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As is well known, a good way to define the internal energy in classical thermodynamics is to define the difference in internal energy of two points as the amount of adiabatic work from one point to the other. From the first law, we know that work is path independent and thus internal energy must be a state function giving us, as a result, a nice definition of heat.

$U$ can only be a state function however if all points in the state space are adiabatically accessible which we know, from the second law, is not the case. In fact, this is one possible formulation of the second law (Caratheodory) although not equivalent to usual formulations.

Keeping all this in mind, can you prove that all points have some adiabatic path between them or must this be postulated?

If so, how do you define adiabatic path? What even is an adiabatic path if heat is not defined until the first law which relies on the idea of an adiabatic path?

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A more precise statement of Caratheodory formulation would be "In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S". This allows the state S to be reached via an adiabatic path from any of the points which are not accessible the other way around. And this is enough to define the internal energy.

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