Equivalence of Two Statement

 

The Clausius statement and the Kelvin-Planck statement of the second law of thermodynamics are equivalent in their consequences. Any device that violates the Kelvin-Planck statement also violates the Clausius statement.

Consider two devices A and B working between a high-temperature reservoir and a low-temperature reservoir. Device A is assumed to transfer Heat (QA,H) to work ( WA, net) and have 100 percent efficiency. That is, 

      WA, net = QA,H

Device A violates the Kelvin-Planck statement.

Device B is a heat pump which receives heat (QB,L) from the low-temperature reservoir, receives work (WA, net) from device A and rejects the total energy of WA, net and QB,L to the high-temperature reservoir. Heat rejected by device C, which is the combination of devices A and B, to the high-temperature reservoir equals

      (QB,L + WA,net) - QA,H 

            = (QB,L + WA, net) - WA, net 

            = QB,L

Device C receives heat (QB,L) from the low-temperature reservoir and rejects the same amount of heat (QB,L) to the high-temperature reservoir. It violates the Clausius statement.

 

It can also be shown in a similar way that if a device violates the Clausius statement, it also violates the Kelvin-Planck statement. Hence, the Clausius statement and the Kelvin-Planck statement are two equivalent expressions of the second law of thermodynamics.