State function

A state function is any thermodynamic property that is fixed by the present state of a system and is indifferent to its history. Internal energy, temperature, pressure, volume, and entropy are all state functions: two systems in the same state share the same values regardless of how each got there. The change in a state function between two states therefore depends only on the endpoints, and its integral around any closed cycle is exactly zero.

Heat and work are the crucial counterexamples — they are path functions, not state functions. Between the same two states, different processes transfer different amounts of heat and do different amounts of work, even though the change in internal energy is the same. This is why one cannot speak of the heat or work 'contained' in a body; only of the heat and work that crossed its boundary along a particular path.

Mathematically, a state function has an exact differential: dU is path-independent and ∮dU = 0, whereas the small heat δQ and work δW are inexact, their integrals depending on the route. This distinction is the formal backbone of the first and second laws and of the whole machinery of thermodynamic potentials.

The path-independence of internal energy was implicit in the first law of the 1840s and made explicit by Clausius, whose introduction of entropy in 1865 supplied a second, subtler state function at the heart of the second law.