Onsager reciprocal relations

The Onsager reciprocity relations (English Onsager reciprocal relations), also known as Onsagerscher reciprocity theorem, describe the relationship between different flows and forces that occur in a thermodynamic system out of equilibrium. They are in an area where the rivers occurring linearly depend on the acting forces. For this purpose, the described system can not be too far from equilibrium, since only then the concept of local equilibrium comes into play.

As an example of such a system can be used, a metal rod, acts on the motor as a temperature difference. This causes heat transfer from the warmer to the colder portions of the system. Similarly, a voltage causing an electric current to the regions of lower electrical potential. These are direct effects, in which a force causes a specific flow for them. Experimentally it can be shown that temperature differences in the metal cause the heat transfer in addition to an electric current and an electrical voltage to a heat flux results (cross- effects). The Onsagersche reciprocity theorem states that the size of such corresponding ( indirect ) effects are identical. In the example described, the size of the heat transport by a current flow (Peltier Koeffizent ) and the size of the current flow caused by a heat transport ( Seebeck Koeffizent ) are equal.

This already observed by William Thomson and other researchers Kelvin relation was provided by the Norwegian physical chemist and theoretical physicist Lars Onsager on a sound theoretical basis within the framework of thermodynamics of irreversible processes. He developed theory is applicable to any number of force and flow pairs in a system. For the description of these reciprocal relations, he was awarded the 1968 Nobel Prize in Chemistry.

• 2.1 The reciprocity relations
• 2.2 Thermodynamic equilibrium and entropy

Formal description within the framework of thermodynamics of irreversible processes

Physics law has numerous, in which two variables are proportional to each other. Examples of such relationships between flows and forces occurring in a thermodynamic system out of equilibrium are well-known laws of nature in vector form:

• Fourier's law of heat conduction.
• Ohm's law the current line.
• Fick's first law of diffusion:
• Newton's law of friction:

Such linear laws can be generally written in the following form:

With:

The thermodynamic forces and their corresponding flows are derived from a balance equation on the conserved quantities. The product of two variables describes the increase of entropy during a voluntary process requiring ( entropy ).

Cross effects between forces and fluxes

There are a number of phenomena in which a thermodynamic force, not only the faces of the effect described above by the laws, but other processes affected. Examples of such phenomena are the thermoelectric effects, thermomagnetic and galvanomagnetic effects or the interdiffusion of two substances together.

At a flow act in these cases not only the corresponding forces, but also cross forces. This superposition is microscopically readily understandable, since in the flow of this size has to be transported through a medium. For example, the heat is transported by a flow of substances containing this substance. In describing these processes, the advantage of the formalism introduced above, it is clear for two rivers with two corresponding forces gives:

With:

The Kreuzkoeffizenten are equal, DHES applies:

They are in an area that depend linearly on each other in the rivers and acting forces. This assumes that the system described may not be too far from equilibrium, because then comes the concept of microscopic reversibility or a local equilibrium to bear. Formally, then any functional relationship between the physical quantities as the Taylor series is aborted after the first member will be described.

Examples

The reciprocity relations

In a system in which both heat and volume flows occur, the result is a superposition of fluxes and forces. The relationships extend to him

And

Using this equation, the diffusion of the component is described by a temperature gradient ( or Soret thermophoresis effect)

And the heat conduction through the material flow (diffusion thermo effect or Dufour effect)

The Onsager reciprocity relations formulated in this case again the equality of the cross coefficients:

A dimensional analysis shows that both coefficients in the same units of measurement temperature times mass density can be measured.

Thermodynamic equilibrium and entropy

A closed system is not in thermodynamic equilibrium, if its entropy is maximized. So it must be converted into free energy by entropy entropy to reach an equilibrium state with minimum free energy and maximum entropy. In a closed system this can be done only by internal conversion ( dissipative ) processes; the size of the entropy production is then obtained from the continuity equation

Wherein the change of the local entropy by internal processes is the partial derivative with respect to time, according to the divergence of the spot, the local Entropieflussdichte, the local flux density of the internal energy and the absolute temperature.

The partial derivative of the local entropy with respect to time can be expressed by the fundamental equation of Gibbs. This therefore results in a multicomponent system isochoric

The extensive quantities internal energy and molar are conserved quantities; are their continuity equations

And, since the change of amount of substance must be taken into account by means of chemical reactions with the reaction rate,

The Gibbs equation is thus

With the transformation from the vector analysis and the definition of the chemical affinity ( However, it remains to explain why ) is obtained for the entropy production

From this equation can be determined for the variables and conjugate thermodynamic forces. A system is not far from its equilibrium state, it is reasonable to assume a linear relationship between a river and the thermodynamic force. The proportionality factor is called the transport coefficient. In the absence of material flow and a reaction thus follows the Fourier law in the form

And in the absence of heat flow Fick's law as