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Establish the equivalence of Clausius and Kelvin-Planck Statements for second law of thermodynamics.
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Equivalency of Clausius and Kelvin-Planck Statements. Though the Clausius and Kelvin - Planck statements, of 2nd law of thermodynamics appear to be different but it can be shown they are really equivalent statements in all respect. In order to establish the fact of equivalency of these statements we can prove by showing that violation of one statement also leads to the violation of the other statement and vice versa.

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(i) To prove that the violation of Kelvin - Planck statement results into violation of Clausius Statement

Refer Fig. A Consider an engine (PMM-2) which violates the Kelvin-Planck statement by receiving, heat from a single high temperature heat reservoir at T1 and converts completely, into work W1 =Q1

Now introduce a heat pump which is driven by engine. On the expense of work W1 =Q1 supplied by the engine, it extracts heat Q2 from low temperature heat reservoir and heat (Q1 + Q2) to high temperature heat reservoir. Combine the engine and heat pump into one system working between two heat reservoirs as shown in Fig. B.

We found that the sole effect of combined system is to transfer heat Q2 from low temperature heat reservoir, T2 to higher temperature heat reservoir without any external work thus violating the Clausius statement also.

(ii)To prove that violation of Clausius statement results into violation of Kelvin-Planck statement :

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Consider a heat pump which violates the Clausius statement.The heat pump receives heat Q2 from low temperature heat reservoir and it rejects the same heat to high temperature reservoir without any external work.

W = 0

Introduce an engine which receives heat Q1 from high temperature reservoir, rejects heat Q2 and does the work W= Q1 —Q2.

Combine the engine and heat pump into one system working between two heat reservoirs.

We found that the sole effect of combined system is to produce work W= Q1 - Q2 By exchanging heat with only one heat reservoir thus violating Kelvin Plank statement.

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