmonic oscillation. Consider gaseous N O , for which the
spring constant κ=1596 Nt/m.
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a) Calculate the reduced mass µ of N O.
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b) Calculate the vibrational energy level spacing for N O. Based on this number,
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would you expect N O to be thermally excited above the vibrational ground state
at T=300K , at T=2500K?
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c) Assuming N O behaves as an ideal diatomic gas, calculate at T=300K the total
vibrational energy per mole. Justify your method for calculating the vibrational
energy.