Isentropic processes are reversible adiabatic processes

Let $S$ be thermodynamic entropy and $T$ the temperature .

Statement

For any reversible process between equilibrium states,

Consequently:

• If the process is adiabatic ($\delta Q_{\mathrm{rev}}=0$ along the path), then it is isentropic ($dS=0$, hence $\Delta S=0$).
• Conversely, if a reversible process is isentropic ($dS=0$), then $\delta Q_{\mathrm{rev}}=0$ and the process is adiabatic.

More generally, for an adiabatic process not assumed reversible,

with equality if and only if the adiabatic process is reversible.

Key hypotheses

• The system passes through equilibrium states, so $S$ and $T$ are well-defined along the path.
• The process is reversible in the standard thermodynamic sense (no dissipation).
• “Adiabatic” means no heat exchange: $\delta Q=0$.

Key conclusions

• In equilibrium thermodynamics, isentropic is equivalent to reversible adiabatic.
• Any irreversibility in an adiabatic transformation produces entropy: $\Delta S>0$.

Proof idea / significance

This follows immediately from the Clausius characterization of entropy (see Clausius' theorem on entropy ) together with the Clausius inequality . In applications, “isentropic” provides an ideal benchmark (max work output or min work input) for real nearly-adiabatic devices.