## 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
• group Thermodynamic States (Remote)

group Small Group Activity

30 min.

##### Thermodynamic States (Remote)

Little is needed. Some students might be bothered by thinking about entropy if it hasn't been mentioned at all in class. Try doing this activity as a follow-up to the “Changes in Internal Energy" about the first law of thermodynamics.
• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.
• group Ideal Gas Model

group Small Group Activity

30 min.

##### Ideal Gas Model

Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.
• group Heat and Temperature of Water Vapor (Remote)

group Small Group Activity

5 min.

##### Heat and Temperature of Water Vapor (Remote)

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
• group Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
• group Name the experiment

group Small Group Activity

30 min.

##### Name the experiment
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 Energy and Entropy Fall 2021 Students will design an experiment that measures a specific partial derivative.
• group Quantifying Change (Remote)

group Small Group Activity

30 min.

##### Quantifying Change (Remote)

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.
• assignment_ind Partial Derivatives from a Contour Map

assignment_ind Small White Board Question

10 min.

##### Partial Derivatives from a Contour Map
AIMS Maxwell AIMS 21 Students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
• face Energy and heat and entropy

face Lecture

30 min.

##### Energy and heat and entropy
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
• accessibility_new Acting Out the Gradient

accessibility_new Kinesthetic

10 min.

AIMS Maxwell AIMS 21 Static Fields Winter 2021

Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".

Goals:

• 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

Tools:

• 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.

Intro:

• 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.

Keywords
Thermo Partial Derivatives
Learning Outcomes