Onsager's reciprocity relations

The Onsager reciprocity relations (engl. Onsager reciprocal relations ), also known as Onsagerscher reciprocity theorem describing a relationship between different flows and forces encountered in a thermodynamic system out of equilibrium. They apply in a range in which the flows that occur depend linearly on the forces acting. For this purpose, the system described must not be too far from equilibrium, since only then does the concept of a local equilibrium come into play.

A metal rod on which a temperature difference acts as a force can serve as an example of such a system . This causes heat transfer from the warmer to the colder sections of the system. In the same way, an electrical voltage causes an electrical current to the areas with a lower electrical potential . These are direct effects in which a force creates a flow specific to it. Experimentally it can be shown that temperature differences in metal cause not only the heat transfer but also an electric current and an electric voltage leads to a heat flow ( cross effects ). Onsager's reciprocity theorem states that the size of such corresponding (indirect) effects is identical. In the example described, the size of the heat transport caused by a current flow ( Peltier coefficient ) and the size of the current flow caused by heat transport ( Seebeck coefficient ) are the same.

This relation , already observed by William Thomson and other researchers, was put on a sound theoretical basis by the Norwegian physical chemist and theoretical physicist Lars Onsager in the context of the thermodynamics of irreversible processes . The theory he developed is applicable to any number of force and flux pairs in a system. He was awarded the Nobel Prize in Chemistry in 1968 for describing these reciprocal relationships .

Formal description in the context of thermodynamics of irreversible processes

Physics knows numerous laws in which two quantities are proportional to one another . Examples of such relationships between occurring flows and forces in a thermodynamic system outside of equilibrium are known laws of nature in vector representation:

• Fourier's law of heat conduction : . ${\displaystyle {\dot {\vec {q}}}=-\lambda \,\operatorname {grad} \,T}$
• Ohm's law the power line: . ${\displaystyle {\vec {i}}=-\sigma \,\operatorname {grad} \,\phi }$
• First Fick's law of diffusion : ${\displaystyle {\vec {j}}=-D\,\operatorname {grad} \,c}$
• Newton's law of friction : ${\displaystyle {\vec {F}}=-\eta \,A\,\operatorname {grad} \,v_{y}\,}$

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

${\displaystyle {\mathbf {J} {_{k}}}\,=\,L_{k}\,\mathbf {X} {_{k}}}$

With:

the flow of any physical quantity ${\displaystyle {\mathbf {J} {_{k}}}}$ ${\displaystyle k}$

the transport coefficient of this size ${\displaystyle L_{k}}$ ${\displaystyle k}$

${\displaystyle {\mathbf {X} {_{k}}}}$, the corresponding driving force, which is given as a gradient of a scalar quantity. ${\displaystyle k}$

The thermodynamic forces and their corresponding flows are derived from a balance equation using the conservation quantities. The product of the two quantities describes the increase in entropy during a voluntary process ( entropy production ).

Cross effects between forces and flows

There are a number of phenomena in which a thermodynamic force not only shows the effect described by the laws mentioned above, but also influences other processes. Examples for such phenomena are the thermoelectric effects , thermomagnetic and galvanomagnetic effects or the interdiffusion of two substances into one another.

In these cases, not only the corresponding forces act on a river , but also cross forces. This superposition is easy to understand microscopically, since when a quantity flows, it has to be transported through a medium. For example, a flow of material also transports the heat contained in this material. When describing these processes, the advantage of the formalism introduced above becomes clear, for two flows with two corresponding forces the following results:

${\displaystyle {\mathbf {J} {_{k}}}\,=\,L_{kk}\,\mathbf {X} {_{k}}\,+\,L_{ki}\,\mathbf {X} {_{i}}}$

${\displaystyle {\mathbf {J} {_{i}}}\,=\,L_{ik}\,\mathbf {X} {_{k}}\,+\,L_{ii}\,\mathbf {X} {_{i}}}$

With:

${\displaystyle {\mathbf {J} {_{k}}},\,{\mathbf {J} {_{i}}}\ }$Rivers from and ${\displaystyle k}$ ${\displaystyle i}$

${\displaystyle L_{kk},\,L_{ii}}$ direct transport coefficients (diagonal coefficients) - analogous to the coefficients already described.

