Physics 214b-2008 Practice questions for exam 1

You should be able to do each of these problems (including all sub-parts) in about 20 minutes (25 minutes for problem 2). If it takes you longer, you have not mastered the associated material thoroughly enough. Note: Of course, these problems don’t cover exactly the same topics as the questions on the real exam. Therefore, doing well on these practice problems does not insure that you are adequately prepared for the exam. However, doing poorly on these problems does mean that you need to study more. You should consider the questions on part one of assignment 5 as additional practice questions for this exam.

Some integrals you may or may not need:

For the following, the parameter a is assumed >0:

1. Heat capacity of a diatomic molecular gas. As we discussed in class, the molar heat capacity is defined as , where U is the internal energy (i.e. the thermal energy) of the sample. For a diatomic molecule, there are seven distinct ways in which it can move. Assume the axis of the molecule (the line connecting the two atoms) is along the x-axis. The molecule can translate in x, y, or z. It can rotate about the x, y, or z axis. Finally, the two atoms can vibrate relative to the center of the molecule in a breathing mode. We can model this breathing mode as a single simple harmonic oscillator. For reasons that may become clear to you later in this problem, rotation around the x-axis does not occur at any experimentally-accessible temperature; for now simply accept that. Recall that rotational kinetic energy is given by , where is the moment of inertia, and r_i is the distance from the rotation axis to mass m_i.

a) At high temperatures, the molar heat capacity of any gas of diatomic molecules is found to be , where N_A is Avogadro’s number (the number of molecules in a mole). Derive this result from the equipartition theorem.

b) As the temperature is lowered, the molar heat capacity drops in two steps. First, it drops to , and then at a lower temperature it drops to , as shown to the right. (The point labelled “0” temperature really only represents the lowest experimentally-accessible temperature, perhaps 10^-6 K.) Explain why this happens. In your response, you should be able to explain why, as the temperature is lowered, there should be two drops in the heat capacity, and that each of them should be a drop of magnitude k_BT. Let’s call these two drops A and B. I do not expect you to be able to specify the temperature at which the drops should occur, nor do I expect you to be able to say whether drop A or drop B should occur at a higher temperature. Hints: Think about the explanation for the temperature dependence of the heat capacity of a crystal, or of the way that Planck resolved the ultraviolet catastrophe.

c) Extra credit: Speculate about why, at any reasonable temperature, rotation about the x-axis does not occur.

2. Show that the wavefunction obeys the Heisenberg uncertainty relation (the position-momentum relation). This function is already correctly normalized.