Thermodynamics and Chemistry

CHAPTER 4 THE SECOND LAW

4.3 CONCEPTSDEVELOPED WITHCARNOTENGINES 110

„ ƒ‚ ...

(a)

3

Th

3

Tc

4

Th

1

3

Tc

(b)

1

Th

1

„ ƒ‚ ...

(c)

1

Th

1

4

Th

1

3

Tc

(d)

3

Th

3

Tc

Figure 4.6 (a) A Clausius device combined with the Carnot engine of Fig.4.5(a).

(b) The resulting impossible Kelvin–Planck engine.

(c) A Kelvin–Planck engine combined with the Carnot heat pump of Fig.4.5(b).

(d) The resulting impossible Clausius device.

device as shown, and our provisional assumption that the Clausius statement is incorrect
must be wrong. In conclusion, if the Kelvin–Planck statement is correct, then the Clausius
statement must also be correct.
We can apply a similar line of reasoning to the heat engine that the Kelvin–Planck
statement claims is impossible (a “Kelvin–Planck engine”) by seeing what happens if we
assume this engine is actually possible. We combine a Kelvin–Planck engine with a Carnot
heat pump, and make the work performed on the Carnot heat pump in one cycle equal to
the work performed by the Kelvin–Planck engine in one cycle, as shown in Fig.4.6(c). One
cycle of the combined system, shown in Fig.4.6(d), shows the system to be a device that
the Clausius statement says is impossible. We conclude that if the Clausius statement is
correct, then the Kelvin–Planck statement must also be correct.
These conclusions complete the proof that the Clausius and Kelvin–Planck statements
are equivalent: the truth of one implies the truth of the other. We may take either statement
as the fundamental physical principle of the second law, and use it as the starting point for
deriving the mathematical statement of the second law. The derivation will be taken up in
Sec.4.4.

4.3.3 The efficiency of a Carnot engine

Integrating the first-law equation dUD∂qC∂wover one cycle of a Carnot engine, we
obtain

0 DqhCqcCw (4.3.1)

(one cycle of a Carnot engine)

Theefficiencyof a heat engine is defined as the fraction of the heat inputqhthat is returned
as net work done on the surroundings:



def

D

w

qh

(4.3.2)