Conjugate variables (thermodynamics)

In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy or pressure and volume. In fact, all thermodynamic potentials are expressed in terms of conjugate pairs. The product of two quantities that are conjugate has units of energy or sometimes power.

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" that, when unbalanced, 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.

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 to entropy 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 the Onsager reciprocal relations. The present article is concerned only with energy-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 ${\displaystyle pV}$ 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 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.

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