Download - Catalyst

Discussion Section Problem 11.22.11
1) N-hexatriene is an organic chromophore with the formula:
The three double bonds contribute six electrons that are delocalized over the
conjugated chain, which is about 5.8 Angstroms in length. Assume the energies of six
electrons can be calculated using the particle in the box model.
a) Calculate the energy of the highest filled energy level.
Solution: This is n=3.
b) Calculate the energy of the lowest unfilled energy level.
Solution: This is n=4.
c) Calculate the energy change ΔE when the electron makes a transition from the
highest filled energy level to the lowest unfilled energy level.
Solution:
d) Calculate the frequency and wavelength of light that is absorbed by this electron
transition:
Solution:
and
2) Assume argon atoms translate in one dimension.
a) What is the AVERAGE translational kinetic energy per argon atom? Assume
T=1000K
Solution: The average energy per argon atom for translatino in one dimension
is
b) Assume you can model the energy of translation of an argon atom using the
particle in a box model. Assume the box is 1 m in length. For what value of the
quantum number n is the particle in the box energy equal to the average energy
calculated in part a? FYI: the atomic weight of argon is 0.018kgmol-1.
Solution:
c) Based on your answers in parts a and b, how important are quantum effects in the
translation of argon at T=1000K?
Answer: There are 61.3 billion energy levels between the ground state energy (n=1)
and the average energy. So the energy level separation ΔE is very much smaller than
kBT and quantum effects are unimportant.
d) Suppose the box within which argon translations is 0.01 nm in length. Calculate
the n for the particle in the box energy that is equal to kBT/2.
e) Based on your answer in part d, how do quantum effects vary with the size of the
box a? Do quantum effect increase or decrease with box size?
Answer: As the box size decreases and approaches atomic dimensions, the energies
and energy level splittings for particle in the box increase and approach kBT. When
ΔE approaches kBT, quantum effects become important.
f) Based on your answers to parts a-e, discuss why the vibrational heat capacity is
almost zero at T=100K, but the translational heat capacity is 3R/2. Hint: atomic
vibrations have amplitudes of about 0.01 Angstroms.
Answer
For vibrations the amplitude of motion is so small that ΔE>>kBT at low T and heat is
not absorbed. So the heat capacity for vibration is zero. For translations the amplitude
of motion is large…so ΔE<<kBT, heat is absorbed at low T and the heat capacity is
3R/2.
CLASSICAL MECHANICS WORKS WHEN ΔE<<kBT. QUANTIZATION IS
IMPORTANT WHEN ΔE>kBT.