## 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
• 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 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.
• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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?

• accessibility_new Acting Out Charge Densities

accessibility_new Kinesthetic

10 min.

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

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.

assignment Homework

ideal gas internal energy engine Energy and Entropy Fall 2020

A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

In this problem, you may treat air as an ideal gas, which satisfies the equation $pV = Nk_BT$. You may also use the property of an ideal gas that the internal energy depends only on the temperature $T$, i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by $C_V=\frac52Nk_B$, which means the internal energy is given by $U=\frac52Nk_BT$.

Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

1. If the air is initially at room temperature (taken as $20^{o}C$) and is then compressed adiabatically to $\frac1{15}$ of the original volume, what final temperature is attained (before fuel injection)?

2. By what factor does the pressure increase?

• assignment Bottle in a Bottle 2

assignment Homework

##### Bottle in a Bottle 2
heat entropy ideal gas Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

Consider the bottle in a bottle problem in a previous problem set, summarized here.

A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.

The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was $p=106$ Pa and its temperature $T=300$ K. Approximate the helium gas as an ideal gas of equations of state $pV=Nk_BT$ and $U=\frac32 Nk_BT$.

1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

2. Compute the integral $\int \frac{{\mathit{\unicode{273}}} Q}{T}$ and the change of entropy $\Delta S$ between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

• assignment Bottle in a Bottle

assignment Homework

##### Bottle in a Bottle
irreversible helium internal energy work first law Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

The internal energy of helium gas at temperature $T$ is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align}

Consider a very irreversible process in which a small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle?

• 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.

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”.

Orient: 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 $U(S,V)$.

Interpret: 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: Could you use the other surface? If so, how?

Discussion: Does your answer depend on the initial state of the system?}

Clarification: 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).

Discussion: What would the surface look like if your answer did not depend on the state of the system?

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 - you move along a level curve of the internal energy. Along this level curve, both volume and entropy change.

Discussion: 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.

Keywords
Thermo Multivariable Functions
Learning Outcomes