1 Fundamental concepts
Thermodynamic equilibrium is the condition in which a system’s macroscopic properties do not change with time because all driving gradients have disappeared or cancel one another. In this state, there is no net transfer of heat, matter, or mechanical work across the system boundary or within the system. The concept serves as a foundation for classical thermodynamics, where equilibrium states are treated as well-defined reference points.
1.1 Definition of equilibrium
A system is in equilibrium when its observable properties, such as temperature, pressure, composition, and phase distribution, remain constant over time. This does not mean that motion or microscopic activity has ceased; rather, any such activity is balanced so that no overall change is detectable at the macroscopic level. Equilibrium is therefore a condition of balance, not necessarily of rest.
1.2 Macroscopic and microscopic perspectives
From a macroscopic viewpoint, equilibrium is described by constant state variables and the absence of net flows. From a microscopic viewpoint, particles may still move, collide, and exchange energy, but these processes occur in a statistically balanced way. The macroscopic steadiness arises from the collective behavior of vast numbers of microscopic constituents.
1.3 State variables and state functions
Equilibrium states are characterized by state variables such as temperature, pressure, volume, and chemical composition. State functions, including internal energy, entropy, and free energies, depend only on the current state and not on the path taken to reach it. This path independence is a central feature that makes equilibrium thermodynamics predictive and mathematically tractable.
1.4 Equilibrium versus steady state
A steady state can show unchanging macroscopic properties while still involving continuous flows of matter or energy. By contrast, thermodynamic equilibrium requires the complete absence of net transport. In a steady state, external inputs and outputs maintain the condition; in equilibrium, no such sustaining flux is needed.
2 Conditions for thermodynamic equilibrium
Thermodynamic equilibrium requires several forms of balance to hold at the same time. These include thermal, mechanical, chemical, and, when relevant, phase equilibrium. If any one of these is violated, the system can evolve toward a more stable state.
2.1 Thermal equilibrium
Thermal equilibrium exists when no net heat flows between parts of a system or between the system and its surroundings. It is the condition associated with uniform temperature and the absence of a thermal gradient capable of driving energy transfer.
2.1.1 Temperature equality
When two regions are in thermal equilibrium, they share the same temperature. If temperatures differ, heat flows from the hotter region to the cooler one until equality is reached. Temperature equality is therefore a practical marker of thermal balance.
2.1.2 Zeroth law of thermodynamics
The zeroth law states that if system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then A and C are also in thermal equilibrium. This principle establishes temperature as a meaningful and transitive property, making thermometry possible.
2.2 Mechanical equilibrium
Mechanical equilibrium is present when pressures and forces are balanced so that there is no tendency for macroscopic motion or deformation. In such a state, the system experiences no unbalanced mechanical stress that would cause expansion, compression, or flow.
2.2.1 Pressure balance
If a system contains regions of differing pressure, matter will move from high pressure toward low pressure until the difference is eliminated or countered by other forces. Mechanical equilibrium requires that pressure be uniform in the absence of external fields or that any variations be exactly compensated by those fields.
2.2.2 Absence of unbalanced forces
A system in mechanical equilibrium has no net force causing acceleration of its parts. This includes the cancellation of surface forces, body forces, and stresses within solids or fluids. When force balance fails, the configuration changes until a stable arrangement is reached.
2.3 Chemical equilibrium
Chemical equilibrium concerns the balance of chemical species and reactions. It is reached when no net change in composition occurs because forward and reverse processes proceed at equal rates.
2.3.1 Equality of chemical potentials
Chemical potential measures the tendency of a species to move, react, or redistribute. At equilibrium, the chemical potential of each species is equal across all regions or phases in which that species is present. Differences in chemical potential drive diffusion and reaction until balance is achieved.
2.3.2 No net reaction flow
In a chemically equilibrated reaction mixture, reactants and products continue to interconvert microscopically, but the overall composition remains constant. The forward and reverse reaction rates are equal, so there is no net progress in either direction.
2.4 Phase equilibrium
Phase equilibrium occurs when multiple phases of a substance coexist without net transfer of mass from one phase to another. Common examples include liquid and vapor in a sealed container or solid and liquid at the melting point.
2.4.1 Coexistence of phases
Coexisting phases remain distinct while sharing compatible thermodynamic conditions. Their temperatures, pressures, and relevant chemical potentials match in a way that prevents one phase from entirely replacing the other. This balance can persist for long periods if the system is undisturbed.
