Equilibrium states and thermodynamic coordinates

An equilibrium state has no continuing macroscopic evolution under its stated constraints. Thermal equilibrium excludes a temperature difference that would drive heat flow through an allowed contact. Mechanical equilibrium excludes an unbalanced force on a movable boundary. Chemical equilibrium excludes a material transfer or reaction affinity that would drive change. A rigid insulated wall can prevent exchanges, so two separately equilibrated compartments need not have equal pressure or temperature until the relevant constraint is removed.

Here a closed system exchanges no matter, an isolated system exchanges neither matter nor energy, and an adiabatic boundary prevents heat exchange but can still transmit work. These definitions avoid conflating insulation with complete isolation. We initially consider a homogeneous single-phase fluid at rest, neglecting gravity, electric fields, interfaces, and long-lived internal variables. More elaborate materials require additional coordinates. A nonequilibrium gas with strong gradients generally needs spatial fields rather than one global temperature and pressure.

For a fixed amount of a simple fluid in one phase, two independent equilibrium coordinates usually determine its state locally. Pressure, volume, and temperature obey an equation of state, so they cannot all be chosen independently. An ideal gas is a useful dilute-gas model with $pV=\nu RT$, where $\nu$ is amount in moles and R is the molar gas constant. A real fluid can depart strongly from this relation near condensation or at high density.

An equation of state is not a complete caloric description. The ideal-gas relation alone specifies how pressure changes with temperature and volume, but it does not supply a numerical heat capacity or an absolute internal energy. Those require additional thermodynamic information. In a two-phase region, temperature and pressure are linked by coexistence and the phase fraction becomes relevant to total volume or energy. Thus the phrase “two coordinates” always includes assumptions about phase, composition, and the chosen domain.

Joining identical noninteracting copies of a homogeneous sample doubles extensive quantities such as volume, amount, internal energy, and entropy. It leaves intensive quantities such as temperature and pressure unchanged. Density is intensive because both mass and volume scale together. This scaling concerns comparable equilibrium copies in the macroscopic limit, where surface and interaction corrections between pieces are negligible. Doubling a container’s volume at fixed particle number is a different operation and need not preserve pressure.

The zeroth law says thermal equilibrium is transitive: if two systems each equilibrate with the same thermometer, they equilibrate with one another when placed in suitable thermal contact. It makes temperature a consistent label for thermal equivalence classes. A thermometer must couple weakly enough, or be accounted for carefully enough, that it does not significantly alter the intended state. Sharing temperature does not imply equal energy; heat capacities, amounts of matter, and material properties can differ.