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## Analyzing a Simply Cycle Using a $pV$ Curve: Instructor's Guide

### Main Ideas

- First and second thermodynamic laws
- Work
- Heat
- Entropy
- Isobaric processes
- Isochoric processes
- Isothermic processes

### Students' Task

*Estimated Time: 65 minutes*

Small groups of students are given a thermodynamic process described by a $pV$ curve for an ideal gas and asked to determine the total work, heat, and change in internal energy for one full cycle and on each leg of the cycle. The small groups can also be asked to find the change in entropy for one full cycle and on each leg of the curve. If any group finishes early, students can also be asked to calculate the efficiency of this engine and see how it compares to the Carnot efficiency.

### Prerequisite Knowledge

- Experience with the first and second thermodynamic laws.
- Previous classes in calculus; the ability to integrate is essential.
- The ability to interpret $pV$ curves.

### Props/Equipment

- Tabletop Whiteboard with markers

### Activity: Introduction

Before beginning this activity, a brief lecture on determining the work and heat transfer of a process using the first law of thermodynamics is useful for putting students in the right mindset. Students are then placed into small groups and a $pV$ curve is drawn that follows the following description:

“An ideal gas begins at initial conditions $V_{0}$, $p_{0}$ and $T_{0}$. We first allow the gas to expand to twice its original volume at fixed temperature. We then cool it at fixed pressure until it returns to its original volume. Finally, we heat it up at fixed volume until it returns to the original pressure and temperature.”

Then, after stating that the $pV$ curves describe a monoatomic ideal gas, write down on the board

$$U=\frac{3}{2}Nk_{B}T$$

and

$$pV=Nk_{B}T \; \; \; . $$

Explain that the work, heat, and internal energy for each leg can be found using these two equations, the first and second thermodynamic laws, and the thermodynamic identity; to make these equations useful, however, the groups must first look for information in the scenario's description.

### Activity: Student Conversations

Students find this activity surprisingly challenging. Some of the issues students will encounter include:

- Some students persist in believing that work is a state function, and mistakenly assume that they can find the work on the third leg by subtraction.
- Some students fail to recognize that the temperature can be extracted from the pressure and volume.
- Using the first law to find heat for each leg can be challenging.
- Actually performing the integrals (which give logarithms) is also tricky for many of our students.

### Activity: Wrap-up

If time permits, compute the work, heat, and internal energy for the cycle so students can see the correct answers for the activity. The integrals for this activity are not very difficult; however, students must find the necessary information in the given expressions and problem description. For example, the fact that the first leg of the cycle is isothermic indicates that the internal energy remains constant; this implies that the work and heat transfer for the first leg will be equal in magnitude and have the proper signs to make $dU=0$ in the first law of thermodynamics.