Equivalence of Kelvin–Planck and Clausius Statement

Violation of Kelvin–Plank Statement by Violating Clausius Statement

From Figure 1.13 (a) let us assume that a heat pump receives heat Q_L from low temperature reservoir at T_L and supplies it to high temperature sink at T_H without any external work, thus violating the Clausius statement. A larger quantity of heat (Q_H + Q_L) is supplied to heat engine (by high temperature source at T_H) which produces net work output, W_net, equal to Q_H and rejects an amount of heat, Q_L, to low temperature reservoir. The composite of two devices is shown in Figure 1.13 (b). It shows that the heat pump helps the heat Q_L to flow from low temperature reservoir to high temperature reservoir, whereas the heat engine supplies back heat Q_L from high temperature reservoir to low temperature thermal energy reservoir. The equivalent system receives heat Q_H from high temperature reservoir and produces an equivalent amount of work, as shown is Figure 1.13 (c).

Figure 1.13 Violation of Kelvin–Plank Statement by Violating Clausius Statement

From Figure 1.13 (c), it can be observed that a heat engine receives heat from single reservoir and produces an equivalent amount of work. It is therefore a perpetual motion machine II (PMM-2), which violates the Kelvin–Plank’s statement for second law of thermodynamics. Thus, violation of Clausius statement leads to violation of Kelvin Plank’s statement, and we can say, the two statements are equivalent.