# Conjugate variables

In thermodynamics, conjugate variables or “conjugate pairs” are sets of intensive X and extensive x variables whose product Xdx has the dimensions of energy.  A central example of a conjugate pair is pressure-volume work, where the multiplication of the intensive variable pressure P by the extensive variable volume dV equates to an amount of spatial work energy dW done. Other conjugate pairs are listed below:

 Intensive Variable Extensive Variable Energy Function Product Person Pressure P Volume dV pressure-volume work δW pdV Clapeyron (1834) Temperature T Entropy dS internal work (transformational content energy) δQ TdS Clausius (1865) Chemical potential μ Particle number dn species transfer work μdn Gibbs (1876) Force F Length dx stress-strain work Fdx Hooke (1660) Electromotive force ε Charge de electrical work εde Gibbs (1876) - Helmholtz (1882) Surface tension γ(superficial tension σ) Surface area dA(area of surface considered s) surface work γdA (σδs) Gibbs (1876) Gravitational potential ψ Mass dm gravitation work ψdm Electric field E Electric dipole moment dp electric polarization Edp Magnetic field B Magnetic moment dm magnetic polarization Bdm

The general use of the conjugate pairs perspective is that one can quantify the internal energy of a system as the sum of the conjugate variables. In short, with any extensity xi (extensive variable) it is always possible to associate a tension variable Xi (intensive variable): $X_i = \frac{\partial U}{\partial x_i}$

which is called the "conjugate", whereby, according to the first law, the change in internal energy dU of a system is given by the summation of the product of the conjugate pairs: $dU = \sum_{i=1}^k X_i dx_i$

The right side of this expression is what is called a Pfaffian form. To give a simple example, in the process whereby an indefinitely small quantity of heat dQ (which according to German physicist Rudolf Clausius is equal to the product TdS) is imparted to a body, thus causing a certain amount of pressure-volume work to be done, in accordance with Boerhaave's law, the change in the internal energy will be the heat added less the work done: $dU = TdS - PdV \,$

which is the first and the second law of thermodynamics combined into an analysis of the process. 

History
It is difficult to track down the origin of this topic, although it might be derived from the homogeneous function of Swiss mathematician Leonhard Euler. 

One of the first to summarize this as a “work principle” seems to have been Danish physical chemist Johannes Bronsted who in a 1946 monograph, reprinted in 1955 as Principles and Problems in Energetics, summarized the main topics in thermodynamics in terms of energetics.  In particular, he stated that
the overall work ∆W performed by a system is the sum of contributions due to transport of extensive quantities ∆Ki across a difference of "conjugated potentials" Pi1 - Pi2: $\Delta W = \sum_{i=1}^{k} (P_{i1} - P_{i2}) \Delta K_i$

in which Pi1 - Pi2 may be T1 - T2 (thermal potential difference), μ1 - μ2 (chemical potential difference), or ψ1 - ψ2 (electric potential difference) and ∆Ki will be ∆S (quantity of entropy), ∆n (quantity of substance), or ∆e (quantity of electricity), respectively .

References
1. (a) Attard, Phil. (2002). Thermodynamics and Statistical Mechanics: Equilibrium by Entropy Maximization (pg. 409). Academic Press.
(b) Alberty, Robert, A. (2003). Thermodynamic of Biochemical Reactions (table 2.1: Conjugate Properties involved in Various Kinds of Work, pg. 32). Hoboken, New Jersey: John Wiley & Sons, Inc.
2. Brønsted, Johannes. (1955). Principles and Problems in Energetics. Interscience.
3. Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (2nd ed). London: Macmillan & Co.
4. Kirkwood, J.G. and Oppennheim, Irwin. (1961). Chemical Thermodynamics (pgs. 8, appendix A-2). McGraw-Hill Book Co. Inc.