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## Internal Energy of a Diatomic Ideal Gas: Instructor's Guide

### Main Ideas

- Students will gain experience in the use of quantum mechanical expressions to solve a thermodynamic problem.
- Students will have to think critically about what constraints limiting conditions put upon the calculation they are assigned.
- Students will gain experience in using substitutions to pull the units from integrals and simplify their computations
- Students will have the opportunity to practice truncating terms in an expansion expression.

### Students' Task

*Estimated Time: 100 minutes*

Given the internal energy of an ideal diatomic molecule, where

$$U=N\left(\sum_{n_{x}n_{y}n_{z}}P_{n_{x}n_{y}n_{z}}\frac{\hbar ^{2} \pi^{2}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)}{2 m L^{2}}+\sum_{lm}P_{lm}\frac{\hbar ^{2}l\left(l+1\right)}{2I}+\sum_{n_{v}}P_{n_{v}}\left(n+\frac{1}{2}\right)\hbar \omega_{0}\right) \; \; \; , $$

- Small groups of students are asked to calculate the rotational, translational, or vibrational energy for an ideal diatomic molecule.
- Each group is also given a limiting case for their particular energy and must use this limiting case to make the energy solvable.
- The small groups will have to remove the units from integrals to make them more easily analyzable and computable.

### Prerequisite Knowledge

- Experience with quantum operators and the energy eigenvalue equation.
- Past work with rotational, translational, and vibrational energy expressions.
- Proficient in calculus computations, particularly integrating techniques for more complex integrals.
- Ability to integrate in both cartesian and polar coordinates.
- Skilled in series expansions and choosing when to truncate a series.
- Experience with using limiting cases to make approximations in calculations.

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student (not strictly required, but helpful)

### Activity: Introduction

Begin by discussing the $U = 3/2 k_BT$ that students have seen for a monatomic ideal gas. Make sure students recognize how useful this has been in the various problems they have solved, and explain that they are going to work out the internal energy of a *diatomic* ideal gas. Since air is mostly a diatomic ideal gas, this should be recognizably important. This introduction is particularly important because of the mathematical complexity of this problem, which can easily leave students drowning and not recognizing what they are doing or why.

This activity is best performed after showing students how to find the expression for the total internal energy of an ideal diatomic gas. Place the students into small groups and assign them to either translational, vibrational, or rotational energy. Ask the groups to find the contribution to the internal energy of the diatomic molecule from their term. Also, assign each group a limiting case on the temperature of the system for their energy. The limiting cases that are reasonably computable with the tools students have at this level are as follows.

*Translational kinetic energy*

$\beta \frac{\hbar ^{2} \pi ^{2}}{2 m L^{2}} \gg 1 $

*Vibrational kinetic energy*

$\beta \hbar \omega_{0} \gg 1 \; \; or \; \; \beta \hbar \omega_{0} \ll 1$

*Rotational kinetic energy*

$\beta \frac{\hbar ^2}{2I} \gg 1 \; \; or \; \; \beta \frac{\hbar ^2}{2I} \ll 1$

There are a few more points that should be made *prior* to having students go ahead on this project.

One point that is helpful to make here is the idea that students should take the things with dimensions out of their integrals as early as possible. Often the integrals themselves contain no physical parameters, in which case the scaling of the solution (e.g. with temperature) can be found without performing the integral. Once one is 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.

A second feature that should be briefly discussed (either in the introduction or soon after they get stuck on it) is the treatment of summation as an integral where there are only very small changes between neighboring summands.

Also note that the low-temperature solutions are *way* easier than the high-T solutions, so it might be worth having everyone start with the low-T limit and only afterwards move to the high-T limit. The low-T solutions are tricky because students don't think to just truncate the Boltzmann sum, but once they realize that (or are told they can do so), it's mathematically very easy.

### Activity: Student Conversations

This problem is mathematically very challenging for students, even though to faculty members it likely seems essentially straightforward. Students struggle to recognize what they should be doing.

This is also an opportunity to discuss the meaning of limiting cases. Our students have essentially only addressed limiting cases by taking Taylor expansions, and in this case Taylor expansions don't help for *either* of the limiting cases, so expect some frustrated students. This is a chance to talk about what limiting cases mean, and in particular to address the fact that only a dimensionless quantity can truly be “large”. When we say “high-T” or “low-T”, we always mean that T is either much greater or much less than something else with units of temperature.

Part-way through, you will probably 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.

### Activity: Wrap-up

In the wrap-up, groups should present their results to the class for both the low-temperature and high-temperature limits. It might be worth getting students to work out what the low and high temperature limits are for nitrogen (but I haven't done this yet).

In the low temperature limit, groups should recognize that they can keep just one or two terms of the summation. Make sure that the class recognizes that internal energy is *always* going to be see that the internal energy is always going to be exponential in $\beta$, which will mean the heat capacity drops to zero exponentially.

Any group that has a high temperature limiting case and has to compute integrals should have removed the “physics” from the exponential terms. After transforming the sums into integrals, students should perform the proper variable substitution that will remove all of the constants from the exponential and leave only a dimensionless husk. Typically, some students will complain that this is an extraneous step; however, if only a dimensionless integral is left, then the units and important thermodynamic terms can immediately be analyzed without even computing the integral; the answer for the internal energy would only change by a constant number factor after computing the integral.

Once the groups have presented their limiting cases, you should summarize each limit, and sketch out the heat capacity as a function of temperature. This is an important result, as it is the students' first mathematical experience that the internal energy is *not* in general proportional to the temperature.