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. [1] 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. [3]

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. [4]

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. [2] 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.