3900_2011_PS2 - Problem Set 2 Chem 3900: Physical Chemistry...

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Unformatted text preview: Problem Set 2 Chem 3900: Physical Chemistry II Spring 2011 Due: Class, Friday, Feb 11th Please make sure to write your name and the recitation you’ll attend. When you print out your work, try to minimize the number of pages. You can do this by printing multiple pages per sheet and/or by using a duplex printer. Problem 1. The first problem examines the electronic partition function for atomic gases. The lowest electronic levels of atomic S are: Level Degeneracies gj Energy Ej/eV 1 5 0.000 2 3 0.049 3 1 0.071 4 5 1.145 i) Write out the partition function for this system. ii) In the same graph, plot the probabilities of the different levels pj versus T (K) for the following temperature ranges: 0 ≤ T ≤ Tmax, Tmax = 300, 3000, 30000, 300000 K. (It is most efficient to make Table of plots with Tmax = 3 × 10k, k = 2 – 5). iii) At some temperature the probability p4 becomes larger than p3 and p2 even though it is at higher energy. Why is this? iv) Excited neon gas is the lasing medium in a helium‐neon laser. The lasing transition is from the 3s level (degeneracy 1) to the 2p level (degeneracy 3) of neon resulting in the emission of 632.8 nm light. In an operating laser the neon gas exhibit what is called population inversion, where the population of level 3s is larger than level 2p, say p3s = 1.1 p2p, even though level 3s is higher than 2p and has lower degeneracy. In a Boltzmann treatment what temperature does this correspond to? Is this physical? Explain. Problem 2. The second problem examines the polyatomic gas SO2. i) Write out the partition function for this system. ii) First derive then plot the average molar energy versus T for the temperature range 0 ≤ T ≤ 2500 K. 1 iii) How is this energy distributed among the different molecular motions: translation, rotation, and vibration? How is this reflected in the temperature dependence? iv) Show that the molar constant volume heat capacity Cv contains a temperature independent part and a temperature dependent part. v) Plot the molar constant volume heat capacity Cv versus T for the same temperature range as above. vi) Give a brief qualitative explanation of the dependence of Cv on T. Problem 3. Consider the following four systems: i) H20 at ambient conditions (room temperature and 1 atm) ii) CH4 at ambient conditions (room temperature and 1 atm) iii) A gas of free electrons at 100 K iv) Rubidium‐87 at 100 nK i) ii) iii) iv) Which of these four systems is best described by Maxwell‐Boltzmann statistics? Explain. Which of these four systems is best described by Bose‐Einstein statistics? Explain. Which of these four systems is best described by Fermi‐Dirac statistics? Explain. Which of these four systems is not well described by independent particle statistics? Explain. Problem 4. Central to Statistical Thermodynamics is the partition function Q(N,V,T). Explain in no more than 4 sentences each: i) The physical interpretation of the partition function. ii) The significance of the partition function to statistical thermodynamics 2 ...
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