2.4.2 Saturation conditions
Saturation describes the state in which a phase, such as vapor, is in equilibrium with another phase, such as liquid. Under saturation conditions, evaporation and condensation occur at equal rates. Similar ideas apply to melting, freezing, sublimation, and deposition.
3 Thermodynamic potentials and equilibrium criteria
Thermodynamic potentials provide compact criteria for determining equilibrium under different constraints. Depending on what is held fixed, equilibrium corresponds to the extremum of an appropriate potential. These functions are especially useful for predicting spontaneous change.
3.1 Entropy maximization
For an isolated system, equilibrium corresponds to the maximum possible entropy consistent with the constraints. This principle expresses the tendency of accessible microstates to become more numerous and evenly distributed. The entropy maximum identifies the most probable macroscopic configuration.
3.2 Internal energy minimization
When entropy and volume are held fixed, equilibrium can be described as the minimum of internal energy. This formulation is useful in constrained settings where the system cannot exchange heat or do work with its surroundings. The minimum-energy state is then the stable arrangement.
3.3 Helmholtz free energy
The Helmholtz free energy is appropriate for systems at constant temperature and volume. At equilibrium under these conditions, the Helmholtz free energy is minimized. This criterion is widely used in chemistry and statistical mechanics because it combines energetic and entropic effects into a single quantity.
3.4 Gibbs free energy
For systems at constant temperature and pressure, equilibrium corresponds to the minimum of Gibbs free energy. This criterion is especially important in chemistry and phase change, where pressure is often externally fixed. A decrease in Gibbs free energy indicates a direction of spontaneous evolution.
3.5 Chemical potential and equilibrium
Chemical potential serves as the driving quantity for material redistribution and reaction. Equilibrium requires that the chemical potential of each component be uniform in the appropriate sense. When this is not the case, matter flows or reactions proceed until balance is restored.
4 Stability and metastability
Not every equilibrium-like state is equally robust. Stability describes how a system responds to small disturbances, while metastability refers to a long-lived but not truly minimal configuration. These ideas help distinguish genuine equilibrium from states that only appear settled.
4.1 Stable equilibrium
A stable equilibrium returns to its original state after a small perturbation. Nearby configurations have higher thermodynamic potential or lower entropy under the relevant constraints, so the system resists change. This restoring tendency is a hallmark of true equilibrium.
4.2 Metastable states
Metastable states persist because the system must overcome an energy barrier before transforming to a more stable configuration. Such states may last a very long time, even though they are not the absolute thermodynamic minimum. Supercooled liquids and supersaturated solutions are common examples.
4.3 Instability and spontaneous change
An unstable state amplifies small disturbances rather than suppressing them. Once a threshold is crossed, the system changes spontaneously toward a new configuration with lower free energy or higher entropy, depending on the constraints. Instability often precedes phase separation or structural reorganization.
4.4 Fluctuations near equilibrium
Even at equilibrium, small random fluctuations occur due to microscopic motion. These variations are usually tiny relative to macroscopic scales and average out over time. Near critical points or in very small systems, however, fluctuations can become significant and may strongly influence behavior.
5 Equilibrium in different types of systems
The equilibrium description depends on whether the system exchanges energy, matter, or both with its environment. The classification into isolated, closed, and open systems clarifies which conditions apply and which potentials are relevant. System uniformity also affects how equilibrium is achieved.
5.1 Isolated systems
An isolated system exchanges neither energy nor matter with its surroundings. Its equilibrium state is determined by entropy maximization at fixed total energy, volume, and composition. This idealized case is central to the most fundamental form of thermodynamic reasoning.
5.2 Closed systems
A closed system can exchange energy but not matter with its surroundings. Such systems often approach equilibrium with the environment while maintaining fixed mass. The relevant equilibrium criteria depend on whether heat transfer, work, or both are permitted.
5.3 Open systems
Open systems exchange both energy and matter with their surroundings. True thermodynamic equilibrium is harder to maintain in open settings because external inputs can continuously disturb the state. Nevertheless, local equilibrium can still be a useful approximation in many practical situations.
5.4 Homogeneous and heterogeneous systems
A homogeneous system has uniform composition and phase throughout, making equilibrium analysis relatively straightforward. A heterogeneous system contains distinct regions or phases, each of which must satisfy equilibrium conditions at the interfaces. Such systems often require simultaneous treatment of thermal, mechanical, chemical, and phase balances.
