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31 December 2011

The first first law assignment -- qualitative justifications of signs

The drawing shows a PV diagram in which a gas expands at constant pressure from A to B, and then goes from B to C at constant volume.  Determine the signs of ΔU, Q, and W for each of the two processes.  Justify your answers.

This is the first PV diagram question which I assign in my honors or AP course.  We have discussed the definitions of the variables in the first law, and how to determine the value of each variable from the PV diagram.*  

*Including the fact that the value of Q cannot be determined directly from the diagram without using the first law.

The solution, in the language and logical order that I prefer:


ΔU is positive, because the product of P and V is larger at point B than at point A.  
W is negative, because the volume increased.
Q must be positive by the first law, Q = ΔU - W, (+) = (+) - (-)


ΔU is positive, because the product of P and V is larger at point C than at point B.
W is zero, because the volume did not change (or because there is no area under the curve from B to C)
must be positive by the first law, Q = ΔU - W, (+) = (+) - (0)

Note that I'm not yet asking for any quantitative answers.  That's too much for the first problem set.  I try to get my class totally comfortable identifying facts, assigning signs, and using the correct vocabulary for each term before I ask for numerical answers.  

Also, look how straightforward the answers.  ΔU is (3/2)PV; W is the area under the curve; and Q is determined from the first law.  It takes a lot of effort on my part to get students disciplined enough to used this approach.  They invariably want to, somehow, somewhere, talk about "molecules moving around:"  "Q is positive because when the pressure increases, the molecules have to move around a lot faster, leading to more heat."  Such a statement is worse than nonsense.  PV diagrams refer to macroscopic systems, and must be interpreted with reference to relevant equations and facts, only.  

This year, anticipating the difficulty of convincing students to use a disciplined, macroscopic approach to the first law of thermodynamics, I promised that the penalty for any reference to "molecules moving around" in a first law justification would earn double points off.  And sure enough, I had a student who lost double credit on this very problem.  But only one this year...


  1. Not being a physics teacher, I found this question incomprehensible, because Q, U, and W are undefined variables. P and V were easy, because you showed a diagram with "pressure" and "volume", which were concepts I already had. Q, U, and W had no immediate referents for me

    I realize that your students have just been given those variables, but perhaps some confusion comes from talking about the quantities as if they were permanently tied to particular variable names.

    Would it help the students learn if you used the names (heat, work, energy) along with, or in place of, the letters in the question? It certainly would have helped me understand the question.

  2. The variables were defined that day in class. You are correct that a test question would redefine. Here is the post that is crystal clear about the meanings of variables in the first law of thermodynamics:


  3. Greg,
    I'm not sure why you wouldn't want students to couple their understandings of PV diagrams with a model of what is happening with the gas molecules inside the piston.

    The explanation I'd like to see is something like:
    From A-B
    the piston expanded at constant pressure. This means the piston clearly did work on the surroundings (W+). We also know that to maintain this constant pressure at a higher volume, the internal energy of the system must increase (Delta U > 0). Since energy must be conserved, we have to add energy to this system from the surroundings, so Q must be positive.

    If students were able to say that in order to maintain the same pressure at a larger volume the molecules must be moving faster, and therefore the internal energy of the gas must be larger, I'd consider it a big win. If they were to say that the momentum transfer must larger since the surface area of the walls are now larger in order to maintain a constant pressure, I'd probably pass out.

    With students starting out with the first law, I find they have lots of trouble understanding the meaning of the symbols they're dealing with, so I'd encourage them to describe those symbols in words as much as possible.

  4. Mr. Burk, I certainly like your explanation in the second paragraph. Brand-new thermodynamics students, I think, don't see the obviousness of the statement that Delta U is positive -- I'd prefer them to cite U = 3/2 PV as justification. But fundamentally, your expected explanation doesn't differ much from mine.

    Now, given that clear explanation, I want to THEN get students to relate the internal energy U to the average kinetic energy of the gas molecules. Once everyone is clear that U has increased, and that U (not Q or W) is related to average molecular speed, great.

    The reason I'm avoiding microscopic issues is that the vast majority of the time, EVERYTHING from novice thermodynamics students becomes "molecules moving faster." Heat added? Molecules move faster. Work done on or by the gas? Molecules move faster. Pressure increases? Molecules move faster.

    My own approach, which is successful but of course not the only way to do things, is as you say in your last statement: get students to understand the macroscopic meaning of the symbols by describing in words as much as possible. Then, and only then, do I go into the specifics of kinetic theory.

  5. But how do they get the equation U=3/2 PV? Isn't the only way to see that to derive it from a microscopic model?

  6. I initially state U = 3/2 nRT as fact, without derivation. Well after everyone can do basic first law problems, I can show the source of pressure, internal energy, etc. from the kinetic model.

    I spent many years deriving U=3/2 PV to blank looks, and getting students attempt to reproduce the derivation rather than use the result on straightforward problems. I've been far more successful working macroscopic-to-microscopic rather than the other way around.

    (Sure, as an experienced physicist, *I* understand U = 3/2 PV as merely a derived result from the assumptions of kinetic theory. But my students aren't that sophisticated -- yet.)

  7. Some food for thought. The following was posted by Matt Greenwolfe to the Modeling listserv back in 2008:

    "I have a thermodynamics unit that I really like, and have thought of creating a modeling thermo course for it. It uses the VPython programming language, developed by Bruce Sherwood and Ruth Chabay from NC State. VPython simplifies scientific programming so that students just have to program the physics and VPython automatically creates a 3-D display of the result. So I have students write a program for balls bouncing around inside a box and making elastic collisions with the walls and each other. Then they can perform computational experiments to find Pressure (calculated from the impulse delivered to the walls) equals (2/3)*(number of balls)*(average KE per ball)/Volume. The computational experiments run exactly like a modeling paradigm lab. Then we do real laboratory experiments to find P = k_B*T*N/Vol, which would be the same result as the computational experiment so long as (2/3)*(average KE) = k_B*T. So they end up telling me what k_B means.

    "This all takes awhile, but I find that before using VPython I was only successful at getting the top third or so of my class to really construct a model of an ideal gas and apply it to solve problems. The rest were doing too much of plugging into equations. Now I can say with certainty that 100% of the class develops a model of an ideal gas and can relate the model to the average quantities like temperature, pressure, etc. This beginning makes all the difference when moving on to more advanced applications like heat engines and entropy."