Biocurious is a weblog about biology, quantified.

The three-cylinders problem

by PhilipJ on 20 January 2006

I’m the TA for the undergrad thermodynamics course being offered this year, and they’ve just handed back their first problem set. The first problem (the so-called three-cylinders problem) is this:

Three identical cylinders are filled with unknown quantities of ideal gases. The cylinders are closed with identical frictionless pistons of mass M. Cylinders A and B are in thermal equilibrium with the room at 20 oC, and cylinder C is kept at a temperature of 80 oC. Is the pressure of Nitrogen gas in cylinder A greater than, less than, or equal to the pressure of the Hydrogen gas in cylinder B? Is the pressure of Hydrogen gas in cylinder C greater than, less than, or equal to the pressure of Hydrogen gas in cylinder B? Why?

Seems like a fairly straightforward problem, right? Well, based on the study below [1], of ~250 introductory physics students asked this problem, only 15% got the correct answer (which I’ll reveal to those who might be confused in the comments section after a few people post their thoughts!). Testing ~65 thermal physics students (who were physics majors), only 40% got the question entirely correct.

I had the pleasure of seeing Lillian McDermott (of the education physics research group at the University of Washington who did these studies) give a talk on these results, and its quite clear that the current methods used to try and teach physics concepts are inadequate. Do any of the other sciences have problems that stem from the very fundamentals? That only 40% of physics majors taking a class in thermodynamics got that answer correct is kinda scary [though my own class fared a little better, ~66% (N=25) had the correct answer].

[1] C. H. Kautz et al, Student understanding of the ideal gas law, Part I: A macroscopic perspective, American Journal of Physics 73 1055 (2005), the first of a pair of papers on the physics education of the ideal gas law.



  1. Uncle Al    3134 days ago    #
    If the frictionless pistons are of the same mass/area (“identical”), then absent an epsilon correction for variation of gravity with height… If they sit there at equilibrium then the pressure inside is the pressure outside plus supported piston weight/area. All internal pressures are identical. The minutia are irrelevant.

    Just be glad the mob can read. Only about 30% of college graduates can make sense of a newspaper editorial page. Diversity works!
  2. PhilipJ    3134 days ago    #
    Al, as you can imagine, is right. The point made by the article is that people often jump to the ideal gas law and try to think about volumes and so on, when a simple free body diagram gives you the right answer without getting confused about what’s inside.
  3. Hogg    3134 days ago    #
    Physics is particularly prone to these kinds of mistakes because physics is almost always taught out of 1000-page textbooks that make no distinction between what’s fundamental, simple, and important, and what’s not! Students memorize equations rather than think about fundamentals. But I don’t blame the students; it is part of the cultural idea of what is “physics” for which we all deserve blame. See, eg, arXiv:physics/0412107.
  4. Andre    3133 days ago    #
    Thanks for the paper, Hogg. I especially like the discussion of dimensional analysis. From my experience with first year physics (and chemistry) texts dimensional analysis is presented as something trivial that you do after you’ve solved a problem to see if the units work out instead of something you use to develop insight into new difficult problems. That’s really unfortunate.

    On the subject of textbook costs, have you ever taught from a free textbook?
  5. Abie    3133 days ago    #
    The tragedy in the figures you gave, is that most students probably view the problem as a “trick(y) question”.
    The famous “It’ not fair, you gave us misleading informations!” line.
    They’re upset when they realise that sometimes teachers could – just could – ask them to think for themselves, in front of something that is not even remotely half complex as a real life problem.
    I wish it would be a problem only in physics, but the same is true everywhere, event outside the hard sciences field.
  6. ab    3133 days ago    #
    Thank god (and Mike Mackey, who taught me foundation chem) for dimensional analysis – when you have to calculate how many mL of a soln of 40mg/mL of dopamine to add to a 100mL bag of dextrose such that a pre-determined infusion rate (mL/h) delivers a desired dose (mcg/kg/min) to a new-born baby (we need that dopamine drip stat!), you’re really happy about dimensional analysis. Even though a dear friend (a chemist) calls it dementia analysis…..(Andre and Marcel, you know who I mean!)
  7. PhilipJ    3132 days ago    #
    Abie – students also have a tendancy to overthink the problem, which I’ll never understand (I have a tendancy to underthink how hard everything is going to be in my research…)

    One complaint I got from a number of students was “but are the pistons moving?!”, which if they were would make the problem orders of magnitude more difficult.

    ab – I am depressed to say that whenever I do anything in the wet lab, I stand there like a buffoon for long periods of time trying to figure out how much of an x mg/mL solution goes into a y μL reaction such that the final concentration is z. It is embarassingly hard!
  8. dete    2392 days ago    #

    I’m not a physicist, but I have a quibble.

    I agree that the pressure at the TOP of each tank would be identical (modulo Uncle Al’s gravitational epsilon), but the pressure at the BOTTOM of each tank would also include the weight of gas in each tank. That would imply to me that the average pressure within each tank would not be equal, but would be largest on the left, and least on the right.

    Indeed, I would go so far as to suggest that this is more than a quibble, and that even on a moderately large scale, the effect would be measurable.


  9. Dood    1395 days ago    #

    I haven’t studied or dutifully applied physics in 22 years but I think that the two most important words in the problem are “unknown” (as in quantities of gases) and “frictionless” (as in pistons). If we are to believe that the pistons holding our gases in their cylinders are stationary in the positions in which we see them in the image, then the pressures in each cylinder can only be the same because the effect of the pressure in the cylinder from the gases has been balanced (equalised?) with the outside air pressure regardless of how much the mass “m” is for our pistons. Regardless of the temperature and initial volume of the gases inserted into the cylinders, the pressure should be the same… I THINK!!! Can someone smarter than me please put me out of my misery?!? Cheers…


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