6 Phase transitions at equilibrium
Phase transitions describe changes from one phase to another under equilibrium conditions. They occur when the thermodynamic potentials of competing phases become comparable and the system can shift between distinct states. The nature of the transition depends on how properties change across the boundary.
6.1 First-order phase transitions
First-order transitions involve discontinuous changes in first derivatives of the thermodynamic potentials, such as entropy or volume. They typically include latent heat and coexistence between phases over a range of conditions. Melting and boiling are familiar examples.
6.2 Second-order phase transitions
Second-order, or continuous, transitions show no latent heat and no jump in the first derivatives of the free energy. Instead, response functions such as heat capacity or compressibility may become singular or sharply enhanced. These transitions are associated with subtle changes in ordering.
6.3 Critical points
A critical point marks the end of a coexistence line, beyond which the distinction between two phases disappears. Near this point, the properties of the phases become increasingly similar, and fluctuations grow large. The behavior there often differs markedly from that in ordinary equilibrium states.
6.4 Coexistence curves and phase diagrams
Phase diagrams map the equilibrium relations among phases as functions of variables such as temperature, pressure, and composition. Coexistence curves indicate the conditions under which multiple phases can exist together. These diagrams are practical tools for identifying stable and metastable regions.
7 Statistical mechanics interpretation
Statistical mechanics explains thermodynamic equilibrium in terms of the distribution of microscopic states. Macroscopic equilibrium emerges because some microscopic arrangements are overwhelmingly more probable than others. This framework connects observables to probabilities and counting methods.
7.1 Microstates and macrostates
A microstate specifies the exact microscopic configuration of a system, while a macrostate describes it through bulk variables. Many microstates can correspond to the same macrostate. Equilibrium is associated with macrostates that can be realized by the largest number of microstates.
7.2 Boltzmann entropy
Boltzmann entropy links entropy to the logarithm of the number of compatible microstates. This relation provides a statistical meaning for the tendency toward equilibrium. A higher entropy state is one that is accessible in many more microscopic ways.
7.3 Canonical and grand canonical ensembles
The canonical ensemble describes systems at fixed temperature, volume, and particle number, while the grand canonical ensemble allows particle exchange as well. These ensembles provide probabilistic descriptions of equilibrium under different constraints. They are widely used to compute average properties and fluctuations.
7.4 Partition functions
The partition function summarizes the statistical weights of all accessible states. From it, one can derive thermodynamic quantities such as free energy, entropy, and pressure. It is a central mathematical object in the equilibrium treatment of many-body systems.
8 Applications and examples
Equilibrium concepts apply across gases, liquids, solids, chemical mixtures, and astronomical systems. In each case, the same general principles determine whether the macroscopic state is stable and what variables characterize it. The details differ, but the underlying logic remains consistent.
8.1 Ideal gas at equilibrium
An ideal gas in a container reaches equilibrium when its temperature and pressure are uniform and its molecules are distributed according to statistical laws. Collisions among molecules produce a stable distribution of speeds and directions. The ideal-gas model provides a simple setting for studying equilibrium behavior.
8.2 Liquid-vapor equilibrium
In a sealed vessel containing liquid and vapor, equilibrium is established when evaporation and condensation occur at equal rates. The pressure of the vapor depends on temperature and the identity of the substance. This equilibrium underlies boiling, evaporation, and condensation phenomena.
8.3 Chemical reaction equilibria
Many reactions do not proceed to complete conversion but instead settle into a balance between reactants and products. At equilibrium, the composition reflects the relative stability of the species and the constraints on the system. Equilibrium constants summarize these relations in compact form.
8.4 Thermal contact between bodies
When two bodies at different temperatures are placed in contact, heat flows from the warmer body to the cooler one until both reach a common temperature. The final temperature depends on their heat capacities and the amount of energy exchanged. This process illustrates the approach to thermal equilibrium.
8.5 Equilibrium in astrophysical and planetary systems
Large-scale natural systems can also approach equilibrium over appropriate time and length scales. Stars, planetary atmospheres, and interstellar matter may exhibit local or partial equilibrium in regions where conditions change slowly. In such contexts, equilibrium concepts help describe temperature distributions, chemical composition, and phase behavior.