Entropy is a state function

Entropy, internal energy and temperature are all state functions: their value depends only on the current state of the system, not on the history that brought it there. Two gases in the same state $(P, V, T, n)$ have the same entropy, regardless of how they got there.

Why this is crucial

Even in an irreversible transformation, $\Delta S_\text{gas}$ can be calculated with the formulas from this section, as if going from A to B along an imagined reversible path. The result is the same, because $S$ depends only on the endpoints A and B, not on the path.

This property is what makes the formulas for the gas, the solid and the thermostat universally applicable. Many real processes are violently irreversible — a gas exploding into a vacuum, two bodies at different temperatures suddenly brought into contact — and for them there is no well-defined “path” in the $P$-$V$ diagram: they pass through non-equilibrium states that cannot even be drawn. And yet, since the initial state A and the final state B are well-defined equilibrium states, we can imagine a fictitious reversible path connecting them and calculate $\Delta S$ along it. The value obtained is the true entropy change of the real process.

A useful analogy is with gravitational potential energy: the difference in altitude between two points on a mountain does not depend on the path chosen to climb it. In the same way, $\Delta S$ between two states does not depend on the transformation chosen to connect them.

There is, however, a fundamental difference from energy. The internal energy of the gas does not change depending on whether the process is reversible or irreversible: it is always a “neutral” state function. The entropy of the gas is also a state function, but the total entropy (gas plus surroundings) clearly distinguishes the two cases: it stays unchanged only in reversible processes, and increases in irreversible ones. It is precisely this distinction that is the content of the second law, the subject of the next section.

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Topics: Entropy and the second law Concepts: Entropy · Internal energy · Temperature

Related exercises: Heat from hot to cold: proof · T ratio · Melting 1 kg of ice: ΔS