Principle II:

All reversible heat engines operating between the two given thermal reservoirs have the same efficiency. The efficiency of reversible heat engine does not depend on the working fluid, it depends only on the temperature of the reservoirs between which it operates.

To prove the proposition, let us assume that the efficiency of the reversible engine is greater than the efficiency of the reversible engine .

Figure 19.3

Refer to Figure 19.3. The engine absorbs energy as heat from the constant temperature thermal reservoir at , does work and rejects energy as heat to the reservoir at . The engine absorbs energy as heat from the reservoir at , does work and rejects energy as heat to the reservoir at . Then and

(19.4)

and

(19.5)

By assumption,

Then,

(19.6)

Therefore,

Since is a reversible engine, it can be made to execute the cycle in the reversed order. That is, when work is performed on the device, it absorbs energy as heat, from the reservoir at and rejects energy as heat to the reservoir at . Since, , can be run as a heat pump utilizing part of the work done by . The combination of the two devices is also shown in the figure.

The net work done by the device is given by

(19.7)

• The resulting device absorbs energy as heat from the reservoir at .
  
  
• Does not require any interaction with the second reservoir.
  
  
• Delivers an equivalent amount of work.

This is in violation of the Kelvin-Planck statement of the second law of thermodynamics. Hence the assumption that , is incorrect. Therefore,

(19.8)

Nov let us assume that the reversible engine is more efficient then the reversible engine . Then the reversible engine can be run as a heat pump, utilizing the part of the work done by . By following the similar argument as the earlier case, we can arrive at the result that,

(19.9)

Hence, it can be concluded that

(19.10)

Stated in works: All reversible engines operating between the two given thermal reservoirs have the same efficiency.