Can two different points can be connected by multiple adiabatic curves?

Question

I was watching this Thermodynamics lecture and I have a question on the 1st law. More exactly on how different adiabatic curves can connect the same initial and final states. See the diagram drawn at 59:50.

The idea is that you can go from $x_{i}$ to $x_{f}$ in the space of state variables on different adiabatic paths. Could this be done in a reversible process, or should the processes involved be irreversible?

I cannot visualize how two different points can be connected by multiple adiabatic curves in a simple $(p, V)$ space in reversible processes.

Edit: One answer could be that one path requires changes in $(p, V)$ space and another path involves changes in other pair of conjugate variables.

Answer 1

If you have only two (2) independent variables, say $p$ and $V$ describing your thermostatic state then between two sates can only be one reversible adiabatic curve. Note though that even for an ideal gas defined by the constitutive equation $pV=\frac{m}{M}RT$ there really are not two but three (3) independent variables, any three from the four $p,V,m,T$, for example, $p,V,m$ or $T,V,p$ etc.

When you allow the mass $m$ variable, dependent or independent, the system is called open but there are other possibilities for your system to have three or more independent variables, say, include electric or magnetic polarization and the ensuing interaction. Once you do that you will have adiabatic surfaces and not just adiabatic lines, and then between two equilibrium points on the same adiabatic surface, that is two equilibrium states having the same entropy but otherwise arbitrary, there can be an infinity of connecting reversible and adiabatic processes between them.

Answer 2

A reversible and irreversible adiabatic process cannot connect the same two equilibrium states. Since entropy is a thermodynamic property, the change in entropy of a system between the same two equilibrium states has to be the same for any path between the same two states, reversible or irreversible.

For a reversible adiabatic process the entropy change between two equilibrium states is zero. An irreversible process generates entropy in the system so the change in entropy of the system is greater than zero for an irreversible adiabatic process.

The only way the irreversible adiabatic process could theoretically connect the same two equilibrium states is if it were able to transfer the generated entropy to the surroundings in the form of heat. Since heat is not possible for an adiabatic process, the irreversible adiabatic process cannot connect the same two equilibrium states as the reversible adiabatic process. In effect, the entropy generated in an irreversible adiabatic process is "trapped" in the system.

Hope this helps.