## Activity: 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.
What students learn
• That thermal system have at least two independent variables. You have to specify changes in both to determine how other (dependent) quantities will change.
• Media

Orient Yourself to the Physical System & the Graph: The surface represents measurements of internal energy on a kilogram of water vapor in a piston (a graduated cylinder with a moveable top). The purple surface graph is $U(S,V)$.

Make Predictions: Starting at the red star, as you increase the volume of the system, what happens to the internal energy of the water vapor?

Answer: The internal energy could increase, decrease, or stay the same. It depends on how the volume is increasing (and what is happening to the entropy).

Student Ideas: Most students will interpret “increasing volume” as “hold the entropy fixed”. This is, of course, a possibility, but it is not the only possible way to increase the volume. From their math classes, many students will have the idea that for a partial derivative, everthing else must be held constant.

Discussion: Alternative Graph Could you use the other surface? If so, how?

Discussion: Does it matter where you start? Does your answer depend on the initial state of the system

Helping Students Understand the Graphs: Students may need guidance to recognize the meaning of the contour lines, and the fact that they can use either surface to answer this question (although one is much easier that the other).

Consider a Special Case: Is it possible to change the volume without changing the internal energy? (Is your answer consistent with your answer to the previous prompt?)

Answer: Yes, it is possible - you move along a level curve of the internal energy. Along this level curve, both volume and entropy change.

Discussion: Checking Consistency Some students will happily answer “internal energy increases” to the first prompt and “yes it is possible to change the volume and not change the internal energy” to the second prompt without resolving the inconsistency between the two answers.

Signs of Partial Derivatives: Using the purple surface graph, determine if the following derivatives are positive, negative, or zero.

$\left(\frac{\partial U}{\partial V}\right)_S \quad \quad \quad \quad \left(\frac{\partial U}{\partial V}\right)_T \quad \quad \quad \quad \left(\frac{\partial U}{\partial V}\right)_p$

SUMMARY PAGE
Goals
• For functions of two independent variables, you need to specify changes in both independent variables to determine how the function changes.
• Can lead into a lecture about partial derivatives and specifying a path.

Time Estimate: 15 minutes

Tools/Equipment

• A Purple $U(S,V)$ Plastic Surface for each group
• Dry-erase markers

Intro

• Students need to be oriented to the surfaces and what they represent.

Whole Class Discussion:

• The main take-away is that for functions of two variables, you need to specify a change in both the independent variables to know the change in the function.
• You can ask student groups to share an example of a path where the volume increases and the internal energy: increases, decreases, and is constant.
• This activity leads into ideas about (1) how the number of independent variable you have in a thermal system corresponds to the number of ways of getting energy into or out of a system, (2) you need to specify the path for a partial derivative, and (3) for thermal system, you actually CANNOT “hold all the other variables 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.
• 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 Squishability'' of Water Vapor (Contour Map)

group Small Group Activity

30 min.

##### “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.
• 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 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.
• assignment Ideal gas calculations

assignment Homework

##### Ideal gas calculations
Ideal gas Entropy Sackur-Tetrode Thermal and Statistical Physics 2020

Consider one mole of an ideal monatomic gas at 300K and 1 atm. First, let the gas expand isothermally and reversibly to twice the initial volume; second, let this be followed by an isentropic expansion from twice to four times the original volume.

1. How much heat (in joules) is added to the gas in each of these two processes?

2. What is the temperature at the end of the second process?

3. Suppose the first process is replaced by an irreversible expansion into a vacuum, to a total volume twice the initial volume. What is the increase of entropy in the irreversible expansion, in J/K?

• group Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 23 (11 years)

Integration Sequence

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy 2021 (2 years)

The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.*

The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between $p$, $V$ and $T$. You may take the number of molecules $N$ to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.
1. What is the change in entropy of the gas? How do you know this?

2. What is the change in temperature of the gas?

• 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 Multivariable Functions
Learning Outcomes