# Conjugate Variables

In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume. In fact all thermodynamic potentials are expressed in terms of conjugate pairs.

For a mechanical 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 certain generalized "forces" which, when imbalanced, cause certain generalized "displacements", and the product of the two is the energy transferred as a result. These forces and their associated displacements are called conjugate variables. The thermodynamic force is always an intensive variable and the displacement is always an extensive variable, yielding an extensive energy transfer. The intensive (force) variable is the derivative of the internal energy with respect to the extensive (displacement) variable, while all other extensive variables are held constant.

The thermodynamic square? can be used as a tool to recall and derive some of the thermodynamic potentials based on conjugate variables.

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 called conjugate variables. For example, consider the pV conjugate pair. The pressure P acts as a generalized force: Pressure differences force a change in volume dV, 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 an intensive variable? and the displacement is always an extensive 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 we consider 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. 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 potentials of each phase is equal, is equilibrium obtained. Wikipedia, Conjugate variables

3.12 - Reciprocating Duality
3.13 - Reciprocals and Proportions of Motions and Substance
7.3 - Law of Love - Reciprocal Interchange of State on Multiple Subdivisions
7.6 - Reciprocal Disintegration and Creation
7.7 - Reciprocal States of Matter and Energy
12.00 - Reciprocating Proportionality
13.14 - Principle of Reciprocity
Figure 3.10 - Temperature Accumulates in the North and Cools in the South Reciprocally
Figure 12.13 - Some Multi-Dimensional as Inverse and Direct Reciprocal Relationships
Figure 13.14 - Equilibrium as Reciprocal Forces
One Balanced Whole and Two Reciprocating Dynamics
reciprocal
Reciprocating Proportionality
Rhythmic Balanced Interchange