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:

A-B:  

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, (+) = (+) - (-)

B-C:

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)

Q 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...

Get the vocabulary right in the first law of thermodynamics!

Q and U from a They Might Be
Giants wiki

W represents work done on a gas. W cannot increase or decrease. W can be positive or negative; negative W means work is done BY a gas.

Q represents heat added to a gas. Q cannot increase or decrease. Q can be positive or negative; negative Q means heat is REMOVED from a gas.

U represents internal energy of a gas. U can increase or decrease.

ΔU represents the change in internal energy. ΔU cannot increase or decrease.  ΔU can be positive or negative; negative ΔU means the internal energy DECREASES.
This was all posted to a class folder yesterday, after I graded a problem about a PV diagram and the first law of thermodynamics.  The 4 students who used illegal phrasology* such as "W decreases" lost credit.  I thought it was worth a classwide reminder with our exam coming up today...

* penalty: 5 yards and loss of down

First Law of Thermodynamics and the Sign of Each Term

The first law of thermodynamics is a statement of energy conservation.  In equation form, it states that

ΔU = Q+W

where each variable has the following meaning:

ΔU is the change in the gas's internal energy
Q is the heat ADDED to the gas
W is the work done ON the gas.

A typical test question will show a pressure-vs-volume graph, and expect the student to use the graph to determine a value, or at least a sign, for each of the quantities.  Therefore, it's worth memorizing the method of determining each variable from a PV diagram:

ΔU: Read the axes of the graph, and ΔU = (3/2)PV.
Q: CANNOT BE FOUND FROM THE GRAPH!
W: Look at the area under the graph.

The quiz/poll thingie I had up last week is shown to the right.  This question only asks for the signs of the three variables. 

Based on the definition ΔU = (3/2)PV, ΔU is positive when the gas's temperature increases, and negative when the temperature drops.  (Why?  Look at the equation, and remember that PV = nRT.) 

The sign of W can be trickier.  Work is done ON the gas when the gas's volume DECREASES. Think of a piston compressing the gas... a force is applied ON the gas over some distance.  So we consider W to be a positive quantity when a gas compresses, and a negative quantity when a gas expands. 

And finaly, the sign of Q can only be determined from the first law equation.  One must find ΔU and W and plug in.

And now, the answer to this week's quiz:

Start with ΔU.  This process starts and ends at the same spot on the PV diagram -- the product of PV does not change. Therefore, ΔU is zero -- there is no net change in internal energy.


Now look at W.  Consider each sub-process individually.  In process A-B, the gas expands, so W is negative.  In process B-C, there is no volume change, so no work is done on or by the gas.  And in process C-A, the gas's volume decreases, so W is positive.  But what is the sign of W for the overall process A-B-C-A?  There is more area under the graph in the expansion from A-B than in the compression from C-A.  So, the net value of W is negative -- net work is done BY the gas.
 
And finally, use the first law to find the sign of Q.  ΔU = Q+W.  Solving for Q, we find that Q = ΔU - W.  Since ΔU was zero and W was negative, then algebraically, Q must be positive.  This means that, in net, heat is added to the gas.
 
As a side note -- this process represents a cycle of a heat engine.  Despite what happens in each individual process, IN NET, some amount of added heat is converted into some amount of work done by a gas.  That's what is meant by a heat engine.
 
More polls soon...
 
GCJ