The entropic statement

The first law tells us which processes are possible from the point of view of energy, but it is not enough to explain why transformations always go in one direction only. This direction is dictated by the second law, which in its entropic formulation has a form of extraordinary simplicity.

Principle — Second law (entropic formulation)

In an isolated system, the total entropy (the sum of the entropies of all the components) can never decrease: $\ev{\Delta S_\text{tot} \geq 0}$ Equality holds if and only if the transformation is reversible.

Two words must be taken seriously. The first is total: $\Delta S_\text{tot}$ is the sum of the entropy changes of all the components involved — the gas, the solid bodies, the thermostats, any machines. It is not the entropy of a single part, which can perfectly well decrease.

The second is isolated: the system under consideration must exchange neither heat nor work with the outside. If a system is not isolated, to apply the principle it is enough to widen the boundaries to include everything it interacts with — in the limit, the entire universe. Hence the equivalent and famous formulation: the entropy of the universe can never decrease.

Isolated

Isolated means: no heat and no work exchanged with the outside. It is the condition that makes $\Delta S_\text{tot} \geq 0$ valid.

The $\geq$ inequality hides two profoundly different regimes: the equality sign ($\Delta S_\text{tot} = 0$) describes ideal reversible processes, the greater-than sign ($\Delta S_\text{tot} > 0$) describes all real irreversible processes. The precise distinction between the two is the subject of the next note.

Links

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

Related exercises: Worked exercise — the coffee cools down · Hot shower: ΔS of the universe · Heat from hot to cold: proof