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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.