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Thermodynamic potentials: definitions, common forms, and applications

An overview of thermodynamic potentials—energy-like functions (U, A, H, G, Ω), their natural variables, how they are formed via Legendre transforms, and their roles in equilibrium and practical calculations.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 16, 2026

Overview

Thermodynamic potentials are energy-like scalar functions used to describe the state and spontaneous evolution of macroscopic systems in thermodynamics. They have the dimensions of energy and are constructed so that their natural variables match the constraints maintained on a system. In practice, different potentials are convenient when different quantities are held fixed: for example temperature, pressure, volume, particle numbers, or chemical potentials. The word "potential" emphasizes that these functions act as driving quantities: differences or gradients of a potential determine the direction of irreversible change, much as a difference in potential energy can drive mechanical motion.

Definitions and natural variables

The most fundamental thermodynamic potential is the internal energy, commonly denoted U. Its natural variables are entropy, volume and the set of particle numbers: U = U(S,V,{N_i}). Entropy S, volume V, and particle numbers N_i are often called the natural extensive parameters or parameters of the state. Other potentials are obtained from U by Legendre transforms that replace some extensive variables by their conjugate intensive variables (for example replacing S by temperature T or V by pressure p). These conjugate pairs include T with temperature, S with entropy, p with pressure, and V with volume. The set of particle numbers N_i is sometimes treated explicitly in formulations as in U(S,V,{N_i}) to emphasize matter exchange.

$N_{i}$

Common potentials and their forms

Internal energy U(S,V,{N_i}): the primary energy function from which others are derived.
Helmholtz free energy A (or F): A = U − T S, natural variables (T,V,{N_i}); useful for systems at fixed temperature and volume and for computing the maximum reversible work extractable other than work of volume change.
Enthalpy H: H = U + p V, natural variables (S,p,{N_i}); convenient for processes at constant pressure, common in chemistry and engineering.
Gibbs free energy G: G = H − T S = U + pV − TS, natural variables (T,p,{N_i}); central to chemical equilibrium and phase stability at constant temperature and pressure.
Grand potential (Landau potential) Ω: Ω = U − T S − ∑ μ_i N_i (or Ω = A − ∑ μ_i N_i), natural variables (T,V,{μ_i}); used when particle exchange with reservoirs at fixed chemical potentials μ_i is allowed.

$N_{i}$

How potentials are used and equilibrium criteria

Each potential provides a criterion for spontaneous change under its natural constraints: at fixed entropy and volume U is minimized subject to conservation laws; at fixed temperature and volume, the Helmholtz free energy A tends to decrease and reaches a minimum at equilibrium; at fixed temperature and pressure, the Gibbs free energy G is minimized. These minimization principles make potentials powerful tools for predicting equilibrium compositions, phase transitions, and the sign of reversible work. In statistical mechanics the same potentials arise naturally as logarithms of partition functions, linking microscopic models to macroscopic thermodynamic behavior.

History, development and practical importance

The use of potentials grew during the 19th century as scientists and engineers sought energy-based descriptions for heat, work and chemical change. Foundational contributions by authors such as Clausius, Helmholtz and Gibbs clarified the relationships among energy, entropy and other state variables. Today thermodynamic potentials provide compact, general formulations across physics, chemistry, materials science and engineering: for example, Gibbs free energy is routinely used to compute chemical equilibria, phase diagrams and electrochemical cell voltages, while the grand potential is the natural tool in open-system statistical mechanics.

Notable distinctions and practical notes

Some points to keep in mind: potentials are not unique—different Legendre transforms produce functions suited to different constraints. The choice of natural variables determines which independent derivatives give physically meaningful quantities: for instance, (∂G/∂T)_p = −S and (∂G/∂p)_T = V. When particle numbers are important, chemical potentials μ_i appear as conjugate variables and govern matter exchange. For more background on foundational concepts see thermodynamic system descriptions and general discussions of energy. Further reading and resources: temperature and thermal contact, entropy and disorder, pressure in fluid systems, and volume work considerations. Technical introductions and applications are available through textbooks and detailed reviews (potential energy context).

For concise formulas and examples of how to compute changes in each potential from measurable quantities, introductory texts and specialized articles can be consulted via academic and educational portals referenced at thermodynamics and related links above.

