Physical Chemistry , 1st ed.

cycle is shown in Figure 3.2. The steps of a Carnot cycle are, for an ideal

gaseous system:

1. Reversible isothermal expansion. In order for this to occur, heat must be
   absorbed from the high-temperature reservoir. We shall define this
   amount of heat as q 1 (labeled as qinin Figure 3.1) and the amount of
   work performed by the system as w 1.


2. Reversible adiabatic expansion. In this step,q0, but since it is expan-
   sion, work is done by the engine. The work is defined as w 2.


3. Reversible isothermal compression. In order for this step to be isother-
   mal, heat must leave the system. It goes into the low-temperature reser-
   voir and will be labeled q 3 (this is labeled as qoutin Figure 3.1). The
   amount of work in this step will be called w 3.


4. Reversible adiabatic compression. The system (that is, the engine) is re-
   turned to its original conditions. In this step,qis 0 again, and work is
   done on the system. This amount of work is termed w 4.
   Since the system has returned to the original conditions, by definition of
   a state function,U0 for the overall process. By the first law of thermo-
   dynamics,
   U 0 q 1 + w 1 + w 2 + q 3 + w 3 + w 4 (3.1)
   Another way of writing this is to consider the entire work performed by the
   cycle, as well as the entire heat flow of the cycle:
   wcyclew 1 + w 2 + w 3 + w 4 (3.2)
   qcycleq 1 + q 3 (3.3)
   so that
   0 qcycle+ wcycle
   qcyclewcycle (3.4)
   We now define efficiency eas the negative ratio of the work of the cycle to the
   heat that comes from the high-temperature reservoir:

e

w

q

cy

1

cle (3.5)

Efficiency is thus a measure of how much heat going into the engine has been

converted into work. The negative sign makes efficiency positive, since work

done bythe system has a negative value but heat coming intothe system has a

positive value. We can eliminate the negative sign by substituting for wcycle

from equation 3.4:

e

qc

q

y

1

cleq^1

q

+

1

q 3

1 + 

q

q

3

1

 (3.6)

Since q 1 is heat going into the system, it is positive. Since q 3 is heat going out

of the system (into the low-temperature reservoir of Figure 3.1), it is negative.

Therefore, the fraction q 3 /q 1 will be negative. Further, it can be argued that the

heat leaving the engine will never be greater than the heat entering the engine.

That would violate the first law of thermodynamics, that energy cannot be

created. Therefore the magnitude q 3 /q 1 will never be greater than 1, but it will

always be less than or (if no work is done) equal to 1. Combining all these

statements, we conclude that

The efficiency of an engine will always be between 0 and 1.

3.3 The Carnot Cycle and Efficiency 69

Step 4

Step 1

A

D

B

C

Step 2

Step 3

Volume

Pressure

Figure 3.2 A representation of the Carnot
cycle performed on a gaseous system. The steps
are: (1) Reversible isothermal expansion. (2)
Reversible adiabatic expansion. (3) Reversible
isothermal compression. (4) Reversible adiabatic
compression. The system ends up at the same
conditions it started at; the volume inside the
four-sided figure is representative of the pV
work performed by the cycle.