Thermodynamics and Chemistry

CHAPTER 4 THE SECOND LAW

4.3 CONCEPTSDEVELOPED WITHCARNOTENGINES 108

(a)

4

Th

1

3

Tc

(b)

4

Th

1

3

Tc

Figure 4.5 (a) One cycle of a Carnot engine that does work on the surroundings.

(b) The same system run in reverse as a Carnot heat pump.

Figures4.5–4.7use the following symbols: A square box represents a system (a

Carnot engine or Carnot heat pump). Vertical arrows indicate heat and horizontal

arrows indicate work; each arrow shows the direction of energy transfer into or out of

the system. The number next to each arrow is an absolute value ofq/J orw/J in the

cycle. For example, (a) shows 4 joules of heat transferred to the system from the hot

reservoir, 3 joules of heat transferred from the system to the cold reservoir, and 1 joule

of work done by the system on the surroundings.

and pushes the piston from the closed end toward the open end of the cylinder; (2) the

supply valve closes and the steam expands in the cylinder until its pressure decreases

to atmospheric pressure; (3) an exhaust valve opens to release the steam either to the

atmosphere or to a condenser; (4) the piston returns to its initial position, driven either

by an external force or by suction created by steam condensation.

The energy transfers involved in one cycle of a Carnot engine are shown schematically
in Fig.4.5(a). When the cycle is reversed, as shown in Fig.4.5(b), the device is called a
Carnot heat pump. In each cycle of a Carnot heat pump,qhis negative andqcis positive.
Since each step of a Carnot engine or Carnot heat pump is a reversible process, neither
device is an impossible device.

4.3.2 The equivalence of the Clausius and Kelvin–Planck statements

We can use the logical tool ofreductio ad absurdumto prove the equivalence of the Clausius
and Kelvin–Planck statements of the second law.
Let us assume for the moment that the Clausius statement is incorrect, and that the de-
vice the Clausius statement claims is impossible (a “Clausius device”) is actually possible.
If the Clausius device is possible, then we can combine one of these devices with a Carnot
engine as shown in Fig.4.6(a) on page 110. We adjust the cycles of the Clausius device and
Carnot engine to transfer equal quantities of heat from and to the cold reservoir. The com-
bination of the Clausius device and Carnot engine is a system. When the Clausius device
and Carnot engine each performs one cycle, the system has performed one cycle as shown
in Fig.4.6(b). There has been a transfer of heat into the system and the performance of an
equal quantity of work on the surroundings, with no other net change. This system is a heat
engine that according to the Kelvin–Planck statement is impossible.
Thus, if the Kelvin–Planck statement is correct, it is impossible to operate the Clausius