Same gas, two different transformations

There is a crucial pedagogical point hidden in thermodynamic diagrams: the same gas can be taken from the same initial state $A$ to the same final state $B$ in completely different ways, one reversible and one irreversible. The natural question is: what changes and what stays the same between the two paths?

The answer separates two categories of quantities. The internal energy $E_\text{int}$ and the entropy of the gas $S_\text{gas}$ are state functions: they depend only on the state of the gas, not on the path taken to reach it. So $\Delta E_\text{int}$ and $\Delta S_\text{gas}$ are identical on the two paths, because $A$ and $B$ are the same. In contrast, the heat $Q$ and the work $L$ exchanged depend on the path: they differ from one process to another, even for the same endpoints.

Summary — From A to B: same gas, same change of state, but...

Reversible: $Q,\ L_\text{gas} \neq 0$, with $\Delta S_\text{tot} = 0$. Irreversible (free expansion): $Q,\ L_\text{gas} = 0$, with $\Delta S_\text{tot} > 0$. In both cases $\Delta E_\text{int}$ and $\Delta S_\text{gas}$ are equal, because they are state functions.

This distinction has a powerful practical consequence. In an irreversible process there is no well-defined path in the $P$-$V$ diagram: there is no curve to integrate. But precisely because entropy is a state function, we can get round the obstacle.

The reversible case as a "useful path"

When $\Delta S_\text{gas}$ must be calculated for an irreversible process, where there is no well-defined path in the $P$-$V$ diagram, one can imagine a fictitious reversible path between the same initial and final states, because entropy is a state function. It is a practical technique: “treat the problem as if it were reversible” to calculate $\Delta S$, but remember that the real one is not.

Linked notes

Topics: Termodinamica Concepts: Primo principio della termodinamica · Energia interna · Entropia Objects: Gas ideale

Related exercises: Esercizio svolto — espansione isoterma reversibile vs irreversibile · Problema — Doccia mattutina · Azoto riscaldato in un cilindro rigido