This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(July 2019) (Learn how and when to remove this message)

Inthermodynamics, theinternal energy of asystem is expressed in terms of pairs ofconjugate variables such astemperature andentropy,pressure andvolume, orchemical potential andparticle number. In fact, allthermodynamic potentials are expressed in terms of conjugate pairs. The product of two quantities that are conjugate hasunits of energy or sometimespower.

For amechanical system, a small increment of energy is the product of a force times a small displacement. A similar situation exists in thermodynamics. An increment in the energy of a thermodynamic system can be expressed as the sum of the products of certaingeneralized "forces" that, when unbalanced, cause certaingeneralized "displacements", and the product of the two is the energy transferred as a result. These forces and their associated displacements are calledconjugate variables. The thermodynamic force is always anintensive variable and the displacement is always anextensive variable, yielding an extensive energy transfer. The intensive (force) variable is thederivative of the internal energy with respect to the extensive (displacement) variable, while all other extensive variables are held constant.

Thethermodynamic square can be used as a tool to recall and derive some of thethermodynamic potentials based on conjugate variables.

In the above description, the product of two conjugate variables yields an energy. In other words, the conjugate pairs are conjugate with respect to energy. In general, conjugate pairs can be defined with respect to any thermodynamic state function. Conjugate pairs with respect toentropy are often used, in which the product of the conjugate pairs yields an entropy. Such conjugate pairs are particularly useful in the analysis of irreversible processes, as exemplified in the derivation of theOnsager reciprocal relations.

Just as a small increment of energy in a mechanical system is the product of a force times a small displacement, so an increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" which, when unbalanced, cause certain generalized "displacements" to occur, with their product being the energy transferred as a result. These forces and their associated displacements are calledconjugate variables.^[1] For example, consider the${\displaystyle pV}$ conjugate pair. The pressure${\displaystyle p}$ acts as a generalized force: Pressure differences force a change in volume${\displaystyle \mathrm{d}V}$, and their product is the energy lost by the system due to work. Here, pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables. In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heat transfer. The thermodynamic force is always anintensive variable and the displacement is always anextensive variable, yielding an extensive energy. The intensive (force) variable is the derivative of the (extensive) internal energy with respect to the extensive (displacement) variable, with all other extensive variables held constant.

The theory of thermodynamic potentials is not complete until one considers the number of particles in a system as a variable on par with the other extensive quantities such as volume and entropy. The number of particles is, like volume and entropy, the displacement variable in a conjugate pair. The generalized force component of this pair is thechemical potential. The chemical potential may be thought of as a force which, when imbalanced, pushes an exchange of particles, either with the surroundings, or between phases inside the system. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds liquid water and water vapor, there will be a chemical potential (which is negative) for the liquid which pushes the water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate, and the chemical potential of each phase is equal, is equilibrium obtained.

The most commonly considered conjugate thermodynamic variables are (with correspondingSI units):

Thermal parameters:

• Temperature:${\displaystyle T}$  (K)
• Entropy:${\displaystyle S}$  (J K⁻¹)

Mechanical parameters:

• Pressure:${\displaystyle p}$  (Pa= J m⁻³)
• Volume:${\displaystyle V}$  (m³ = J Pa⁻¹)

or, more generally,

• Stress:${\displaystyle {\sigma }_{ij}}$ (Pa= J m⁻³)
• Volume ×Strain:${\displaystyle V\times {\epsilon }_{ij}}$ (m³ = J Pa⁻¹)

Material parameters:

• chemical potential:${\displaystyle \mu }$ (J)
• particle number:${\displaystyle N}$   (particles or mole)

For a system with different types${\displaystyle i}$ of particles, a small change in the internal energy is given by:

${\displaystyle \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V+\sum _i{\mu }_i\mathrm{d}{N}_i,}$

where${\displaystyle U}$ is internal energy,${\displaystyle T}$ is temperature,${\displaystyle S}$ is entropy,${\displaystyle p}$ is pressure,${\displaystyle V}$ is volume,${\displaystyle {\mu }_i}$ is the chemical potential of the${\displaystyle i}$-th particle type, and${\displaystyle N_i}$ is the number of${\displaystyle i}$-type particles in the system.

