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## Name the Experiment (Introduction): Instructor's Guide

### Main Ideas

- Partial derivatives
- Physical representation
- Thermodynamic variables
- Practicing changing certain variables while holding others constant

### Students' Task

*Estimated Time: 15 min*

Students will design an experiment that measures a specific partial derivative.

### Prerequisite Knowledge

- The ability to interpret partial derivatives.
- The ability to physically interpret partial derivatives.

### Props/Equipment

- Tabletop Whiteboard with markers

### Activity: Introduction

Write a thermodynamic derivative on the board, and ask the students to describe the experiment that you would perform in order to measure it, and draw a picture of the apparatus.

Partials that should be considered for this activity:

Simple 3D | $\left(\frac{\partial V}{\partial p}\right)_T\;\left(\frac{\partial V}{\partial T}\right)_p$ |

Simple 1D | $\left(\frac{\partial L}{\partial \tau}\right)_T\;\left(\frac{\partial L}{\partial T}\right)_\tau$ |

Simple adiabatic | $\left(\frac{\partial T}{\partial V}\right)_S\;\left(\frac{\partial V}{\partial p}\right)_S$ |

First Law (challenging) | $\left(\frac{\partial U}{\partial T}\right)_V\;\left(\frac{\partial U}{\partial P}\right)_S$ |

A particularly challenging pair of derivatives are $\left(\frac{\partial p}{\partial S}\right)_T$ and $\left(\frac{\partial V}{\partial S}\right)_T$. In particular, the idea of “heating” something at constant temperature is quite counterintuitive. It may help to invoke the example of melting ice, in which you are heating the ice, but it stays at zero centigrade.

### Activity: Student Conversations

- It's worthwhile inverting derivatives from time to time. Students who can understand $\left(\frac{\partial p}{\partial S}\right)_V$ may struggle with $\left(\frac{\partial S}{\partial p}\right)_V$, even though one is just the inverse of the other.

- The first time entropy is varied in a derivative, students may be very confused as to how one could either measure a change in entropy or induce a change in entropy.

- Students can confuse what is meant by “insulated system” and “isolated system.”

### Activity: Wrap-up

If many or most of the groups had trouble with a particular concept, it's worth bringing everyone together to discuss this. As well, if there was a particular group that had a unique solution, it is worth showing to the class as well. Here is a narrative for this activity.

### Extensions

This activity is the initial activity of the Name the Experiment sequence in the context of thermodynamics. It is strongly recommended that this activity is done prior to any of the others.

- Follow-up activities:
- Name the Experiment: Changing Entropy: In this small group activity, students connect derivatives containing length and tension with measurements of a rubber band.
- Name the Experiment: Maxwell Relations: In this small group activity, students must determine whether it is easier to experimentally measure a given partial derivative or an equivalent found using Maxwell relations.