${\displaystyle L_{ik},\,L_{ki}}$ Cross coefficients - describe the overlapping effects between the rivers

${\displaystyle {\mathbf {X} {_{k}}},\,{\mathbf {X} {_{i}}}\ }$corresponding thermodynamic forces for the quantities and ${\displaystyle k}$ ${\displaystyle i}$

The cross coefficients are equal, i. H. the following applies:

${\displaystyle L_{ik}\,=\,L_{ki}}$ ( Reciprocity relationship )

They apply in an area in which flows and forces are linearly dependent on one another. This assumes that the system described must not be too far from equilibrium, since the concept of microscopic reversibility or local equilibrium then comes into play. Formally, any functional relationship between the physical quantities is described as a Taylor series , which is broken off after the first term.

Examples

The reciprocity relations

In a system in which both heat and volume flows occur, there is a superposition of flows and forces. Relationships expand too

${\displaystyle \mathbf {J} _{u}=L_{uu}\,\nabla {\frac {1}{T}}-L_{uk}\,\nabla {\frac {\mu _{k}}{T}}}$

and

${\displaystyle \mathbf {J} _{k}=L_{ku}\,\nabla {\frac {1}{T}}-L_{kk}\,\nabla {\frac {\mu _{k}}{T}}.}$

With these equations, the diffusion of the component is described by a temperature gradient ( thermophoresis or Soret effect) ${\displaystyle \ k}$

${\displaystyle \mathbf {J} _{k}\propto L_{ku}\,\nabla {\frac {1}{T}}}$

and the heat conduction through the material flow (diffusion thermal effect or Dufour effect)

${\displaystyle \mathbf {J} _{u}\propto L_{uk}\,\nabla {\frac {\mu _{k}}{T}}.}$

In this case, Onsager's reciprocity relationships again formulate the equality of the cross coefficients:

${\displaystyle \ L_{uk}=\ L_{ku}.}$

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

Thermodynamic equilibrium and entropy production

A closed system is not in thermodynamic equilibrium if its entropy is not maximal. Free energy has to be converted into entropy through entropy production in order to reach a state of equilibrium with minimum free energy and maximum entropy. In a closed system, this transformation can only take place through internal ( dissipative ) processes; the size of the entropy production then results from the continuity equation ${\displaystyle \ F}$ ${\displaystyle \ S}$ ${\displaystyle \ \sigma }$

${\displaystyle \sigma \;{\stackrel {\textrm {def}}{=}}\;{\frac {d_{\mathrm {i} }s}{dt}}={\frac {\partial s}{\partial t}}+\nabla \cdot \mathbf {J} _{s}={\frac {\partial s}{\partial t}}+\nabla \cdot {\frac {\mathbf {J} _{u}}{T}}}$,

where the change in local entropy density due to internal processes is the partial derivative with respect to time , the divergence with respect to location, the local entropy flux density , the local flux density of the internal energy and the absolute temperature . ${\displaystyle \ d_{\mathrm {i} }s}$ ${\displaystyle \ \partial /\partial t}$ ${\displaystyle \ \nabla \cdot }$ ${\displaystyle \mathbf {J} _{s}}$ ${\displaystyle \mathbf {J} _{u}}$ ${\displaystyle \ T}$

The partial derivative of the local entropy density with respect to time can be expressed by Gibb's fundamental equation. This results in an isochoric multi-component system

${\displaystyle {\frac {\partial s}{\partial t}}={\frac {1}{T}}{\frac {\partial u}{\partial t}}-\sum _{k}{\frac {\mu _{k}}{T}}{\frac {\partial \rho _{k}}{\partial t}}.}$

The extensive quantities internal energy and quantity of matter are conservation quantities ; their equations of continuity are ${\displaystyle \ U}$ ${\displaystyle \ n_{k}}$

${\displaystyle {\frac {\partial u}{\partial t}}+\nabla \cdot \mathbf {J} _{u}=0}$

and, since the change in the amount of substance due to chemical reactions with the reaction rate must be taken into account, ${\displaystyle \ n_{k}}$ ${\displaystyle \ j}$ ${\displaystyle \ \sum _{j}\nu _{jk}v_{j}}$

${\displaystyle {\frac {\partial \rho _{k}}{\partial t}}+\nabla \cdot \mathbf {J} _{k}+\sum _{j}\nu _{jk}v_{j}=0.}$

The Gibbs equation thus becomes

${\displaystyle {\frac {\partial s}{\partial t}}=-{\frac {1}{T}}\nabla \cdot \mathbf {J} _{u}+\sum _{k}{\frac {\mu _{k}}{T}}\left(\nabla \cdot \mathbf {J} _{k}+\sum _{j}\nu _{jk}v_{j}\right)}$.

