The Thermodynamics of Irreversible Processes

Onsager [1] developed an approach to the study of non-equilibrium processes which is based on elementary thermodynamic concepts. This provides a useful background to the subjects discussed in this chapter and helps show the connection between various processes occurring in a system undergoing an irreversible change. 

The fundamental quantity describing the irreversible process is the flux vector Ji for species i. In the case of mass transfer, it describes a flow in a given direction in space in units of moles or grams crossing unit area per second. It can also be described as the product of the local concentration c,- times the velocity v,- at which molecules or ions are moving. Thus, 

The latter definition is conceptually helpful when the concentration is changing with position in the system. One may write a similar relationship for an energy flux, that is 

The change in the flux with position in the system may be related to the time derivative of the local concentration by applying Gauss theorem. Consider a system with volume V and surface area A. Suppose that substance i is flowing out of this volume. Then the rate of substance i leaving in moles per second can be found by integrating the flux /, over the surface area A, so that 

This is Gauss theorem stated with respect to mass transfer. 

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

If Onsager s great achievement with the thermodynamics of irreversible processes met with initial indifference, Onsager s next feat created a sensation ill the scientific world. In a discussion remark in 1942, he disclosed that he had solved exactly the two-dimensional Ismg model, a model of a ferro-magnet, and showed that it had a phase transition with a specific heat that rose to infinity at the transi-... 

Equilibrium thermodynamics was developed about 150 years ago. It is concerned only with the achievement of an equilibrium state, without taking into account the time which a system requires for the transition from an initial to a final state. Thus, only the thermodynamics of irreversible processes can be used to describe processes which lead to the formation of self-organising systems. Here, the time factor, and thus also the rate at which material reactions occur, is taken into account. Evolutionary processes are irreversibly coupled with temporal sequences, so that classical thermodynamics no longer suffices to describe them (Schuster and Sigmund, 1982). 

For most problems one needs to know how the elements of the second-order shear tensor are related to the velocity gradients and the coefficient of viscosity. It may be shown from the thermodynamics of irreversible processes (G12, C12, Bll) that for a Newtonian fluid the diagonal and nondiagonal elements of t have the form... 

A final remark should be made as to the validity of eq. (2.13). This equation suggests the existence of a set of independent relaxation mechanisms. A general proof for the existence of such mechanisms could be given for visco-elastic solids in terms of the thermodynamics of irreversible processes (52) at small deviation from equilibrium. For liquid systems, however, difficulties arise from the fact that in these systems displacements occur which are not related to the thermodynamic functions. 

The forces Fk involve gradients of intensive properties (temperature, electrochemical potential). The Ljk are called phenomenological coefficients and the fundamental theorem of the thermodynamics of irreversible processes, due originally to Onsager (1931a, b), is that when the fluxes and forces are chosen to satisfy the equation... 

Any dynamic system becomes stable eventually and comes to the rest point, i.e. attains its equilibrium or steady state. For closed systems, a detailed equilibrium is achieved at this point. This is not so simple as it would seem, as substantiated by a principle of the thermodynamics of irreversible processes. At a point of detailed equilibrium not only does the substance concentration remain unchanged (dcjdt = 0), but also the rate of each direct reaction is balanced by that of its associated reverse counterpart... 

Let us now provide a brief summary of the application of the thermodynamics of irreversible processes to diffusion [6,12,21,22],... 

Diffusion is defined as the process in which components are transported from one part of a mixture to another as a result of random molecular motion. Phenomenologically, it can be most rigorously treated by the proper application of the thermodynamics of irreversible processes [de Geoot (1951) Hooyman (1955) Gosting (1956)]. 

According to the thermodynamics of irreversible processes, the mutual diffusion coefficient D may be a function of penetrant concentration ct, position x, and time t. In the present chapter we shall discuss sorption behavior of systems in which D varies with cx only, and shall use the notation D (cx) to indicate this condition. It is assumed that the sample film is so thin that diffusion takes place effectively in the direction of its thickness. At the beginning of an absorption or a desorption experiment the film is conditioned so that Cj is uniform everywhere in it. This initial concentration is denoted by cf. Then we have... 

Refs. [i] Prigogine I-Autobiography http //nobelprize.org/chemistry/ laureates/1977/prigogine [ii] Prigogine I (1967) Introduction to the thermodynamics of irreversible processes. Wiley Interscience, New York [iii] Prigogine I (1962) Non-equilibrium statistical mechanics. Wiley Interscience, New York [iv] Glansdorff P, Prigogine I (1971) Thermodynamic theory of structure stability and fluctuations. Wiley, London... 

