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Clausius theorem and entropy

For reversible cycles, the cyclic integral ∮ δQ_rev/T vanishes, implying the existence of entropy as a state function with dS = δQ_rev/T.

Clausius theorem and entropy

Statement

For a system undergoing a reversible cyclic process through equilibrium states, the Clausius integral satisfies

\oint \frac{\delta Q_{\mathrm{rev}}}{T} = 0.

Consequently, there exists a state function $S$ (the thermodynamic entropy ) such that for any two equilibrium states $A,B$ and any reversible path $\gamma$ from $A$ to $B$ ,

S(B)-S(A) = \int_{\gamma} \frac{\delta Q_{\mathrm{rev}}}{T},

so the integral is path independent among reversible paths and defines $S$ up to an additive constant.

More generally, for an arbitrary (possibly irreversible) cyclic process,

\oint \frac{\delta Q}{T} \le 0,

which is the Clausius inequality .

Key hypotheses and conclusions

Hypotheses

The process is quasi-static/reversible so that $T$ is well-defined along the path via temperature .
The system passes through equilibrium states (so thermodynamic state variables are meaningful).
The second law of thermodynamics holds.

Conclusions

The differential form $\delta Q_{\mathrm{rev}}/T$ is an exact differential: $dS = \delta Q_{\mathrm{rev}}/T$ along reversible changes.
Entropy differences are intrinsic to the endpoints (state function property).
Irreversibility manifests as strict inequality in the cyclic Clausius integral and, equivalently, nonnegative entropy production.

Cross-links to definitions

Entropy: thermodynamic entropy .
Temperature (integrating factor): temperature .
Second-law context: second law and Kelvin–Planck–Clausius equivalence .
Mathematical structure (exactness/integrating factors): exact differential criterion and integrating factor lemma .
Engine viewpoint connection: Carnot theorem .

Proof idea / significance

Idea (from Carnot cycles). Using Carnot’s theorem , one shows that for reversible processes between fixed endpoint states, the quantity $\int \delta Q_{\mathrm{rev}}/T$ is independent of the chosen reversible path. Path independence implies the existence of a state function whose differential along reversible paths is $\delta Q_{\mathrm{rev}}/T$ . Extending to arbitrary cycles yields the inequality form.

Significance. Clausius’ theorem is the precise bridge from the operational second law (impossibility statements) to a new state function, entropy, which then organizes equilibrium thermodynamics, stability criteria, and the construction of thermodynamic potentials.