Physical meaning

An extreme value (not always a minimum) of a thermodynamic potential indicates thermodynamic equilibrium.

Thus, after the connection of one closed system to another, a thermodynamic equilibrium has been established as soon as the entropy of the total system is maximal. In this case, all intensive parameters of the two systems are also equal in each case:

$\begin{align} T_1 & = T_2\\ p_1 & = p_2\\ \mu_1 & = \mu_2 \end{align}$

Moreover, thermodynamic potentials summarize the equations of state of the system, since these are accessible by differentiating a thermodynamic potential according to its dependent variables.

Description

The internal energy U(S,V,N) and the functions arising from it by means of the Legendre transformation are thermodynamic potentials (except for the exception , see below). The Legendre transformation transforms the potentials

From entropy to temperature , since ∂ ${\frac {\partial U}{\partial S}}=T$
from volume to pressure , since ∂ ${\frac {\partial U}{\partial V}}=-p$
From particle number to chemical potential μ $\mu$ , since ∂ ${\frac {\partial U}{\partial N}}\,=\mu$

Given these 3 pairs of variables, there are 2^3 = 8 possible thermodynamic potentials:

Thermodynamic potentials
Name (alternative name)	Formula symbol	natural variables (intensive bold)	Characteristic function f	Interrelationships
Inner energy			${\mathrm d}U=TdS-pdV+\mu {d}N$	$U(S,V,N)=TS-pV+{\sum _{{i=1}}^{{K}}}\,\mu _{i}N_{i}.$ (Euler equation for internal energy).
Free energy (Helmholtz potential)		$\boldsymbol{T},V,N$	${\mathrm d}F=-SdT-pdV+\mu {d}N$
Enthalpy		$S,\boldsymbol{p},N$	${\mathrm d}H=TdS+Vdp+\mu {d}N$	$H=U(S,V,N)\qquad \ +pV$
Gibbs energy (Free enthalpy)		$\boldsymbol{T},\boldsymbol{p},N$	${\mathrm d}G=-SdT+Vdp+\mu {d}N$	${\begin{aligned}G&=U(S,V,N)-TS+pV\\&=H(S,p,N)-TS\\&=F(T,V,N)\qquad \quad +pV\end{aligned}}$
-		$S,V,\boldsymbol{\mu}$	${\mathrm d}R=TdS-pdV-N{d}\mu$	$R=U(S,V,N)\qquad \qquad \quad -\mu N$
Great Cannon Potential	$\Omega$	$\boldsymbol{T},V,\boldsymbol{\mu}$	${\mathrm d}\Omega =-S{d}T-p{d}V-N{d}\mu$	${\begin{aligned}\Omega &=U(S,V,N)-TS\qquad \ -\mu N\\&=R(S,V,\mu )-TS\\&=F(T,V,N)\qquad \qquad \quad -\mu N\end{aligned}}$
-		$S,\boldsymbol{p},\boldsymbol{\mu}$	${\mathrm d}J=TdS+Vdp-N{d}\mu$	${\begin{aligned}J&=U(S,V,N)\qquad \quad +pV-\mu N\\&=R(S,V,\mu )\qquad \quad +pV\\&=H(S,p,N)\qquad \quad \qquad -\mu N\end{aligned}}$
-		$\boldsymbol{T},\boldsymbol{p},\boldsymbol{\mu}$ Not useful, since the Gibbs-Duhem relation $T,p,\mu$ forbids the independent default of the variables	${\mathrm d}K=-SdT+Vdp-N{d}\mu$	${\begin{aligned}K&=U(S,V,N)-TS+pV-\mu N=0\\&=J(S,p,\mu )-TS\\&=\Omega (T,V,\mu )\qquad \quad +pV\\&=G(T,p,N)\qquad \quad \qquad -\mu N\end{aligned}}$

All meaningful thermodynamic potentials arising from U(S,V,N) by Legendre transformation provide the same complete information about a system. However, the simplest description of the system provides only one of the potentials, depending on the ensemble; this is extremal at equilibrium.

One way to remember the thermodynamic potentials with their natural variables is the Guggenheim square.