Reversible and irreversible

The second law not only forbids total entropy from decreasing, but also provides a sharp, quantitative criterion for distinguishing the two fundamental types of transformation.

Operational definition of reversibility

A transformation is reversible if $\Delta S_\text{tot} = 0$, irreversible if $\Delta S_\text{tot} > 0$.

This is an operational definition: it says nothing about speed, friction or technical detail, but reduces everything to the calculation of a single number. It is enough to calculate the total entropy change of the process (system plus surroundings) and look at its value.

The physical meaning of the two cases is profound. A reversible transformation is an idealisation: a process so delicate that it can be exactly inverted, returning both the system and its surroundings to their starting state, without leaving any trace. No real process truly is; it is a limit that can be approached by making transformations infinitely slow and free of friction (quasi-static).

An irreversible transformation, by contrast, produces entropy: once it has happened, there is no way of returning exactly to the initial state of the universe. The system can be brought back, but only at the cost of further increasing the entropy of the surroundings. Every cup of coffee that cools, every gas that expands, every instance of friction that heats a surface is an entropy debt that the universe never settles.

$\Delta S_\text{gas} = 0$ does not mean reversible

Be careful not to confuse the two levels. In a transformation, the entropy of the gas may well stay unchanged or even decrease; what the second law constrains is the total entropy. A free expansion into a vacuum has $Q = 0$ but $\Delta S_\text{tot} > 0$: it is irreversible even though it exchanges no heat.

The reversible limiting case, $\Delta S_\text{tot} = 0$, is far from useless: it will be precisely the condition that, in the next chapter, allows us to define the ideal transformations of a gas and to calculate the maximum work extractable from a heat source.

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Topics: Entropy and the second law Concepts: Second law of thermodynamics · Thermodynamic transformations Skills: Entropy balance

Related exercises: Worked exercise — free expansion is not reversible · Gas in reversible adiabatic expansion · Is ΔS equal to zero admissible?