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
    • activity_media/BlueThermoTpFace.jpg
    • activity_media/PurpleThermoSVSide.jpg


  • 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


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


  • 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.

  • group Changes in Internal Energy (Remote)

    group Small Group Activity

    30 min.

    Changes in Internal Energy (Remote)

    Thermo Internal Energy 1st Law of Thermodynamics

    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 Heat and Temperature of Water Vapor (Remote)

    group Small Group Activity

    5 min.

    Heat and Temperature of Water Vapor (Remote)

    Thermo Heat Capacity Partial Derivatives

    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.
  • 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 Ideal Gas Model

    group Small Group Activity

    30 min.

    Ideal Gas Model

    Ideal Gas surfaces thermo

    Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.
  • group A glass of water

    group Small Group Activity

    30 min.

    A glass of water
    Energy and Entropy 2021 (2 years)

    thermodynamics intensive extensive temperature volume energy entropy

    Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
  • assignment Adiabatic Compression

    assignment Homework

    Adiabatic Compression
    ideal gas internal energy engine Energy and Entropy 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 2021 (2 years)

    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).

    3. Discuss your results.

  • group Using $pV$ and $TS$ Plots

    group Small Group Activity

    30 min.

    Using \(pV\) and \(TS\) Plots
    Energy and Entropy 2021 (2 years)

    work heat first law energy

    Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.
  • 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?

  • accessibility_new Acting Out Current Density

    accessibility_new Kinesthetic

    10 min.

    Acting Out Current Density
    Static Fields 2022 (4 years)

    Steady current current density magnetic field idealization

    Ring Cycle Sequence

    Integration Sequence

    Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

Thermo Multivariable Functions
Learning Outcomes