Guide 18-3. Heat Engines and the Carnot Cycle, revised

In section 18.6-8 of your text, there's an inconsistency in the way the author treats the sign convention for the quantity Q. As you know, Q is taken to be positive when heat goes into a system. In the textbook's discussion of heat engines, Qh represents heat added to the system from the hot reservoir, and Qc represents heat expelled from the system to the cold reservoir. If one uses the standard convention, then Qc would have to represent a negative number. Yet the author says to treat Qc as a magnitude, hence, positive. Shame on the author for changing his sign convention halfway through the chapter. If he wanted to deal in magnitudes only, then he should have written the equations using Version 1 below. Version 2 is also consistent with the prior sign convention for Q. We use the same equation numbers below that the textbook uses starting on page 601. In Version 1, note the use of the absolute value signs around Qc. In Version 2, the absolute value signs are dropped in favor of changing the sign in front of Qc.

Equation Number

Inconsistent
textbook
equation

Consistent with
sign convention,
Version 1		 Consistent with
sign convention,
Version 2
18-10 W = Qh - Qc Wtot = Qh - |Qc| Wtot = Qh + Qc
18-12 e = 1 - Qc/Qh e = 1 - |Qc|/Qh e = 1 + Qc/Qh
middle of p.603 Qc/Qh = Tc/Th |Qc|/Qh = Tc/Th -Qc/Qh = Tc/Th

Note also that we changed the generic W symbol to Wtot. Wtot represents the net work done by the system (the working fluid of the heat engine) on the environment in one complete cycle. During part of the cycle, the system will do work on the environment while during another part of the cycle the environment will do work on the system. Since the purpose of a heat engine is to do net work on the environment, the work done by the system on the environment will be greater than the work done by the environment on the system; hence, the total work will be positive.

Equation 18.13, e = 1 - Tc/Th, is fine as it stands, because it doesn't include Q. The two temperatures, being expressed in kelvins, must be greater than 0. Efficiency is always a positive quantity.

For comparison, the equation for efficiency given on the AP exam is: e = |W/Qh|. Note the use of the absolute value symbol so that there will be no confusion about signs. So in this case you don't have to worry about signs, but you do have to realize that W = Wtot.


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