## 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

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.

The contour map represents measurements on a kilogram of water vapor in an insulated piston (a cylindrical thermos with a moveable top).

Estimate the “Squishability”: 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 contour map and estimate the squishability of water vapor:

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

Experimental Design: 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?

Defining the Experiment: 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?

Discussion: Variables in Thermodynamics 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.

Consider Other Experiments: 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?

Answer: Once two thermodynamic variables are fixed, the state is fully determined so changes cannot be measured. Answer: Squishability is related to the change in volume, so an experiment to measure this quantity must incorporate a change in volume.

SUMMARY PAGE
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:

• Squishability contour map
• 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:

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

• Discussion: Extensive vs. Intensive 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.

• 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 Quantifying Change

group Small Group Activity

30 min.

##### Quantifying Change

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.
• group Heat and Temperature of Water Vapor

group Small Group Activity

30 min.

##### Heat and Temperature of Water Vapor

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 Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

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

Warm-Up

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 consider whether the thermo surfaces reflect the properties of an ideal gas.
• assignment Heat of vaporization of ice

assignment Homework

##### Heat of vaporization of ice
Vaporization Heat Thermal and Statistical Physics 2020 The pressure of water vapor over ice is 518 Pa at $-2^\circ\text{C}$. The vapor pressure of water at its triple point is 611 Pa, at 0.01$^\circ\text{C}$ (see Estimate in $\text{J mol}^{-1}$ the heat of vaporization of ice just under freezing. How does this compare with the heat of vaporization of water?
• group Gravitational Potential Energy

group Small Group Activity

60 min.

##### Gravitational Potential Energy

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.
• group Gravitational Force

group Small Group Activity

30 min.

##### Gravitational Force

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.

Keywords
Thermo Partial Derivatives
Learning Outcomes