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## Clausius theorem
## Additional recommended knowledgeThe equality holds in the reversible case and the '<' is in the irreversible case. The reversible case is used to introduce the function state entropy. This is because in cyclic process the variation of a state function is zero. First we have to prove the lemma: "any reversible process can be replaced by a combination of reversible isothermal and adiabatic processes".
Consider a reversible process A corollary of this theorem is that any reversible cycle can be replaced by a series of Carnot cycles. Suppose each of these Carnot cycles absorbs heat T and rejects heat _{1}^{i}dQ at _{2}^{i}T.
Then, for each of these engines, we have _{2}^{i}dQ (i.e. _{1}^{i}/dQ_{2}^{i} = −T_{1}^{i}/T_{2}^{i}dQ).
The negative sign is included as the heat lost from the body has a negative value.
Summing over a large number of these cycles, we have, in the limit,
_{1}^{i}/T_{1}^{i} + dQ_{2}^{i}/T_{2}^{i} = 0This means that the quantity Further, using Carnot's principle, for an irreversible cycle, the efficiency is less than that for the Carnot cycle, so that As the heat is transferred out of the system in the second process, we have, assuming the normal conventions for heat transfer, So that, in the limit we have, The above inequality is called the |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Clausius_theorem". A list of authors is available in Wikipedia. |