Page 100 - Physical Chemistry

lev38627_ch03.qxd 2/29/08 3:12 PM Page 81

81
Since q is negative and q is positive, the efficiency is less than 1. Section 3.2
H
C
To further simplify the analysis, we assume that the heat q is absorbed from a hot Heat Engines
H
reservoir and that q is emitted to a cold reservoir, each reservoir being large enough
C
to ensure that its temperature is unchanged by interaction with the engine. Figure 3.2
Hot reservoir at t H
is a schematic diagram of the heat engine.
Since our analysis at this point will not require specification of the temperature q H
scale, instead of denoting temperatures with the symbol T (which indicates use of the
ideal-gas scale; Sec. 1.5), we shall use t (tau). We call the temperatures of the hot and
cold reservoirs t and t . The t scale might be the ideal-gas scale, or it might be based Heat w
C
H
on the expansion of liquid mercury, or it might be some other scale. The only restric- Engine
tion we set is that the t scale always give readings such that the temperature of the hot q C
reservoir is greater than that of the cold reservoir: t t . The motivation for leav-
C
H
ing the temperature scale unspecified will become clear in Sec. 3.6.
Cold reservoir at t C
Carnot’s Principle
We now use the second law to prove Carnot’s principle: No heat engine can be more Figure 3.2
efficient than a reversible heat engine when both engines work between the same pair
of temperatures t and t . Equivalently, the maximum amount of work from a given A heat engine operating between
C
H
two temperatures. The heat and
supply of heat is obtained with a reversible engine. work quantities are for one cycle.
To prove Carnot’s principle, we assume it to be false and show that this assump- The widths of the arrows indicate
tion leads to a violation of the second law. Thus, let there exist a superengine whose that q w q .
C
H
efficiency e super exceeds the efficiency e rev of some reversible engine working between
the same two temperatures as the superengine:
e super 7 e rev (3.4)
where, from (3.1),
w super w rev
e super , e rev (3.5)
q H,super q H,rev

Let us run the reversible engine in reverse, doing positive work w rev on it, thereby
causing it to absorb heat q C,rev from the cold reservoir and emit heat q H,rev to the hot
reservoir, where these quantities are for one cycle. It thereby functions as a heat pump,
or refrigerator. Because this engine is reversible, the magnitudes of the two heats and
the work are the same for a cycle of operation as a heat pump as for a cycle of opera-
tion as a heat engine, except that all signs are changed. We couple the reversible heat
pump with the superengine, so that the two systems use the same pair of reservoirs
(Fig. 3.3).

Hot reservoir at t H

Reversible
heat engine Superengine
acting as a
heat pump
Net
work
output
Figure 3.3

Cold reservoir at t C Reversible heat pump coupled
with a superengine.