Entropy in Adiabatic Transformations

By Prof. Marco Ruzzi.

Theoretical background

The entropy balance for the system and its surroundings is:

\[\Delta S_{\mathrm{tot}}=\Delta S+\Delta S_{\mathrm{surr}}.\]

For a reversible transformation ΔS_tot=0. For an irreversible transformation ΔS_tot>0. In the adiabatic processes considered here there is no heat exchange with the surroundings; their entropy change is zero.

Exercise 4 — Reversible adiabatic process

Determine the entropy change of a system undergoing a reversible adiabatic transformation. Compare it with an irreversible adiabatic transformation and explain why the result is consistent with entropy being a state function.

Solution

Reversibility gives:

\[\Delta S_{\mathrm{tot}}=0.\]

For the adiabatic process, q_surr=0 and the surroundings are taken at constant volume, so:

\[\Delta S_{\mathrm{surr}}=0.\]

Consequently:

\[\Delta S_{\mathrm{tot}}=\Delta S+0=0.\] \[\Delta S=0.\]

A reversible adiabatic transformation therefore does not produce a positive entropy change.

Irreversible adiabatic transformations

For an irreversible adiabatic transformation the surroundings still exchange no heat, and their entropy remains unchanged under the same assumptions. The entropy balance is now:

\[\Delta S_{\mathrm{tot}}=\Delta S+0>0.\] \[\Delta S>0.\]

The contrast between ΔS>0 for an irreversible adiabatic process and ΔS=0 for a reversible one might seem inconsistent with S being a state function. There is no contradiction. Starting from the same initial state, the irreversible adiabatic process cannot reach the same final state as a reversible adiabatic process: their entropy changes differ, and therefore their endpoints differ.