Activity: Free expansion

Energy and Entropy 2021 (2 years)
Students will determine the change in entropy (positive, negative, or none) for both the system and surroundings in three different cases. This is followed by an active whole-class discussion about where the entropy comes from during an irreversible process.
What students learn Students struggle with understanding that entropy can be created. It's an extensive quantity, and is the only one that isn't normally conserved, so that makes it pretty weird. We (professors) don't always realize how very weird this is, and students don't have the vocabulary to explain it to us, and are often afraid to try.
  • Media
    • activity_media/piston-full-300K_WK9eDUd.svg
    • activity_media/piston-half-300K_BZCT77I.svg
    • activity_media/adiabatic-piston-full_1ABc14C.svg
    • activity_media/adiabatic-piston-half-300K_C1pLCRc.svg
    • activity_media/free-expansion-full-300K_OgvbRhD.svg
    • activity_media/free-expansion-half-300K_kzFfzj4.svg


I always tell them that this quiz will only be graded on whether they did it, in hopes of reducing their level of fear. The “activity” really begins after they've finished the quiz. I put a table on the board of the various \(\Delta S\) and ask them to share their answers to each question with their little white boards (but raising hands would be okay). I make someone volunteer to explain why they chose what they did in each case. Students naturally will start asking questions, and the challenge will be in getting them to stop... and in trying to provide answers that will satisfy them.

Consider the three processes described below.
Process #1
Five moles of an ideal gas are initially confined in a one-liter cylinder with a movable piston, at a temperature of 300 K. Slowly the gas expands against the movable piston, while the cylinder is in contact with a thermal reservoir at 300 K. The temperature of the gas remains constant at 300 K while the volume increases to two liters.
Process #2
A thin plastic sheet divides an insulated two-liter container in half. Five moles of the same ideal gas are confined to one half of the container, at a temperature of 300 K. The other half of the container is a vacuum. The plastic divider is suddenly removed and the gas expands to fill the container. No work is done on or by the gas. The final temperature of the gas is also 300 K.
Process #3
The same cylinder as in process #1 is thermally insulated and then allowed to slowly expand, starting at 300 K, to twice its original size (two liters).
#1 Isothermal expansion #2 Free expansion #3 Adiabatic expansion
  1. Are \(\Delta S_\text{isothermal}\), \(\Delta S_\text{free}\) and \(\Delta S_\text{adiabatic}\), the change in entropy of the gas for each process, positive, negative, or zero? Please explain your reasoning.
  2. Is \(\Delta S_\text{isothermal}\) greater than, less than, or equal to \(\Delta S_\text{free}\)? How do each of these compare with \(\Delta S_\text{adiabatic}\)? Please explain.
  3. Are \(\Delta S_\text{surr-isothermal}\), \(\Delta S_\text{surr-free}\) and \(\Delta S_\text{surr-adiab}\), the change in entropy of the surroundings for each process, positive, negative, or zero? Please explain.

Student Conversations

One confusion that comes up is how to find \(\Delta S\). Sometimes we use \(\int dQ/T\) (for reversible processes), but sometimes we don't do so, and instead argue based on the initial and final states.

The main confusion, however, is where the entropy comes from in the case of the free expansion. One correct answer is to explain that entropy is simply created when something irreversible is done, which is a natural consequence of the Second Law. I also explain that entropy is *not* something that is conserved. This is troubling to them, and the same question repeats... which possibly means I don't have the best answer for it. Students wonder if entropy can be “real”, if it can be created willy-nilly like this, so I end up emphasizing that it *is* real, and that it can be measured.

One student asked if entropy could be measured *directly*, to which I answered that it can't be measured directly, but neither can energy. In both cases one is forced to measure other properties and infer the energy or entropy. But that this doesn't mean that we aren't actually measuring them.

Finally, after we've talked about the properties of entropy, students are liable to ask what entropy really *is*. Eventually I relent and give a preview of the statistical interpretation.


Some groups of students will have many questions that will threaten to push the group past the time limit. At some point, it may be best to simply state what the answers actually are, and that if students are still confused as to why, they should contact you. During the discussion, students may have made compelling arguments for incorrect solutions, and it is important to point out where their logic was flawed, so as to ensure students end up with a solid understanding.

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    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.

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    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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    2. Why does it take only two variables to define the state?
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    Suppose that a system of \(N\) atoms of type \(A\) is placed in diffusive contact with a system of \(N\) atoms of type \(B\) at the same temperature and volume.

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    2. If the atoms are identical (\(A=B\)), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

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    Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

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    2. Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

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Learning Outcomes