Activity: “Squishability" of Water Vapor (Contour Map)

Students determine the “squishibility” (an extensive compressibility) by taking \(-\partial V/\partial P\) holding either temperature or entropy fixed.
What students learn
  • The value of the derivative depends on the path. In other words, what you hold constant in a partial derivative matters.
  • For any particular graph, not all partial derivatives correspond to a slope
  • Media
    • activity_media/thermo_squishability_blacksquare_tpu_sv_horizlabels_axis_lg.png
    • activity_media/thermo_squishability_blacksquare_tpu_sv_horizlabels_axis_lg.pdf


  • The value of a derivative depend on what you hold constant (you get different values if you hold \(T\) or \(S\) constant.)
  • Derivatives are ratios of small changes.
  • On this graph \((\partial V/\partial p)_S\) is a slope; \((\partial V/\partial p)_S\).

Time Estimate: 30 minutes


  • Purple \(U(S,V)\) plastic surface for each group
  • Squishability contour maps
  • Student handout for each student
  • A personal or shared writing space for each student to write/draw/sketch.


  • No intro is needed

Whole Class Discussion:

  • How did you calculate the squishability.

  • On this graph \((\partial V/\partial p)_S\) is a slope; \((\partial V/\partial p)_T\) is not.

“Squishability” of Water Vapor

Working in small groups (2 or 3 people), solve as many of the problems below as possible. Try to resolve questions within the group before asking for help. Each group member should then write up solutions in their own words.

Estimate: The purple plastic surface represents measurements on a kilogram of water vapor in an insulated piston (a cylindrical thermos with a moveable top). There is a matching contour map with labels.

Imagine a thermodynamic quantity, the “squishability”, which is the negative rate of change of the volume of a fluid as the pressure changes.

Pick a point on the surface and estimate the squishability of water vapor:

  1. with temperature held constant
  2. with entropy held constant

Need a plot P vs V with level curves for T and S, or one of the plots from the other activities.

Experiment: Design an experiment to measure the squishability of water vapor at constant temperature and describe your experiment. What data would you collect and how would you use it to calculate the squishability?

Students should describe what they are going to change, measure, and hold constant and the physical method they would use for each of them.

In your experiment, what variables are you considering to be independent? What variables are dependent?

Students typically will say that the independent variable is the one that you change, and the dependent variable is the one that you measure. In thermodynamics, variables that are held constant are often considered independent. Because you may be able to perform the experiment again at a different constant value.

Explore: What would happen if you tried to measure the squishability with both temperature and entropy fixed? Alternatively, what would happen if you tried to measure the squishability in a container that cannot change size?

WCD: Does the squishability depend on how much water vapor you have? Yes - it is extensive. Note: we've invented squishability, but it is similar to compressibility, -\(\beta = \frac{1}{V}\frac{dV}{dP}\), an intensive quantity.

Thermo Partial Derivatives
Learning Outcomes