Here, the temperature, pressure, and chemical potential are the generalized forces, which drive the generalized changes in entropy, volume, and particle number respectively. These parameters all affect theinternal energy of a thermodynamic system. A small change${\displaystyle \mathrm{d}U}$ in the internal energy of the system is given by the sum of the flow of energy across the boundaries of the system due to the corresponding conjugate pair. These concepts will be expanded upon in the following sections.

While dealing with processes in which systems exchange matter or energy, classical thermodynamics is not concerned with therate at which such processes take place, termedkinetics. For this reason, the termthermodynamics is usually used synonymously withequilibrium thermodynamics. A central notion for this connection is that ofquasistatic processes, namely idealized, "infinitely slow" processes. Time-dependent thermodynamic processes far away from equilibrium are studied bynon-equilibrium thermodynamics. This can be done through linear or non-linear analysis ofirreversible processes, allowing systems near and far away from equilibrium to be studied, respectively.

As an example, consider the${\displaystyle pV}$ conjugate pair. Thepressure acts as a generalized force – pressure differences force a change involume, and their product is the energy lost by the system due tomechanical work. Pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.

The above holds true only for non-viscous fluids. In the case ofviscous fluids andplastic andelastic solids, the pressure force is generalized to thestress tensor, and changes in volume are generalized to the volume multiplied by thestrain tensor.^[2] These then form a conjugate pair. If${\displaystyle {\sigma }_{ij}}$ is theij component of the stress tensor, and${\displaystyle {\epsilon }_{ij}}$ is theij component of the strain tensor, then the mechanical work done as the result of a stress-induced infinitesimal strain${\displaystyle {\epsilon }_{ij}}$ is:

${\displaystyle \delta w=V\sum _{ij}{\sigma }_{ij}\mathrm{d}{\epsilon }_{ij}}$

or, usingEinstein notation for the tensors, in which repeated indices are assumed to be summed:

${\displaystyle \delta w=V{\sigma }_{ij}\mathrm{d}{\epsilon }_{ij}}$

In the case of pure compression (i.e. no shearing forces), the stress tensor is simply the negative of the pressure times theunit tensor so that

${\displaystyle \delta w=V(-p{\delta }_{ij})\mathrm{d}{\epsilon }_{ij}=-\sum _{k}pV\mathrm{d}{\epsilon }_{kk}}$

Thetrace of the strain tensor (${\displaystyle {\epsilon }_{kk}}$) is the fractional change in volume so that the above reduces to${\displaystyle \delta w=-p\mathrm{d}V}$ as it should.

In a similar way,temperature differences drive changes inentropy, and their product is the energy transferred byheating. Temperature is the driving force, entropy is the associated displacement, and the two form a pair of conjugate variables. The temperature/entropy pair of conjugate variables is the onlyheat term; the other terms are essentially all various forms ofwork.

Thechemical potential is like a force which pushes an increase inparticle number. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds water and water vapor, there will be a chemical potential (which is negative) for the liquid, pushing water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate is equilibrium obtained.

• Generalized coordinate andgeneralized force: analogous conjugate variable pairs found in classical mechanics.
• Intensive and extensive properties
• Bond graph

• Lewis, Gilbert Newton; Randall, Merle (1961).Thermodynamics. Revised by Kenneth S. Pitzer and Leo Brewer (2nd ed.).New York City: McGraw-Hill Book.ISBN 9780071138093.{{cite book}}:ISBN / Date incompatibility (help)
• Callen, Herbert B. (1998).Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons.ISBN 978-0-471-86256-7.

• Thermodynamic properties

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from July 2019
• All articles lacking in-text citations
• Use shortened footnotes from May 2021
• CS1 errors: ISBN date