So I've been doing a course in Thermodynamics for a while and one of the questions that occasionally struck me was "How do I practically calculate the difference of internal energy between two arbitrary states, A and B?"
Thermodynamics by H.B. Callen provides a really good insight into this and according to him, Joule, after his experiments concluded that- "In an adiabatic system, any two equilibrium states can definitely be connected by some mechanical process and the work done in that mechanical process is the internal energy difference $\Delta U$ between them".
I'll start with a simple example of a "container with a piston" system with an "ideal gas" inside.
Let's start with some equilibrium state $(P_0,V_0)$ with adiabatic walls, so no exchange of heat is possible. So, I can expand or contract the system to take it to some state $(P,V)$ which satisfies $PV^{\gamma}=P_0 {V_0}^{\gamma}=\kappa$ (constant).
Now, if I talk about the equilibrium state $(P_0,2V_0)$, this state too is an equilibrium state but it lies on some different adiabat along which states $(P,V)$ satisfy $PV^{\gamma}=P_0 {(2V_0)}^{\gamma}=\kappa'$ (a different constant).
Suppose we have to find the $\Delta U$ between$(P_0,V_0)$ and $(P_0,2V_0)$. But, if we have an adiabatic system, we can't jump between different adiabats i.e. I can't find a mechanical process which takes $(P_0,V_0)$ to $(P_0,2V_0)$. I have explained this problem through a diagram too.
If I can't find a mechanical process connecting these states, then I definitely can't find the internal energy difference between them which seems odd because we do that everytime in various thermodynamics problems. So, if I were to be an experimentalist, how would I go about finding the difference in internal energy between these two states?
Edit: I changed the example which I gave and added an image to further clarify the problem.