In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy, pressure and volume, or chemical potential and particle number.
In fact, all thermodynamic potentials are expressed in terms of conjugate pairs.
An increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" that, when unbalanced, cause certain generalized "displacements", and the product of the two is the energy transferred as a result.
In the above description, the product of two conjugate variables yields an energy.
In general, conjugate pairs can be defined with respect to any thermodynamic state function.
Such conjugate pairs are particularly useful in the analysis of irreversible processes, as exemplified in the derivation of the Onsager 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.
, 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.
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 the chemical 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.
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 corresponding SI units): For a system with different types
These parameters all affect the internal energy of a thermodynamic system.
This can be done through linear or non-linear analysis of irreversible processes, allowing systems near and far away from equilibrium to be studied, respectively.
The pressure acts as a generalized force – pressure differences force a change in volume, and their product is the energy lost by the system due to mechanical work.
Pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.
In the case of viscous fluids and plastic and elastic solids, the pressure force is generalized to the stress tensor, and changes in volume are generalized to the volume multiplied by the strain tensor.
In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heating.
Temperature is the driving force, entropy is the associated displacement, and the two form a pair of conjugate variables.
The chemical potential is like a force which pushes an increase in particle number.