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 Δ*U, 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:**

A-B:

Δ

*U*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*= Δ

*U*-

*W,*(+) = (+) - (-)

B-C:

Δ

*U*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)

*Q*must be positive by the first law,

*Q*= Δ

*U*-

*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. Δ

*U*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...