The fundamental thermodynamic identity

Consider a homogeneous simple compressible substance with a fixed composition and no additional mechanical work coordinates. Along a reversible infinitesimal comparison, the heat input is $T\,dS$ and pressure-volume work on the system is $-p\,dV$. Combining them gives $dU=T\,dS-p\,dV$ at fixed amount. Because U and S are state functions, this differential relation defines the local equilibrium energy surface independent of which real process later connects nearby states. It is not a new energy source or a separate conservation law.

Allowing the amount $\nu$ of one species to vary adds chemical potential times amount change. Here chemical potential is per mole, with units of joules per mole, rather than per particle. For several species, include a sum of such terms. A reaction changes species amounts subject to its stoichiometric constraints. The identity alone is not an open-flow energy balance: material entering a pipe can also transport enthalpy and kinetic energy, which require an explicitly chosen control-volume description.

Suppose an isolated ideal gas freely expands. Its energy change is zero and its entropy increases, while its volume increases. The equilibrium identity can consistently connect the neighboring states of a reversible comparison surface through canceling entropy and volume terms. It does not assign $T\,dS$ as actual heat during the free expansion. The actual process is not a path of homogeneous equilibrium states, and its entropy balance includes production that the reversible heat formula does not represent.

A fundamental equation $U(S,V,\nu)$ contains more information than the pressure equation of state alone. Differentiating it supplies thermal, mechanical, and chemical responses, while second derivatives generate compatibility relations. An arbitrary proposed pressure law and energy law need not arise from a consistent smooth U. Their derivatives must satisfy the thermodynamic identities. This makes the fundamental relation both a compact representation and a diagnostic test for phenomenological material models.

For an ordinary macroscopic bulk phase, scaling entropy, volume, and amount together scales U by the same factor. Euler’s theorem on homogeneous functions then gives the displayed relation. Differentiating that relation produces the fundamental differential plus an extra combination of changes in temperature, pressure, and chemical potential. The extra combination must vanish. Thus these three intensive quantities cannot be varied independently within a one-component phase: specifying two generally fixes the third locally.

The assumptions behind this reasoning matter. A small droplet can have significant surface energy, and a self-gravitating system can be nonadditive, so naive extensive scaling needs modification. We retain the ordinary bulk thermodynamic regime. Even there, energy and entropy reference choices affect some absolute potentials while leaving consistently computed observable differences intact. The practical habit is to specify the independent coordinates and composition first, then take derivatives with the remaining coordinates explicitly held fixed.