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The second property to be demonstrated is that all reversible engines operating between the same two reservoirs have the same efficiency. To this, stat with the two engines D and E of Figure 4.10 (a), which are operating between two common heat reservoirs at temperatures
Figure 4.10 (a) Two uncoupled engines D and E working between the same reservoirs. (b) The engines, With D working reverse.
Suppose the cycle of D is so that it operates as a refrigerator, and the two engines are coupled such that the work output of E is used to drive D, as shown in Figure 4.10(b). Since
equivalent to a spontaneous transfer of heat from the cold reservoir to the hot reservoir, a process second law does not allow. The original assumption must therefore be wrong, and it is impossible to construct an irreversible engine such that E is more efficient than the reversible engine D.
Now it is quite easy to demonstrate that the efficiencies of all reversible engines operating between the same reservoirs are equal. Suppose that D and E are reversible engines. If they are as shown in Figure 4.10(b), the efficiency of E cannot be greater than the efficiency of D, or second law would violated. If both engines are then reversed, the same reasoning implies that the efficiency of D cannot be greater than the efficiency of E. Combining these results leads to the conclusion that all reversible engines working between same two reservoirs have the same efficiency.

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Chapter 4 Solutions
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