In thermodynamics,

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

which is called the "conjugate", whereby, according to the first law, the change in internal energy

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

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

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

in which

References

1. (a) Attard, Phil. (2002).

(b) Alberty, Robert, A. (2003).

2. Brønsted, Johannes. (1955).

3. Clausius, Rudolf. (1879).

4. Kirkwood, J.G. and Oppennheim, Irwin. (1961).

Further reading

● Foss, O. (1948). “The Brønsted Work Principle: Thermoelectric, Galvanic and Thermo-Chemical”,

● LaMer, V.K., Foss, O., and Reiss, H. (1949). “Some New Prodcedures in Thermodynamic Theory Inspired by the Recent Work of J.N. Bronsted.”

● MacRae, D. (1955). “Brønsted's Work Principle

● MacDougal, F.H. (1940). “Brønsted's Criticism of Classical Thermodynamics.”

● Thims, Libb. (2007).

External links

● Conjugate variables (thermodynamics) – Wikipedia.

**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):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: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: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*: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.Further reading

● Foss, O. (1948). “The Brønsted Work Principle: Thermoelectric, Galvanic and Thermo-Chemical”,

*The Rockefeller Institute.*● LaMer, V.K., Foss, O., and Reiss, H. (1949). “Some New Prodcedures in Thermodynamic Theory Inspired by the Recent Work of J.N. Bronsted.”

*Annals of the New York Academy of Sciences.*● MacRae, D. (1955). “Brønsted's Work Principle

**and Gibbs' Treatment of Electromotive Force.”***The Journal of Physical Chemistry*● MacDougal, F.H. (1940). “Brønsted's Criticism of Classical Thermodynamics.”

*The Journal of Physical Chemistry.*● Thims, Libb. (2007).

*Human Chemistry (Volume Two)*, (keyword: conjugate variables, pg. 643). (preview), (Google books). Morrisville, NC: LuLu.External links

● Conjugate variables (thermodynamics) – Wikipedia.