Thermodynamics and Chemistry

CHAPTER 4 THE SECOND LAW

4.3 CONCEPTSDEVELOPED WITHCARNOTENGINES 111

„ ƒ‚ ...

(a)

4

Th

1

3

Tc

5

Th

1

4

Tc

(b)

1

Th

1

Tc

„ ƒ‚ ...

(c)

3

Th

1

2

Tc

4

Th

1

3

Tc

(d)

1

Th

1

Tc

Figure 4.7 (a) A Carnot engine of efficiencyD1=4combined with a Carnot engine

of efficiencyD1=5run in reverse.

(b) The resulting impossible Clausius device.

(c) A Carnot engine of efficiencyD1=3combined with the Carnot engine of effi-

ciencyD1=4run in reverse.

(d) The resulting impossible Clausius device.

By substituting forwfrom Eq.4.3.1, we obtain

D 1 C

qc

qh

(4.3.3)

(Carnot engine)

Becauseqcis negative,qhis positive, andqcis smaller in magnitude thanqh, the efficiency
is less than one. The example shown in Fig.4.5(a) is a Carnot engine withD1=4.
We will be able to reach an important conclusion regarding efficiency by considering a
Carnot engine operating between the temperaturesThandTc, combined with a Carnot heat
pump operating between the same two temperatures. The combination is a supersystem, and
one cycle of the engine and heat pump is one cycle of the supersystem. We adjust the cycles
of the engine and heat pump to produce zero net work for one cycle of the supersystem.
Could the efficiency of the Carnot engine be different from the efficiency the heat pump
would have when run in reverse as a Carnot engine? If so, either the supersystem is an
impossible Clausius device as shown in Fig.4.7(b), or the supersystem operated in reverse
(with the engine and heat pump switching roles) is an impossible Clausius device as shown
in Fig.4.7(d). We conclude thatall Carnot engines operating between the same two tem-
peratures have the same efficiency.

This is a good place to pause and think about the meaning of this statement in light of

the fact that the steps of a Carnot engine, being reversible changes, cannot take place in

a real system (Sec.3.2). How can an engine operate that is not real? The statement is

an example of a common kind of thermodynamic shorthand. To express the same idea

more accurately, one could say that all heat engines (real systems) operating between

the same two temperatures have the samelimitingefficiency, where the limit is the

reversible limit approached as the steps of the cycle are carried out more and more

slowly. You should interpret any statement involving a reversible process in a similar

fashion: a reversible process is an idealizedlimitingprocess that can be approached

but never quite reached by a real system.