Assignment
Show your work and state clearly all assumptions made (particularly important for the "order of magnitude" calculations)!
1. Consider the Redlich-Kwong EOS.
a. Derive an expression for the work required to carry out an adiabatic, reversible compression from molar volume V1 to molar volume V2 for a gas that obeys the Redlich- Kwong equation of state.
b. Derive an expression for the heat required to carry out an isothermal, reversible expansion from molar volume V2 to molar volume V3 for a gas that obeys the Redlich- Kwong equation of state.
2. The use of atomic force microscopy has been highly valuable to better understand the properties of DNA, which encodes all life on this planet. Through AFM, researchers have been able to quantitatively measure the mechanical properties of DNA. It was found that the double helix is highly flexible, with the ability to adopt multiple conformations. Consider DNA as a chain of N small, rigid units of length λ that can adopt any angle with respect to any other (a freely jointed chain). The length λ is called the persistence length and corresponds to ~130 base pairs, extending ~45 nm.
a. Suppose that this DNA molecule resists extension from equilibrium through a restoring force F = -kFx, where x is the displacement of the end of the chain from the equilibrium value and kF is the force constant. Plot this force as a function of x, where kF =10-5 kg/s2.
Write an expression for the work that must be done to extend this DNA molecule by x.
b. In reality, DNA cannot adopt any conformation. Instead, each longer unit of length λ can only adopt an angle of 0o or 180o with respect to the adjacent units (a one-dimensional, freely jointed chain). In this case, the restoring force of a chain extended by x = n λ is given by
F = (kBT/2??) ln {(1 + v)/(1 - v)}; v = n/N
What is the magnitude of the force that must be applied to extend a DNA molecule with N = 200 by 90 nm?
c. Plot this restoring force vs x, where x can be positive or negative.
d. How is the variation in restoring force with distance different from that predicted in (a)?
e. Write an expression for the work involved in extending this DNA molecule considering the restoring force in (b) and calculate the work involved in extending the DNA molecule from v = 0 to v = 1.
f. Can you make any approximations for the restoring force if v << 1? Write this expression out for F.
g. Plot the restoring force in (f) vs x, where x can be positive or negative.
h. How is the variation in restoring force with distance that you obtain in (f) different from that predicted in (c) or (a)? Explain your answer.
3. Buckminsterfullerene can be considered the most symmetric molecule in the world.
a. What is the point group of buckminsterfullerene?
b. The vibrational modes of buckminsterfullerene are of the following symmetries and energies:
|
Symmetry
|
Energy (cm-1)
|
|
Au
|
976
|
|
T1u
|
525, 578, 1180, 1430
|
|
T2u
|
354, 715, 1037, 1190, 1540
|
|
Gu
|
345, 757, 776, 963, 1315, 1410
|
|
Hu
|
403, 525, 667, 738, 1215, 1342, 1566
|
How many vibrational modes have a vibrational temperature θV < 1000 K?
c. What is the molar constant volume heat capacity at 10 K? 100 K? 1000 K?
d. How do your calculated values compare to experiment? Cite your source and justify any differences.
4. A magnetic field can influence the heat capacity of a molecule. Consider NO, which is a ground- state doublet.
a. Estimate the constant-volume heat capacity of NO at 1 atm at room temperature using statistical mechanics.
b. How much does that heat capacity change if we now consider the Earth's magnetic field at room temperature? At 4.2 K?
c. How much does that heat capacity change if we consider the magnetic field of a relatively standard 600 MHz NMR at room temperature? At 4.2 K?
5. If 70% of Asia is covered by foliage (i.e., plants) that uses photosynthesis to produce glucose, calculate the annual enthalpy change from these processes. How many cars in the US does that "compensate" for if we assume an average commute distance of 20 miles? Cite your sources.