## 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

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.

• 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.
• 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 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.
• assignment Calculation of $\frac{dT}{dp}$ for water

assignment Homework

##### Calculation of $\frac{dT}{dp}$ for water
Clausius-Clapeyron Thermal and Statistical Physics 2020 Calculate based on the Clausius-Clapeyron equation the value of $\frac{dT}{dp}$ near $p=1\text{atm}$ for the liquid-vapor equilibrium of water. The heat of vaporization at $100^\circ\text{C}$ is $2260\text{ J g}^{-1}$. Express the result in kelvin/atm.
• assignment Vapor pressure equation

assignment Homework

##### Vapor pressure equation
phase transformation Clausius-Clapeyron Thermal and Statistical Physics 2020 Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that $\Delta V \approx V_g$. Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.
1. Solve for $\frac{dp}{dT}$ in terms of the pressure of the vapor and the latent heat $L$ and the temperature.

2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

• group Name the experiment

group Small Group Activity

30 min.

##### Name the experiment
Energy and Entropy 2021 (3 years)

Student groups design an experiment that measures an assigned partial derivative. In a compare-and-contrast wrap-up, groups report on how they would measure their derivatives.
• 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.
• face Phase transformations

face Lecture

120 min.

##### Phase transformations
Thermal and Statistical Physics 2020

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

Keywords
Thermo Partial Derivatives
Learning Outcomes