We engage in a systematic treatment of the thermodynamics of irreversible processes in the above configurations. Consider first the isothermal case summarized by the constraints (a) J7 - VXT - VyT - 0 isothermal conditions are now maintained along x and y, and no current is allowed to flow... 

Transference numbers are quantities which are treated in the thermodynamics of irreversible processes. In a continuous system, the average velocity Vi of a species i related to a reference velocity w, describes the diffusional motion of the species i. The diffusion current density Ji represents in moles/cm sec the flow of species i in unit time perpendicular to a surface of unit area which by itself is moving with velocity... 

We show below that the Onsager relation (Eq. (24.37)) is the special case of the more general Onsager relation between sedimentation potential and electrophoretic mobility derived on the basis of the thermodynamics of irreversible processes [14—16]. According to the thermodynamics of irreversible processes, we may generally write... 

Earlier [26,27,43,46] a phenomenological approach, based on the premise that the thermodynamics of irreversible processes [29] joined with Nemst-Planck equations for ion fluxes, would be useful was applied to the solution of intraparticle diffusion controlled ion exchange (IE) of fast chemical reactions between B and A counterions and the fixed R groups of the ion exchanger. In the model, diffusion within the resin particle, was considered the slow and sole controlling step. 

Modern opinion views the Nernst-Plank theory as a special case of applying the thermodynamics of irreversible processes to ion exchange. It may also be argued theoretically and experimentally that the observed characteristics of ion exchange rate behaviour can only be fully explained by including chemical reaction as a flux-coupling mechanism as well as the diffusion potential. From a research standpoint it is most probable that future theoretical advances in ion exchange kinetics will result from the further application of non-equilibrium thermodynamics. 

Haase, R., The Thermodynamics of Irreversible Processes, Addison-Wesley, London, England, 1969. Haase, R. and Siry, M., Diffusion im kritischen Entmischungsgebiet binarer fliissiger Systeme, ... 

The thermodynamics of irreversible processes should be set up from the scratch as a continuum theory, treating the state parameters of the theory as field variables [32]. This is also the way in which classical fluid mechanic theory is formulated. Therefore, in the computational fluid dynamics literature, the transport phenomena and the extensions of the classical thermodynamic relations are both interpreted as closures of the fluid dynamic theory. The validity of the thermodynamic relations for fluid dynamic systems has been approached from the viewpoint of the kinetic theory of gases [13]. However, any Arm distinction between irreversible thermodynamics and fluid mechanics... 

The relationship between ionic conductivity and Onsager s theory can now be presented in terms of the electrochemical potential. By expressing the force leading to the transport of ions in terms of the gradient of jr,-, one finds important relationships between the diffusion coefficients of the ions, and the molar conductivity and mobility. Furthermore, when the force has the correct Newtonian units, one is also in a position to calculate the rate of entropy production. On the basis of the thermodynamics of irreversible processes, the relationship between the flux of ion i and the force Vp,- is... 

In this article we shall not utilize the generalized Fokker-Planck equations 6 which have been successfully used to calculate coefficients of viscosity and thermal conductivity.13 14 Rather, we shall find it more convenient to proceed directly from the Liouville equation. To obtain an expression for the contribution of the intermolecular forces to the heat flux, we shall postulate a plausible generalization of the usual phenomenological equations of the thermodynamics of irreversible processes to the space of molecular pairs. Although we shall not prove it here, it may be shown that the same results can also be obtained (with greater labor) from the Fokker-Planck equations ... 

The last term is characteristic of the thermodynamics of irreversible processes. Its magnitude becomes positive if the system s processes are irreversible. Typical irreversible processes are the adsorption or desorption of surfactants at liquid interfaces. The derivative of the second term of Eq. (2C.2), as local entropy production is... 

Light-scattering experiments are not relaxation experiments like the foregoing but instead, as we have seen, involve fluctuation phenomena and time-correlation functions. In connection with the development of the thermodynamics of irreversible processes, Onsager (1931) proposed the principle that spontaneous fluctuations in A(t, t) regress back to equilibrium according to the same relaxation equations that de-... 

Onsager relations - An important set of equations in the thermodynamics of irreversible processes. They express the symmetry between the transport coefficients describing reciprocal processes in systems with a linear dependence of flux on driving forces. 

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