Equivalence of Kelvin-Planck and Clausius Statements

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 Q_H taken from the reservoir at T_H into work. If such an impossible heat engine is assumed to exist it will violate the Clausius statement. Consider a refrigerator pumping Q_L heat from the low temperature reservoir at T_L to the reservoir at higher temperature T_H. Combined with the Kelvin-Planck violator, the arrangement is pumping Q_L heat from T_L to T_H, without any external agency. Hence it violate the Clausius statement.