Clausius Inequality

\[\oint \frac{dQ}{T} \le 0\] This relationship gives us a way to determine if a cyclical process is reversible or irreversible. The cyclic integral value is equal to zero for reversible cycle; and less than zero for irreversible cycle.

4.3 Entropy

The Clausius inequality forms the basis for the definition of a new property called entropy. Entropy is a state function. \[dS = \frac{dQ_{\text{rev}}}{T}\] The entropy change of a system during a process can be calculated: \[\Delta S = S_2- S_1 = \int_1^2 \frac{dQ_{\text{rev}}}{T}\] To perform this integral, one needs to know the relation between \(Q\) and \(T\) during the process.

Note that the cyclic integral of \(\oint dQ / T\) will give us the entropy change only if the integration is carried out along an internally reversible path between two states. For irreversible processes, we may imagine a reversible process between the two states (initial and final) and calculate the entropy change (since entropy is a property).

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