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## Activity Name: Instructor's Guide

### Main Ideas

Students get practice with statistical mechanics, and gain some understanding of the ideal gas, including the idea that the internal energy is *not* in general proportional to temperature.

### Students' Task

*Estimated Time: 2 1/2 hours*

Students work out the various terms of the internal energy of a diatomic ideal gas in high- and low-temperature limits. This is a really long activity (when taken with the accompanying lecture) but seems to be really worthwhile.

### Prerequisite Knowledge

Students should be familiar with Boltzmann statistics and finding the internal energy from the

### Props/Equipment

- Tabletop Whiteboard with markers

### Activity: Introduction

Introduce the quantum mechanical results for the eigenenergies of the particle in a box, the rigid rotor and the simple harmonic oscillator. (This takes me about 10 or 15 minutes.) Go on to work out how the probabilities and partition functions work out for non-interacting particles and separating energies. The result is that we can treat each separable term in the energy separately, and we don't need to handle all the molecules at once. Try to make the connection with the concept of “separation of variables” that was covered in the central forces paradigm.

### Activity: Student Conversations

Part-way through, you will need to bring the class together for a discussion of how to handle the high-temperature and low-temperature limits. The low-temperature case is actually quite simple (although students rarely realize this), since you can just keep the largest Boltzmann factors. For the high-temperature limit, you need to argue that the neighboring eigenstates are very close to one another, and the summation can be replaced by an integral.

Another point I like to make here is the idea that students should take the things with dimensions out of their integrals as early as possible. Once they are able to do this, the definite integral is just a number, and the dependence on temperature and other parameters is known, even before performing the integral itself.

### Activity: Wrap-up

Groups share their results with the class. There are five problems to be solved, high-T translational energy, low-T rotations, high-T rotations, low-T vibrations and high-T vibrations.

Once we have all the answers, we can discuss the equipartition theorem, and the fact that any system with a gap (a lowest-energy excitation) will have an exponential drop off in its heat capacity below that energy. We can sketch out the heat capacity, and talk a bit about that, e.g. where the regimes fall.