Thermodynamic Equilibrium

Thermodynamic equilibrium is a state of a thermodynamic system in which macroscopic properties remain unchanged under fixed external conditions and no spontaneous change is possible within the imposed constraints. It is a central concept of thermodynamics, providing the reference states used to define material properties and analyze physical processes. Equilibrium does not mean that microscopic motion stops: particles continue moving and interacting, while their collective properties have stationary statistical descriptions. (damtp.cam.ac.uk)

Conditions for equilibrium

Full equilibrium requires the simultaneous satisfaction of several conditions, insofar as the system permits the corresponding exchanges.

Thermal equilibrium exists when bodies able to exchange energy as heat have the same temperature, so there is no net heat transfer between them. The zeroth law of thermodynamics states that bodies individually in thermal equilibrium with a third body are also in thermal equilibrium with each other. This transitive relationship provides the basis for temperature measurement. (damtp.cam.ac.uk)

Mechanical equilibrium requires balanced forces and the absence of unbalanced stresses that would produce macroscopic motion. For fluid compartments separated by a freely movable, planar piston, with negligible external forces, equilibrium requires equal pressure. Pressure need not be spatially uniform in every equilibrium system: under gravity, a stationary fluid can have a pressure gradient that balances its weight. Thus, equilibrium requires force balance rather than universal uniformity of every property. (damtp.cam.ac.uk)

Material and chemical equilibrium require that permitted particle transfers and chemical reactions have no net thermodynamic driving force. Without external fields, phases exchanging a given species must have equal chemical potentials for that species. For a reaction with stoichiometric coefficients $\nu_i$ , taken positive for products and negative for reactants, chemical equilibrium requires

\sum_i \nu_i\mu_i=0.

Forward and reverse reactions may continue at equal rates; constant composition therefore does not imply molecular inactivity. (damtp.cam.ac.uk)

Constraints and thermodynamic criteria

Equilibrium is always defined relative to constraints. An impermeable partition prevents matter exchange, while an insulating wall prevents heat exchange. Two subsystems can consequently remain stationary despite differences that would disappear if the relevant barrier were removed. Partial equilibrium means that some permitted processes have equilibrated while others have not. (damtp.cam.ac.uk)

The second law of thermodynamics supplies an extremum criterion. For an isolated system with fixed internal energy, volume, and conserved amounts of matter, stable equilibrium maximizes entropy over the accessible states. An allowed spontaneous relaxation increases total entropy until no further increase is possible. The qualification “accessible” matters because conservation laws and physical barriers restrict the states being compared. (ocw.mit.edu)

Other boundary conditions lead to equivalent free-energy criteria. For a closed system at fixed temperature and volume, stable equilibrium minimizes Helmholtz free energy,

F=U-TS.

At fixed temperature and pressure, it minimizes Gibbs free energy,

G=U+pV-TS.

These statements assume the appropriate material constraints and, in their simplest form, only pressure–volume work. They concern variations among allowed configurations at the specified conditions, not minimization with respect to every variable independently. (damtp.cam.ac.uk)

Statistical interpretation

Statistical mechanics connects equilibrium properties to microscopic states. An isolated equilibrium system is conventionally described by a microcanonical ensemble, with equal probabilities assigned to accessible microstates in the specified energy range. A system weakly coupled to a thermal reservoir instead has the canonical probability distribution

P_i=\frac{e^{-E_i/(k_{\mathrm B}T)}}{Z}, \qquad Z=\sum_i e^{-E_i/(k_{\mathrm B}T)}.

Here $E_i$ is a microstate’s energy, $k_{\mathrm B}$ is the Boltzmann constant, and $Z$ is the partition function. The exponential weighting is the Boltzmann distribution. (damtp.cam.ac.uk)

An equilibrium average can remain constant even though individual microscopic configurations change. Fluctuations are therefore compatible with equilibrium. For many ordinary macroscopic systems, relative fluctuations are small because of the large number of constituent particles, making thermodynamic quantities effectively reproducible at the macroscopic scale. (damtp.cam.ac.uk)

Phase coexistence and metastability

Equilibrium does not require a single homogeneous phase. Liquid water and water vapor can coexist when thermal, mechanical, and material equilibrium conditions are satisfied. At a phase transition, the competing phases have equal relevant chemical potentials; their proportions may differ while their intensive equilibrium conditions remain the same. These relations underlie equilibrium phase diagrams. (damtp.cam.ac.uk)

A metastable state differs from globally stable equilibrium. It occupies a local free-energy minimum and can resist small disturbances, although another accessible state has lower free energy. Barriers to forming the stable phase can delay conversion. Supercooled liquids illustrate this distinction: apparent persistence alone does not establish globally stable equilibrium. Thermodynamic stability and the rate of reaching a stable state are separate questions. (damtp.cam.ac.uk)

Steady states and local equilibrium

A stationary macroscopic state is not necessarily an equilibrium state. A system maintained between hot and cold reservoirs may sustain a constant heat current while its temperature profile remains unchanged. This is a nonequilibrium steady state: continuing exchange with the surroundings maintains the flow. Equilibrium, by contrast, has no such net dissipative transport. (ocw.mit.edu)

Local thermodynamic equilibrium is an approximation in which small regions of a globally nonequilibrium system are assigned equilibrium thermodynamic properties. Temperature, pressure, and composition may vary between regions, even though equilibrium relations are applied within each one. This permits thermodynamic descriptions of transport and spatially varying systems without asserting that the system as a whole is in equilibrium. (doi.org)