In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
Orient Yourself to the Physical System & the Graph: The plot shows various thermodynamic quantities for water vapor in a piston (cylindrical thermos with a movable top) at different states. From state (point) \(A\) to state (point) \(B\), estimate the following quantities:
| Verbal Description | Symbol | Estimate (with Units) |
| Change in volume: | \(\Delta V_{A\rightarrow B}\) | |
| Change in entropy: | \(\Delta S_{A\rightarrow B}\) | |
| Change in temperature: | \(\Delta T_{A\rightarrow B}\) | |
| Change in pressure: | \(\Delta P_{A\rightarrow B}\) | |
| Change in internal energy: | \(\Delta U_{A\rightarrow B}\) | |
Under what circumstances would you be willing to label these quantities with `d's instead of \(\Delta\)'s? For example, \(dV\) instead of \(\Delta V\).
Determine a Rate: Pick two of the variables in the table and determine the rate of change of one with respect to the other from \(A\) to \(B\). What experiment could you do to measure this rate?
Inverting Your Rate: Determine the reciprocal of the rate you calculated. Brainstorm a meaningful name for this rate.
Reversing the Path: How does the rate you previously calculated change if instead you went from state \(B\) to state \(A\)?