Only one unique adiabatic path between two states

Question

My Textbook says:

Two states A and B can never lie both on a reversible as will as irreversible adiabatic path. There lies only one unique adiabatic path linkage between two states A and B.

I don't understand what the book means. Why cannot there be both reversible and irreversible adiabatic paths between two states? And if only one path is possible, then how do we know which one?

What I understood is: In adiabatic process: ΔU=W. Now since ΔU is state function, it will mean that Wirreversible=Wreversible, which is not possible. But mathematical proof aside, is there any example/physical explanation to this?

Answer 1

For an infinitesimal adiabatic process $dS_\mathrm{surroundings} = 0$ since no heat is exchanged with the surroundings. But $dS_\mathrm{system} = 0$ is only true for a reversible process, since only then $dS_\mathrm{system} = dq/T$. For an irreversible process $dS_\mathrm{system} \neq dq/T =0$. Therefore

$$dS_{\mathrm{irrev}}\neq dS_{\mathrm{rev}}=0$$

This means that the state accessed with a reversible process through an integral change $\int dS_{\mathrm{rev}}=0$ cannot be accessed directly through an irreversible adiabatic process, for which $\Delta S = \int dS_{\mathrm{irrev}} \neq 0$.

Now since S is a state function, the terminal states of the two processes must therefore differ (QED).

As regards the second part (how do we know which is the reversible), you can use the approach just outlined to conclude that for the reversible process $\Delta S =0$.

Answer 2

Suppose that in a given transformation, ${\Delta U}$ is defined, and cannot be changed. But the same ${\Delta U}$ is the sum of two terms $Q$ and $W$ that can be changed and modified arbitrarily. If one of these terms, here the heat $Q$, is chosen to be zero, the other term $W$ cannot vary and stays equal to $W_{rev}$.