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Estimated Time: 15 min
Students will design an experiment that measures a specific partial derivative.
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.
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.
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.