## Activity: 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 Small Group Activity schedule 30 min. build Graph with empty $T$ and $p$ axes, Graph of $S$ and $V$ contours on $T$ and $p$ axes, (optional, for demonstration) $U(T,p)$ plastic surface (blue) description Student handout (PDF)
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Thermo
What students learn
• The number of pieces of information you need to specify a state / the number of independent variables is equal to the number of ways you can get energy into or out of the system (2 in the simplest thermo examples).
• Which variables are independent and can be chosen for convenience.
• You cannot hold two variables constant and change the state if there are only two degrees of freedom.
• Media
• 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_ind Curvilinear Coordinates Introduction

assignment_ind Small White Board Question

10 min.

##### Curvilinear Coordinates Introduction
AIMS Maxwell Fall 21 Central Forces Spring 2021 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

Curvilinear Coordinate Sequence

First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles $\theta$ and $\phi$. Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.
• group Leibniz Notation

group Small Group Activity

5 min.

##### Leibniz Notation
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics; unlike standard Leibniz notation, this notation explicitly specifies constant variables. Students are guided in linking the variables from a contextless Leibniz-notation partial derivative to their proper variable categories.
• group Charged Sphere

group Small Group Activity

30 min.

##### Charged Sphere

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
• 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 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 Flux through a Cone

group Small Group Activity

30 min.

##### Flux through a Cone
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

Integration Sequence

Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ .
• accessibility_new Curvilinear Basis Vectors

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
AIMS Maxwell Fall 21 Central Forces Spring 2021 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
• group Using Tinker Toys to Represent Spin 1/2 Quantum Systems

group Small Group Activity

10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems

Arms Sequence for Complex Numbers and Quantum States

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases ($x$, $y$, and $z$). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
• accessibility_new Acting Out Charge Densities

accessibility_new Kinesthetic

10 min.

##### Acting Out Charge Densities
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear $\lambda$, surface $\sigma$, and volume $\rho$ charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.

Goals

• The number of pieces of information you need to specify a state/the number of independent variables is equal to the number of ways you can get energy into or out of the system (2 in the simplest thermo examples)
• Which variables are independent can be chosen for convenience.
• You cannot hold two variables constant and change the state if there are only two degrees of freedom.

Time Estimate: 30 minutes

Tools/Equipment

• Graph with empty $T$ and $p$ axes
• Graph of $S$ and $V$ contours on $T$ and $p$ axes
• (optional, for demonstration) $U(T,p)$ plastic surface (blue)

Intro

• Little is needed. Some students might be bothered by thinking about entropy if it hasn't been mentioned at all in class.
• This activity is a good follow-up to the “Changes in Internal Energy" activity about the 1st law of thermodynamics.

Whole Class Discussion:

• Get students to articulate that they needed 2 pieces of information to identify a state, but it didn't need to be the numbers located on the perpendicular axes - it could have been any two pieces of information.
• Introduce the language that simple thermal systems have 2 degrees of freedom (number of independent variables). The number of degrees of freedom is equal to the number of ways of getting energy into or out of the system. For simple thermal systems, you get energy into or out of the system through heat or work (1st law of thermodynamics)
• Changes in state also require specifying two changes - you can only hold 1 variable constant and change the state! If you hold two variables constant, you're stuck!
• There is freedom to choose which variables you want to be independent. “Choices” are usually related to what is controlled in an experiment.
• Two common ways to think about 2D graphs with 2 independent variables: with 1 independent variable on the horizontal axis, a dependent variable on the vertical axis, and the other independent variable labeling the curve (more common in thermo) OR with both independent variables on the axes and a dependent variables as a surface with levels curves on the graph (more common in E&M and mechanics)

Orient: Imagine that you have water vapor in a container where you can control the temperature of the gas. The temperature and pressure axes show possible values of temperature and pressure for 1kg of water vapor.

1. Without pointing to it or marking it, have one member of your group select an arbitrary location (a “state”) on the page.
2. That person should now describe their state in words so that another member of the group can mark it.
3. How many pieces of information do you need to specify the state?

It may be helpful to give an example of a state on the surface, or a full example of playing this “game”.
Prep: Need a contour map with $U(T,p)$ with $T$ and $p$ axes only, no states or other contours.
Goal: Gets at the idea that you only need two variables to specify the state. Cartesian or polar coordinates might come up as another example---talk about how axes in this case have different dimensions --- generally true in thermo.

Coordinate: Now imagine you did an experiment where you measured the pressure, $p$ of the water vapor as you varied the temperature, $T$, for several fixed values of entropy, $S$, and volume, $V$, and plotted these curves on the same set of axes. The new contour map shows these plots.

• Locate your state on the new graph.
• Specify your state in as many ways as possible.

Prep: Need a contour map with $U(T,p)$ with $S, V$ contours, no states.
Goal: Gets at the idea that you can specify the state with any two of the four variables.
Note: Students might try to over-specify the state. Ask how many of the 3 or 4 variables you can independently pick. Maybe as a WCD have 3 groups each specify a different variable

Explore: Choose a second, nearby state and mark it on the graph. In as many different ways as you can, describe how to get from your old state to your new state.

Goal: Gets at the idea that you need to specify changes in two variables (“constant” is a specified change of zero)

Discussion: How many pieces of information did you have to give to describe a path? Could you have given less information?

Can you find a nearby state where the path involves holding one of the thermodynamic variables constant? 2 variables? 3 variables?

Goal: Gets at the idea that you can specify one variable to be constant. If you try to specify 2 (or 3) variables to be constant, then the new state is the same as the old state (you haven't gone anywhere).

New Representation: Can you find the states you have been considering on these alternate representations:

1. A rubber sheet that can be stretched and squished so that the $S$ and $V$ contours are straight and perpendicular to each other square.

Goal: The set of numbers that describes a state is unique and can be located on different representations of the system.

2. A plastic surface whose height represents the internal energy of the system.

Students will need to be told what the axes represent.

Prep: Could have a demonstration $U(T,p)$ surface to show students.

Goal: The set of numbers that describes a state is unique and can be located on different representations of the system.

Author Information
Raising Physics to the Surface
Keywords
Thermo
Learning Outcomes