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# Equivalence of Kelvin-Planck and Clausius Statements

The Clausius and Kelvin-Planck statements of the second law are entirely equivalent. This equivalence can be demonstrated by showing that the violation of either statement can result in violation of the other one.

Equivalence of Kelvin-Planck and Clausius Statements

The Clausius and Kelvin-Planck statements of the second law are entirely equivalent. This equivalence can be demonstrated by showing that the violation of either statement can result in violation of the other one.

Referring to Figure 4.4(a) the device marked Clausius violator is pumping Q1 amount of heat from a low temperature reservoir at T1 to a high temperature reservoir at T2 without the aid of any external agency. This is an impossible arrangement.

If such an arrangement is possible it would also violate Kelvin-Planck statement. Let a heat engine operating between the same reservoirs at T2 and T1 take in Q2 as heat input at T2. It converts a part of this heat into work and rejects heat Q3to the sink at T1. Since the Clausius violator is rejecting the same quantity Q2at T2, it can be supplied directly into the heat engine so that the reservoir at T2 can be eliminated. This combination as shown in Figure 4.4 (b) is producing continuous work with a single reservoir at T1. Hence it violates the Kelvin-Planck statement.

Figure 4.4 Illustration of the equivalence of Clausius and Kelvin-Planck’s   statemen

Referring to Figure 4.5 a Kelvin-planck violator is converting all heat QH taken from the reservoir at TH into work. If such an impossible heat engine is assumed to exist it will violate the Clausius statement. Consider a refrigerator pumping QL heat from the low temperature reservoir at TL to the reservoir at higher temperature TH. Combined with the Kelvin-Planck violator, the arrangement is pumping QL heat from TL to TH, without any external agency. Hence it violate the Clausius statement.

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