With the transformation from the vector analysis and the definition of the chemical affinity results for the entropy production ${\displaystyle \nabla \cdot (g\mathbf {J} )=\mathbf {J} \cdot \nabla g+g\nabla \cdot \mathbf {J} }$ ${\displaystyle \ A_{j}=-\sum _{k}\nu _{jk}\mu _{k}}$

${\displaystyle \sigma =\mathbf {J} _{u}\cdot \nabla {\frac {1}{T}}-\sum _{k}\mathbf {J} _{k}\cdot \nabla {\frac {\mu _{k}}{T}}-\sum _{j}{\frac {A_{j}v_{j}}{T}}}$.

It has been identified with the term: ${\displaystyle \nabla \cdot \mathbf {J} _{s}}$

${\displaystyle \nabla \cdot \mathbf {J} _{s}=\nabla \cdot \left({\frac {\mathbf {J} _{u}}{T}}-\sum _{k}{\frac {\mathbf {J} _{k}\mu _{k}}{T}}\right)}$.

Can from this equation to the variables and conjugate thermodynamic forces and be determined. If a system is not far from its state of equilibrium, it makes sense to assume a linear relationship between a flow and the thermodynamic force. The proportionality factor is called the transport coefficient. In the absence of a material flow and a reaction, the Fourier law follows in the form ${\displaystyle \ u}$ ${\displaystyle \ \rho _{k}}$ ${\displaystyle \ \mathbf {X} _{u}=\nabla \,{\frac {1}{T}}}$ ${\displaystyle \mathbf {X} {_{k}}=-\nabla \,{\frac {\mu _{k}}{T}}}$ ${\displaystyle \ L}$

${\displaystyle \mathbf {J} _{u}=L_{u}\,\nabla \,{\frac {1}{T}}\!}$

and in the absence of a heat flow Fick's law as

${\displaystyle \mathbf {J} _{k}=-L_{k}\,\nabla \,{\frac {\mu _{k}}{T}}\!}$.

Individual evidence

1. ↑ Description of the practical experiment F7: The diffusion thermal effect (PDF; 405 kB) Institute for Physical Chemistry (IPC) at the Karlsruhe Institute of Technology (www.ipc.kit.edu). December 1, 2016. Accessed May 5, 2019.
2. ^ Nobel lecture Lars Onsager: The motion of ions: principles and concepts ( English , PDF; 129 kB) The Nobel Foundation (nobelprize.org). December 11, 1968. Archived from the original on August 9, 2017. Retrieved May 5, 2019.
3. ↑ Dilip Kondepudi, Ilya Prigogine: Modern thermodynamics: From heat engines to dissipative structures . Ed .: Wiley, John & Sons. 1st edition. 1998, ISBN 978-0-471-97394-2 , pp. 345 .

literature

• Sybren Ruurds de Groot , Peter Mazur : Non-Equilibrium Thermodynamics . Dover Publications, 1985, ISBN 0-486-64741-2
• Bogdan Baranowski: Non-Equilibrium Thermodynamics in Physical Chemistry . Leipzig: German publishing house for basic industry, 1975.
• D. Kondepudi, Ilya Prigogine : Modern Thermodynamics: From Heat Engines to Dissipative Structures . Wiley & Sons, 1998. ISBN 0-471-97394-7 , pp. 351ff.
• Aharon Katchalsky , Peter F. Curran: Nonequilibrium Thermodynamics in Biophysics. Harvard University Press, Cambridge 1965, ISBN 0-674-62550-1
• Lars Onsager: Reciprocal Relations in Irreversible Processes. I . In: Physical Review . tape 37 , no. 4 , 1931, p. 405-426 , doi : 10.1103 / PhysRev.37.405 ( PDF ).