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## Using $pV$ and $TS$ Plots: Instructor's Guide

### Main Ideas

- Work
- Heat transfer
- Internal energy
- Entropy
- The simple cycle
- Consequences of holding particular variables constant

### Students' Task

*Estimated Time: 35 minutes *

- Students are placed into small groups and given handout with a $pV$ plot that describes the cycle of a process. Each group must determine whether the work and heat transfer for a single cycle and on each leg of the $pV$ plot are positive, negative, or zero.
- On the same handout, a $TS$ plot can be found. The groups perform the same analysis for a single cycle and on each leg of the $TS$ plot.
- Students must think about what kind of physical system these plots could describe.

### Prerequisite Knowledge

- Knowledge of the first and second laws.
- Experience with the thermodynamic identity.
- Experience with calculus, particularly computing integrals.

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student

### Activity: Introduction

Before starting this activity, a lecture on calculating the heat or work in a thermodynamic process using the first thermodynamic law and the thermodynamic identity is recommended. To help students think about how to find the signs of the work and heat , the instructor can write down on the board

$$ dU = \delta Q + \delta W$$

and

$$ dU = TdS - pdV \; \; \; , $$

where $\delta$ is an inexact differential.

### Activity: Student Conversations

- Some students will claim that because we return to the original state, the net work (or possibly net heat) must be zero.
- Many students will not recognize that on a $pV$ graph, the work is just (the negative of) the area under the curve. While I wouldn't want them to think of this as a definition of heat, it is a necessary and useful tool to have in their toolbox.
- Although the math using $TS$ diagram is effectively identical to that with the $pV$ diagram (with heat and work swapped, and a few minus signs), students will still find the $TS$ diagram challenging.
- It is helpful to ask students to describe what is happening in each stage (e.g. “it is being compressed at fixed pressure… so probably its temperature is dropping, unless it is something weird like ice”).
- Some students assume that temperature is fixed when working on the $pV$ plot, because it isn't listed.
- Many students will try to invoke or use the ideal gas law.

### Activity: Wrap-up

Bring the class back together and have a group argue if the work is positive or negative on each leg of either the $pV$ or $TS$ curve (instructor's preference of which to analyze first). Help groups resolve any inconsistencies in answers for the work and heat of each leg.

Many students become confused as to if work is being done on or by the system in these plots because of the minus sign associated with the thermodynamic identity term containing $pdV$. Be sure to point out that if the volume of the system increases, then work is being done by the system; show this by analyzing a leg of the $pV$ curve where the pressure remains constant and the volume increases and display that the total work of the system would be negative.

The analysis of the $TS$ curve should be nearly identical to the $pV$ curve. Be sure to note that the minus sign that appears in the work term is no longer present in the heat expression.