Clausius inequality

Consider a cyclic process in which a system exchanges heat $\delta Q$ with its surroundings (see heat ). Let $T_{\mathrm{b}}$ denote the boundary temperature at which each heat element crosses the system boundary (e.g., the temperature of a thermal reservoir supplying that heat), understood on the absolute temperature scale . The Clausius inequality states that

Equality holds if and only if the cycle is reversible ; strict inequality indicates a genuinely irreversible cycle.

A useful special case is a cycle exchanging finite heats $Q_i$ with reservoirs at fixed temperatures $T_i$:

Physical interpretation

The inequality expresses the content of the second law in a form that directly compares heat transfers at different temperatures: heat exchanged at lower temperature carries a larger “entropy weight” $\delta Q/T$. The strictness of the inequality measures how much irreversibility is present in the cycle.

Key relations

• Entropy as a state function. For a reversible process between equilibrium states $A$ and $B$, the integral of $\delta Q/T_{\mathrm{b}}$ is path-independent. This motivates defining the entropy change by

  $$S(B)-S(A)=\int_A^B \frac{\delta Q_{\mathrm{rev}}}{T},$$

  where the subscript “rev” emphasizes evaluation along a reversible path (so the boundary and system temperatures coincide).

• Inequality for general processes. For any process between equilibrium states,

  $$S(B)-S(A)\ge \int_A^B \frac{\delta Q}{T_{\mathrm{b}}},$$

  with equality only in the reversible limit.

• Entropy production form. Writing

  $$\Delta S - \int \frac{\delta Q}{T_{\mathrm{b}}} = S_{\mathrm{gen}},$$

  the Clausius inequality is equivalent to $S_{\mathrm{gen}}\ge 0$.

• Adiabatic implication. If the system is thermally insulated by an adiabatic wall so that $\delta Q=0$, then $\Delta S\ge 0$ for a closed system, with equality only for a reversible adiabatic (isentropic) change.