Kelvi Plank and Clausius Statement Equivalence

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EQUIVALENCE OF THE KELVIN-PLANK AND CLAUSIUS STATEMENTS
 
Two statements are said to be equivalent if the truth statement implies the trust of the other, and vacation of one statement implies the violation of other. Thus, there could be two methods to show the equivalence.  

Method 1: Violation of Kelvin-Plank statement by violating Clausius statement: From Fig. (2.4.9) (a) let us assume that a heat pump receives heat Q2 from low temperature reservoir at T2 and supplies it to high temperature sink at T1 without any external work, thus violating the Clausius statement. A larger quantity of heat (Q1 + Q2) is supplied to heat engine (by high temperature source at T1)  

Which produces net work output Wnet equal to Q1 and rejects an amount of heat Q2 to low temperature reservoir.  
The composite of two devices is shown in Fig. (b). It shows that the heat pump helps the heat Q2 to flow from low temperature reservoir to high temperature reservoir, whereas, the heat engine supplier back heat Q2 from high temperature reservoir to low temperature thermal energy reservoir. The equivalent system receives heat Q1 from high temperature reservoir and produces an equivalent amount of work, as shown is Fig. (c).  

From Fig.(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 of type 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 Planks’s statement, and we can say, the two statements are equivalent.  

Method 2: Violation of Clausius statement by violating Kelvin-Plank’s statement:  
From fig. (a) let us assume that a heat engine receives heat Q1 from high temperature reservoir and convert it into work rejecting any heat to sink, thus violating Kelvin Plank’s statement.  
Refrigerator’s receives heat Q2 from low temperature reservoir and supplies an...
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