• Partial derivatives
• Physical representation
• Thermodynamic variables
• Practicing changing certain variables while holding others constant
• Maxwell relations
• Symmetry of mixed partials
• Thermodynamic internal energies

Estimated Time: 50 minutes (for both parts)

Students are placed into small groups and given thermodynamic partial derivatives that are difficult to measure directly. The groups must use the Maxwell relations to find a partial that is more easily measured and then design an experiment to measure the new partial.

• The ability to interpret partial derivatives.
• The ability to physically interpret partial derivatives.
• Understanding of the first and second thermodynamic laws.
• Familiarity with the symmetry of mixed partials.
• Experience using Maxwell relations.

• Tabletop Whiteboard with markers

We have found this activity is most successful following a lecture or activity that has students practicing using Maxwell Relations in a mathematical context. Then, place the students in small groups and write the thermodynamic partials that are not easily measured on the board (typically an expression with entropy in it). If desired, the instructor can also create more partial derivatives and give each group their own unique partial. For a challenge, the instructor can flip the partial derivatives upside down; this forces the students to first invert the partial before using Maxwell relations to find a more easily measured partial. Some recommended partials for this activity are:

$$ \left(\frac{\partial \tau}{\partial L}\right)_{T} \left(\frac{\partial \tau}{\partial T}\right)_{L} \left(\frac{\partial V}{\partial T}\right)_{p} \left(\frac{\partial T}{\partial p}\right)_{V} \left(\frac{\partial T}{\partial V}\right)_{S} \left(\frac{\partial S}{\partial T}\right)_{L} \; \; \; $$

Most of these partial derivatives can already be measured directly; assign each group a partial derivative ask the groups to find another way to measure the partial experimentally. The instructor can choose to mention or not mention the use of Maxwell relations before students get to work depending on what level of challenge is desired. Also, the instructor can invert the partial derivatives to increase the level of challenge even more.

• One challenge students have is that some of the derivatives provided are actually the inverse of a derivative that shows up in a Maxwell relation. This provides a bit more challenge in figuring out which Maxwell relation to use, and the instructor needs to pay attention and give hints as needed.

• The first two (isothermal) derivatives are much harder to imagine measuring directly than the second two, since it is hard to imagine isothermally heating a system, so it is important to make sure that students from both categories share their conclusions.

If each group was given a separate partial to measure, have groups present their experiments to the class. It is crucial to emphasize how important the Maxwell relations are for experimentally measuring difficult partials; because of the equivalent expressions given by Maxwell relations, changes in indirectly measured variables such as entropy can be easily measured by observing the change in a directly measured variable. Most importantly, be sure to ask each group if they think it would be easier to experimentally measure the partial derivative given directly or to experimentally measure the equivalent partial found with Maxwell relations. Be sure to mention to the class that Maxwell relations do not always give a more easily measured partial derivative, and that students will have to think critically to decide when a Maxwell relation is helpful or hindering.

This activity is the third of the Name the Experiment sequence in the context of thermodynamics. Prior to this activity, it is highly recommended to do the original Name the Experiment: Introduction activity.

• Preceding activities:
  • Name the Experiment: Introduction: This small group activity has students connect thermodynamic derivatives with experiments.
  • Name the Experiment: Changing Entropy: In this small group activity, students connect derivatives containing length and tension with measurements of a rubber band.