Page 101 - Physical Chemistry

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Chapter 3 We shall run the superengine at such a rate that it withdraws heat from the hot
The Second Law of Thermodynamics reservoir at the same rate that the reversible heat pump deposits heat into this reser-
voir. Thus, suppose 1 cycle of the superengine absorbs 1.3 times as much heat from
the hot reservoir as 1 cycle of the reversible heat pump deposits into the hot reservoir.
The superengine would then complete 10 cycles in the time that the heat pump com-
pletes 13 cycles. After each 10 cycles of the superengine, both devices are back in their
original states, so the combined device is cyclic.
Since the magnitude of the heat exchange with the hot reservoir is the same for
the two engines, and since the superengine is by assumption more efficient than the
reversible engine, the equations of (3.5) show that the superengine will deliver more
work output than the work put into the reversible heat pump. We can therefore use part
of the mechanical work output of the superengine to supply all the work needed to run
the reversible heat pump and still have some net work output from the superengine left
over. This work output must by the first law have come from some input of energy to
the system of reversible heat pump plus superengine. Since there is no net absorption
or emission of heat to the hot reservoir, this energy input must have come from a net
absorption of heat from the cold reservoir. The net result is an absorption of heat from
the cold reservoir and its complete conversion to work by a cyclic process.
However, this cyclic process violates the Kelvin–Planck statement of the second
law of thermodynamics (Sec. 3.1) and is therefore impossible. We were led to this im-
possible conclusion by our initial assumption of the existence of a superengine with
e super e . We therefore conclude that this assumption is false. We have proved that
rev
e1any engine2 e1a reversible engine2 (3.6)
for heat engines that operate between the same two temperatures. (To increase the effi-
ciency of a real engine, one can reduce the amount of irreversibility by, for example,
using lubrication to reduce friction.)
Now consider two reversible heat engines, A and B, that work between the same
two temperatures with efficiencies e rev,A and e rev,B . If we replace the superengine in the
above reasoning with engine A running forward and use engine B running backward
as the heat pump, the same reasoning that led to (3.6) gives e rev,A e rev,B . If we now
interchange A and B, running B forward and A backward, the same reasoning gives
e rev,B e rev,A . These two relations can hold only if e rev,A e rev,B .
We have shown that (1) all reversible heat engines operating between the same
two temperatures have the same efficiency e , and (2) this reversible efficiency is the
rev
maximum possible for any heat engine that operates between these temperatures, so
e irrev e rev (3.7)
These conclusions are independent of the nature of the working substance used in the
heat engines and of the kind of work, holding also for non-P-V work. The only as-
sumption made was the validity of the second law of thermodynamics.

Calculation of e rev
Since the efficiency of any reversible engine working between the temperatures t and
H
t is the same, this efficiency e rev can depend only on t and t :
H
C
C
e rev f1t , t 2 (3.8)
C
H
The function f depends on the temperature scale used. We now find f for the ideal-
gas temperature scale, taking t T. Since e rev is independent of the nature of the
working substance, we can use any working substance to find f. We know the most
about a perfect gas, so we choose this as the working substance.
Consider first the nature of the cycle we used to derive (3.8). The first step in-
volves absorption of heat q from a reservoir whose temperature remains at T . Since
H
H