Onsager reciprocal relations

In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

"Reciprocal relations" occur between different pairs of forces and flows in a variety of physical systems. For example, consider fluid systems described in terms of temperature, matter density, and pressure. In this class of systems, it is known that temperature differences lead to heat flows from the warmer to the colder parts of the system; similarly, pressure differences will lead to matter flow from high-pressure to low-pressure regions. What is remarkable is the observation that, when both pressure and temperature vary, temperature differences at constant pressure can cause matter flow (as in convection) and pressure differences at constant temperature can cause heat flow. Perhaps surprisingly, the heat flow per unit of pressure difference and the density (matter) flow per unit of temperature difference are equal. This equality was shown to be necessary by Lars Onsager using statistical mechanics as a consequence of the time reversibility of microscopic dynamics (microscopic reversibility). The theory developed by Onsager is much more general than this example and capable of treating more than two thermodynamic forces at once, with the limitation that "the principle of dynamical reversibility does not apply when (external) magnetic fields or Coriolis forces are present", in which case "the reciprocal relations break down".[1]

Though the fluid system is perhaps described most intuitively, the high precision of electrical measurements makes experimental realisations of Onsager's reciprocity easier in systems involving electrical phenomena. In fact, Onsager's 1931 paper[1] refers to thermoelectricity and transport phenomena in electrolytes as well known from the 19th century, including "quasi-thermodynamic" theories by Thomson and Helmholtz respectively. Onsager's reciprocity in the thermoelectric effect manifests itself in the equality of the Peltier (heat flow caused by a voltage difference) and Seebeck (electric current caused by a temperature difference) coefficients of a thermoelectric material. Similarly, the so-called "direct piezoelectric" (electric current produced by mechanical stress) and "reverse piezoelectric" (deformation produced by a voltage difference) coefficients are equal. For many kinetic systems, like the Boltzmann equation or chemical kinetics, the Onsager relations are closely connected to the principle of detailed balance[1] and follow from them in the linear approximation near equilibrium.

Experimental verifications of the Onsager reciprocal relations were collected and analyzed by D. G. Miller[2] for many classes of irreversible processes, namely for thermoelectricity, electrokinetics, transference in electrolytic solutions, diffusion, conduction of heat and electricity in anisotropic solids, thermomagnetism and galvanomagnetism. In this classical review, chemical reactions are considered as "cases with meager" and inconclusive evidence. Further theoretical analysis and experiments support the reciprocal relations for chemical kinetics with transport.[3] Kirchhoff's law of thermal radiation is another special case of the Onsager reciprocal relations applied to the wavelength-specific radiative emission and absorption by a material body in thermodynamic equilibrium.

For his discovery of these reciprocal relations, Lars Onsager was awarded the 1968 Nobel Prize in Chemistry. The presentation speech referred to the three laws of thermodynamics and then added "It can be said that Onsager's reciprocal relations represent a further law making a thermodynamic study of irreversible processes possible."[4] Some authors have even described Onsager's relations as the "Fourth law of thermodynamics".[5]

Classical cross effects
Potential/Flux Heat Electric Diffusion Deformation
Temperature Thermal conduction Seebeck effect Soret effect (thermophoresis) Thermoelasticity
Voltage Peltier effect Ohm's law Electromigration Piezoelectricity
Chemical potential Dufour effect Galvanic

cell

Fick's law Osmosis
Stress Thermoelasticity Piezoelectricity Osmosis Hooke's law
  1. ^ a b c Onsager, Lars (1931-02-15). "Reciprocal Relations in Irreversible Processes. I." Physical Review. 37 (4). American Physical Society (APS): 405–426. doi:10.1103/physrev.37.405. ISSN 0031-899X.
  2. ^ Miller, Donald G. (1960). "Thermodynamics of Irreversible Processes. The Experimental Verification of the Onsager Reciprocal Relations". Chemical Reviews. 60 (1). American Chemical Society (ACS): 15–37. doi:10.1021/cr60203a003. ISSN 0009-2665.
  3. ^ Yablonsky, G. S.; Gorban, A. N.; Constales, D.; Galvita, V. V.; Marin, G. B. (2011-01-01). "Reciprocal relations between kinetic curves". EPL (Europhysics Letters). 93 (2). IOP Publishing: 20004. arXiv:1008.1056. doi:10.1209/0295-5075/93/20004. ISSN 0295-5075. S2CID 17060474.
  4. ^ The Nobel Prize in Chemistry 1968. Presentation Speech.
  5. ^ Wendt, Richard P. (1974). "Simplified transport theory for electrolyte solutions". Journal of Chemical Education. 51 (10). American Chemical Society (ACS): 646. doi:10.1021/ed051p646. ISSN 0021-9584.

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