Activity: Ideal Gas Model

Students consider whether the thermo surfaces reflect the properties of an ideal gas.
  • group Small Group Activity schedule 30 min. build Ideal Gas Contour Maps, handout for each student, purple and blue thermo surfaces for instructor, personal or shared writing space for each student description Student handout (PDF)
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    Ideal Gas surfaces thermo
What students learn
  • Equation and definition of an Ideal Gas
  • Students explore what it would mean for something to be an ideal gas or a good approximation of an ideal gas
  • Students might discuss the difference between temperature and internal energy
  • Can discuss degrees of freedom
  • Media
    • activity_media/ideal_gas_model_2_svu_tpu.pdf
    • activity_media/BlueThermoTpFace_iDluixD.jpg
    • activity_media/PurpleThermoSVFace.jpg

The internal energy of an ideal gas is described by the equation:

\[ U(p,T)=\left(\frac{\# dof}{2}\right) NkT \]

The surfaces represent the internal energy of water vapor:

  • The blue surface shows the internal energy as a function of temperature and pressure, \(U(T,p)\), with volume and entropy contours etched into the surface.
  • The purple surface shows the internal energy as a function of volume and entropy, \(U(S,V)\), with pressure and temperatures contours etched into the surface.

By examining the plastic surfaces, how can you tell if an ideal gas model is a good model for water vapor?

Answer: The biggest indicator on the p-T surface is if the surface is a plane. On the contour map, this corresponds to the internal energy contours being nearly parallel to the volume axis (constant at constant temperature) AND having the same, constant spacing between them. Also, volume contours are nearly linear (ideal gas law). For an ideal gas on the S-V graph, isotherms should be parallel to and never cross the U contours, i.e. on the purple surface the isotherms are level curves.

Discussion: Sketching a 2D graph of U vs T for an Ideal Gas When working with a contour map, students might benefit from sketching a graph of internal energy vs.temperature for an ideal gas, then sketching a plot of internal energy vs.temperature for water vapor, using the values from the contour map.

Discussion: Linearity The instructor might model the blue surface, describing the features of the surface, including its near-flatness. (Although it is not a perfectly flat, and this can be demonstrating by placing the surface upside down on a flat table, against a small whiteboard, etc.)

Discussion: Applications of the Model One can discuss degrees of freedom and gasses/situations that are typically modeled as ideal.

If you give the students the scaling and the number of particles, they could be able to figure out the approximate degrees of freedom for water vapor (canonically, 9).
  • 1kg water vapor is 55 mols of water
  • Internal Energy 2cm \(\rightarrow\) 170. kJ
    Temperature 2cm \(\rightarrow\) 70 K

  • Equation and definition of an Ideal Gas
  • Students explore what it would mean for something to be an ideal gas or a good approximation of an ideal gas
  • Students might discuss the difference between temperature and internal energy
  • Can discuss degrees of freedom

Time Estimate: 15-30 minutes

Tools and Equipment:

  • Purple and Blue Thermo surfaces for each group
  • Remote Option: Ideal Gas Contour Maps
  • Student handout for each student
  • A personal or shared writing space for each student to write/draw/sketch


  • Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy

Whole Class Discussion:

  • A whole class discussion should elicit student ideas about how to determine if a system fits the ideal gas model perfectly, and when the ideal gas model is a “good" fit.
  • Compare and contrast how the ideal gas relationships appear on the blue vs purple surface. In particular, U \(\propto\) T - blue surface is a plane; on the purple (and blue) surface the temperature and internal energy contours are parallel.

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  • 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?

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    The concentration of potassium \(\text{K}^+\) ions in the internal sap of a plant cell (for example, a fresh water alga) may exceed by a factor of \(10^4\) the concentration of \(\text{K}^+\) ions in the pond water in which the cell is growing. The chemical potential of the \(\text{K}^+\) ions is higher in the sap because their concentration \(n\) is higher there. Estimate the difference in chemical potential at \(300\text{K}\) and show that it is equivalent to a voltage of \(0.24\text{V}\) across the cell wall. Take \(\mu\) as for an ideal gas. Because the values of the chemical potential are different, the ions in the cell and in the pond are not in diffusive equilibrium. The plant cell membrane is highly impermeable to the passive leakage of ions through it. Important questions in cell physics include these: How is the high concentration of ions built up within the cell? How is metabolic energy applied to energize the active ion transport?

    David adds
    You might wonder why it is even remotely plausible to consider the ions in solution as an ideal gas. The key idea here is that the ideal gas entropy incorporates the entropy due to position dependence, and thus due to concentration. Since concentration is what differs between the cell and the pond, the ideal gas entropy describes this pretty effectively. In contrast to the concentration dependence, the temperature-dependence of the ideal gas chemical potential will not be so great.

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    assignment Homework

    Adiabatic Compression
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    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?

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    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?

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Author Information
Raising Physics to the Surface
Ideal Gas surfaces thermo
Learning Outcomes