Irreversible thermodynamics reciprocal relations

L. Onsager (Yale) discovery of the reciprocity relations bearing his name, which are fundamental for the thermodynamics of irreversible processes. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

Although irreversible thermodynamics neatly defines the driving forces behind associated flows, so far it has not told us about the relationship between these two properties. Such relations have been obtained from experiment, and famous empirical laws have been established like those of Fourier for heat conduction, Fick for simple binary material diffusion, and Ohm for electrical conductance. These laws are linear relations between force and associated flow rates that, close to equilibrium, seem to be valid. The heat conductivity, diffusion coefficient, and electrical conductivity, or reciprocal resistance, are well-known proportionality constants and as they have been obtained from experiment, they are called phenomenological coefficients Li /... 

In the absence of gradients of salt concentration and temperature, flows of water and electric current in bentonite clay are coupled through a set of linear phenomenological equations, derived from the theory of irreversible thermodynamics (Katchalsky and Curran, 1967), making use of Onsager s Reciprocal Relations (Groenevelt, 1971) ... 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical (i.e., xy = X/,-). This is known as a reciprocal relation. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Thus, the tensor above can be rewritten as... 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Formally, it will be even necessary to make corrections already in the starting flux equations. The detailed formulation of linear irreversible thermodynamics also includes coupling terms (cross terms) obeying the Onsager reciprocity relation. They take into account that the flux of a defect k may also depend on the gradient of the electrochemical potential of other defects. This concept has been worked out, in particular, for the case of the ambipolar transport of ions and electrons.230... 

Onsager s reciprocal relations of irreversible thermodynamics [27-30] imply that if temperature gradients give rise to diffusion velocities (thermal diffusion), then concentration gradients must produce a heat flux. This reciprocal cross-transport process, known as the Dufour effect, provides another additive contribution to q. It is conventional to express the concentration gradients in terms of differences in diffusion velocities by using the diffusion equation, after which it is found that the Dufour heat flux is [5]. 

Equation 3.3.7 expresses the Onsager reciprocal relations (ORR), named after Lars Onsager who first established the principles of irreversible thermodynamics (Onsager, 1931). The ORR have been the subject of many journal papers receiving support as well as criticism, the latter from, in particular, Coleman and Truesdell (1960) and Truesdell (1969). We shall assume the validity of the ORR in the development that follows. 

Thermodynamics for non-equilibrium processes is referred to as irreversible thermodynamics. The scientific field of irreversible thermodynamics was established during the early 1900 s. There are three major reasons why irreversible thermodynamics is important for non-equilibrium systems. In the first place special attention is paid to the validity of the classical thermodynamic relations outside equilibrium (i.e., simple systems). In the second place the theory gives a description of the coupled transport processes (i.e., the Onsager reciprocal relations). In the third place the theory quantifies the entropy that is produced during transport. Irreversible thermodynamics can also be used to assess the second law efficiency of how valuable energy resources are exploited. 

An irreversible process involves the natural movement of a system from a non-equilibrium state to an equilibrium state without intervention, thus it is a spontaneous process. Basically thermodynamics can tell us the direction in which a process will occur, but can say nothing about the speed (rate) of the process. Onsager might be counted as the founder of the field with his papers from in 1931 entitled Reciprocal relations in irreversible processes, Phys. Rev., vol. 37, pp. 405-426 Phys. Rev., vol. 38, pp. 2265-2279. 

It is understood that these relations are derived adopting several relations from irreversible thermodynamics, e.g., the second law of thermodynamics, the Gibbs-Duhem relation, the linear law and the Onsager reciprocal relations [39, 22, 62, 18, 5]. 

In irreversible thermodynamics Onsager reciprocity relations are (usually) postulated which in our context (4.514), (4.515) are... 

Electrophoresis and sedimentation potential also offer a test of predictions of thermodynamics of irreversible processes, provided these are supplemented by classical analysis of the data. Few measurements of sedimentation potential have been reported [1] and the theories due to Kruyt [2], Debye and Huckel [3] and Henry [4] are not in complete agreement. The thermodynamics of irreversible processes [5] may be helpful since the theory does not depend on any model. In the present chapter it is intended (i) to test linear phenomenological relations, (ii) to test the Onsager s reciprocal relation and (iii) to examine the validity of conflicting theories of electrophoresis. 

We should also mention that the normal solution of the Boltzmann equation discussed here, together with the //-theorem discussed in the previous section, can be used to provide a derivation of the principles of nonequilibrium thermodynamics. For mixtures, one can show that the various diffusion coefficients that occur in the Navier-Stokes equations can be expressed in a form where Onsager reciprocal relations are satisfied. However, both for mixtures and for pure gases the relation between the normal solution and irreversible thermodynamics only holds if one does not go beyond in the -expansion of the distribution function. ... 

Miller, D.G., 1960a. Errata— Thermodynamics of irreversible processes. The experimental verification of the Onsager reciprocal relations Chem. Rev. 60, 593. 

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